Noether's Theorems Applications in Mechanics and Field Theory Atlantis Studies in Variational Geometry

Atlantis Studies in Variational Geometry Series Editors: Demeter Krupka · Huafei Sun

Gennadi Sardanashvily Noether's Theorems Applications in Mechanics and Field Theory Atlantis Studies in Variational Geometry

Volume 3

Series editors Demeter Krupka, University of Hradec Kralove, Hradec Kralove, Czech Republic Huafei Sun, Beijing Institute of Technology, Beijing, China More information about this series at http://www.atlantis-press.com Gennadi Sardanashvily

Noether’s Theorems Applications in Mechanics and Field Theory Gennadi Sardanashvily Moscow State University Moscow Russia

ISSN 2214-0700 ISSN 2214-0719 (electronic) Atlantis Studies in Variational Geometry ISBN 978-94-6239-170-3 ISBN 978-94-6239-171-0 (eBook) DOI 10.2991/978-94-6239-171-0

Library of Congress Control Number: 2016932506

© Atlantis Press and the author(s) 2016 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher.

Printed on acid-free paper To my wife Aida Karamysheva Professor, molecule biologist

Preface

Noether’s first and second theorems are formulated in a very general setting of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Lagrangian theory generally is characterized by a hierarchy of nontrivial Noether and higher-stage Noether identities and the corresponding gauge and higher-stage gauge symmetries which characterize the degeneracy of a Lagrangian system. By analogy with Noether identities of differential operators, they are described in the homology terms. In these terms, Noether’s inverse and direct second theorems associate to the Koszul–Tate graded chain complex of Noether and higher-stage Noether identities the gauge cochain sequence whose ascent gauge operator provides gauge and higher-stage gauge symmetries of Grassmann-graded Lagrangian theory. If these symmetries are algebraically closed, an ascent gauge operator is generalized to a nilpotent BRST operator which brings a gauge cochain sequence into a BRST complex and provides the BRST extension of original Lagrangian theory. In the present book, the calculus of variations and Lagrangian formalism are phrased in algebraic terms of a variational bicomplex on an infinite order jet manifold that enables one to extend this formalism to Grassmann-graded Lagrangian systems of even and odd variables on graded bundles. Cohomology of a graded variational bicomplex provides the global solutions of the direct and inverse problems of the calculus of variations. In this framework, Noether’s direct first theorem is formulated as a straightforward corollary of the global variational formula. It associates to any Lagrangian symmetry the conserved symmetry current whose total differential vanishes on-shell. Proved in a very general setting, so-called Noether’s third theorem states that a conserved symmetry current along any gauge symmetry is reduced to a superpotential, i.e., it is a total differential on-shell. This also is the case of covariant Hamiltonian formalism on smooth fibre bundles seen as the particular Lagrangian one on phase Legendre bundles.

vii viii Preface

Lagrangian formalism on smooth fibre bundles and graded bundles provides the comprehensive formulations both of classical field theory and nonrelativistic mechanics. Non-autonomous nonrelativistic mechanics is adequately formulated as particular Lagrangian and Hamiltonian theory on a configuration bundle over the time axis. Conserved symmetry currents of Noether’s first theorem in mechanics are integrals of motion, but the converse need not be true. In Hamiltonian mechanics, Noether’s inverse first theorem states that all integrals of motion come from symmetries. In particular, this is the case of energy functions with respect to different reference frames. The book presents a number of physically relevant models: superintegrable Hamiltonian systems, the global Kepler problem, Yang–Mills gauge theory on principal bundles, SUSY gauge theory, gauge gravitation theory on natural bundles, topological Chern–Simons field theory and topological BF theory, exemplifying a reducible degenerate Lagrangian system. Our book addresses to a wide audience of theoreticians, mathematical physicists and mathematicians. With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. We have tried to give the necessary mathematical background, thus making our exposition self-contained. For the sake of convenience of the reader, a number of relevant mathematical topics are compiled in appendixes.

Moscow Gennadi Sardanashvily October 2015 Contents

1 Calculus of Variations on Fibre Bundles ...... 1 1.1 Inﬁnite Order Jet Formalism ...... 1 1.2 Variational Bicomplex ...... 7 1.2.1 Cohomology of the Variational Bicomplex ...... 9 1.3 Lagrangian Formalism ...... 12 2 Noether’s First Theorem ...... 17 2.1 Lagrangian Symmetries...... 17 2.2 Gauge Symmetries: Noether’s Direct Second Theorem ...... 20 2.3 Noether’s First Theorem: Conservation Laws...... 23 3 Lagrangian and Hamiltonian Field Theories ...... 27 3.1 First Order Lagrangian Formalism ...... 27 3.2 Cartan and Hamilton–De Donder Equations...... 30 3.3 Noether’s First Theorem: Energy-Momentum Currents ...... 32 3.4 Conservation Laws in the Presence of a Background Field . . . . 34 3.5 Covariant Hamiltonian Formalism ...... 36 3.6 Associated Lagrangian and Hamiltonian Systems ...... 41 3.7 Noether’s First Theorem: Hamiltonian Conservation Laws. . . . . 47 3.8 Quadratic Lagrangian and Hamiltonian Systems ...... 49 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics...... 59 4.1 Geometry of Fibre Bundles over R...... 60 4.2 Lagrangian Mechanics. Integrals of Motion ...... 63 4.3 Noether’s First Theorem: Energy Conservation Laws ...... 67 4.4 Gauge Symmetries: Noether’s Second and Third Theorems . . . . 71 4.5 Non-autonomous Hamiltonian Mechanics ...... 73 4.6 Hamiltonian Conservation Laws: Noether’s Inverse First Theorem ...... 80 4.7 Completely Integrable Hamiltonian Systems ...... 84

ix x Contents

5 Global Kepler Problem...... 93 6 Calculus of Variations on Graded Bundles ...... 103 6.1 Grassmann-Graded Algebraic Calculus ...... 103 6.2 Grassmann-Graded Differential Calculus ...... 106 6.3 Differential Calculus on Graded Bundles ...... 109 6.4 Grassmann-Graded Variational Bicomplex...... 121 6.5 Grassmann-Graded Lagrangian Theory ...... 127 6.6 Noether’s First Theorem: Supersymmetries ...... 129 7 Noether’s Second Theorems ...... 135 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems ...... 136 7.2 Noether’s Inverse Second Theorem ...... 145 7.3 Gauge Supersymmetries: Noether’s Direct Second Theorem . . . 148 7.4 Noether’s Third Theorem: Superpotential ...... 152 7.5 Lagrangian BRST Theory ...... 155 8 Yang–Mills Gauge Theory on Principal Bundles ...... 163 8.1 Geometry of Principal Bundles ...... 163 8.2 Principal Gauge Symmetries ...... 171 8.3 Noether’s Direct Second Theorem: Yang–Mills Lagrangian . . . . 173 8.4 Noether’s First Theorem: Conservation Laws...... 175 8.5 Hamiltonian Gauge Theory ...... 177 8.6 Noether’s Inverse Second Theorem: BRST Extension ...... 179 9 SUSY Gauge Theory on Principal Graded Bundles ...... 183 10 Gauge Gravitation Theory on Natural Bundles ...... 189 10.1 Relativity Principle: Natural Bundles ...... 189 10.2 Equivalence Principle: Lorentz Reduced Structure ...... 191 10.3 Metric-Afﬁne Gauge Gravitation Theory ...... 194 10.4 Energy-Momentum Gauge Conservation Law ...... 197 10.5 BRST Gravitation Theory ...... 199 11 Chern–Simons Topological Field Theory ...... 201 12 Topological BF Theory...... 207

Glossary ...... 211

Appendix A: Differential Calculus over Commutative Rings ...... 213

Appendix B: Differential Calculus on Fibre Bundles...... 227 Contents xi

Appendix C: Calculus on Sheaves ...... 259

Appendix D: Noether Identities of Differential Operators ...... 271

References ...... 281

Index ...... 287

Introduction

Noether’s theorems are well known to treat symmetries of Lagrangian systems. Noether’s first theorem associates to a Lagrangian symmetry the conserved symmetry current whose total differential vanishes on-shell. The second ones provide the correspondence between Noether identities and gauge symmetries of a Lagrangian system. We refer the reader to the brilliant book of Yvette Kosmann-Schwarzbach [84] for the history and references on this subject. Our book aims to present Noether’s theorems in a very general setting of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. We however start with even Lagrangian formalism on smooth fibre bundles (Chap. 2), and focus ourselves especially on first-order Lagrangian theory (Chap. 3) because the most physically relevant models (nonrelativistic mechanics, gauge field theory, gravitation theory, etc.) are of this type. Lagrangian theory of even (commutative) variables on an n-dimensional smooth manifold X conventionally is formulated in terms of fibre bundles over X and jet manifolds of their sections [15, 53, 108, 133, 143], and it lies in the framework of general technique of nonlinear differential equations and operators [25, 53, 85]. This formulation is based on the categorial equivalence of projective C1ðXÞ- modules of finite rank and vector bundles over X in accordance with the classical Serre–Swan theorem, generalized to noncompact manifolds (Theorem A.10). The calculus of variations and Lagrangian formalism on a fibre bundle Y ! X can be adequately formulated in algebraic terms of the variational bicomplex (1.21) of differential forms on an infinite order jet manifold J1Y of sections of Y ! X [3, 15, 56, 61, 108, 133, 143]. In this framework, finite order Lagrangians and Euler–Lagrange operator are defined as elements (1.32)–(1.33) of this bicomplex (Sect. 1.3). The cohomology of the variational bicomplex (Sect. 1.2) provides a solution of the global inverse problem of the calculus of variations (Theorems 1.15–1.16), and states the global variational formula (1.36) for Lagrangians and Euler–Lagrange operators (Theorem 1.17).

xiii xiv Introduction

In these terms, Noether's first theorem (Theorem 2.7) and Noether’s direct second theorem (Theorem 2.6) are straightforward corollaries of the global decomposition (1.36). Noether’s first theorem associates to any symmetry of a Lagrangian L (Definition 2.7) the conserved symmetry current (2.21) whose total differential vanishes on-shell (Sect. 2.3). One can show that a conserved symmetry current itself is a total differential on-shell if it is associated to a gauge symmetry (Theorem 2.8). Treating gauge symmetries of Lagrangian theory, one is traditionally based on an example of Yang–Mills gauge theory of principal connections on the principal bundle P ! X (8.3) where gauge symmetries are vertical principal automorphisms of this bundle (Sect. 8.2). They are represented by global sections of the associated group bundle (8.43) and, thus, look like symmetries depending on parameter functions. This notion of gauge symmetries is generalized to Lagrangian theory on an arbitrary fibre bundle Y ! X (Definition 2.3) and on graded bundles (Definition 7.3). Given a gauge symmetry of a Lagrangian system on fibre bundles, Noether’s direct second theorem (Theorem 2.6) states that its Euler–Lagrange operator obeys the corresponding Noether identities (2.17) (Sect. 2.2). A problem is that any Euler– Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and nontrivial ones. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones and so on. Thus, there is a hierarchy of nontrivial Noether and higher-stage Noether identities which characterizes the degeneracy of a Lagrangian theory (Sect. 7.1). A Lagrangian system is called degenerate if it admits nontrivial Noether identities and reducible if there exist nontrivial higher-stage Noether identities. We follow the general analysis of Noether and higher-stage Noether identities of differential operators on fibre bundles when trivial and nontrivial Noether identities are described by boundaries and cycles of a certain chain complex [61, 123]. This description involves Grassmann-graded objects. In a general setting, we therefore consider Grassmann-graded Lagrangian systems of even and odd variables (Chap. 6). Different geometric models of odd variables either on graded manifolds or supermanifolds are discussed [28, 30, 46, 107, 134]. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras [7, 61, 131]. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. We follow the Serre–Swan theorem for graded manifolds (Theorem 6.2) [11, 61]. It states that, if a graded commutative C1ðXÞ-ring is generated by a projective C1ðXÞ-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is a smooth manifold X. Accordingly, we describe odd variables on a smooth manifold X in terms of graded bundles over X [61, 134]. Let us recall that a graded manifold is a local-ringed space (Definition C.1), characterized by a smooth body manifold Z and some structure sheaf A of Grassmann algebras on Z [7, 61, 134]. Its sections form a graded commutative Introduction xv

C1ðZÞ-ring A of graded functions on a graded manifold ðZ; AÞ. It is called the structure ring of ðZ; AÞ. The differential calculus on a graded manifold is defined as the Chevalley–Eilenberg differential calculus over its structure ring (Sect. 6.2). By virtue of the well-known Batchelor theorem (Theorem 6.1), there exists a vector bundle E ! Z with a typical fibre V such that the structure sheaf A of ðZ; AÞ is Ã isomorphic to a sheaf AE of germs of sections of the exterior bundle Ê of the dual EÃ of E whose typical fibre is the Grassmann algebra ^V Ã [7, 14]. This Batchelor’s isomorphism is not canonical. In applications, it however is fixed from the beginning. Therefore, we restrict our consideration to graded manifolds ðZ; AEÞ, termed the simple graded manifolds, modelled over vector bundles E ! Z (Definition 6.4). Let us note that a manifold Z itself can be treated as a trivial simple graded ð ; 1Þ fi Â R ! manifold Z CZ modelled over a trivial bre bundle Z Z whose structure ring of graded functions is reduced to a commutative ring C1ðZÞ of smooth real functions on Z (Remark 6.1). Accordingly, a configuration fibre bundle Y ! X in Lagrangian theory of even variables can be regarded as the graded bundle (6.29)of trivial graded manifolds (Remark 6.4). It follows that, in a general setting, one can define a configuration space of Grassmann-graded Lagrangian theory of even and odd variables as being the graded bundle ðX; Y; AFÞ (6.32) over a trivial graded ð ; 1Þ ! ! manifold X CX modelled over the smooth composite bundle F Y X (6.33) (Sect. 6.5). If Y ! X is a vector bundle, this is the particular case of graded vector bundles in [73, 107] whose base is a trivial graded manifold. By analogy with the calculus of variations and Lagrangian theory on smooth fibre bundles, Grassmann-graded Lagrangian theory on a graded bundle ðX; Y; AFÞ (Sect. 6.5) comprehensively is phrased in terms of the Grassmann-graded variational bicomplex (6.54) of graded exterior forms on a graded infinite order jet 1 manifold ðJ Y; AJ1FÞ [6, 11, 12, 61, 134, 137]. Graded Lagrangians and Euler– Lagrange operator are defined as elements (6.69) and (6.70) of this graded bicomplex. The cohomology of a Grassmann-graded variational bicomplex (Theorems 6.9–6.10) provides a solution of the global inverse problem of the calculus of variations (Theorem 6.11), and states the global variational formula (6.73) for graded Lagrangians and Euler–Lagrange operators (Theorem 6.13). In these terms, Noether’s first theorem is formulated in a very general setting as a straightforward corollary of the global variational formula (6.73) (Sect. 6.6). It associates to any supersymmetry of a graded Lagrangian L (Definition 6.9) the conserved supersymmetry current (6.84) whose graded total differential vanishes on-shell (Theorem 6.20). One can show that a conserved supersymmetry current along a gauge supersymmetry is a graded total differential on-shell (Theorem 7.11). Given a gauge supersymmetry of a graded Lagrangian, Noether’s direct second theorem (Theorem 7.10) states that an Euler–Lagrange operator obeys the corresponding Noether identities. As was mentioned above, a problem is that any Euler– Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and nontrivial ones, and that there is a hierarchy of Noether and higher-stage Noether identities. xvi Introduction

We follow the general analysis of Noether identities and higher-stage Noether identities of differential operators on fibre bundles (Appendix D). If a certain homology regularity condition (Definition 7.2) holds, one can associate to a Grassmann-graded Lagrangian system the exact Koszul–Tate chain complex (7.28) possessing the boundary Koszul–Tate operator (7.26) whose nilpotentness is equivalent to all complete nontrivial Noether identities (7.12) and higher-stage Noether identities (7.29) (Theorem 7.5) [11, 12, 61, 134, 137]. It should be noted that the notion of higher-stage Noether identities has come from that of reducible constraints. The Koszul–Tate complex of Noether identities has been invented similarly to that of constraints under the regularity condition that Noether identities are locally separated into the independent and dependent ones [6, 44]. This condition is relevant for constraints, defined by a finite set of functions which the inverse mapping theorem is applied to. However, Noether identities of differential operators, unlike constraints, are differential equations (Appendix D). They are given by an infinite set of functions on a Fréchet manifold of infinite order jets where the inverse mapping theorem fails to be valid. Therefore, the regularity condition for the Koszul–Tate complex of constraints is replaced with the above mentioned homology regularity condition. Noether’s inverse second theorem formulated in homology terms (Theorem 7.9) associates to the Koszul–Tate chain complex (7.28) the cochain sequence (7.38) with the ascent operator (7.39), called the gauge operator, whose components are complete nontrivial gauge and higher-stage gauge supersymmetries of Grassmann-graded Lagrangian theory [12, 61, 134, 137]. The gauge operator unlike the Koszul–Tate one is not nilpotent, unless gauge symmetries are Abelian (Remark 7.12). Therefore, an intrinsic definition of nontrivial gauge and higher-stage gauge symmetries meets difficulties. Another problem is that gauge symmetries need not form an algebra [48, 60, 63]. However, we can say that gauge symmetries are algebraically closed in a sense if the gauge operator admits a nilpotent extension, termed the BRST (Becchi–Rouet–Stora– Tyitin) operator (Sect. 7.5). If the BRST operator exists, the above mentioned cochain sequence is brought into the BRST complex. The Koszul–Tate and BRST complexes provide a BRST extension of original Lagrangian theory by Grassmann-graded ghosts and Noether antifields. This extension is a preliminary step towards the BV (Batalin–Vilkovisky) quantization of reducible degenerate Lagrangian theories [6, 13, 44, 50, 63]. For the purpose of applications in field theory and mechanics, Chap. 3 of the book addresses first-order Lagrangian and covariant Hamiltonian theories on fibre bundles since the most of relevant field models is of this type. We restrict our consideration to classical symmetries represented by the projectable vector fields u (3.24) on a configuration bundle Y ! X (Sect. 3.3). Since the corresponding symmetry current (3.29) is linear in a vector field u, one usually deals with the following types of symmetries. (i) If u is a vertical vector field, the corresponding symmetry current is the Noether one (3.32). This is the case of so-called internal symmetries Introduction xvii

(ii) Let τ be a vector field on X. Then its lift γτ (B.27) onto Y (Definition B.2) provides the corresponding energy-momentum current (3.33) (Definition 3.3). We usually have either the horizontal lift Γτ (B.56) by means of a connection Γ on Y ! X, e.g., in Yang–Mills gauge theory (Sect. 8.4), or the functorial lift eτ (Definition B.28) which is an infinitesimal general covariant transformation. This is the case of gauge gravitation theory (Sect. 10.4). Applied to field theory, the familiar symplectic Hamiltonian technique takes a form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space, where canonical variables are field functions at each instant of time [65]. The true Hamiltonian counterpart of first-order Lagrangian theory on a fibre bundle ! μ Y X is covariant Hamiltonian formalism, where canonical momenta pi corre- i i μ spond to jets yμ of field variables y with respect to all base coordinates x . This formalism has been vigorously developed since 1970s in the Hamilton–De Donder, polysymplectic, multisymplectic, k-symplectic, k-cosymplectic and other variants [26, 27, 36, 45, 64, 69, 72, 81, 92, 95, 96, 103, 112, 114, 117]. We follow polysymplectic Hamiltonian formalism on a fibre bundle Y ! X where the Legendre bundle Π (3.6) plays the role of a phase space (Sect. 3.5). A key point is that polysymplectic Hamiltonian formalism on a phase space Π is equivalent to particular first-order Lagrangian theory on a configuration space Π ! X. This fact enables us to describe symmetries of Hamiltonian field theory similarly to those in Lagrangian formalism (Sect. 3.7). One can formulate non-autonomous nonrelativistic mechanics as particular field theory on fibre bundles Q ! R over the time axis R [62, 98, 121, 135]. Its velocity space is the first-order jet manifold J1Q of sections of a configuration bundle Q ! R, and its phase space Π (3.8) is the vertical cotangent bundle VÃQ of Q ! R (Chap. 4). A difference between mechanics and field theory however lies in the fact that fibre bundles over R always are trivial, and that all connections on these fibre bundles are flat. Consequently, they are not dynamic variables, but characterize nonrelativistic reference frames (Definition 4.1). By virtue of Noether’s first theorem, any symmetry defines a symmetry current which also is an integral of motion in Lagrangian and Hamiltonian mechanics (Theorem 4.4). The converse is not true in Lagrangian mechanics where integrals of motion need not come from symmetries. We show that, in Hamiltonian mechanics, any integral of motion is a symmetry current (Theorem 4.15). One can think of this fact as being Noether’s inverse first theorem. The book presents a number of physically relevant models: commutative and noncommutative integrable Hamiltonian systems (Sect. 4.7), the global Kepler problem (Chap. 5), Yang–Mills gauge theory on principal bundles (Chap. 8), SUSY gauge theory (Chap. 9) on principal graded bundles, gauge gravitation theory on natural bundles (Chap. 10), topological Chern–Simons field theory (Chap. 11), and topological BF theory, exemplifying a reducible degenerate Lagrangian system (Chap. 12). For the sake of convenience of the reader, a number of relevant mathematical topics are compiled in appendixes, thus making the exposition self-contained.

Chapter 1 Calculus of Variations on Fibre Bundles

There are different approaches to the calculus of variations on a fibre bundle Y → X. It can be adequately formulated in algebraic terms of the variational bicomplex (1.21) of differential forms on the infinite order jet manifold J ∞Y of sections of Y → X [3, 15, 56, 61, 108, 133, 143]. In this framework, finite order Lagrangians (Definition 1.3) and Euler–Lagrange operators (Definition 1.4) are introduced as the elements (1.32) and (1.33) of this bicomplex, respectively. The cohomology of the variational bicomplex (Theorems 1.12 and 1.13) provides a solution of the global inverse problem of the calculus of variations (Theorems 1.15 and 1.16), and states the global variational decomposition (1.36) for Lagrangians and Euler–Lagrange operators (Theorem 1.17). It should be emphasized that we deal with a variational bicomplex of the differ- ∗ ential graded algebra (henceforth, DGA) O∞Y (1.7) of differential forms of finite jet order on an infinite order jet manifold J ∞Y . They are exterior forms on finite order jet manifolds J r Y modulo the pull-back identification. One also considers both the ∗ variational bicomplex of the DGA Q∞ (1.10) [3, 143] of differential forms of locally finite jet order on J ∞Y and different variants of variational sequences of finite jet order [2, 86, 89, 90, 150].

1.1 Inﬁnite Order Jet Formalism

Let Y → X be a fibre bundle over an n-dimensional smooth manifold X (Definition B.2). Finite order jet manifolds J k Y of a fibre bundle Y → X (Definition B.11) form the inverse sequence

r π π − Y ←− J 1Y ←−···J r−1Y ←−r 1 J r Y ←−··· , (1.1)

πr where r−1 are the afﬁne bundles (B.77) modelled over the vector bundles (B.78). Its inverse limit J ∞Y is deﬁned as a minimal set such that there exist surjections

π ∞ : ∞ → ,π∞ : ∞ → ,π∞ : ∞ → k , J Y X 0 J Y Y k J Y J Y (1.2)

π ∞ = π k ◦ π ∞ < obeying the relations r r k for all admissible k and r k.Aninverse limit of the inverse system (1.1) always exists. It consists of those elements

r k (...,zr ,...,zk ,...), zr ∈ J Y, zk ∈ J Y, k = π k ( ) of the Cartesian product J Y which satisfy the relations zr r zk for all k k > r. One can think of elements of J ∞Y as being inﬁnite order jets of sections of Y → X identiﬁed by their Taylor series at points of X.AsetJ ∞Y is provided with the inverse limit topology. This is the coarsest topology such that the surjections π ∞ r (1.2) are continuous. Its base consists of inverse images of open subsets of r = ,... π ∞ ∞ J Y , r 0 , under the maps r . With the inverse limit topology, J Y is a paracompact Fréchet (complete metrizable, but not Banach) manifold modelled over λ i i a locally convex vector space of formal number series {a , a , aλ, ···}[143]. One can π ∞ ∞ → r show that surjections r are open maps admitting local sections, i.e., J Y J Y λ i are continuous bundles (Remark B.2). A bundle coordinate atlas {UY ,(x , y )} of Y → X provides J ∞Y with a manifold coordinate atlas ∂ μ ∞ −1 λ i i x i {(π ) (U ), (x , y )} ≤|Λ|, y = dμ y . (1.3) 0 Y Λ 0 λ+Λ ∂x λ Λ

Definition 1.1 One calls J ∞Y , possessing the above mentioned properties, the infinite order jet manifold. Theorem 1.1 A fibre bundle Y is a strong deformation retract of the infinite order jet manifold J ∞Y [3, 56]. Proof To show that Y is a strong deformation retract of J ∞Y , let us construct a ∞ homotopy from J Y to Y in an explicit form. Let γ(k), k ≤ 1, be global sections of an affine jet bundles J k Y → J k−1Y . Then we have a global section

λ i λ i i i ∞ γ : Y (x , y ) → (x , y , yΛ = γ(|Λ|)Λ ◦ γ(|Λ|−1) ◦···◦γ(1)) ∈ J Y (1.4)

π ∞ : ∞ → of an open surjection 0 J Y Y . Let us consider a map

∞ λ i i λ i i ∞ [0, 1]×J Y (t, x , y , yΛ) → (x , y , yΛ) ∈ J Y, (1.5) i i i λ i i yΛ = fk (t)yΛ + (1 − fk (t))γ(k)Λ(x , y , yΣ ), |Σ| < k =|Λ|, where fk (t) is a continuous monotone real function on [0, 1] such that 0, t ≤ 1 − 2−k , f (t) = k 1, t ≥ 1 − 2−(k+1). 1.1 Inﬁnite Order Jet Formalism 3

A glance at the transition functions (1.3) shows that, although written in a coordinate form, this map is globally deﬁned. It is continuous because, given an open subset ⊂ k (π ∞)−1( ) ⊂ ∞ Uk J Y , the inverse image of an open set k Uk J Y is an open subset

( , ]×(π ∞)−1( ) ∪ ( , ]×(π ∞ )−1(π k [ ∩ γ ( k−1 )]) ∪ tk 1 k Uk tk−1 1 k−1 k−1 Uk (k) J Y ···∪[ , ]×(π ∞)−1(π k [ ∩ γ ◦···◦γ ( )]) 0 1 0 0 Uk (k) (1) Y

∞ of [0, 1]×J Y , where [tr , 1]=supp fr . Then the map (1.5) is a desired homotopy from J ∞Y to Y which is identiﬁed with its image under the global section (1.4). Theorem 1.2 By virtue of the Vietoris–Begle theorem [23], there is an isomorphism

H ∗(J ∞Y ; R) = H ∗(Y ; R) between the cohomology of J ∞Y with coefficients in the constant sheaf R and that of Y . The inverse sequence (1.1) of jet manifolds yields the direct sequence of differ- O∗ = O∗( r ) ential graded algebras (DGAs, Definition A.5) r J Y of exterior forms on finite order jet manifolds

π ∗ π 1∗ πr ∗ O∗( ) −→ O∗( ) −→0 O∗ −→···O∗ −→r−1 O∗ −→··· , X Y 1 r−1 r (1.6)

πr ∗ where r−1 are the pull-back monomorphisms. Its direct limit

→ O∗ = O∗ ∞Y lim r (1.7)

(Definition A.1) exists and consists of all exterior forms on finite order jet manifolds ∗ modulo the pull-back identification. In accordance with Theorem A.4, O∞Y is a DGA which inherits operations of the exterior differential d and the exterior product ∧ O∗ of exterior algebras r . ∗ ∗ If there is no danger of confusion, we further denote O∞ = O∞Y . ∗ ∗ Theorem 1.3 The cohomology H (O∞) of the de Rham complex

0 d 1 d 0 −→ R −→ O∞ −→ O∞ −→··· (1.8)

O∗ ∗ ( ) of a DGA ∞ equals the de Rham cohomology HDR Y of a ﬁbre bundle Y [2, 15]. Proof By virtue of Theorem A.8, the operation of taking homology groups of cochain ∗ complexes commutes with the passage to a direct limit. Since a DGA O∞ is a direct O∗ ∗(O∗ ) limit of DGAs r , its cohomology H ∞ is isomorphic to the direct limit of a direct sequence

∗ ( ) −→ ∗ ( 1 ) −→···−→ ∗ ( r ) −→··· HDR Y HDR J Y HDR J Y (1.9) 4 1 Calculus of Variations on Fibre Bundles

∗ ( r ) r of the de Rham cohomology groups HDR J Y of ﬁnite order jet manifolds J Y . In accordance with Remark B.15, all these groups equal the de Rham cohomology ∗ ( ) ∗(O∗ ) HDR Y of Y , and so is its direct limit H ∞ . ∗ Theorem 1.4 Anyclosedformφ ∈ O∞ is decomposed into a sum φ = σ + dξ, where σ is a closed form on Y .

∗ One can think of elements of O∞ as being differential forms on an inﬁnite order jet manifold J ∞Y as follows. Let G∗ be a sheaf of germs of exterior forms on J r Y ∗ r G G∗ πr and r a canonical presheaf of local sections of r . Since r−1 are open maps, there is the direct sequence of presheaves

∗ π 1∗ ∗ πr ∗ ∗ G −→0 G ···−→r−1 G −→··· . 0 1 r

∗ ∞ ∗ Its direct limit G∞ is a presheaf of DGAs on J Y .LetQ∞ be a sheaf of DGAs of ∗ ∞ germs of G∞ on J Y . A structure module

∗ ∗ Q∞ = Γ(Q∞) (1.10)

∗ ∗ of global sections of Q∞ is a DGA such that, given an element φ ∈ Q∞ and a point z ∈ J ∞Y , there exist an open neighborhood U of z and an exterior form φ(k) on some ﬁnite order jet manifold J k Y so that

φ| = π ∞∗φ(k)| . U k U

∗ Therefore, one can think of Q∞ as being an algebra of locally exterior forms on ∗ ∗ ﬁnite order jet manifolds. In particular, there is a monomorphism O∞ → Q∞.

Theorem 1.5 A paracompact topological space J ∞Y admits the partition of unity 0 by elements of a ring Q∞ [143].

∗ Since elements of a DGA Q∞ are locally exterior forms on ﬁnite order jet manifolds, the following Poincaré lemma holds.

∗ Theorem 1.6 Given a closed element φ ∈ Q∞, there exists a neighborhood U of ∞ each point z ∈ J Y such that φ|U is exact.

∗ ∗ Theorem 1.7 The cohomology H (Q∞) of the de Rham complex

0 d 1 d 0 −→ R −→ Q∞ −→ Q∞ −→··· (1.11)

∗ of a DGA Q∞ equals the de Rham cohomology of a ﬁbre bundle Y [3, 143]. 1.1 Inﬁnite Order Jet Formalism 5

Proof Let us consider the de Rham complex of sheaves

0 d 1 d 0 −→ R −→ Q∞ −→ Q∞ −→··· (1.12) on J ∞Y . By virtue of Theorem 1.6, it is exact at all terms, except R. Being sheaves of 0 r Q∞-modules, the sheaves Q∞ are ﬁne and, consequently, acyclic because a paracom- ∞ 0 pact space J Y admits the partition of unity by elements of a ring Q∞ (Theorems 1.5, C.7 and C.8). Thus, the complex (1.12) is a resolution of the constant sheaf R on ∞ ∗ ∗ J Y . In accordance with abstract de Rham Theorem C.6, cohomology H (Q∞) of the complex (1.11) equals the cohomology H ∗(J ∞Y ; R) of J ∞Y with coefﬁcients in the constant sheaf R. Since Y is a strong deformation retract of J ∞Y , there is the isomorphism (B.79) and, consequently, a desired isomorphism

∗(Q∗ ) = ∗ ( ). H ∞ HDR Y

∗ ∗ ∗ Due to a monomorphism O∞ → Q∞, one can restrict O∞ to the coordinate chart (1.3) where horizontal forms dxλ and contact one-forms

i i i λ θΛ = dyΛ − yλ+Λdx

0 ∗ ∞ make up a local basis for a O∞-algebra O∞. Though J Y is not a smooth manifold, ∗ elements of O∞ are exterior forms on ﬁnite order jet manifolds and, therefore, their ∗ coordinate transformations are smooth. Moreover, there is a decomposition O∞ = k,m ∗ 0 k,m ⊕O∞ of O∞ into O∞-modules O∞ of k-contact and m-horizontal forms together with the corresponding projections

∗ k,∗ m ∗ ∗,m hk : O∞ → O∞ , h : O∞ → O∞ .

In particular, the projection h0 takes a form

∗ 0,∗ λ λ i h0 : O∞ → O∞ , h0(dx ) = dx , h0(θΛ) = 0. (1.13)

It is called the horizontal projection. ∗ Accordingly, the exterior differential on O∞ is decomposed into a sum d = dV + dH of the vertical differential

◦ m = m ◦ ◦ m, (φ) = θ i ∧ ∂Λφ, φ ∈ O∗ , dV h h d h dV Λ i ∞ and the total differential

λ dH ◦ hk = hk ◦ d ◦ hk , dH ◦ h0 = h0 ◦ d, dH (φ) = dx ∧ dλ(φ), 6 1 Calculus of Variations on Fibre Bundles where = ∂ + i ∂ + i ∂Λ dλ λ yλ i yλ+Λ i (1.14) 0<|Λ| are the inﬁnite order total derivatives. These differentials obey nilpotent conditions

dH ◦ dH = 0, dV ◦ dV = 0, dH ◦ dV + dV ◦ dH = 0, (1.15)

∗,∗ and bring O∞ into a bicomplex. 0 0 0 Let us consider the O∞-module dO∞ of derivations of a real ring O∞ (Deﬁnition A.4).

0 0 1 ∗ Theorem 1.8 The derivation module dO∞ is isomorphic to the O∞-dual (O∞) of 1 a module of one-forms O∞ [59].

∗ 0 Proof At first, let us show that O∞ is generated by elements df, f ∈ O∞. It suffices 1 0 to justify that any element of O∞ is a finite O∞-linear combination of elements df, 0 1 f ∈ O∞. Indeed, every φ ∈ O∞ is an exterior form on some finite order jet manifold r ∞( r ) O1 J Y . By virtue of Serre–Swan Theorem A.10, a C J Y -module r of one-forms on J r Y is a projective module of finite rank, i.e., φ is represented by a finite C∞(J r Y )- ∞ r 0 1 ∗ linear combination of elements df, f ∈ C (J Y ) ⊂ O∞. Any element Φ ∈ (O∞) 0 1 yields a derivation ϑΦ ( f ) = Φ(df) of a real ring O∞. Since a module O∞ is 0 1 ∗ generated by elements df, f ∈ O∞, different elements of (O∞) provide different 0 1 ∗ 0 derivations of O∞, i.e., there is a monomorphism (O∞) → dO∞.Bythesame 0 0 formula, any derivation ϑ ∈ dO∞ sends df → ϑ(f ) and, since O∞ is generated by 1 0 elements df, it defines a morphism Φϑ : O∞ → O∞. Moreover, different derivations 0 1 ∗ ϑ provide different morphisms Φϑ . Thus, we have a monomorphism dO∞ → (O∞) 0 1 ∗ and, consequently, an isomorphism dO∞ = (O∞) .

∗ The proof of Theorem 1.8 gives something more. A DGA O∞ is the minimal ∗ 0 Chevalley–Eilenberg differential calculus O A over a real ring A = O∞ of smooth real functions on ﬁnite order jet manifolds of Y → X (Deﬁnition A.7). Let ϑφ, 0 1 ∗ ϑ ∈ dO∞, φ ∈ O∞, denote the interior product. Extended to a DGA O∞, the interior product obeys a rule

ϑ(φ ∧ σ) = (ϑφ) ∧ σ + (−1)|φ|φ ∧ (ϑσ).

1 0 Restricted to the coordinate chart (1.3), O∞ is a free O∞-module generated by λ i 0 1 ∗ 0 one-forms dx , θΛ. Since dO∞ = (O∞) , any derivation of a real ring O∞ takes a coordinate form ϑ = ϑλ∂ + ϑi ∂ + ϑi ∂Λ, λ i Λ i (1.16) 0<|Λ| where

∂Λ( j ) = ∂Λ j = δ j δΛ i yΣ i dyΣ i Σ 1.1 Inﬁnite Order Jet Formalism 7

λ i i up to permutations of multi-indices Λ and Σ [59]. Its coefﬁcients ϑ , ϑ , ϑΛ are local smooth functions of ﬁnite jet order possessing the transformation law

∂x λ ∂y i ∂y i ϑ λ = ϑμ,ϑ i = ϑ j + ϑμ, ∂xμ ∂y j ∂xμ ∂y i ∂y i ϑ i = Λ ϑ j + Λ ϑμ. (1.17) Λ j Σ ∂ μ |Σ|≤|Λ| ∂yΣ x

0 Any derivation ϑ (1.16)ofaringO∞ yields a derivation (called the Lie derivative) ∗ Lϑ of a DGA O∞ given by relations

Lϑ φ = ϑdφ + d(ϑφ), (1.18)

Lϑ (φ ∧ φ ) = Lϑ (φ) ∧ φ + φ ∧ Lϑ (φ ).

Remark 1.1 In particular, the total derivatives (1.14) are deﬁned as the local deriva- O0 φ = φ tions of ∞ and the corresponding local Lie derivatives dλ Ldλ of a DGA ∗ O∞.

1.2 Variational Bicomplex

∗,∗ In order to transform a bicomplex O∞ into the variational one, let us consider the ∗,n following two operators acting on O∞ [53, 147]. (i) There exists an R-module endomorphism

1 ρ = ρ ◦ h ◦ hn : O∗>0,n → O∗>0,n, (1.19) k k ∞ ∞ k>0 ρ(φ) = (− )|Λ|θ i ∧[ (∂Λφ)],φ∈ O>0,n, 1 dΛ i ∞ 0≤|Λ| possessing the following properties.

>0,n Theorem 1.9 For any φ ∈ O∞ ,aformφ − ρ(φ) is locally dH -exact on each coordinate chart (1.3). Theorem 1.10 The operator ρ (1.19) obeys a relation

>0,n−1 (ρ ◦ dH )(ψ) = 0,ψ∈ O∞ .

It follows from Theorems 1.9 and 1.10 that ρ (1.19) is a projector, i.e., ρ ◦ ρ = ρ. (ii) One deﬁnes a variational operator

∗,n ∗+1,n δ = ρ ◦ d : O∞ → O∞ . (1.20) 8 1 Calculus of Variations on Fibre Bundles

Theorem 1.11 The variational operator δ (1.20) is nilpotent, i.e., δ ◦ δ = 0, and it obeys a relation δ ◦ ρ = δ.

k,n Let us denote Ek = ρ(O∞ ).

∗ Deﬁnition 1.2 Provided with the operators dH , dV , ρ and δ, a DGA O∞ is decomposed into a variational bicomplex

......

dV 6 dV 6 dV 6 −δ 6 ρ 1,0 dH 1,1 dH 1,n 0 → O∞ → O∞ → ··· O∞ → E1 → 0

dV 6 dV 6 dV 6 −δ 6 0 dH 0,1 dH 0,n 0,n (1.21) 0 → R → O∞ → O∞ → ··· O∞ ≡ O∞ 6 6 6 d d d 0 → R → O0(X) → O1(X) → ··· On(X) → 0 6 6 6 00 0

This bicomplex possesses the following cohomology [55, 56, 122].

Theorem 1.12 The second row from the bottom and the last column of the variational bicomplex (1.21) make up the variational complex

δ δ 0 dH 0,1 dH 0,n 0 → R → O∞ −→ O∞ ···−→ O∞ −→ E1 −→ E2 −→··· . (1.22)

Its cohomology is isomorphic to the de Rham cohomology of a ﬁbre bundle Y :

Proof See the proof below.

Theorem 1.13 The rows of contact forms of the variational bicomplex (1.21)are exact sequences.

Proof See the proof below.

Let us note that the cohomology isomorphism (1.23) gives something more. Due to the relations dH ◦ h0 = h0 ◦ d and δ ◦ ρ = δ, we have a cochain morphism

n−1 d n d n+1 d n+2 ··· → O∞ → O∞ → O∞ → O∞ → ··· h h ρ ρ 0 ? 0 ? ? ? δ δ 0,n−1 dH 0,n ··· → O∞ → O∞ → E1 → E2 −→··· 1.2 Variational Bicomplex 9

∗ of the de Rham complex (1.8)ofaDGAO∞ to its variational complex (1.22). By virtue of Theorems 1.3 and 1.12, the corresponding homomorphism of their ∗ cohomology groups is an isomorphism. Then the splitting of a closed form φ ∈ O∞ in Theorem 1.4 leads to the following decompositions.

0,m Theorem 1.14 Any dH -closed form φ ∈ O ,m< n, is represented by a sum

m−1 φ = h0σ + dH ξ, ξ ∈ O∞ , (1.24) where σ is a closed m-form on Y . Any δ-closed form φ ∈ Ok,n is split into

0,n−1 φ = h0σ + dH ξ, k = 0,ξ∈ O∞ , (1.25) 0,n φ = ρ(σ) + δ(ξ), k = 1,ξ∈ O∞ , (1.26)

φ = ρ(σ) + δ(ξ), k > 1,ξ∈ Ek−1, (1.27) where σ is a closed (n + k)-form on Y .

1.2.1 Cohomology of the Variational Bicomplex

This section is devoted to the proof of Theorems 1.12 and 1.13. At ﬁrst, we obtain ∗ the corresponding cohomology of the DGA Q∞ (1.10). For this purpose, one can use abstract de Rham Theorem C.5 because, as was mentioned above, a paracompact ∞ 0 inﬁnite order jet manifold J Y admits the partition of unity by elements of Q∞,but 0 ∗ ∗ not O∞ [3, 143]. After that, we show that cohomology of O∞ ⊂ Q∞ equals that of ∗ Q∞ [55, 56, 122]. Let us start with the so called algebraic Poincaré lemma [108, 147].

Lemma 1.1 If Y is a contractible bundle Rn+p → Rn, the variational bicomplex (1.21) is exact at all terms, except R.

Proof The homotopy operators for dV , dH , δ and ρ are given by the formulas (5.72), (5.109), (5.84) in [108] and (4.5) in [147], respectively.

∗ ∗ ∞ Let Q∞ be a sheaf of germs of differential forms φ ∈ O∞ on J Y . It is decom- ∗,∗ ∗ ∗ posed into the variational bicomplex Q∞ . A DGA Q∞ of global sections of Q∞ also ∗,∗ is decomposed into the variational bicomplex Q∞ similar to the bicomplex (1.21). Let us consider a variational subcomplex

δ δ 0 dH 0,1 dH 0,n 0 → R → Q∞ −→ Q∞ ···−→ Q∞ −→ E1 −→ E2 −→··· (1.28)

∗,∗ of Q∞ and a subcomplexes of sheaves of contact forms

ρ k,0 dH k,1 dH k,n 0 → Q∞ −→ Q∞ ···−→ Q∞ −→ Ek → 0, k = 1,..., (1.29) 10 1 Calculus of Variations on Fibre Bundles

k,n where Ek = ρ(Q∞ ). By virtue of Lemma 1.1, these complexes are exact at all terms, except R. Since a paracompact space J ∞Y admits the partition of unity by elements of the 0 m,k 0 ring Q∞, the sheaves Q∞ of Q∞-modules are fine (Theorem C.8) and, consequently, acyclic (Theorem C.7). Let us show that the sheaves Ek also are fine [56, 61]. Though 0 the R-modules Γ(Ek>1) fail to be Q∞-modules [147], one can use the fact that the k,n 0 sheaves Ek>0 are projections ρ(Q∞ ) of sheaves of Q∞-modules. Let {Ui }i∈I be a ∞ 0 locally finite open cover of J Y and { fi ∈ Q∞} the associated partition of unity. ∞ k,n For any open subset U ⊂ J Y and any section ϕ of the sheaf Q∞ over U, let us put k,n gi (ϕ) = fi ϕ. The endomorphisms gi of Q∞ yield the R-module endomorphisms

ρ = ρ ◦ : −→in Qk,n −→gi Qk,n −→ gi gi Ek ∞ ∞ Ek of sheaves Ek . They possess the properties required for Ek to be a ﬁne sheaf. Indeed, ∈ ⊂ for each i I, supp fi Ui provides a closed set such that gi is zero outside this set, while a sum gi is the identity morphism. i∈I Consequently, all sheaves, except R, in the complexes (1.28) and (1.29) are acyclic. Therefore, these complexes are resolutions of the constant sheaf R and the zero sheaf on J ∞Y , respectively. Let us consider the corresponding subcomplexes

δ δ 0 dH 0,1 dH 0,n 0 → R → Q∞ → Q∞ ···→ Q∞ → Γ(E1) → Γ(E2) →··· , (1.30) ρ k,0 dH k,1 dH k,n 0 → Q∞ → Q∞ ···→ Q∞ → Γ(Ek ) → 0, k = 1,..., (1.31)

∗ of DGA Q∞. In accordance with abstract de Rham Theorem C.6, cohomology of the complex (1.30) equals the cohomology of J ∞Y with coefﬁcients in the constant sheaf R, while the complex (1.31) is exact. Since Y is a strong deformation retract of J ∞Y , cohomology of the complex (1.30) equals the de Rham cohomology of Y (Remark B.15). Thus, the following has been proved.

Lemma 1.2 The cohomology of the variational complex (1.30) equals the de Rham cohomology of a ﬁbre bundle Y . All the complexes (1.31) are exact.

Now, let us show the following.

∗ ∗ ∗ Lemma 1.3 A subalgebra O∞ ⊂ Q∞ has the same dH - and δ-cohomology as Q∞.

Let the common symbol D stand for dH and δ. Bearing in mind the decompositions ∗ (1.24)–(1.27), it suffices to show that, if an element φ ∈ O∞ is D-exact in an algebra ∗ ∗ Q∞, then it is so in an algebra O∞. Lemma 1.1 states that, if Y is a contractible bundle and a D-exact form φ on J ∞Y is ∗ ∗ ∞ of finite jet order [φ] (i.e., φ ∈ O∞), there exists a differential form ϕ ∈ O∞ on J Y such that φ = Dϕ. Moreover, a glance at the homotopy operators for dH and δ shows that the jet order [ϕ] of ϕ is bounded by an integer N([φ]), depending only on the 1.2 Variational Bicomplex 11 jet order of φ. Let us call this fact the finite exactness of an operator D. Lemma 1.1 ∞ shows that the finite exactness takes place on J Y |U over any domain U ⊂ Y .Let us prove the following.

Lemma 1.4 Given a family {Uα} of disjoint open subsets of Y , let us suppose that ∞ | the ﬁnite exactness takes place on J Y Uα over every subset Uα from this family. ∞ Then it is true on J Y over the union ∪ Uα of these subsets. α

φ ∈ O∗ ∞ (π ∞)−1(∪ ) Proof Let ∞ be a D-exact form on J Y . The ﬁnite exactness on 0 Uα φ = ϕ (π ∞)−1( ) [ϕ ] < ([φ]) holds since D α on every 0 Uα and α N . Lemma 1.5 Suppose that the ﬁnite exactness of an operator D takes place on J ∞Y over open subsets U, V of Y and their non-empty overlap U ∩ V . Then it also is true ∞ on J Y |U∪V .

∗ ∞ Proof Let φ = Dϕ ∈ O∞ be a D-exact form on J Y . By assumption, it can be ϕ (π ∞)−1( ) ϕ (π ∞)−1( ) ϕ brought into the forms D U on 0 U and D V on 0 V , where U and ϕV are differential forms of bounded jet order. Let us consider their difference ϕ − ϕ (π ∞)−1( ∩ ) U V on 0 U V .ItisaD-exact form of bounded jet order

[ϕU − ϕV ] < N([φ]) which, by assumption, can be written as ϕU − ϕV = Dσ where σ also is of bounded jet order [σ] < N(N([φ])). Lemma 1.6 below shows that σ = σU + σV where σU σ (π ∞)−1( ) (π ∞)−1( ) and V are differential forms of bounded jet order on 0 U and 0 V , respectively. Then, putting

ϕ |U = ϕU − DσU ,ϕ|V = ϕV + DσV ,

φ ϕ (π ∞)−1( ) ϕ (π ∞)−1( ) we have the form , equal to D U on 0 U and D V on 0 V , respec- ϕ −ϕ (π ∞)−1( ∩ ) φ = ϕ tively. Since the difference U V on 0 U V vanishes, we obtain D (π ∞)−1( ∪ ) on 0 U V where ϕ | = ϕ ϕ = U U ϕ | = ϕ V V is of bounded jet order [ϕ ] < N(N([φ])).

∗ Lemma 1.6 Let U and V be open subsets of a bundle Y and σ ∈ G∞ a differential form of bounded jet order on

(π ∞)−1( ∩ ) ⊂ ∞ . 0 U V J Y

Then σ is decomposed into a sum σU +σV of differential forms σU and σV of bounded (π ∞)−1( ) (π ∞)−1( ) jet order on 0 U and 0 V , respectively. 12 1 Calculus of Variations on Fibre Bundles

Proof By taking the smooth partition of unity on U ∪ V subordinate to the cover {U, V } and passing to the function with support in V , one gets a smooth real function f on U ∪ V which equals 0 on a neighborhood of U \ V and 1 on a neighborhood of \ ∪ (π ∞)∗ (π ∞)−1( ∪ ) V U in U V .Let 0 f be the pull-back of f onto 0 U V . A differential ((π ∞)∗ )σ (π ∞)−1( ) form 0 f equals 0 on a neighborhood of 0 U and, therefore, can be (π ∞)−1( ) σ extended by 0 to 0 U . Let us denote it U . Accordingly, a differential form ( − (π ∞)∗ )σ σ (π ∞)−1( ) σ = σ + σ 1 0 f has an extension V by0to 0 V . Then U V is a desired decomposition because σU and σV are of the jet order which does not exceed that of σ .

To prove the finite exactness of D on J ∞Y , it remains to choose an appropriate cover of Y . A smooth manifold Y admits a countable cover {Uξ } by domains Uξ , ξ ∈ N, and its refinement {Uij}, where j ∈ N and i runs through a finite set, such that Uij ∩ Uik =∅, j = k [67]. Then Y has a finite cover {Ui =∪j Uij}. Since the finite exactness of an operator D takes place over any domain Uξ , it also holds over any member Uij of the refinement {Uij} of {Uξ } and, in accordance with Lemma 1.4, over any member of the finite cover {Ui } of Y . Then by virtue of Lemma 1.5,the finite exactness of D takes place on J ∞Y over Y . Similarly, one can show that:

k,n Lemma 1.7 Restricted to O∞ , an operator ρ remains exact.

Lemmas 1.2 and 1.7 result in Theorems 1.12 and 1.13.

1.3 Lagrangian Formalism

Lagrangian theory on ﬁbre bundles is formulated in terms of a variational bicomplex (Deﬁnition 1.2)asfollows.

Definition 1.3 A finite order Lagrangian is defined as horizontal densities

0,n L = L ω ∈ O∞ (1.32) of the variational complex (1.22) (see the notation (B.32)).

Deﬁnition 1.4 The variational operator δ : O0,n → ,δ= E = E θ i ∧ ω = (− )|Λ| (∂ΛL )θ i ∧ ω, ∞ E1 L L i 1 dΛ i (1.33) 0≤|Λ| is called the Euler–Lagrange operator, and its coefﬁcients Ei (1.33) are said to be the variational derivatives. 1.3 Lagrangian Formalism 13

The Lagrangian L (1.32)istermedvariationally trivial if it is δ-closed, i.e., δ(L) = 0. The following is a corollary of Theorem 1.14. Theorem 1.15 The ﬁnite order Lagrangian L (1.32) is variationally trivial if and only if

0,n−1 L = h0σ + dH ξ, ξ ∈ O∞ , where σ is a closed n-form on Y .

In particular, a variationally trivial Lagrangian necessarily is dH -exact if the de n ( ) Rham cohomology group HDR Y of Y is trivial. A variational operator δ : → ,δ(E ) = [∂ΛE θ j ∧ θ i + (− )|Λ|θ j ∧ (∂ΛE θ i )]∧ω, E1 E2 j i Λ 1 dΛ j i 0≤|Λ| is called the Helmholtz–Sonin map. An element E ∈ E1 is said to be the Euler– Lagrange-type operator if it satisﬁes the Helmholtz condition δ(E ) = 0, i.e., if it is δ-closed. The following is a corollary of Theorem 1.14, too.

Theorem 1.16 An Euler–Lagrange-type operator E ∈ E1 reads

0,n E = δL + ρ(σ), L ∈ O∞ , where σ is a closed (n + 1)-form on Y .

In particular, any Euler–Lagrange-type operator E ∈ E1 is the Euler–Lagrange E = δ n+1( ) one L (1.33) if the de Rham cohomology group HDR Y of Y is trivial. For instance, this is the case of an afﬁne bundle Y → X (Remark B.14). Certainly, every Euler–Lagrange-type operator locally is an Euler–Lagrange operator over some open subset of Y . Remark 1.2 Theorems 1.15 and 1.16 provide a solution of the so called global inverse problem of the calculus of variations. This solution agrees with that of [2] obtained by computing cohomology of a variational sequence of bounded jet order, but without minimizing an order of a Lagrangian (see also particular results of [87, 149]). A solution of the global inverse problem of the calculus of variations in the case of a ∗ graded differential algebra Q∞ (1.10) has been found in [3, 143] (Theorem 1.2). A glance at the expression (1.33) shows that, if a Lagrangian L (1.32)isofr-order, its Euler–Lagrange operator EL is of 2r-order. Its kernel

2r EL = Ker EL ⊂ J Y (1.34) is called the Euler–Lagrange equation. It is locally given by the equalities 14 1 Calculus of Variations on Fibre Bundles E = (− )|Λ| (∂ΛL ) = . i 1 dΛ i 0 (1.35) 0≤|Λ|

However, it may happen that the Euler–Lagrange equation (1.34) is not a differential equation in the strict sense of Deﬁnition B.12 because Ker EL need not be a closed subbundle of J 2r Y → X. The Euler–Lagrange equation (1.35) traditionally is derived from the variational formula dL = δL − dH ΞL (1.36) of the calculus of variations. In formalism of a variational bicomplex, this formula is a corollary of Theorem 1.13.

Theorem 1.17 The exactness of the row of one-contact forms of the variational 1,n bicomplex (1.21)atatermO∞ relative to the projector ρ provides a global R- module splitting

1,n 1,n−1 O∞ = E1 ⊕ dH (O∞ ).

In particular, any Lagrangian L admits the decomposition (1.36).

n Deﬁned with accuracy to a dH -closed summand, a form ΞL ∈ O∞ in the variational formula (1.36) reads λ μλ i λνs ...ν i Ξ = L +[(∂ L − dμ F )θ + F 1 θ ]∧ωλ, (1.37) L i i i νs ...ν1 s=1 ν ...ν ν ...ν μν ...ν ν ...ν k 1 = ∂ k 1 L − k 1 + ψ k 1 , = , ,..., Fi i dμ Fi i k 2 3

ν ...ν (ν ν )...ν ψ k 1 ψ k k−1 1 = where i are local functions such that i 0. It is readily observed that the form ΞL (1.37) possesses the following properties:

• h0(ΞL ) = L, i • h0(ϑdΞL ) = ϑ Ei ω for any derivation ϑ (1.16).

Consequently, ΞL is a Lepage equivalent of a Lagrangian L.

Remark 1.3 Following the terminology of ﬁnite order jet formalism [53, 64, 88, 89], n we call an exterior n-form ρ ∈ O∞ the Lepage form if, for any derivation ϑ (1.16), λ i the density h0(ϑdρ) depends only on the restriction of ϑ to a derivation ϑ ∂λ +ϑ ∂i ∞ 0 of the subring C (Y ) ⊂ O∞. The Lepage forms constitute a real vector space. In particular, closed n-forms and (2 ≤ k)-contact n-forms are Lepage forms. Given a Lagrangian L, a Lepage form ρ is called the Lepage equivalent of L if h0(ρ) = L. Any Lepage form ρ is a Lepage equivalent of the Lagrangian h0(ρ). Conversely, any r-order Lagrangian possesses a Lepage equivalent of (2r − 1)-order [64]. The 1.3 Lagrangian Formalism 15

Lepage equivalents of a Lagrangian L constitute an afﬁne space modelled over a ν ...ν ψ k 1 = vector space of contact Lepage forms. In particular, one can locally put i 0 in the formula (1.37).

∗ Deﬁnition 1.5 We agree to call a pair (O∞Y, L) the Lagrangian system on a ﬁbre bundle Y .

Hereafter, base manifolds X of ﬁbre bundles are assumed to be oriented and connected.

Chapter 2 Noether’s First Theorem

We here are concerned with Lagrangian theory on ﬁbre bundles (Chap.1). In this case, Noether’s ﬁrst theorem (Theorem 2.7) and Noether’s direct second theorem (Theorem 2.6) are corollaries of the global variational formula (1.36).

2.1 Lagrangian Symmetries

Noether’s theorems deal with infinitesimal transformations of Lagrangian systems. ∗ Definition 2.1 Given a Lagrangian system (O∞Y, L) (Definition1.5), its infinites- 0 imal transformations are defined to be contact derivations of a real ring O∞Y [59, 61]. 0 The derivation ϑ ∈ dO∞Y (1.16)istermedcontact if the Lie derivative Lϑ (1.18) ∗ along ϑ preserves an ideal of contact forms of a DGA O∞Y , i.e., the Lie derivative Lϑ of a contact form is a contact form. Theorem 2.1 The derivation ϑ (1.16) is contact if and only if it takes a form ϑ = υλ∂ + υi ∂ + [ (υi − i υμ) + i υμ]∂Λ. λ i dΛ yμ yμ+Λ i (2.1) 0<|Λ|

Proof The expression (2.1) results from a direct computation similar to that of the ﬁrst part of Bäcklund’s theorem [76]. One can then justify that local functions (2.1) satisfy the transformation law (1.17). Comparing the expressions (2.1) and (B.81) enables one to regard the contact derivation ϑ (2.1) as the inﬁnite order jet prolongation

ϑ = J ∞υ (2.2)

© Atlantis Press and the author(s) 2016 17 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_2 18 2 Noether’s First Theorem of its restriction λ i υ = υ ∂λ + υ ∂i (2.3) to a ring C∞(Y ). Since coefﬁcients υλ and υi of υ (2.3) generally depend on jet i coordinates yΛ of bounded jet order 0 < |Λ|≤N, one calls υ (2.3)thegeneralized vector ﬁeld. It can be represented as a section of the pull-back bundle

J N Y × TY → J N Y. Y

Let ϑ and ϑ be contact derivations (2.1) whose restrictions to a real ring C∞(Y ) are generalized vector ﬁelds υ and υ, respectively. Certainly, their Lie bracket [ϑ, ϑ ] is a contact derivation. Its restriction to C∞(Y ) reads

λ λ i i [υ,υ ]J =[ϑ(υ ) − ϑ (υ )]∂λ +[ϑ(υ ) − ϑ (υ )]∂i . (2.4)

We call it the bracket of generalized vector ﬁelds [υ,υ ]J. It obeys a relation

∞ ∞ ∞ [J υ, J υ ]=J ([υ,υ ]J). (2.5)

If υ and υ are vector fields on Y , their bracket (2.4) is the familiar Lie one. The contact derivation ϑ (2.1)issaidtobeprojectable, if the generalized vector λ λ field υ (2.3) projects onto a vector field υ ∂λ on X, i.e., its components ϑ depend only on coordinates on X. In particular, it is readily observed that the bracket (2.4) of projectable generalized vector fields also is projectable. Every contact derivation ϑ (2.1) admits the canonical splitting ϑ = ϑ + ϑ = υλ + ∞υ = υλ +[υi ∂ + υi ∂Λ], H V dλ J V dλ V i dΛ V i (2.6) 0<|Λ| υ = υ + υ = υλ + (υi − i υμ)∂ = υλ + υi ∂ , H V dλ yμ i dλ V i (2.7) into the horizontal and vertical parts ϑH and ϑV , respectively [59].

Theorem 2.2 Any vertical contact derivation ϑ = ∞υ = υi ∂ + υi ∂Λ,υ= υi ∂ , J i dΛ i i 0<|Λ| obeys the relations

ϑdH φ =−dH (ϑφ), (2.8) ∗ Lϑ (dH φ) = dH (Lϑ φ), φ ∈ O∞Y. (2.9)

Proof It is easily justiﬁed that, if φ and φ satisfy the relation (2.8), then φ ∧ φ does i well. Then it sufﬁces to prove the relation (2.8) when φ is a function and φ = θΛ. 2.1 Lagrangian Symmetries 19

The result follows from the equalities

i i i i λ i λ i ϑθΛ = υΛ, dH (υΛ) = υλ+Λdx , dH θλ = dx ∧ θλ+Λ, ◦ vi ∂Λ = vi ∂Λ ◦ . dλ Λ i Λ i dλ

The relation (2.9) is a corollary of the equality (2.8).

∗ Given a Lagrangian system (O∞Y, L), let us consider the Lie derivative Lϑ L (1.18) of its Lagrangian L along the contact derivation ϑ (1.9). The global decomposition (1.36) results in the following corresponding splitting of Lϑ L.

0,n Theorem 2.3 Given a Lagrangian L ∈ O∞ Y , its Lie derivative Lϑ L along the contact derivation ϑ (2.6) fulﬁls the ﬁrst variational formula

Lϑ L = υV δL + dH (h0(ϑΞL )) + L dV (υH ω), (2.10) where ΞL is the Lepage equivalent (1.37) of L and h0 is the horizontal projection (1.13).

Proof The formula (2.10) comes from the variational formula (1.36) and the relations (2.8)–(2.9)asfollows:

Lϑ L = ϑdL + d(ϑL) =[ϑV dL − dV L ∧ υH ω]+[dH (υH L) +

dV (L υH ω)]=ϑV dL + dH (υH L) + L dV (υH ω) =

υV δL − ϑV dH ΞL + dH (υH L) + L dV (υH ω) =

υV δL + dH (ϑV ΞL + υH L) + L dV (υH ω), where

ϑV ΞL = h0(ϑV ΞL ), υH L = h0(υH ΞL ) since ΞL − L is a one-contact form and υH = ϑH .

Deﬁnition 2.2 The generalized vector ﬁeld υ (2.7)onY is called the Lagrangian symmetry (or, shortly, the symmetry) of a Lagrangian L if the Lie derivative LJ ∞υ L ∞ of L along the contact derivation J υ (2.2)isdH -exact, i.e.,

LJ ∞υ L = dH σ, (2.11) where σ is a horizontal (n − 1)-form.

Remark 2.1 Certainly, the jet prolongation J ∞υ of a generalized vector field υ in the expression (2.11) always is of finite jet order because a Lagrangian L is of finite order (Sect. 3.3). Therefore, we further use the notation J ∗υ for the finite order jet prolongation of υ whose order however is nor specified. 20 2 Noether’s First Theorem

Theorem 2.4 A glance at the expression (2.10) shows the following. (i) A generalized vector field υ is a Lagrangian symmetry only if it is projectable. (ii) Any projectable generalized vector field is a symmetry of a variationally trivial Lagrangian. (iii) A projectable generalized vector field υ is a Lagrangian symmetry if and only if its vertical part υV (2.7)iswell. (iv) A projectable generalized vector field υ is a symmetry if and only if the density υV δLisdH -exact. Remark 2.2 In accordance with the standard terminology, symmetries represented by generalized vector fields (2.3) are called generalized symmetries because they depend on derivatives of variables. Generalized symmetries of differential equations and Lagrangian systems have been intensively investigated [25, 40, 59, 76, 85, 108]. Accordingly, by symmetries one means only those represented by vector fields υ = u on Y . We agree to call them classical symmetries.

Remark 2.3 Owing to the relation (2.5), the bracket (2.4) of Lagrangian symmetries is a Lagrangian symmetry. It follows that symmetries constitute a real Lie algebra ∗ GL with respect to this bracket, and their jet prolongation υ → J υ (2.2) provides a 0 monomorphism of this Lie algebra to the Lie algebra dO∞Y (1.16). Remark 2.4 Let υ be a classical symmetry of a Lagrangian L, i.e., it is a vector field on Y . Then the relation LJ ∗υ EL = δ(LJ ∗υ L) (2.12) holds [53, 108]. It follows that υ also is a symmetry of the Euler–Lagrange operator EL of L (Definition4.5), i.e., LJ ∗υ EL = 0, and as a consequence it is an infinitesimal symmetry of the Euler–Lagrange equation EL (1.34) (Definition 4.4). However, the equality (2.12) fails to be true in the case of generalized symmetries.

2.2 Gauge Symmetries: Noether’s Direct Second Theorem

As was mentioned above, the notion of gauge symmetries comes from Yang–Mills gauge theory on principal bundles (Sect.8.2). It is generalized to Lagrangian theory on an arbitrary ﬁbre bundle Y → X as follows [9, 10].

Deﬁnition 2.3 Let E → X be a vector bundle and E(X) a C∞(X)-module of sections of E → X.Letζ be a linear differential operator on E(X) (Deﬁnition A.3) with values into a vector space GL of symmetries of a Lagrangian L (Remark2.3). Elements uξ = ζ(ξ) (2.13) of Im ζ are termed the gauge symmetries of a Lagrangian L parameterized by sections ξ of E → X. The latter are regarded as the gauge parameters. 2.2 Gauge Symmetries: Noether’s Direct Second Theorem 21

Remark 2.5 A differential operator ζ in Deﬁnition2.3 takes its values into a vector ∞ 0 space GL as a subspace of a C (X)-module dO∞Y seen as a real vector space. The differential operator ζ is assumed to be at least of ﬁrst order (Remark2.7).

Equivalently, the gauge symmetry (2.13) is given by a section ζ of a ﬁbre bundle

(J r Y × J m E) × TY → J r Y × J m E Y Y Y (Deﬁnition B.14) such that uξ = ζ(ξ) = ζ ◦ξ for any section ξ of E → X. Hence, it is a generalized vector ﬁeld uζ on a bundle product Y ×X E represented by a section of the pull-back bundle

J k (Y × E) × T (Y × E) → J k (Y × E), k = max(r, m), X Y X X which lives in TY ⊂ T (Y ×X E). This generalized vector ﬁeld yields the con- ∞ 0 tact derivation J uζ (2.2) of a real ring O∞[Y ×X E] which obeys the following condition.

0,n 0,n • Given a Lagrangian L ∈ O∞ E ⊂ O∞ [Y ×X E], let us consider its Lie derivative

∞ ∗ ∗ = + ( ) LJ uζ L J uζ dL d J uζ L (2.14)

∗ where d is the exterior differential on O∞[Y ×X E]. Then, for any section ξ of ∗ → ξ ∗ E X, the pull-back LJ uζ L is dH -exact. It follows from the first variational formula (2.10) for the Lie derivative (2.14) that the above mentioned condition holds only if uζ is projected onto a generalized vector field on Y and, in this case, if and only if the density (uζ )V EL is dH -exact. Thus, we come to the following equivalent definition of gauge symmetries.

Definition 2.4 Let E → X be a vector bundle. A gauge symmetry of a Lagrangian L parameterized by sections ξ of E → X is defined as a generalized vector field u on Y ×X E such that: ∞ 0 (i) a contact derivation ϑ = J u of a ring O∞[Y ×X E] vanishes on a subring 0 O∞ E, a ∞ (ii) a generalized vector field u is linear in coordinates χΛ on J E, and it is projected onto a generalized vector field on E, i.e., it takes a form ⎛ ⎞ ⎛ ⎞ = ⎝ λΛ( μ)χ a ⎠ ∂ + ⎝ iΛ( μ, j )χ a ⎠ ∂ , u ua x Λ λ ua x yΣ Λ i (2.15) 0≤|Λ|≤m 0≤|Λ|≤m

(iii) the vertical part of u (2.15) obeys the equality

uV δL = dH σ. (2.16) 22 2 Noether’s First Theorem

Theorem 2.5 By virtue of item (iii) of Deﬁnition2.4,u(2.15) is a gauge symmetry if and only if its vertical part is so.

Gauge symmetries possess the following particular properties. (i) Let E → X be another vector bundle and ζ a linear E(X)-valued differential ∞ operator on a C (X)-module E (X) of sections of E → X. Then uζ (ξ ) = (ζ ◦ ζ )(ξ ) also is a gauge symmetry of L parameterized by sections ξ of E → X.It factorizes through the gauge symmetries uχ (2.13). (ii) The conserved symmetry current Ju (2.21) associated to a gauge symmetry in accordance with Noether’s ﬁrst theorem (Theorem 2.7) is reduced to a superpotential (Theorem 2.8). (iii) Noether’s direct second theorem (Theorem2.6) associates to a gauge symmetry of a Lagrangian L the Noether identities (NI) of its Euler–Lagrange operator. Theorem 2.6 Let u (2.15) be a gauge symmetry of a Lagrangian L, then its Euler– Lagrange operator δL obeys the NI (2.17).

Proof The density (2.16) is variationally trivial and, therefore, its variational derivatives with respect to variables χ a vanish, i.e., E = (− )|Λ| [( iΛ − i λΛ)E ]= η( i − i λ)Λ E = a 1 dΛ ua yλua i ua yλua dΛ i 0 (2.17) 0≤|Λ| 0≤|Λ|

(see Remark7.6 for the notation). In accordance with Deﬁnition D.1, the equalities (2.17) are the NI for the Euler–Lagrange operator δL.

Remark 2.6 If the gauge symmetry u (2.15) is of second jet order in gauge parameters, i.e., = ( i χ a + iμχ a + iνμχ a )∂ , uV ua ua μ ua νμ i (2.18) the corresponding NI (2.17) take a form

i E − ( iμE ) + ( iνμE ) = . ua i dμ ua i dνμ ua i 0 (2.19)

Let us note that the NI (2.17) need not be independent (Sect. 6.2).

Remark 2.7 A glance at the expression (2.19) shows that, if a gauge symmetry is independent of derivatives of gauge parameters (i.e., a differential operator ζ in Deﬁnition2.3 is of zero order), then all variational derivatives of a Lagrangian equals zero, i.e., this Lagrangian is variationally trivial. Therefore, such gauge symmetries usually are not considered.

Remark 2.8 The notion of gauge symmetries can be generalized as follows. Let a differential operator ζ in Deﬁnition2.3 need not be linear. Then elements of Im ζ are called the generalized gauge symmetry. However, Noether’s direct second The- orem2.6 is not relevant to generalized gauge symmetries because, in this case, an Euler–Lagrange operator satisﬁes the identities depending on gauge parameters. 2.2 Gauge Symmetries: Noether’s Direct Second Theorem 23

It follows from Noether’s direct second Theorem2.6 that gauge symmetries of Lagrangian field theory characterize its degeneracy. A problem is that any Lagrangian possesses gauge symmetries and, therefore, one must separate them into the trivial and nontrivial ones. Moreover, gauge symmetries can be reducible, i.e., Ker ζ = 0. Another problem is that gauge symmetries need not form an algebra [48, 60, 63]. The Lie bracket [uφ, uφ ] of gauge symmetries uφ, uφ ∈ Im ζ is a symmetry, but it need not belong to Im ζ . To solve these problems, we follow a different definition of gauge symmetries (Definitions 7.4–7.5) as those associated to nontrivial NI by means of Noether’s inverse second Theorem 7.9. They are parameterized by Grassmann- graded ghosts, but not gauge parameters (Remark7.7).

2.3 Noether’s First Theorem: Conservation Laws

∗ Let (O∞Y, L) be a Lagrangian system (Deﬁnition 1.5). The following is Noether’s ﬁrst theorem.

Theorem 2.7 Let the generalized vector field υ (2.7) be a symmetry of a Lagrangian L (Definition2.2), i.e., let it obey the equality (2.11). Then the first variational formula (2.10) restricted to the Euler–Lagrange equation EL (1.34) takes a form of the weak conservation law on-shell

0 ≈−dH (−h0(ϑΞL ) + σ) ≈−dH Jυ (2.20) of a symmetry current

μ Jυ = Jυ ωμ =−h0(ϑΞL ) + σ (2.21) along a generalized vector ﬁeld υ. It is called the Lagrangian conservation law.

The weak conservation law (2.20) leads to a differential conservation law

λ ∂λ(Jυ ◦ s) = 0 on classical solutions s of the Euler–Lagrange equation (1.35) (Deﬁnition B.13). This differential conservation law, in turn, yields an integral conservation law ∗ s Jυ = 0, (2.22)

∂ M where M is an n-dimensional compact submanifold of X with a boundary ∂ M.

Remark 2.9 Of course, the symmetry current Jυ (2.21) is deﬁned with the accuracy to a dH -closed term. For instance, if we choose a different Lepage equivalent ΞL 24 2 Noether’s First Theorem

(1.37) in the variational formula (1.36), the corresponding symmetry current differs from Jυ (2.21)inadH -exact term. This term is independent of a Lagrangian, and it does not contribute to the integral conservation law (2.22).

Obviously, the symmetry current Ju (2.21) is linear in a generalized vector ﬁeld υ. Therefore, one can consider a superposition of symmetry currents

Jυ + Jυ = Jυ+υ , Jcυ = cJυ , c ∈ R, associated to different symmetries υ and that of weak conservation laws (2.20). A symmetry υ of a Lagrangian L is called exact if the Lie derivative LJ ∗υ L of L ∗ along J υ vanishes, i.e., LJ ∗υ L = 0. In this case, the ﬁrst variational formula (2.10) takes a form

∗ 0 = υV δL + dH (h0(J υΞL )), and results in the weak conservation law (2.20):

∗ 0 ≈ dH (h0(J υΞL )) ≈−dH Jυ , (2.23)

∗ of the symmetry current Jυ =−h0(J υΞL ). i For instance, let υ = υ ∂i be a vertical generalized vector ﬁeld on Y → X.Ifit is an exact symmetry of L, the weak conservation law (2.23) takes a form

∗ 0 ≈−dH (J υΞL ). (2.24)

Deﬁnition 2.5 The equality (2.24) is called the Noether conservation law of a Noether current ∗ J =−J υΞL . (2.25)

If a Lagrangian L admits the gauge symmetry u (2.15), the weak conservation law (2.20) of the corresponding symmetry current Ju (2.21) holds. We call it the gauge conservation law. Because gauge symmetries depend on parameter variables and their jets, all gauge conservation laws possess the following peculiarity.

Theorem 2.8 If u (2.15) is a gauge symmetry of a Lagrangian L, the corresponding conserved symmetry current Ju (2.21) along u takes a form

μ νμ Ju = W + dH U = (W + dνU )ωμ, (2.26)

νμ where the term W vanishes on-shell, and U = U ωνμ is a horizontal (n − 2)-form.

Proof Theorem 2.8 is a particular variant of Theorem7.11.

Deﬁnition 2.6 AtermU in the expression (2.26) is called the superpotential. 2.3 Noether’s First Theorem: Conservation Laws 25

If a symmetry current admits the decomposition (2.26), one says that it is reduced to a superpotential [39, 61, 66, 128]. If a symmetry current J reduces to a superpotential, the integral conservation law (2.22) becomes tautological.

Remark 2.10 Theorem 2.8 generalizes the result in [66] for gauge symmetries u λ λ λ whose gauge parameters χ = u are components of a projection u ∂λ of u onto X.

Remark 2.11 In mechanics on a conﬁguration bundle over R (Chap. 4), the whole conserved current (2.26) along a gauge symmetry vanishes on-shell (Theorem4.7).

Sometimes, Theorem2.8 is called Noether’s third theorem on a superpotential.

Chapter 3 Lagrangian and Hamiltonian Field Theories

We focus our attention on first order Lagrangian and polysymplectic Hamiltonian theories on fibre bundles since the most of relevant field models is of this type [53, 61, 133]. These theories are equivalent in the case of hyperregular Lagrangians (Definition3.1), and one can state comprehensive relations between them for semiregular and almost regular Lagrangians (Sect. 3.6). Moreover, polysymplectic Hamil- tonian formalism on the Legendre bundle Π (3.6) is equivalent to particular first order Lagrangian theory on a configuration space Π that enables us to describe symmetries of covariant Hamiltonian theory similarly to those in Lagrangian formalism (Sect. 3.7).

3.1 First Order Lagrangian Formalism

Let π : Y → X be a fibre bundle over a (1 < n)-dimensional base X (see Chap. 4 for the case of n = 1). Let Y be provided with an atlas of bundle coordinates (xλ, yi ). In Lagrangian formalism on a configuration space. Y → X (Sect.1.3), a first order Lagrangian is defined as a horizontal density

n L = L ω : J 1Y → ∧ T ∗ X (3.1) on the first order jet manifold J 1Y of Y . Its pull-back onto higher order jet manifolds J ∗Y can be considered (Definition 1.3). The corresponding second-order Euler–Lagrange operator (1.33) (Definition 1.4) reads

2 ∗ n ∗ EL : J Y → T Y ∧ (∧ T X), E = (∂ L − π λ)θ i ∧ ω, πλ = ∂λL . L i dλ i i i (3.2)

© Atlantis Press and the author(s) 2016 27 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_3 28 3 Lagrangian and Hamiltonian Field Theories

Its kernel deﬁnes the second order Euler–Lagrange equation (1.35):

(∂ − ∂λ)L = , i dλ i 0 (3.3) on a conﬁguration space Y → X. Given a ﬁrst order Lagrangian L, let us consider the vertical tangent morphism (B.17): 1 1 n ∗ VL : VY J Y −→ VY J Y × ∧ T X (3.4) Y Y

1 over Y to L (3.1), where VY J Y denotes the vertical tangent bundle (Deﬁnition B.3) of J 1Y → Y . Since J 1Y → Y is an afﬁne bundle modelled over the vector bundle (B.47), we have the canonical vertical splitting

1 1 ∗ VY J Y = J Y ×(T X ⊗ VY). Y Y

Accordingly, the vertical tangent map VL (3.4) yields a linear morphism

n J 1Y ×(T ∗ X ⊗ VY) −→ J 1Y ×(∧ T ∗ X) Y Y Y over J 1Y and the corresponding morphism

n L : J 1Y → V ∗Y ⊗(∧ T ∗ X) ⊗ TX (3.5) Y Y over Y , where V ∗Y is the vertical cotangent bundle of Y → X (Deﬁnition B.4). The latter is termed the Legendre map associated to a Lagrangian L. A ﬁbre bundle n Π = V ∗Y ⊗(∧ T ∗ X) ⊗ TX (3.6) Y Y over Y is named the Legendre bundle. It is a composite bundle Π → Y → X, called the composite Legendre bundle and provided with the adapted holonomic coordinates ( λ, i , λ) x y pi possessing transition functions ε j λ λ ∂x ∂y ∂x μ p = det ν p . (3.7) i ∂x ∂ yi ∂xμ j

λ ◦ = π λ With respect to these coordinates, the Legendre map (3.5) reads pi L i . Remark 3.1 There is the canonical isomorphism

− Π = ∗ ∧(n∧1 ∗ ), ( λ) → λ i ω , V Y T X pi pi dy λ (3.8) Y where {dyi } are ﬁbre bases for the vertical cotangent bundle V ∗Y of Y → X. 3.1 First Order Lagrangian Formalism 29

Certainly, the Legendre map (3.5) need not be a bundle isomorphism. Its range

1 NL = L(J Y ) ⊂ Π (3.9) is termed the Lagrangian constraint space.

Deﬁnition 3.1 A Lagrangian L is said to be: • hyperregular if the Legendre map L is a diffeomorphism; • (∂μ∂ν L ) = regular if L is a local diffeomorphism, i.e., det i j 0; −1 • semiregular if the inverse image L (q) of any point q ∈ NL is a connected submanifold of J 1Y ; • almost regular if the Lagrangian constraint space NL is a closed imbedded subbundle iN : NL → Π of a Legendre bundle Π → Y and a Legendre map

1 L : J Y → NL (3.10)

is a fibred manifold (Definition B.1) with connected fibres (i.e., a Lagrangian L is semiregular).

Given a ﬁrst order Lagrangian L, its Lepage equivalents ΞL (1.37) in the variational formula (1.36) read

Ξ = + (π λ − ψμλ)θ i ∧ ω + ψλμθ i ∧ ω , L L i dμ i λ i μ λ (3.11)

ψμλ =−σ λμ where i i are skew-symmetric local functions on Y . These Lepage equivalents constitute an afﬁne space modelled over a vector space of dH -exact one-contact Lepage forms

ρ =− ψμλθ i ∧ ω + ψλμθ i ∧ ω . dμ i λ i μ λ

Let us choose the Poincaré–Cartan form

= L ω + π λθ i ∧ ω HL i λ (3.12) as the origin of this afﬁne space because it is deﬁned on J 1Y . In a general setting, one also considers other Lepage equivalents of L [91, 92]. The Poincaré–Cartan form (3.12) takes its values into a subbundle

n−1 J 1Y ×(T ∗Y ∧( ∧ T ∗ X)) Y Y

n of ∧ T ∗ J 1Y . Hence, it deﬁnes a bundle morphism

− 1 ∗ n 1 ∗ HL : J Y → ZY = T Y ∧ ( ∧ T X) (3.13) 30 3 Lagrangian and Hamiltonian Field Theories over Y whose range 1 Z L = HL (J Y ) (3.14) is an imbedded subbundle iL : Z L → ZY of the ﬁbre bundle ZY → Y .This morphism is called the homogeneous Legendre map. Accordingly, the ﬁbre bundle ZY → Y (3.13)issaidtobethehomogeneous Legendre bundle. It is equipped with ( λ, i , λ, ) the holonomic coordinates x y pi p possessing transition functions ε j λ λ ∂x ∂y ∂x μ p = det ν p , (3.15) i ∂x ∂ yi ∂xμ j ε j i ∂x ∂y ∂ y μ p = det ν p − p . ∂x ∂ yi ∂xμ j

With respect to these coordinates, the morphism HL (3.13) reads

( μ, ) ◦ = (π μ, L − i π μ). pi p HL i yμ i

A glance at the transition functions (3.15) shows that ZY (3.13) is a one- dimensional afﬁne bundle πZΠ : ZY → Π (3.16) over the Legendre bundle Π (3.8) modelled over the pull-back vector bundle

n Π × ∧ T ∗ X → Π. X

Moreover, the Legendre map L (3.5) is exactly the composition of morphisms

1 L = πZΠ ◦ HL : J Y → Π. Y

3.2 Cartan and Hamilton–De Donder Equations

Let us note that the Euler–Lagrange equation (3.3) does not exhaust all equations considered in ﬁrst order Lagrangian theory. Being a Lepage equivalent of L, the Poincaré–Cartan form HL (3.12)alsoisa Lepage equivalent of the ﬁrst order Lagrangian

= ( ) = (L + (i − i )π λ)ω, ( i ) = i λ, L h0 HL yλ yλ i h0 dy yλdx (3.17)

1 1 λ i i i i on the repeated jet manifold J J Y , coordinated by (x , y , yλ,yμ, yμλ).TheEuler– Lagrange operator for L (called the Euler–Lagrange–Cartan operator) reads 3.2 Cartan and Hamilton–De Donder Equations 31

E : 1 1 → ∗ 1 ∧ (∧n ∗ ), L J J Y T J Y T X E =[(∂ L − π λ + ∂ π λ(j − j )) i + ∂λπ μ(j − j ) i ]∧ω, L i dλ i i j yλ yλ dy i j yμ yμ dyλ = ∂ + i ∂ + i ∂μ. dλ λ yλ i yλμ i

E ⊂ 1 1 1 Its kernel Ker L J J Y is the Cartan equation on a conﬁguration space J Y , which is locally given by equalities

∂λπ μ(j − j ) = , i j yμ yμ 0 (3.18) ∂ L − π λ + (j − j )∂ π λ = . i dλ i yλ yλ i j 0 (3.19) E | = E Since L J 2Y L , the Cartan equations (3.18) and (3.19) are equivalent to the Euler–Lagrange equation (3.3) on integrable sections of J 1Y → X. These equations are equivalent if a Lagrangian is regular. The Cartan equations (3.18) and (3.19)on sections s : X → J 1Y are equivalent to the relation

∗ s (u dHL ) = 0, (3.20) which is assumed to hold for all vertical vector ﬁelds u on J 1Y → X.

Deﬁnition 3.2 The homogeneous Legendre bundle ZY (3.13) admits the canonical exterior form Ξ = ω + λ i ∧ ω . Y p pi dy λ (3.21)

It is termed the multisymplectic Liouville form

Accordingly, the imbedded subbundle Z L (3.14)ofZY is provided with the pull- Ξ = ∗ Ξ back De Donder form L iL Y . There is the equality = ∗Ξ = ∗( ∗ Ξ ). HL HL L HL iL Y (3.22)

By analogy with the Cartan equation (3.20), the Hamilton–De Donder equation for sections r of Z L → X is written as

∗ r (u dΞL ) = 0, (3.23) where u is an arbitrary vertical vector ﬁeld on Z L → X. Then the following holds [64]. Theorem 3.1 Let the homogeneous Legendre map HL be a submersion. Then a section sofJ1Y → X is a solution of the Cartan equation(3.20) if and only if HL ◦ s is a solution of the Hamilton–De Donder equation (3.23), i.e., the Cartan and Hamilton–De Donder equations are quasi-equivalent. Remark 3.2 The Legendre bundle Π (3.6) and the homogeneous Legendre bundle ZY (3.13) play the role of a phase space and homogeneous phase space in covariant Hamiltonian ﬁeld theory, respectively (Sect. 3.5). 32 3 Lagrangian and Hamiltonian Field Theories

3.3 Noether’s First Theorem: Energy-Momentum Currents

We restrict our study of symmetries of first order Lagrangian field theory to classical symmetries, represented by projectable vector fields on a fibre bundle Y → X.This is the case of all basic classical field models. Let λ μ i μ j u = u (x )∂λ + u (x , y )∂i (3.24) be a projectable vector field on a fibre bundle Y → X. Its canonical decomposition (2.7) into the horizontal and vertical parts over J 1Y reads

λ i i i λ u = u H + uV = u (∂λ + yλ∂i ) + (u ∂i − yλu ∂i ).

Given the first order Lagrangian L (3.1), it is sufficient to consider the first order jet prolongation (B.52): 1 = λ∂ + i ∂ + ( i − i ∂ μ)∂λ, J u u λ u i dλu yμ λu i of u (3.24) onto J 1Y . Then the first variational formula (2.10) takes a form

LJ 1u L = uV EL + dH (h0(u HL )), (3.25) where ΞL = HL is the Poincaré–Cartan form (3.12). Its coordinate expression reads

∂ λL +[ λ∂ + i ∂ + ( i − i ∂ μ)∂λ]L = λu u λ u i dλu yμ λu i (3.26) ( i − i λ)E − [π λ( μ i − i ) − λL ]. u yλu i dλ i u yμ u u

If u is a symmetry of L (Deﬁnition 2.2), we obtain the weak Lagrangian conservation law (2.20): ≈− [π λ( μ i − i ) − λL + σ λ]≈− J λ, 0 dλ i u yμ u u dλ u (3.27) of the symmetry current (2.21): J = J λω =[π λ( μ i − i ) − λL + σ λ]ω , u u λ i u yμ u u λ (3.28) on J 1Y along the vector ﬁeld u (3.24). If u is an exact symmetry, the symmetry current (3.28) reads J = J λω =[π λ( μ i − i ) − λL ]ω . u u λ i u yμ u u λ (3.29)

Remark 3.3 If we choose a different Lepage equivalent ΞL (3.11) in the ﬁrst variational formula (Remark 2.9), the corresponding symmetry current differs from Ju (3.28)inadH -exact term ψ = (ψμλ( i − i ν ))ω . dμ i u yν u λ 3.3 Noether’s First Theorem: Energy-Momentum Currents 33

As was mentioned above, the symmetry current Ju (3.28) is linear in a vector ﬁeld u. Therefore, one can consider a superposition of symmetry currents

Ju + Ju = Ju+u , Jcu = cJu, c ∈ R, (3.30) and a superposition of weak conservation laws (3.27) associated to different symmetries u. i For instance, let v = v ∂i be a vertical vector ﬁeld on Y → X. If it is an exact symmetry of L, the weak conservation law (3.27) takes a form of the Noether conservation law (2.24): ≈ (π λvi ) 0 dλ i (3.31) of the Noether current (2.25): J λ =−π λvi . v i (3.32)

This is the case of so called internal symmetries in field theory. λ In contrast to a case of internal symmetries, let τ = τ ∂λ be a vector field on X and γτ its lift (B.27) onto Y (Definition B.2).

Deﬁnition 3.3 A symmetry current

J λ = π λ(τ μ i − (γ τ)i ) − τ λL γτ i yμ (3.33) along γτ (B.27) is called the energy-momentum current.Ifγτ (B.27) is an exact symmetry of a Lagrangian L,theenergy-momentum conservation law

≈− [π λ(τ μ i − (γ τ)i ) − τ λL ] 0 dλ i yμ (3.34) holds.

In particular, given the connection Γ (B.55) on a ﬁbre bundle Y → X,letΓτ (B.56) be the horizontal lift of a vector ﬁeld τ on X. The corresponding energy- momentum current (3.33) along Γτ reads

J = τ μJ λ ω = τ μ(π λ( i − Γ i ) − δλ L )ω . Γτ Γ μ λ i yμ μ μ λ

λ Its coefﬁcients JΓ μ are components of the tensor ﬁeld

J = J λ μ ⊗ ω , J λ = π λ( i − Γ i ) − δλ L , Γ Γ μdx λ Γ μ i yμ μ μ (3.35) termed the energy-momentum tensor relative to a connection Γ [42, 100, 119]. If Γτ (B.56) is an exact symmetry of a Lagrangian L, we obtain the energy-momentum conservation law (3.34):

≈− [π λτ μ( i − Γ i ) − δλ τ μL ]. 0 dλ i yμ μ μ 34 3 Lagrangian and Hamiltonian Field Theories

For instance, let a ﬁbre bundle Y → X admit a ﬂat connection Γ . By virtue of i Theorem B.15, there exist bundle coordinates such that Γλ = 0, and the corresponding energy-momentum tensor (3.35) takes a form

J λ = π λ i − δλ L 0 μ i yμ μ of the familiar canonical energy–momentum tensor. In particular, this is just the case of nonrelativistic mechanics on fibre bundles over R where τ = ∂t is the canonical vector field on R and JΓ (3.35) is the energy function EΓ (4.41) with respect to a reference frame Γ (Sect.4.3). The lift γτ, considered for an arbitrary vector field τ on X, exemplifies a gauge transformation in accordance with Definition 2.4. It is parameterized by vector fields τ as sections of the vector bundle TX → X. Then it should be emphasized that, since the horizontal lift Γτ (B.56) is independent of derivatives of components of a vector field τ, it can not be an exact symmetry of a Lagrangian L for any vector field τ, unless a Lagrangian L is variationally trivial (Remark 2.7). At the same time, a canonical functorial lift τ, e.g., the τ (B.29) (Definition B.6) depends on derivatives of gauge parameters τ, and can become a gauge symmetry. In this case, the energy-momentum current (3.33) is reduced to a superpotential (Theorem 2.8). This is just the case of gravitation theory on natural bundles (Sect. 10.4). Any projectable vector field u (3.24) can be represented as a sum

λ u = γ(u ∂λ) + v (3.36)

λ λ of some lift γ(u )∂λ) (B.27) of its projection u ∂λ onto X and a vertical vector ﬁeld v.Letu (3.36) be an exact symmetry and Ju the corresponding symmetry current (3.29). In view of superposition (3.30), it falls into a sum

Ju = Jγτ + Jv of the energy-momentum current Jγτ (3.33) and the Noether one Jv (3.32).

3.4 Conservation Laws in the Presence of a Background Field

In Lagrangian field theory on a fibre bundle Y → X,bybackground fields are meant classical fields which do not obey Euler–Lagrange equations. Let these fields be represented by global sections h of a fibre bundle Σ → X endowed with bundle coordinates (xλ,σm ). 3.4 Conservation Laws in the Presence of a Background Field 35

For instance, Lagrangians of gauge and matter fields (Chap. 8), except the topological ones (Chap. 11), depend on a pseudo-Riemannian world metric (Remark B.12) as a background field. In order to formulate Lagrangian field theory in the presence of background fields, let us consider a bundle product

Ytot = Σ × Y → X (3.37) X coordinated by (xλ,σm , yi ) and its jet manifold

1 1 1 J Ytot = J Σ × J Y. (3.38) X

Let L be a first order Lagrangian on the jet manifold (3.38). It can be considered as a total Lagrangian of field theory on the configuration space (3.37) where background fields are treated as dynamic variables. Given a global section h of Σ → X, we obtain the pull-back Lagrangian

1 ∗ Lh = (J h) L (3.39) on J 1Y . It can be regarded as a Lagrangian of first order field theory on a configuration space Y in the presence of a background field h. Let us consider the variational formula (1.36) for a total Lagrangian L:

i m dL − δL − dH Ξ = 0,δL = (Ei θ + Emθ ) ∧ ω.

Its pull-back 2 ∗ (J h) (dL − δL − dH Ξ) = dLh − δLh − dH Ξh (3.40) is exactly the variational formula for the Lagrangian Lh (3.39) in the presence of a background ﬁeld h. The corresponding Euler–Lagrange operator in the presence of a background ﬁeld h reads

2 ∗ 2 ∗ i m 1 ∗ i δLh = (J h) (δL) = (J h) (Ei θ + Emθ ) ∧ ω = (J h) (Ei )θ ω.

The variational formula (3.40) enables us to obtain conservation laws in the presence of a background ﬁeld. Let

λ μ m μ n i μ n j u = u (x )∂λ + u (x ,σ )∂m + u (x ,σ , y )∂i (3.41) be a vector ﬁeld on Ytot (3.37) projected onto Σ and X. Its restriction

u uh : Y × h(X) −→ TY × T Σ −→ TY, X X = = λ∂ + i ∂ , i ( μ, j ) = i ( μ, n( ), j ), uh u u λ uh i uh x y u x h x y (3.42) 36 3 Lagrangian and Hamiltonian Field Theories is a vector ﬁeld on Y . Let us suppose that u (3.41) is an exact symmetry of a total Lagrangian L. The corresponding ﬁrst variational formula (3.25) leads to an equality

= ( m − m λ)∂ L + π λ ( m − m μ) + 0 u yλ u m m dλ u yμ u ( i − i λ)δ L − [∂λL ( μ i − i ) − λL ]. u yλu i dλ i u yμ u u

Putting σ m = hm(x), we obtain an equality

= ( 1 )∗[( m − m λ)∂ L + π λ ( m − m μ)]+ 0 J h u yλ u m mdλ u yμ u ( i − i λ)δ L − [∂λL ( μ i − i ) − λL ]. uh yλu i h dλ i h u yμ uh u h

On the shell δi Lh = 0, this equality is brought into the weak transformation law

≈ ( 1 )∗[( m − m λ)∂ L + π λ ( m − m μ)]− 0 J h u yλ u m mdλ u yμ u (3.43) [∂λL ( μ i − i ) − λL ] dλ i h u yμ uh u h of a symmetry current

J λ = ∂λL (uμ yi − ui ) − uλL uh i h μ h h

i of dynamic fields y along the vector field uh (3.42) in the presence of a background field h.

3.5 Covariant Hamiltonian Formalism

As was mentioned above, there are different variants of covariant Hamiltonian formalism on fibre bundles. We follow polysymplectic Hamiltonian formalism where the Legendre bundle Π (3.6) plays the role of a phase space [54, 61, 118]. If X = R, this is the case of Hamiltonian nonrelativistic mechanics (Chap.4). A key point is that polysymplectic Hamiltonian formalism on a phase space Π is equivalent to particular first order Lagrangian theory on a configuration space Π → X. Given the Hamiltonian form H (3.52)onΠ, the corresponding covariant Hamilton equation (3.61) is the Euler–Lagrange equation of the affine first order Lagrangian L H (3.59) (Theorem 3.4). Moreover, it follows from the equality (3.83) that a Hamiltonian form H possesses the same classical symmetries as a Lagrangian L H . This fact enables us to describe symmetries of Hamiltonian field theory similarly to those in Lagrangian formalism. Treated as a phase space, the Legendre bundle Π (3.6) is endowed with the following polysymplectic structure (Definitions 3.4–3.10). 3.5 Covariant Hamiltonian Formalism 37

Deﬁnition 3.4 There is the canonical bundle monomorphism

+ Θ : Π −→ n∧1 ∗ ⊗ ,Θ= λ i ∧ ω ⊗ ∂ , Y T Y TX Y pi dy λ (3.44) Y Y called the polysymplectic Liouville form on Π.

It should be emphasized that ΘY (3.44)isaTX-valued, but not tangent-valued (i.e., T Π-valued) form on Π. Therefore, standard technique of tangent-valued forms, as like as that of exterior forms (e.g., the exterior differential d) is not applied to ΘY (3.44). At the same time, there is a unique TX-valued (n + 2)-form

Υ = λ ∧ i ∧ ω ⊗ ∂ Y dpi dy λ (3.45) on Π so that a relation ΥY φ = d(ΘY φ) holds for an arbitrary exterior one-form φ on X.

Deﬁnition 3.5 The form ΥY (3.45) is termed the polysymplectic form. Deﬁnition 3.6 A Legendre bundle Π endowed with the polysymplectic form (3.45) is said to be the polysymplectic manifold.

Remark 3.4 Following [69], one sometimes provides Π (3.6) with an exterior form λ ∧ i ∧ ω dpi dy λ which however globally is ill deﬁned because it is not maintained under transition functions (3.7).

By analogy with the notion of Hamiltonian vector fields on a symplectic manifold (Remark 4.17), we define Hamiltonian connections on a polysymplectic manifold. Let J 1Π be the first order jet manifold of a composite Legendre bundle Π → X. ( λ, i , λ, i , λ ) It is equipped with the adapted coordinates x y pi yμ pμi . Definition 3.7 A connection

γ = λ ⊗ (∂ + γ i ∂ + γ μ∂i ) dx λ λ i λi μ (3.46) on Π → X is called the Hamiltonian connection if an exterior form

γ Υ = (∂ + γ i ∂ + γ μ∂i ) ( λ ∧ i ∧ ω) Y λ λ i λi μ dpi dy on J 1Π is exact, i.e.,

γ Υ = λ ∧ i ∧ ω − (γ i λ − γ λ i ) ∧ ω = . Y dpi dy λ λdpi λi dy dHγ (3.47)

Components of a Hamiltonian connection satisfy the conditions

∂i γ j − ∂ j γ i = ,∂γ μ − ∂ γ μ = ,∂γ i + ∂i γ μ = . λ μ μ λ 0 i μj j μi 0 j λ λ μj 0 (3.48) 38 3 Lagrangian and Hamiltonian Field Theories

It is readily observed that, by virtue of the conditions (3.48), the second term in the right-hand side of the equality (3.47) also is an exact form. In accordance with the relative Poincaré lemma [53], this term can be brought into the form dH ∧ ω where H is a function on Π. Then a form Hγ in the expression (3.47) reads

= λ i ∧ ω − H ω. Hγ pi dy λ (3.49)

On another side, let us consider the homogeneous Legendre bundle ZY (3.13) and the afﬁne bundle ZY → Π (3.16). This afﬁne bundle is modelled over the pull-back n ∗ vector bundle Π ×X ∧ T X → Π in accordance with the exact sequence

n ∗ 0 −→ Π × ∧ T X −→ ZY −→ Π −→ 0. (3.50) X

A homogeneous Legendre bundle ZY is provided with the canonical multisymplectic Liouville form ΞY (3.21). Its exterior differential is the multisymplectic form

Ω = Ξ = ∧ ω + λ ∧ i ∧ ω . Y d Y dp dpi dy λ (3.51)

Let h =−H ω be a section of the afﬁne bundle ZY → Π (3.16). A glance at the transformation law (3.15) shows that it is not a density.

Deﬁnition 3.8 By analogy with Hamiltonian mechanics (Sect. 4.5), −h is said to be the covariant Hamiltonian of covariant Hamiltonian formalism. It deﬁnes the pull-back = ∗Ξ = λ i ∧ ω − H ω H h Y pi dy λ (3.52)

(cf. (3.49)) of a multisymplectic Liouville form ΞY onto a Legendre bundle Π which is called the Hamiltonian form on Π.

The following is a straightforward corollary of this deﬁnition.

Theorem 3.2 (i) Hamiltonian forms constitute a non-empty affine space modelled over a linear space of horizontal densities H = Hω on Π → X. (ii) Every connection Γ on a fibre bundle Y → X yields the splitting (B.57) of the exact sequence (3.50) and defines a Hamiltonian form

= Γ ∗Ξ = λ i ∧ ω − λΓ i ω. HΓ Y pi dy λ pi λ (3.53)

(iii) Given a connection Γ on Y → X, every Hamiltonian form H admits the decomposition

= − = λ i ∧ ω − λΓ i ω − Hω. H HΓ HΓ pi dy λ pi λ Γ (3.54)

Remark 3.5 In polysymplectic Hamiltonian formalism, the homogeneous Legen- dre bundle ZY (3.13) plays the role of a homogeneous phase space.IfX = R,a 3.5 Covariant Hamiltonian Formalism 39 phase space Π and a homogeneous phase space ZY are the vertical cotangent bundle V ∗Y and the cotangent bundle T ∗Y , respectively (Sect. 4.5). In this case, the multisymplectic form ΩY (3.51) is exactly the canonical symplectic form (4.58)onthe cotangent bundle T ∗Y of Y . One can generalize item (ii) of Theorem 3.2 as follows. We agree to call any bundle morphism λ i 1 Φ = dx ⊗ (∂λ + Φλ∂i ) : Π → J Y (3.55) Y over Y the Hamiltonian map. In particular, let Γ be a connection on Y → X. Then the composition

λ i 1 Γ = Γ ◦ πΠY = dx ⊗ (∂λ + Γλ ∂i ) : Π → Y → J Y (3.56) is a Hamiltonian map. Theorem 3.3 Every Hamiltonian map (3.55) deﬁnes a Hamiltonian form

=−Φ Θ = λ i ∧ ω − λΦi ω. HΦ Y pi dy λ pi λ (3.57)

Proof Given an arbitrary connection Γ on a ﬁbre bundle Y → X, the corresponding Hamiltonian map (3.56) deﬁnes a form −Γ ΘY which is exactly the Hamiltonian form HΓ (3.53). Since Φ − Γ is a VY-valued basic one-form on Π → X, HΦ − HΓ is a horizontal density on Π. Then the result follows from item (i) of Theorem 3.2. Theorem 3.4 Every Hamiltonian form H (3.52) admits the Hamiltonian connection γH (3.46) which obeys the condition

γH ΥY = dH, (3.58) γ i = ∂i H ,γλ =−∂ H . λ λ λi i

Proof It is readily observed that the Hamiltonian form H (3.52) is the Poincaré– Cartan form (3.12) of a ﬁrst order Lagrangian

= ( ) = ( λ i − H )ω L H h0 H pi yλ (3.59) on the jet manifold J 1Π [53, 81]. The Euler–Lagrange operator (3.2) associated to this Lagrangian reads

1 ∗ n ∗ EH : J Π → T Π ∧ (∧ T X), E =[( i − ∂i H ) λ − ( λ + ∂ H ) i ]∧ω. H yλ λ dpi pλi i dy (3.60)

It is termed the Hamilton operator for H. A glance at the expression (3.60)shows that this operator is an afﬁne morphism over Π of constant rank. It follows that its kernel i = ∂i H , λ =−∂ H yλ λ pλi i (3.61) 40 3 Lagrangian and Hamiltonian Field Theories is an afﬁne closed imbedded subbundle of the jet bundle J 1Π → Π. Therefore, it admits a global section γH which is a desired Hamiltonian connection obeying the relation (3.58).

It follows from the expression (3.49) that, conversely, a Hamiltonian form is associated to any Hamiltonian connection.

Deﬁnition 3.9 The Lagrangian L H (3.59) is called the characteristic Lagrangian of a Hamiltonian system.

Remark 3.6 In fact, the Lagrangian (3.59) is the pull-back onto J 1Π of an exterior 1 form L H on a product Π ×Y J Y . It should be emphasized that, if dim X > 1, there is a set of Hamiltonian connections associated to the same Hamiltonian form H. They differ from each other in soldering forms σ on Π → X which fulﬁll the equation σ ΥY = 0. Every Hamiltonian form H yields a Hamiltonian map

1 1 1 i i H = J πΠY ◦ γH : Π → J Π → J Y, yλ ◦ H = ∂λH , (3.62) which is the same for all Hamiltonian connections γH associated to H. Being a closed imbedded subbundle of the jet bundle J 1Π → X, the kernel (3.61) of the Euler–Lagrange operator EH (3.60) defines a first order Euler–Lagrange equation on Π in accordance with Definition B.12.

Deﬁnition 3.10 It is called the covariant Hamilton equation.

1 Every integral section J r = γH ◦ r of a Hamiltonian connection γH associated to a Hamiltonian form H is obviously a solution of the covariant Hamilton equation (3.61). By virtue of Theorem B.4, if r : X → Π is a global classical solution, there exists an extension of a local section

1 J r : r(X) → Ker EH

1 of the jet bundle J Π → Π over r(X) ⊂ Π to a Hamiltonian connection γH which has r as an integral section. Substituting J 1r in (3.62), we obtain an equality

1 J (πΠY ◦ r) = H ◦ r, (3.63) which is equivalent to the covariant Hamilton equation (3.61).

Remark 3.7 Similarly to the Cartan equation (3.20), the covariant Hamilton equation (3.61) is equivalent to the condition

r ∗(u dH) = 0 (3.64) for any vertical vector ﬁeld u on Π → X. 3.6 Associated Lagrangian and Hamiltonian Systems 41

3.6 Associated Lagrangian and Hamiltonian Systems

Let us study the relations between first order Lagrangian and covariant Hamiltonian formalisms on fibre bundles [54, 61, 118]. We are based on the fact that any Lagrangian L (3.1) on a jet manifold J 1Y defines the Legendre map L (3.5) and its jet prolongation

1 : 1 1 −→ 1Π, ( λ, i , λ ) ◦ 1 = (π λ,i , π λ), J L J J Y J pi yμ pμi J L i yμ dμ i Y = ∂ + j ∂ + j ∂μ, dλ λ yλ j yλμ j and that any Hamiltonian form H (3.52) on a phase space Π deﬁnes the Hamiltonian map H (3.62) and its jet prolongation

1 1 1 1 i i i 1 i i i J H : J Π −→ J J Y,(yμ,yλ, yλμ) ◦ J H = (∂μH , yλ, dλ∂μH ), Y = ∂ + j ∂ + ν ∂ j . dλ λ yλ j pλj ν

Let us start with the case of a hyperregular Lagrangian L, i.e., when the Legendre map L is a diffeomorphism (Deﬁnition 3.1). Then L−1 is a Hamiltonian map. Let us consider a Hamiltonian form

= + −1∗ , H = μ−1i − L ( μ, j , −1 j ), H HL−1 L L pi L μ x y L μ (3.65)

−1 where HL−1 is the Hamiltonian form (3.57) associated to a Hamiltonian map L . Let s be a classical solution of the Euler–Lagrange equation (3.3) for a Lagrangian L. A direct computation shows that L ◦ J 1s is a solution of the covariant Hamilton equation (3.61) for the Hamiltonian form H (3.65). Conversely, if r is a classical solution of the covariant Hamilton equation (3.61) for the Hamiltonian form H (3.65), then s = πΠY ◦ r is a classical solution of the Euler–Lagrange equation (3.3)for L (see the equality (3.63)). It follows that, in the case of hyperregular Lagrangians, covariant Hamiltonian formalism is equivalent to the Lagrangian one. Let now L be an arbitrary Lagrangian on a jet manifold J 1Y .

Deﬁnition 3.11 A Hamiltonian form H is said to be associated to a Lagrangian L if H satisﬁes the relations

L ◦ H ◦ L = L, (3.66) = + ∗ . H HH H L (3.67)

A glance at the relation (3.66) shows that L ◦ H is the projector

μ( ) = ∂μL ( μ, i ,∂j H ( )), ∈ , pi p i x y λ p p NL (3.68) 42 3 Lagrangian and Hamiltonian Field Theories

1 from Π onto the Lagrangian constraint space NL = L(J Y ) (3.9). Accordingly, 1 H ◦ L is the projector from J Y onto H(NL ).

Deﬁnition 3.12 A Hamiltonian form is termed weakly associated to a Lagrangian L if the condition (3.67) holds on a Lagrangian constraint space NL .

Theorem 3.5 If a Hamiltonian map Φ (3.55) obeys a relation

L ◦ Φ ◦ L = L, (3.69)

∗ then a Hamiltonian form H = HΦ + Φ L is weakly associated to a Lagrangian L. If Φ = H, then H is associated to L.

Theorem 3.6 Any Hamiltonian form H weakly associated to a Lagrangian L fulﬁlls a relation | = ∗ | , H NL H HL NL (3.70) where HL is the Poincaré–Cartan form (3.12).

Proof Therelation(3.67) takes a coordinate form

H ( ) = μ∂i H − L ( μ, i ,∂j H ( )), ∈ . p pi μ x y λ p p NL (3.71)

Substituting (3.68) and (3.71)in(3.52), we obtain the relation (3.70).

The difference between associated and weakly associated Hamiltonian forms lies in the following. Let H be an associated Hamiltonian form, i.e., the equality (3.71) holds everywhere on Π. Acting on this equality by the exterior differential, we obtain the relations ∂μH (p) =−(∂μL ) ◦ H(p), p ∈ NL , ∂i H (p) =−(∂i L ) ◦ H(p), p ∈ NL , ( μ − (∂μL )( μ, i ,∂j H ))∂i ∂aH = . pi i x y λ μ α 0 (3.72)

Therelation(3.72) shows that an associated Hamiltonian form (i.e., a Hamiltonian map H) is not regular outside a Lagrangian constraint space NL .

Remark 3.8 Any Hamiltonian form is weakly associated to a Lagrangian L = 0, while associated Hamiltonian forms are only HΓ (3.53).

Remark 3.9 A hyperregular Lagrangian has a unique weakly associated Hamiltonian form (3.65). In the case of a regular Lagrangian L, a Lagrangian constraint space NL is an open subbundle of a Legendre bundle Π → Y .IfNL = Π, a weakly associated Hamiltonian form fails to be deﬁned everywhere on Π in general. At the same time, NL itself can be provided with the pull-back polysymplectic structure with respect to the imbedding NL → Π, so that one may consider Hamiltonian forms on NL . 3.6 Associated Lagrangian and Hamiltonian Systems 43

One can say something more in a case of semiregular Lagrangians (Deﬁnition 3.1).

Theorem 3.7 The Poincaré–Cartan form HL for a semiregular Lagrangian L is −1 constant on the connected inverse image L (p) of any point p ∈ NL .

Proof Let u be a vertical vector ﬁeld on the afﬁne jet bundle J 1Y → Y which takes its values into the kernel of the tangent map T L to L. Then Lu HL = 0.

A corollary of Theorem 3.7 is the following.

Theorem 3.8 Hamiltonian forms weakly associated to a semiregular Lagrangian L coincide with each other on the Lagrangian constraint space NL , and the Poincaré– Cartan form HL (3.12) for L is the pull-back

= ∗ ,(πλ i − L )ω = H ( μ, j ,πμ)ω, HL L H i yλ x y j (3.73) of any such a Hamiltonian form H. Proof Given a vector v ∈ TpΠ,thevalueT H(v) HL (H(p)) is the same for all Hamiltonian maps H satisfying the relation (3.66). Then the result follows from the relation (3.70).

Theorem 3.8 enables us to relate the Euler–Lagrange equation for an almost regular Lagrangian L with the covariant Hamilton equation for Hamiltonian forms weakly associated to L [54, 117, 118].

Theorem 3.9 Let a section r of Π → X be a classical solution of the covariant Hamilton equation (3.61) for a Hamiltonian form H weakly associated to a semiregular Lagrangian L. If r lives in a Lagrangian constraint space NL , a section s = πΠY ◦ rofY→ X satisﬁes the Euler–Lagrange equation (3.3), while its ﬁrst order jet prolongation s = H◦r = J 1s obeys the Cartan equation (3.18) and (3.19). Proof Put s = H ◦ r. Since r(X) ⊂ NL , then

r = L ◦ s, J 1r = J 1L ◦ J 1s.

If r is a classical solution of the covariant Hamilton equation, an exterior form EH 1 ( ) E = ( 1)∗E vanishes at points of J r X . Hence, the pull-back form L J L H vanishes at points J 1s(X). It follows that the section s of a jet bundle J 1Y → X obeys the Cartan equation (3.18) and (3.19). By virtue of the relation (3.63), we have s = J 1s. Hence, s is a classical solution of the Euler–Lagrange equation.

The converse assertion is more intricate [53].

Theorem 3.10 Given a semiregular Lagrangian L, let a section s of the jet bundle J 1Y → X be a solution of the Cartan equation (3.18) and (3.19). Let H be a 44 3 Lagrangian and Hamiltonian Field Theories

Hamiltonian form weakly associated to L so that the associated Hamiltonian map satisﬁes a condition

◦ ◦ = 1(π 1 ◦ ). H L s J 0 s

Then a section

= ◦ , λ = π λ( λ, j , j ), i = i , r L s ri i x s sλ r s of a composite Legendre bundle Π → X is a classical solution of the Hamilton equation (3.61)forH. Being restricted to classical solutions of Euler–Lagrange equations, Theorem 3.10 comes to the following. Theorem 3.11 Given a semiregular Lagrangian L, let a section s of a ﬁbre bundle Y → X be a classical solution of the Euler–Lagrange equation (3.3) (i.e., J 1sisa = π 1 ◦ 1 solution of the Cartan equation (3.18) and (3.19), and s 0 J s). Let H be a Hamiltonian form weakly associated to L, and let H satisfy a relation

H ◦ L ◦ J 1s = J 1s. (3.74)

Then a section r = L ◦ J 1s of a composite Legendre bundle Π → X is a classical solution of the covariant Hamilton equation (3.61)forH. Remark 3.10 Let L = 0. This Lagrangian is semiregular. Its Euler–Lagrange equation comes to the identity 0 = 0. Every section s of a ﬁbre bundle Y → X is a solution of this equation. Given a section s,letΓ be a connection on Y such that s is its integral section. The Hamiltonian form HΓ (3.53) is associated to L = 0, and a Hamiltonian map HΓ satisﬁes the relation (3.74). The corresponding Hamilton equation has a classical solution

= ◦ 1 , i = i , λ = . r L J s r s ri 0

In view of Theorem 3.11, one may try to consider a set of Hamiltonian forms associated to a semiregular Lagrangian L in order to exhaust all classical solutions of the Euler–Lagrange equation for L. Deﬁnition 3.13 Let us say that a set of Hamiltonian forms H weakly associated to a semiregular Lagrangian L is complete if, for each classical solution s of the Euler–Lagrange equation for L, there exists a classical solution r of the covariant Hamilton equation for a Hamiltonian form H from this set such that s = πΠY ◦ r. A complete family of Hamiltonian forms associated to a given Lagrangian need not exist, or it fails to be deﬁned uniquely. For instance, Remark 3.10 shows that the Hamiltonian forms (3.53) constitute a complete family associated to the zero Lagrangian, but this family is not minimal. 3.6 Associated Lagrangian and Hamiltonian Systems 45

By virtue of Theorem 3.11, a set of weakly associated Hamiltonian forms is complete if, for every classical solution s of the Euler–Lagrange equation for L, there is a Hamiltonian form H from this set which fulfills the relation (3.74). In the case of almost regular Lagrangians (Definition 3.1), one can formulate the following necessary and sufficient conditions of the existence of weakly associated Hamiltonian forms. An immediate consequence of Theorem 3.5 is the following.

Theorem 3.12 A Hamiltonian form H weakly associated to an almost regular 1 Lagrangian L exists if and only if the ﬁbred manifold J Y → NL (3.10) admits a global section.

1 Proof A global section of J Y → NL can be extended to a Hamiltonian map Φ : Π → J 1Y which obeys the relation (3.69).

In particular, on an open neighborhood U ⊂ Π of each point p ∈ NL ⊂ Π, there exists a complete set of local Hamiltonian forms weakly associated to an almost regular Lagrangian L. Moreover, one can always construct a complete set of associated Hamiltonian forms [118]. At the same time, a complete set of associated Hamiltonian forms may exist when a Lagrangian is not necessarily semiregular [53]. Given a global section Ψ of the ﬁbred manifold

1 L : J Y → NL , (3.75) let us consider the pull-back form

= Ψ ∗ = ∗ HN HL iN H (3.76) on NL called the constrained Hamiltonian form. By virtue of Theorem 3.7, it does not depend on the choice of a section of the fibred manifold (3.75) and, consequently, ∗ HL = L HN . For sections r of the fibre bundle NL → X, one can write the constrained Hamilton equation ∗ r (u N dHN ) = 0, (3.77) where u N is an arbitrary vertical vector field on NL → X. These equations possess the following important properties.

Theorem 3.13 For any Hamiltonian form H weakly associated to an almost regular Lagrangian L, every classical solution r of the covariant Hamilton equation which lives in the Lagrangian constraint space NL is a solution of the constrained Hamilton equation (3.77). Proof Such a Hamiltonian form H deﬁnes a global section Ψ = H ◦ iN of the = ∗ ﬁbred manifold (3.75). Since HN iN H due to the relation (3.73), the constrained Hamilton equation can be written as

∗( ∗ ) = ∗( | ) = . r u N diN H r u N dH NL 0 (3.78) 46 3 Lagrangian and Hamiltonian Field Theories

Let us note that these equations differ from the Hamilton equation (3.64) restricted to NL . These read ∗( | ) = , r u dH NL 0 (3.79) where r is a section of NL → X and u is an arbitrary vertical vector ﬁeld on Π → X. A solution r of the equation (3.79) obviously satisﬁes the weaker condition (3.78).

Theorem 3.14 The constrained Hamilton equation (3.77) is proved to be equivalent to the Hamilton–De Donder equation (3.23). Proof It is readily observed that L = πZΠ ◦ HL . Hence, the projection πZΠ (3.16) yields a surjection of Z L onto NL . Given a section Ψ of the ﬁbred manifold (3.75), we have a morphism HL ◦ Ψ : NL → Z L . By virtue of Theorem (3.7), this is a surjection such that πZΠ ◦ HL ◦ Ψ = Id NL . Hence, HL ◦Ψ is a bundle isomorphism over Y which is independent of the choice of ∗ a global section Ψ . Combination of (3.22) and (3.76) results in HN = (HL ◦ Ψ) ΞL that leads to a desired equivalence.

This proof gives something more. Namely, since Z L and NL are isomorphic, the homogeneous Legendre map HL fulﬁls the conditions of Theorem 3.1. Then combining Theorems 3.1 and 3.14, we obtain the following.

Theorem 3.15 Let L be an almost regular Lagrangian such that the ﬁbred manifold (3.75) has a global section. A section s of the jet bundle J 1Y → X is a solution of the Cartan equation (3.20) if and only if L ◦ s is a solution of the constrained Hamilton equation (3.77).

Theorem 3.15 also is a corollary of Theorem 3.16 below. The constrained Hamil- tonian form HN (3.76) deﬁnes the constrained Lagrangian

1 ∗ L N = h0(HN ) = (J iN ) L H (3.80)

1 on the jet manifold J NL of the ﬁbre bundle NL → X.

Theorem 3.16 There are the relations

1 ∗ 1 ∗ L = (J L) L N , L N = (J Ψ) L, where L is the Lagrangian (3.17).

The Euler–Lagrange equation for the constrained Lagrangian L N (3.80) is equivalent to the constrained Hamilton equation (3.77) and, by virtue of Theorem 3.16,is quasi-equivalent to the Cartan equation. 3.7 Noether’s First Theorem: Hamiltonian Conservation Laws 47

3.7 Noether’s First Theorem: Hamiltonian Conservation Laws

In order to describe symmetries of covariant Hamiltonian theory, let us use the fact that this theory can be reformulated as particular Lagrangian theory with the characteristic Lagrangian L H (3.59) (Sect. 3.5). Moreover, it follows from the equality (3.83) that a Hamiltonian form H possesses the same classical symmetries as a Lagrangian L H . This fact enables us to analyze symmetries of polysymplectic Hamil- tonian theory similarly to those in Lagrangian formalism (Sect. 3.3) in accordance with Noether’s theorems. We restrict our consideration to classical symmetries defined by projectable vector fields u (3.24) on a fibre bundle Y → X. In accordance with the canonical lift (B.29), every projectable vector field u on Y → X gives rise to a vector field

= μ∂ + i ∂ + (−∂ j λ − ∂ μ λ + ∂ λ μ)∂i u u μ u i i u p j μu pi μu pi λ (3.81) on the Legendre bundle Π and then to the vector ﬁeld

Ju = u + J 1u (3.82)

1 on Π ×Y J Y . Then we have

= = (− i ∂ H − ∂ ( μH ) − λ∂i H + λ∂ i )ω. Lu H LJu L H u i μ u ui λ pi λu (3.83)

It follows that a Hamiltonian form H and the characteristic Lagrangian L H have the same classical symmetries.

Remark 3.11 Given the splitting (3.54) of a Hamiltonian form H, the Lie derivative (3.83) takes a form

= λ([∂ + Γ i ∂ , ] j −[∂ + Γ i ∂ , ]ν Γ j )ω − Lu H p j λ λ i u λ λ i u ν (3.84) μ (∂μu HΓ + u dHΓ )ω, where [., .] is the Lie bracket of vector ﬁelds.

Let us apply the ﬁrst variational formula (3.25) to the Lie derivative LJu L H (3.59) [53]. It reads

− i ∂ H − ∂ ( μH ) − λ∂i H + λ∂ i =−( i − i μ)( λ + ∂ H ) + u i μ u ui λ pi λu u yμu pλi i (−∂ j λ − ∂ μ λ + ∂ λ μ − λ μ)( i − ∂i H ) − i u p j μu pi μu pi pμi u yλ λ [ λ(∂i H μ − i ) − λ( μ∂i H − H )]. dλ pi μ u u u pi μ 48 3 Lagrangian and Hamiltonian Field Theories

On the shell (3.61), this identity takes a form

− i ∂ H − ∂ ( μH ) − λ∂i H + λ∂ i ≈− u i μ u ui λ pi λu (3.85) [ λ(∂i H μ − i ) − λ( μ∂i H − H )]. dλ pi μ u u u pi μ

If LJ 1u L H = 0, we obtain the weak Hamiltonian conservation law

≈− [ λ( μ∂i H − i ) − λ( μ∂i H − H )] 0 dλ pi u μ u u pi μ (3.86) of a Hamiltonian symmetry current Jλ = λ( μ∂i H − i ) − λ( μ∂i H − H ) u pi u μ u u pi μ (3.87) along a vector ﬁeld u. On classical solutions r of the covariant Hamilton equation (3.61), the weak equality (3.86) leads to the differential conservation law ∂ (Jλ( )) = . λ u r 0

There is the following relation between differential conservation laws in Lagrangian and Hamiltonian formalisms.

Theorem 3.17 Let a Hamiltonian form H be associated to an almost regular Lagrangian L. Let r be a classical solution of the covariant Hamilton equation (3.61) for H which lives in the Lagrangian constraint space NL . Let s = πΠY ◦ rbe the corresponding classical solution of the Euler–Lagrange equation for L so that the relation (3.74) holds. Then, for any projectable vector ﬁeld u on a ﬁbre bundle Y → X, we have

1 Ju(r) = Ju(πΠY ◦ r), Ju(L ◦ J s) = Ju(s), (3.88)

1 where Ju is the symmetry current (3.29)onJ Y and Ju is the symmetry current (3.87)onΠ.

Proof The proof follows from the relations (3.68), (3.71) and (3.73).

By virtue of Theorems 3.9–3.11, it follows that: • if Ju in Theorem 3.17 is a conserved symmetry, then the symmetry current Ju (3.88) is conserved on classical solutions of the covariant Hamilton equation which live in a Lagrangian constraint space, • if Ju in Theorem 3.17 is a conserved symmetry current, then the symmetry current Ju (3.88) is conserved on classical solutions s of the Euler–Lagrange equation which obey the condition (3.74). i In particular, let u = u ∂i be a vertical vector ﬁeld on Y → X. Then the Lie derivative Lu H (3.84) takes a form = ( λ[∂ + Γ i ∂ , ] j − H)ω. Lu H p j λ λ i u u d Γ 3.7 Noether’s First Theorem: Hamiltonian Conservation Laws 49

Jλ =− i λ The corresponding Hamiltonian symmetry current (3.87) reads u u pi (cf. Ju (2.25)). λ Let τ = τ ∂λ be a vector ﬁeld on X and Γτ (B.56) its horizontal lift onto Y by means of a connection Γ on Y → X. In this case, the weak identity (3.85) takes a form

−(∂ + Γ j ∂ − λ∂ Γ i ∂ j )H + λ i ≈− J λ , μ μ j pi j μ λ Γ pi Rλμ dλ Γ μ where the Hamiltonian symmetry current (3.87) reads

Jλ = τ μJ λ = τ μ( λ∂i H − δλ ( ν ∂i H − H)). Γ Γ μ pi μ Γ μ pi ν Γ Γ (3.89)

The relations (3.88) show that, on the Lagrangian constraint space NL , the Hamil- tonian symmetry current (3.89) can be treated as the Hamiltonian energy-momentum current relative to a connection Γ . However, Theorem 3.17 fails to provide straightforward relations between symmetries of Lagrangians and associated Hamiltonian forms. In Sect.3.8, we can obtain such relations between symmetries of almost regular quadratic Lagrangians L (3.90) and the corresponding constrained Lagrangians L N (3.80) (Theorem 3.21).

3.8 Quadratic Lagrangian and Hamiltonian Systems

Field theories with almost regular quadratic Lagrangians L (3.90) admit a comprehensive polysymplectic Hamiltonian formulation [53, 54, 61]. There exists a complete set of Hamiltonian forms weakly associated to L (Theorem 3.20). The- orem 3.22 also states that Lagrangian symmetries under a certain condition yield the Hamiltonian ones restricted to a Lagrangian constraint space. At the same time, a Hamiltonian system can possess additional symmetries (Remark 3.15). In Sect.8.5, an example of the almost regular quadratic Yang–Mills Lagrangian (8.62) is analyzed in detail. Given a ﬁbre bundle Y → X, let us consider a quadratic Lagrangian L given by a coordinate expression

1 λμ ν k i j λ ν k i ν k L = a (x , y )yλ yμ + b (x , y )yλ + c(x , y ), (3.90) 2 ij i where a, b and c are local functions on Y . This property is coordinate-independent i owing to the afﬁne transformation law (B.44) of the jet coordinates yλ. The associated Legendre map L (3.5) is given by a coordinate expression

λ ◦ = λμ j + λ, pi L aij yμ bi (3.91) 50 3 Lagrangian and Hamiltonian Field Theories and it is an afﬁne morphism over Y . It yields the corresponding linear morphism

: ∗ ⊗ → Π, λ ◦ = λμ j , a T X VY pi a aij yμ (3.92) Y Y

j where yμ are fibred coordinates on the vector bundle (B.47). Let the Lagrangian L (3.90) be almost regular, i.e., the morphism a (3.92)isof constant rank. Then the Lagrangian constraint space NL (3.91) is an affine subbundle of a Legendre bundle Π → Y , modelled over the vector subbundle N L (3.92)of Π → Y . Hence, NL → Y has a global section s. For the sake of simplicity, let us assume that s = 0 is the canonical zero section of Π → Y . Then N L = NL . Accordingly, the kernel of the Legendre map (3.91) is an affine subbundle of the affine jet bundle J 1Y → Y , modelled over the kernel of the linear morphism a (3.92). Then there exists a connection

Γ : → ⊂ 1 , λμΓ j + λ = , Y Ker L J Y aij μ bi 0 (3.93) on Y → X. Connections (3.93) constitute an afﬁne space modelled over a linear i λ space of soldering forms φ = φλdx ⊗ ∂i on Y → X, satisfying the conditions

λμφ j = aij μ 0 (3.94)

φi λ = and, as a consequence, the conditions λbi 0. If the Lagrangian (3.90) is regular, the connection (3.93) is unique. Remark 3.12 If s = 0, one can consider connections Γ with values into Ker s L.

Amatrixa in the Lagrangian L (3.90) can be seen as a global section of constant rank of a tensor bundle

n 2 ∧ T ∗ X ⊗[∨(TX⊗ V ∗Y )]→Y. Y Y

Then it satisﬁes the following corollary of Theorem B.13.

Theorem 3.18 Given a k-dimensional vector bundle E → Z, let a be a ﬁbre metric of rank r in E. There is a splitting E = Ker a ⊕Z E where E = E/Ker a, and a is a nondegenerate ﬁbre metric in E .

Theorem 3.19 There exists a linear bundle map

σ : Π → ∗ ⊗ , i ◦ σ = σ ij μ, T X VY yλ λμ p j (3.95) Y Y such that a ◦ σ ◦ iN = iN . 3.8 Quadratic Lagrangian and Hamiltonian Systems 51

Proof The map (3.95) is a solution of the algebraic equations

λμσ jk αν = λν . aij μαakb aib (3.96)

By virtue of Theorem 3.18, there exists the bundle splitting

TX∗ ⊗ VY = Ker a ⊕ E (3.97) Y Y and an atlas of this bundle such that transition functions of Ker a and E are independent. Since a is a nondegenerate section of

n 2 ∧ T ∗ X ⊗(∨ E∗) → Y, Y there exist ﬁbre coordinates (y A) on E such that a is brought into a diagonal matrix with nonvanishing components aAA. Due to the splitting (3.97), we have the corresponding bundle splitting

TX⊗ V ∗Y = (Ker a)∗ ⊕ E∗. Y Y

Then a desired map σ is represented by a direct sum σ1 ⊕ σ0 of an arbitrary sections n 2 σ1 of a ﬁbre bundle ∧ TX ⊗Y (∨ Ker a) → Y and a section σ0 of a ﬁbre bundle n 2 AA −1 ∧ TX⊗Y (∨ E ) → Y which has nonvanishing components σ = (aAA) with respect to the coordinates (y A) on E. We have relations

σ0 = σ0 ◦ a ◦ σ0, a ◦ σ1 = 0,σ1 ◦ a = 0. (3.98)

Remark 3.13 Using the relations (3.98), one can write the above assumption, that the Lagrangian constraint space NL → Y admits a global zero section, in the form

μ = μλσ jk ν . bi aij λν bk (3.99)

With the relations (3.93), (3.96) and (3.98), we obtain the splitting

J 1Y = S (J 1Y ) ⊕ F (J 1Y ) = Ker L ⊕ Im(σ ◦ L), (3.100) Y Y i = S i + F i =[ i − σ ik ( αμ j + α)]+[σ ik ( αμ j + α)], yλ λ λ yλ λα akj yμ bk λα akj yμ bk where, in fact, σ = σ0 owing to the relations (3.98) and (3.99). Then with respect to i i the coordinates Sλ and Fλ, the Lagrangian (3.90) reads

1 λμ i j L = a FλFμ + c , (3.101) 2 ij 52 3 Lagrangian and Hamiltonian Field Theories where F i = σ ik αμ( j − Γ j ) λ 0λαakj yμ μ (3.102) for some (Ker L)-valued connection Γ (3.93)onY → X. Thus, the Lagrangian (3.90), written in the form (3.101), factorizes through the covariant differential relative to any such connection.

Remark 3.14 Let us note that, in gauge theory of principal connections (Chap. 7), we have the canonical (independent of a Lagrangian) variant (8.38) of the splitting (3.100) where F is the strength form (8.36). The Yang–Mills Lagrangian (8.62)of gauge ﬁeld theory is exactly of the form (3.101) where c = 0.

Field theories with almost regular quadratic Lagrangians admit comprehensive Hamiltonian formulation. Let L (3.90) be an almost regular quadratic Lagrangian brought into the form (3.101), σ = σ0 +σ1 the linear map (3.95), and Γ the connection (3.93). Similarly to the splitting (3.100) of a jet bundle J 1Y → Y , we have the following decomposition of a phase space:

Π = R(Π) ⊕ P(Π) = Ker σ0 ⊕ NL , (3.103) Y Y λ = Rλ + Pλ =[ λ − λμσ jk α]+[ λμσ jk α]. pi i i pi aij μα pk aij μα pk (3.104)

The relations (3.98) lead to the equalities

σ jk Rα = ,σjk Pα = , RλF i = . 0μα k 0 1μα k 0 i λ 0 (3.105)

Relative to the coordinates (3.104), the Lagrangian constraint space NL (3.91)is given by the equations Rλ =[ λ − λμσ jk α]= . i pi aij μα pk 0 (3.106)

Let the splitting (3.97) be provided with the adapted ﬁbre coordinates (ya, y A) such that the matrix function a (3.92) is brought into a diagonal matrix with nonvanishing components aAA. Then the Legendre bundle Π (3.103) is endowed with the dual (nonholonomic) ﬁbre coordinates (pa, pA) where pA are coordinates on the Lagrangian constraint manifold NL , given by the equalities pa = 0. Relative to σ σ AA = ( )−1,σaa = these coordinates, 0 becomes a diagonal matrix 0 aAA 0 0, while σ Aa = σ AB = = i λ = i λ 1 1 0. Let us write pa Maλ pi , pA MAλ pi , where M is a matrix function on Y which obeys the relations

i λμ = ,(−1)aλσ ij = , Maλaij 0 M i 0λμ 0 (3.107) i ( ◦ σ )λj = j ,(−1)Aμ i = μν σ ki . MAλ a 0 iμ MAμ M j MAλ a jk 0νλ 3.8 Quadratic Lagrangian and Hamiltonian Systems 53

Let us consider an afﬁne Hamiltonian map

Φ = Γ + σ : Π → 1 ,Φi = Γ i + σ ij μ, J Y λ λ λμ p j (3.108) and a Hamiltonian form

(Γ, σ ) = + Φ∗ = λ i ∧ ω − H 1 HΦ L pi dy λ (3.109) i λ 1 ij λ μ ij λ μ λ λ i [Γλ p + σ p p + σ p p − c ]ω = (R + P )dy ∧ ωλ − i 2 0λμ i j 1λμ i j i i λ λ i 1 ij λ μ ij λ μ [(R + P )Γλ + σ P P + σ R R − c ]ω. i i 2 0λμ i j 1λμ i j

Theorem 3.20 The Hamiltonian forms H(Γ, σ1) (3.109) parameterized by connections Γ (3.93) are weakly associated to the Lagrangian (3.90), and they constitute a complete set.

Proof By the very definitions of Γ and σ, the Hamiltonian map (3.108) satisfies the condition (3.66). Then H(Γ, σ1) is weakly associated to L (3.90) in accordance with Theorem 3.5. Let us write the first Hamilton equation (3.61) for a section r of the composite Legendre bundle Π → X. It reads

1 J s = (Γ + σ)◦ r, s = πΠY ◦ r. (3.110)

Due to the surjections S and F (3.100), the Hamilton equation (3.110) is brought into the two parts

S ◦ 1 = Γ ◦ ,∂i − σ ik ( αμ∂ j + α) = Γ i ◦ , J s s λr 0λα akj μr bk λ s (3.111) F ◦ 1 = σ ◦ ,σik ( αμ∂ j + α) = σ ik α. J s r 0λα akj μr bk λαrk (3.112)

Let s be an arbitrary section of Y → X, e.g., a classical solution of the Euler– Lagrange equation. There exists the connection Γ (3.93) such that the relation (3.111) holds, namely, Γ = S ◦ Γ where Γ is a connection on Y → X which has s as an integral section. It is easily seen that, in this case, the Hamiltonian map (3.108) satisﬁes the relation (3.74)fors. Hence, the Hamiltonian forms (3.109) constitute a complete set. It is readily observed that, if σ1 = 0, then Φ = H(Γ ), and the Hamiltonian forms H(Γ, σ1 = 0) (3.109) are associated to the Lagrangian (3.90). For different σ1, we have different complete sets of Hamiltonian forms (3.109). Hamiltonian forms H(Γ, σ1) and H(Γ ,σ1) (3.109) of such a complete set differ φi Rλ φ from each other in the term λ i , where are the soldering forms (3.94). This term vanishes on the Lagrangian constraint space (3.106). Accordingly, the covariant Hamilton equation for different Hamiltonian forms H(Γ, σ1) and H(Γ ,σ1) (3.109) differs from each other in the equations (3.111). These equations are independent of momenta and play the role of gauge-type conditions. 54 3 Lagrangian and Hamiltonian Field Theories

Since the Lagrangian constraint space NL (3.106) is an imbedded subbundle of Π → Y , all Hamiltonian forms H(Γ, σ1) (3.109) deﬁne a unique constrained Hamiltonian form HN (3.76)onNL which reads

∗ λ i λ i 1 ij λ μ H = i H(Γ, σ ) = P dy ∧ ωλ −[P Γλ + σ P P − c ]ω. N N 1 i i 2 0λμ i j

In view of the relations (3.105), the corresponding constrained Lagrangian L N (3.80) 1 on J NL takes a form

λ i 1 ij λ μ L = h (H ) = (P Fλ − σ P P + c )ω. (3.113) N 0 N i 2 0λμ i j

1 It is the pull-back onto J NL of a Lagrangian

λ i i λ i 1 ij λ μ 1 ij λ μ L (Γ,σ ) = R (Sλ − Γλ ) + P Fλ − σ P P − σ R R + c H 1 i i 2 0λμ i j 2 1λμ i j

1 on J Π for any Hamiltonian form H(Γ, σ1) (3.109). 1 In fact, the constrained Lagrangian L N (3.113) is deﬁned on a product NL ×Y J Y Π : Π → (Remark 3.6). Since the phase space (3.103) is a trivial bundle pr2 NL over the Lagrangian constraint space NL , one can consider the pull-back

λ i 1 ij λ μ LΠ = (P Fλ − σ P P + c )ω (3.114) i 2 0λμ i j

1 of the constrained Lagrangian L N (3.113) onto Π ×Y J Y . Let us study symmetries of the constrained Lagrangians L N (3.113) and LΠ (3.114). We aim to show that, under certain conditions, they inherit symmetries of an original Lagrangian L (Theorems 3.21 and 3.22). i Let a vertical vector ﬁeld u = u ∂i on Y → X be an exact symmetry of the Lagrangian L (3.101), i.e.,

= ( i ∂ + i ∂λ)L ω = . LJ 1u L u i dλu i 0 (3.115)

Since 1 i i i k k J u(yλ − Γλ ) = ∂k u (yλ − Γλ ), (3.116) one easily obtains from the equality (3.115) that

k ∂ λμ + ∂ k λμ + λμ∂ k = . u kaij i u akj aik j u 0 (3.117)

It follows that the summands of the Lagrangian (3.101) are separately invariant, i.e.,

1 ( λμF i F j ) = , 1 ( ) = k ∂ = . J u aij λ μ 0 J u c u k c 0 (3.118) 3.8 Quadratic Lagrangian and Hamiltonian Systems 55

The equalities (3.102), (3.116) and (3.117) give the transformation law

1 ( λμF j ) =−∂ k λμF j . J u aij μ i u akj μ (3.119)

The relations (3.98) and (3.117) lead to the equality

λμ[ k ∂ σ jn − ∂ j σ kn − σ jk ∂ n] αν = . aij u k 0μα k u 0μα 0μα k u anb 0 (3.120)

Let us compare symmetries of the Lagrangian L (3.101) and the constrained Lagrangian L N (3.113). Given the Legendre map L (3.91) and the tangent morphism

: 1 → , ˙ = ( ˙i ∂ +˙k ∂ν )( i λμF j ), T L TJ Y TNL pA y i yν k MAλaij μ let us consider the map

1 1 λ i i T L ◦ J u : J Y (x , y , yλ) → (3.121) i ∂ + ( k ∂ + ∂ k ∂ν )( i λμF j )∂ A = u i u k ν u k MAλaij μ i ∂ +[ k ∂ ( i ) λμF j + i 1 ( λμF j )]∂ A = u i u k MAλ aij μ MAλ J u aij μ i ∂ +[ k ∂ ( i ) λμF j − i ∂ k λμF j ]∂ A = u i u k MAλ aij μ MAλ i u akj μ i ∂ +[ k ∂ ( ◦ σ )μi Pλ − ( ◦ σ )μi ∂ k Pλ]∂ j ∈ , u i u k a 0 jλ i a 0 jλ i u k μ TNL where the relations (3.107) and (3.119) have been used. Let us assign to a point ( λ, i , Pλ) ∈ x y i NL some point

( λ, i , i ) ∈ −1( λ, i , Pλ) x y yλ L x y i (3.122) and then an image of the point (3.122) under the morphism (3.121). We get a map

v : ( λ, i , Pλ) → i ∂ +[ k ∂ ( ◦ σ )μi Pλ − ( ◦ σ )μi ∂ k Pλ]∂ j N x y i u i u k a 0 jλ i a 0 jλ i u k μ (3.123) which is independent of the choice of a point (3.122). Therefore, it is a vector field on the Lagrangian constraint space NL . This vector field gives rise to a vector field

v = i ∂ +[ k ∂ ( ◦ σ )μi Pλ − ( ◦ σ )μi ∂ k Pλ]∂ j + i ∂λ J N u i u k a 0 jλ i a 0 jλ i u k μ dλu i (3.124)

1 on NL ×Y J Y .

Theorem 3.21 The Lie derivative LJvN L N of the constrained Lagrangian L N (3.113) along the vector ﬁeld JvN (3.124) vanishes.

Proof One can show that v (Pλ) =−∂ k Pλ N i i u k (3.125) 56 3 Lagrangian and Hamiltonian Field Theories

Rλ = v (L ) = on the constraint manifold i 0. Then the invariance condition J N N 0 falls into the three equalities

v (σ ij PλPμ) = , v (PλF i ) = , v ( ) = . J N 0λμ i j 0 J N i λ 0 J N c 0 (3.126)

The latter is exactly the second equality (3.118). The ﬁrst equality (3.126)issatisﬁed due to the relations (3.120) and (3.125). The second one takes a form

v (Pλ( i − Γ i )) = . J N i yλ λ 0

It holds owing to the relations (3.116) and (3.125).

Thus, any exact vertical symmetry u of the Lagrangian L (3.101) yields the exact symmetry vN (3.123) of the constrained Lagrangian L N (3.113). Turn now to symmetries of the Lagrangian LΠ (3.114). Since LΠ is the pull-back 1 of L N onto Π ×Y J Y , its symmetry must be an appropriate lift of the vector ﬁeld vN (3.123) onto Π. Given a vertical vector ﬁeld u on Y → X, let us consider its canonical lift (3.81):

= i ∂ − ∂ j λ∂i , u u i i u p j λ (3.127) onto a Legendre bundle Π. It readily observed that the vector ﬁeld u is projected onto the vector ﬁeld vN (3.123). Let us additionally suppose that the one-parameter group of automorphisms of Y generated by u preserves the splitting (3.100), i.e., u obeys a condition

k ∂ (σ im νμ) + σ im νμ∂ k − ∂ i σ km νμ = . u k 0λν amj 0λν amk j u k u 0λν amj 0 (3.128)

The relations (3.116) and (3.128) lead to a transformation law

1 i i j J u(Fμ) = ∂ j u Fμ. (3.129)

Theorem 3.22 If the condition (3.128) holds, the vector ﬁeld u(3.127) is an exact symmetry of the Lagrangian LΠ (3.114) if and only if u is an exact symmetry of the Lagrangian L (3.101).

Proof Due to the condition (3.128), the vector ﬁeld u (3.127) preserves the splitting (3.103), i.e.,

(Pλ) =−∂ k Pλ, (Rλ) =−∂ k Rλ. u i i u k u i i u k

The vector ﬁeld u gives rise to the vector ﬁeld (3.82):

= i ∂ − ∂ j λ∂i + i ∂λ, Ju u i i u p j λ dλu i 3.8 Quadratic Lagrangian and Hamiltonian Systems 57

1 on Π ×Y J Y , and we obtain the Lagrangian symmetry condition

( i ∂ − ∂ i λ∂ j + i ∂λ)L = . u i j u pi λ dλu i Π 0 (3.130)

It is readily observed that the ﬁrst and third terms of the Lagrangian LΠ are separately invariant due to the relations (3.118) and (3.129). Its second term is invariant owing to the equality (3.120). Conversely, let the invariance condition (3.130) hold. It falls into the independent equalities

(σ ij λ μ) = , ( λF i ) = , i ∂ = , Ju 0λμ pi p j 0 Ju pi λ 0 u i c 0 (3.131) i.e., the Lagrangian LΠ is invariant if and only if its three summands are separately invariant. One obtains at once from the second condition (3.131) that the quantity F is transformed as the dual of momenta p. Then the ﬁrst condition (3.131)shows that the quantity σ0 p is transformed by the same law as F . It follows that the term aFF in the Lagrangian L (3.101) is transformed as a(σ0 p)(σ0 p) = σ0 pp, i.e., it is invariant. Then this Lagrangian is invariant due to the third equality (3.131).

Remark 3.15 A Lagrangian LΠ may possess symmetries which do not come from symmetries of an original Lagrangian L. They are represented by vector ﬁelds on Π which are not the canonical lift (3.127) of vector ﬁelds on Y (Remark 4.16).

Chapter 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

By mechanics throughout the book is meant classical non-autonomous nonrelativistic mechanics subject to time-dependent coordinate and reference frame transformations. This mechanics is formulated adequately as Lagrangian and Hamil- tonian theory on fibre bundles Q → R over the time axis R [62, 98, 121, 135]. Since equations of motion of mechanics almost always are of first and second order, we restrict our consideration to first order Lagrangian and Hamiltonian theory. Its velocity space is the first order jet manifold J 1 Q of sections of a configuration bundle Q → R, and its phase space Π (3.8) is the vertical cotangent bundle V ∗ Q of Q → R. This formulation of mechanics is similar to that of classical field theory on fibre bundles over a smooth manifold X of dimension n > 1 (Chap. 3). A difference between mechanics and field theory however lies in the fact that fibre bundles over R always are trivial, and that all connections on these fibre bundles are flat. Con- sequently, they are not dynamic variables, but characterize nonrelativistic reference frames (Definition4.1). In Lagrangian mechanics, Noether’s first theorem (Theorem 2.7) is formulated as a straightforward corollary of the first variational formula (4.30). It associates to any classical Lagrangian symmetry υ (4.27) the conserved current (4.32) whose total differential vanishes on-shell. In particular, an energy function relative to a reference frame is the symmetry current (4.41) along a connection Γ on a configuration bundle Q → R which characterizes this reference frame (Definition4.1). A key point is that, in Lagrangian mechanics, any conserved current is an integral of motion (Theorem 4.4), but the converse need not be true (e.g., the Rung–Lenz vector (5.6) in a Lagrangian Kepler model). Hamiltonian formulation of non-autonomous nonrelativistic mechanics is similar to covariant Hamiltonian field theory on fibre bundles [61, 137] in the particular case of fibre bundles over R [62, 121, 136]. A key point is that a non-autonomous Hamiltonian system on a phase space V ∗ Q is equivalent to a particular first order Lagrangian system with the characteristic Lagrangian (4.76)onV ∗ Q as a configuration space. This fact enables one to apply Noether’s first theorem to study symmetries © Atlantis Press and the author(s) 2016 59 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_4 60 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics in Hamiltonian mechanics (Sect. 4.6). In particular, we show that, since Hamiltonian symmetries are vector fields on a phase space V ∗ Q (Definition4.7), any integral of motion in Hamiltonian mechanics (Definition 4.6) is some conserved symmetry current (Theorem4.15). Therefore, it may happen that symmetries and the corresponding integrals of motion define a Hamiltonian system in full. This is the case of commutative and noncommutative completely integrable systems (Sect. 4.7).

4.1 Geometry of Fibre Bundles over R

This Section summarizes peculiarities of geometry of fibre bundles over R [62, 98]. Let π : Q → R be a fibred manifold (Definition B.1) whose base is regarded as the time axis R parameterized by the Cartesian coordinate t with transition functions t = t+const. Relative to the Cartesian coordinate t, the time axis R is provided with the global standard vector field ∂t and the global standard one-form dt which also is a global volume form on R. The symbol dt also stands for any pull-back of the standard one-form dt onto a fibre bundle over R.

Remark 4.1 Point out one-to-one correspondence between the vector ﬁelds f ∂t ,the densities fdt and the real functions f on R. Roughly speaking, we can neglect the contribution of T R and T ∗R to some expressions. In particular, the canonical imbedding (B.48) of J 1 Q takes the form (4.6).

In order that the dynamics of a mechanical system can be defined at any instant t ∈ R, we further assume that a fibred manifold Q → R is a fibre bundle (Definition B.2) with a typical fibre M.

Remark 4.2 In accordance with Remark B.11, a fibred manifold Q → R is a fibre bundle if and only if it admits an Ehresmann connection Γ , i.e., the horizontal lift Γ∂t onto Q of the standard vector field ∂t on R is complete.

Given bundle coordinates (t, qi ) on a ﬁbre bundle Q → R, the ﬁrst order jet man- 1 → R ( , i , i ) ifold J Q of Q is provided with the adapted coordinates t q qt possessing transition functions (B.44) which read

= + ., i = i ( , j ), i = (∂ + j ∂ ) i . t t const q q t q qt t qt j q (4.1)

In mechanics on a conﬁguration space Q → R, the jet manifold J 1 Q plays the role of a velocity space.

Remark 4.3 By virtue of Theorem B.7, any ﬁbre bundle Q → R is trivial. Its different trivializations ψ : Q = R × M (4.2) 4.1 Geometry of Fibre Bundles over R 61 differ from each other in ﬁbrations Q → M. Given the trivialization (4.2) coordinated by (t,qi ), there is a canonical isomorphism

1(R × ) = R × , i = ˙ i , J M TM qt q (4.3)

i ˙ i that one can justify by inspection of transition functions of coordinates qt and q when transition functions of qi are time-independent. Due to the isomorphism (4.3), every trivialization (4.2) yields the corresponding trivialization of the jet manifold

J 1 Q = R × TM. (4.4)

As a palliative variant, one develops nonrelativistic mechanics on the conﬁguration space (4.2) and the velocity space (4.4) [35, 94]. Its phase space R×T ∗ M is provided with the presymplectic form

∗Ω = ∧ i pr 2 T dpi dq (4.5)

∗ which is the pull-back of the canonical symplectic form ΩT (4.102)onT M.A problem is that the presymplectic form (4.5) is broken by time-dependent transformations.

With respect to the bundle coordinates (4.1), the canonical imbedding (B.48) of J 1 Q takes a form

λ : 1 ( , i , i ) → ( , i , ˙ = , ˙i = i ) ∈ , (1) J Q t q qt t q t 1 q qt TQ (4.6) λ = = ∂ + i ∂ . (1) dt t qt i

From now on, the jet manifold J 1 Q is identified with its image in TQwhich is an affine subbundle of TX modelled over the vertical tangent bundle VQ of a fibre bundle Q → R. Using the morphism (4.6), one can define the contraction

1 × ∗ → × R,(i ; ˙, ˙ ) → λ (˙ +˙ i ) = ˙ + i ˙ , J Q T Q Q qt t qi (1) tdt qi dq t qt qi Q Q

i ˙ ∗ where (t, q , t, q˙i ) are holonomic coordinates on the cotangent bundle T Q. In view of the morphism λ(1) (4.6), any connection

i Γ = dt ⊗ (∂t + Γ ∂i ) (4.7) on a fibre bundle Q → R can be identified with a nowhere vanishing horizontal vector field i Γ = ∂t + Γ ∂i (4.8) on Q which is the horizontal lift Γ∂t (B.56) of the standard vector field ∂t on R by means of the connection (4.7). Conversely, any vector field Γ on Q such that 62 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics dtΓ = 1 defines a connection on Q → R. Therefore, the connections (4.7) further are identified with the vector fields (4.8). The integral curves of the vector field (4.8) coincide with the integral sections for the connection (4.7). Connections on a fibre bundle Q → R constitute an affine space modelled over a vector space of vertical vector fields on Q → R. Accordingly, the covariant differential (B.61), associated to a connection Γ on Q → R, takes its values into the vertical tangent bundle VQof Q → R:

: 1 → , i ◦ = i − Γ i . DΓ J Q VQ q DΓ qt (4.9) Q

A connection Γ on a ﬁbre bundle Q → R is obviously ﬂat.

Theorem 4.1 By virtue of Theorem B.15, every connection Γ on a fibre bundle Q → R defines an atlas of local constant trivializations of Q → R such that the i associated bundle coordinates (t, qΓ ) on Q possess transition functions independent of t, and Γ = ∂t (4.10) with respect to these coordinates. Conversely, every atlas of local constant trivializations of a fibre bundle Q → R determines a connection on Q → R which is equal to (4.10) relative to this atlas.

A connection Γ on a fibre bundle Q → R is said to be complete if the horizontal vector field (4.8) is complete. In accordance with Remark B.11, a connection on a fibre bundle Q → R is complete if and only if it is an Ehresmann connection. The following holds [98].

Theorem 4.2 Every trivialization of a fibre bundle Q → R yields a complete connection on this fibre bundle. Conversely, every complete connection Γ on Q → R defines its trivialization (4.2) such that the horizontal vector field (4.8) equals ∂t relative to the bundle coordinates associated to this trivialization.

It follows from Theorem4.1 that, in mechanics unlike field theory, connections Γ (4.8) on a configuration bundle Q → R fail to be dynamic variables. They characterize reference frames as follows. From the physical viewpoint, a reference frame in mechanics determines a tangent vector at each point of a configuration space Q, which characterizes the velocity of an observer at this point. This speculation leads to the following mathematical definition of a reference frame in mechanics [62, 98, 121].

Deﬁnition 4.1 In nonrelativistic mechanics, a reference frame is the connection Γ (4.8) on a conﬁguration bundle Q → R, i.e., a section of a velocity bundle J 1 Q → Q.

By virtue of this deﬁnition, one can think of the horizontal vector ﬁeld (4.8) associated to a connection Γ on Q → R as being a family of observers, while the corresponding covariant differential (4.9): 4.1 Geometry of Fibre Bundles over R 63

i = ( i ) = i − Γ i , qΓ DΓ qt qt (4.11)

Γ i determines the relative velocity with respect to a reference frame . Accordingly, qt are regarded as the absolute velocities. In accordance with Theorem4.1, any reference frame Γ on a configuration bundle Q → R is associated to an atlas of local constant trivializations, and vice versa. A connection Γ takes the form Γ = ∂t (4.10) with respect to the corresponding i coordinates (t, qΓ ), whose transition functions are independent of time. One can think of these coordinates as also being a reference frame, corresponding to the connection (4.10). They are called the adapted coordinates to a reference frame Γ . Thus, we come to the following definition, equivalent to Definition 4.1.

Deﬁnition 4.2 In mechanics, a reference frame is an atlas of local constant trivializations of a conﬁguration bundle Q → R.

i In particular, with respect to the coordinates qΓ adapted to a reference frame Γ , the velocities relative to this reference frame (4.11) coincide with the absolute ones

i = ( i ) = i . qΓ DΓ qΓ t qΓ t

A reference frame is said to be complete if the associated connection Γ is complete. By virtue of Theorem 4.2, every complete reference frame deﬁnes a trivialization of a bundle Q → R, and vice versa.

4.2 Lagrangian Mechanics. Integrals of Motion

As was mentioned above, our exposition is restricted to first order Lagrangian theory on a fibre bundle Q → R [62, 98, 135]. This is a standard case of Lagrangian mechanics. In mechanics, a first order Lagrangian (3.1) is defined as a horizontal density

L = L dt, L : J 1 Q → R, (4.12) on a velocity space J 1 Q. The corresponding second order Euler–Lagrange operator (3.2), termed the Lagrange operator, reads

δ = (∂ L − ∂t L )θ i ∧ , = ∂ + i ∂ + i ∂t . L i dt i dt dt t qt i qtt i (4.13)

Let us further use the notation

π = ∂t L ,π= ∂t ∂t L . i i ji j i (4.14) 64 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

2 The kernel EL = Ker δL ⊂ J Q of the Lagrange operator (4.13) defines a second order Lagrange equation (∂ − ∂t )L = i dt i 0 (4.15) on Q. Its classical solutions (Definition B.13) are (local) sections c of a fibre bundle 2 Q → R whose second order jet prolongations J c = ∂ttc live in EL (4.15). Every first order Lagrangian L (4.12) yields the Legendre map (3.5):

1 ∗ L : J Q −→ V Q, pi ◦ L = πi , (4.16) Q where the Legendre bundle Π = V ∗ Q (3.8) is the vertical cotangent bundle V ∗ Q of i Q → R (Deﬁnition B.4) provided with holonomic coordinates (t, q , pi ).Aswas mentioned above, it plays the role of a phase space of mechanics on a conﬁguration space Q → R. The corresponding Lagrangian constraint space (3.9)is

1 NL = L(J Q). (4.17)

Given a ﬁrst order Lagrangian L, its Lepage equivalent ΞL (3.11) in the variational formula (1.36) is the Poincaré–Cartan form (3.12):

= π i − (π i − L ) HL i dq i qt dt (4.18)

(see the notation (4.14)). This form takes its values into a subbundle

J 1 Q × T ∗ Q ⊂ T ∗ J 1 Q. Q

Hence, it deﬁnes the homogeneous Legendre map (3.13):

1 ∗ HL : J Q → ZY = T Q, (4.19)

1 ∗ whose range (3.14) is an imbedded subbundle Z L = HL (J Q) ⊂ T Q of the ∗ i ˙ homogeneous Legendre bundle ZY = T Q (4.19). Let (t, q , p0 = t, pi =˙qi ) denote holonomic coordinates on T ∗ Q which possess transition functions j j ∂q ∂q p = p j , p = p0 + p j . (4.20) i ∂qi 0 ∂t With respect to these coordinates, the homogeneous Legendre map HL (4.19) reads

( , ) ◦ = (L − i π ,π ). p0 pi HL qt i i

In view of the morphism (4.19), the cotangent bundle T ∗ Q plays the role of a homogeneous phase space of mechanics. 4.2 Lagrangian Mechanics. Integrals of Motion 65

A glance at the transition functions (4.20) shows that the canonical map (B.19):

ζ : T ∗ Q → V ∗ Q, (4.21) is a one-dimensional afﬁne bundle over the vertical cotangent bundle V ∗ Q. Herewith, the Legendre map L (4.16) is exactly the composition of morphisms

1 ∗ L = ζ ◦ HL : J Q → V Q. Q

Remark 4.4 Just as in Sect. 3.2, the Poincaré–Cartan form HL (4.18)alsoisthe = Poincaré–Cartan form HL HL of a ﬁrst order Lagrangian

= ( ) = (L + ( i − i )π ) , ( i ) = i , L h0 HL q(t) qt i dt h0 dq q(t)dt (4.22) on the repeated jet manifold J 1 J 1 Q. The Lagrange operator (4.13)forL (4.22) reads

δ =[(∂ L − π + ∂ π ( j − j )) i + ∂t π ( j − j ) i ]∧ , L i dt i i j q(t) qt dq i j q(t) qt dqt dt = ∂ + i ∂ + i ∂t . dt t qt i qtt i

Its kernel Ker δL ⊂ J 1 J 1 Q deﬁnes the Cartan equation

∂t π ( j − j ) = ,∂L − π + ∂ π ( j − j ) = i j q(t) qt 0 i dt i i j q(t) qt 0 (4.23) on a velocity space J 1 Q.

In mechanics, the Lagrange equation (4.15) as like as the Hamilton one (4.71) is an ordinary differential equation. One can think of its classical solutions s(t) as being a motion in a configuration space Q. In this case, the notion of integrals of motion can be introduced as follows [62, 136]. In a general setting, let an equation of motion of a mechanical system is an r-order differential equation E on a fibre bundle Y → R given by a closed subbundle of the jet bundle J r Y → R in accordance with Definition B.12.

Deﬁnition 4.3 An integral of motion of this mechanical system is deﬁned as a (k < r)-order differential operator Φ on Y such that E belongs to the kernel of an r-order jet prolongation of a differential operator dt Φ, i.e.,

r−k−1( Φ)| = r−k Φ| = , = ∂ + a∂ + a ∂t +··· . J dt E J E 0 dt t yt a ytt a (4.24)

It follows that an integral of motion Φ is constant on classical solutions s of a differential equation E, i.e., there is the differential conservation law

k ∗ k+1 ∗ (J s) Φ = const., (J s) dt Φ = 0. (4.25) 66 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

We agree to write the condition (4.24)asaweak equality

r−k−1 J (dt Φ) ≈ 0, which holds on-shell, i.e., on solutions of a differential equation E by the formula (4.25). In mechanics, we restrict our consideration to integrals of motion Φ which are functions on J k Y . As was mentioned above, equations of motion of mechanics mainly are either of ﬁrst or second order. Accordingly, their integrals of motion are functions on Y = J 0Y or J 1Y . In this case, the corresponding weak equality (4.24) takes a form dt Φ ≈ 0 (4.26) of a weak conservation law of an integral of motion. Integrals of motion can come from symmetries. This is the case both of Lagrangian mechanics on a conﬁguration space Y = Q (Theorems 4.4 and 4.5) and Hamiltonian mechanics on a phase space Y = V ∗ Q (Theorem 4.11).

Definition 4.4 Let an equation of motion of a mechanical system be an r-order differential equation E ⊂ J r Y .Itsinfinitesimal symmetry (or, shortly, a symmetry) is defined as a vector field on J r Y whose restriction to E is tangent to E.

Following Definition 4.4, let us introduce a notion of the symmetry of differential operators in the following relevant case. Let us consider an r-order differential operator on a fibre bundle Y → R which is represented by an exterior form E on J r Y (Definition B.15). Let its kernel Ker E be an r-order differential equation on Y → R.

Theorem 4.3 It is readily justified that a vector field ϑ on J r Y is a symmetry of the equation Ker E in accordance with Definition4.4 if and only if Lϑ E ≈ 0.

Motivated by Theorem4.3, we come to the following notion.

Deﬁnition 4.5 Let E be the above mentioned differential operator. A vector ﬁeld ϑ r on J Y is termed the symmetry of a differential operator E if the Lie derivative Lϑ E vanishes.

By virtue of Theorem 4.3, a symmetry of a differential operator E also is a symmetry of a differential equation Ker E . Let us note that there exist integrals of motion which are not associated to symmetries of an equation of motion, e.g., the Rug–Lenz vector (5.6) in a Lagrangian Kepler system (Chap. 5). 4.3 Noether’s First Theorem: Energy Conservation Laws 67

4.3 Noether’s First Theorem: Energy Conservation Laws

In Lagrangian mechanics, integrals of motion come from symmetries of a Lagrangian (Theorem 4.4) in accordance with Noether’s first theorem (Theorem 2.7). However as was mentioned above, not all integrals of motion are of this type. ∗ Given a Lagrangian system (O∞ Q, L) (Definition1.5) on a fibre bundle Q → 0 R, its infinitesimal transformations are contact derivations ϑ of a real ring O∞ Q (Definition2.1). In accordance with Theorem2.1, a contact derivation is the infinite order jet prolongation J ∞υ (2.2) of the generalized vector field (2.3):

t i i i υ = υ ∂t + υ (t, q , qΛ)∂i , υ = υ + υ = υt (∂ + i ∂ ) + (υi − i υt )∂ , V H t qt i qt i on Q. Just as in first order Lagrangian field theory, we further deal with classical symmetries υ which are vector fields on a configuration space Q. Moreover, not concerned with time-reparametrization, we restrict our consideration to vector fields

t i i t υ = υ ∂t + υ (t, q )∂i ,υ= 0, 1. (4.27)

Herewith, in ﬁrst order Lagrangian mechanics, contact derivations (2.2) can be reduced to the ﬁrst order jet prolongation

ϑ = 1υ = υt ∂ + υi ∂ + υi ∂t J t i dt i (4.28) of the vector ﬁelds υ (4.27). 1 Let L be the Lagrangian (4.12) on a velocity space J Q. Its Lie derivative LJ 1υ L along the contact derivation (4.28) obeys the ﬁrst variational formula (3.25):

LJ 1υ L = υV δL + dH (υHL ), (4.29) where HL is the Poincaré–Cartan form (4.18). Its coordinate expression reads

[υt ∂ + υi ∂ + υi ∂t ]L = (υi − i υt )E + [π (υi − υt i ) + υt L ]. t i dt i qt i dt i qt (4.30)

In accordance with Noether’s first theorem (Theorem2.7), if the vector field υ (4.27) is a symmetry of a Lagrangian L (Definition2.2), a corollary of the first variational formula (4.30) on-shell is the weak Lagrangian conservation law

≈ (υ − σ) ≈ (π (υi − υt i ) + υt L − σ) ≈− J 0 dH HL dt i qt dt dt υ dt (4.31) of a symmetry current

J =−(υ − σ) =−(π (υi − υt i ) + υt L − σ) υ HL i qt (4.32) 68 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics along υ. The symmetry current (4.32) obviously is deﬁned with the accuracy to a constant summand.

Theorem 4.4 It is readily observed that the conserved symmetry current Jυ (4.32) along a classical symmetry is a function on a velocity space J 1 Q, and it is an integral of motion of a Lagrangian system in accordance with Deﬁnition4.3.

Theorem 4.5 If a symmetry υ of a Lagrangian L is classical, this is a symmetry of the Lagrange operator δL(4.13)(Remark2.4) and, as a consequence, an inﬁnitesimal symmetry of the Lagrange equation EL (4.15) (Theorem4.3).

Remark 4.5 Given a Lagrangian L,letL be its partner (4.22) on the repeated jet 1 1 manifold J J Q. Since HL (4.16) is the Poincaré–Cartan form both for L and L,a Lagrangian L does not lead to new conserved symmetry currents.

If a symmetry υ of a Lagrangian L is exact, i.e., LJ 1υ L = 0, the ﬁrst variational formula (4.29) takes a form

0 = υV δL + dH (υHL ). (4.33)

It leads to the weak conservation law (4.31):

0 ≈−dt Jυ , (4.34) of the symmetry current

J =−υ =−π (υi − υt i ) − υt L υ HL i qt (4.35) along a vector ﬁeld υ.

Remark 4.6 The ﬁrst variational formula (4.33) also can be utilized when a Lagrangian possesses exact symmetries, but an equation of motion is a sum

(∂ − ∂t )L + ( , j , j ) = i dt i fi t q qt 0 (4.36) of a Lagrange equation and an additional non-Lagrangian external force. Let us substitute Ei =−fi from this equality in the ﬁrst variational formula (4.33). Then we have the weak transformation law

(υi − i υt ) ≈ J qt fi dt υ of the symmetry current Jυ (4.35) on the shell (4.36).

It is readily observed that the first variational formula (4.30) is linear in a vector field υ. Therefore, one can consider superposition of the equalities (4.30) for different vector fields. 4.3 Noether’s First Theorem: Energy Conservation Laws 69

For instance, if υ and υ are projectable vector fields (4.27), they are projected onto the standard vector field ∂t on R, and the difference of the corresponding equalities (4.30) results in the first variational formula (4.30) for a vertical vector field υ − υ. Conversely, every vector field υ (4.27), projected onto ∂t , can be written as a sum

υ = Γ + v (4.37) of some reference frame (4.8):

i Γ = ∂t + Γ ∂i , (4.38) and a vertical vector field v on Q → R. It follows that the first variational formula (4.30) for the vector field υ (4.27) can be represented as a superposition of those for the reference frame Γ (4.38) and a vertical vector field v. If υ = v is a vertical vector field, the first variational formula (4.30) reads

(vi ∂ + vi ∂t )L = vi E + (π vi ). i dt i i dt i

If v is an exact symmetry of L, we obtain from (4.34) the Noether conservation law i 0 ≈ dt (πi v ) (3.31) of the Noether current

i Jv =−πi v , (4.39) which is a Lagrangian integral of motion by virtue of Theorem4.4.

Remark 4.7 Let us assume that, given a trivialization Q = R × M in bundle coordinates (t, qi ), a Lagrangian L is independent of some coordinate qa. Then a vertical vector ﬁeld v = ∂i is an exact symmetry of L, and we have the conserved Noether current Jv =−πi (4.39) which is an integral of motion.

In the case of the reference frame Γ (4.38), where υt = 1, the ﬁrst variational formula (4.30) reads

(∂ + Γ i ∂ + Γ i ∂t )L = (Γ i − i )E − (π ( i − Γ i ) − L ), t i dt i qt i dt i qt (4.40) where = J = π ( i − Γ i ) − L EΓ Γ i qt (4.41) is called the energy function relative to a reference frame Γ [62, 121]. i With respect to the coordinates qΓ adapted to a reference frame Γ , the ﬁrst variational formula (4.40) takes a form

∂ L =− i E − (π i − L ), t qΓ t i dt i qΓ t (4.42) 70 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics and the EΓ (4.41) coincides with the canonical energy function

= π i − L . EL i qΓ t (4.43)

A glance at the expression (4.42) shows that the vector field Γ (4.38) is an exact symmetry of a Lagrangian L if and only if, written with respect to coordinates adapted to Γ , this Lagrangian is independent of the time t. In this case, the energy function EΓ (4.42) relative to a reference frame Γ is conserved: 0 ≈−dt EΓ .Itisa Lagrangian integral of motion in accordance with Theorem4.4. Since any vector field υ (4.27) can be represented as the sum (4.37) of the reference frame Γ (4.38) and a vertical generalized vector field v, the symmetry current (4.35) along the vector field υ (4.27)isasumJυ = EΓ + Jv of the Noether current Jv (4.39) along a vertical vector field v and the energy function EΓ (4.41) relative to a reference frame Γ . Conversely, energy functions relative to different reference frames Γ and Γ differ from each other in the Noether current (4.39) along a vertical vector field Γ − Γ :

i i EΓ − EΓ = πi (Γ − Γ ) = JΓ −Γ . (4.44)

Remark 4.8 Given a conﬁguration space Q of a mechanical system and the connection Γ (4.38)onQ → R, let us consider a quadratic Lagrangian

1 L = m (t, qk )(qi − Γ i )(q j − Γ j )dt, (4.45) 2 ij t t where mij is a nondegenerate positive-definite fibre metric in the vertical tangent bundle VQ→ Q.Itiscalledthemass tensor. Such a Lagrangian is globally defined i i due to linear transformation laws of the relative velocities qΓ (4.11). Let qΓ be fibre coordinates adapted to a reference frame Γ . Then the Lagrangian (4.45) reads

1 k i j L = m (q )qΓ q dt. (4.46) 2 ij t Γ t

i Since coordinates qΓ possess time-independent transition functions, let us assume that a mass tensor mij is independent of time. In this case, a horizontal vector ﬁeld Γ∂t = Γ = ∂t is an exact symmetry of the Lagrangian L (4.46) that leads to a weak conservation law of the canonical energy function (4.43):

1 k i j E = m (q )qΓ q dt. (4.47) L 2 ij t Γ t

Relative to arbitrary bundle coordinates (t, qi ) on Q, the energy function (4.47) reads

1 k i i j j EΓ = m (t, q )(q − Γ )(q − Γ ). 2 ij t t 4.3 Noether’s First Theorem: Energy Conservation Laws 71

This is an energy function relative to a reference frame Γ .

Remark 4.9 Let us consider a one-dimensional motion of a point mass m0 subject to friction. It is described by the dynamic equation

m0qtt =−kqt , k > 0, on a conﬁguration space R2 → R coordinated by (t, q). This equation is a Lagrange equation of a Lagrangian 1 k = 2 , L m0 exp t qt dt 2 m0 termed the Havas Lagrangian [62, 113]. It is readily observed that the Lie derivative of this Lagrangian along a vector ﬁeld

1 k Γ = ∂t − q∂q (4.48) 2 m0 vanishes. Consequently, we have the conserved energy function (4.41) with respect to the reference frame Γ (4.48). This energy function reads 1 k k EΓ = m0 exp t qt qt + q . 2 m0 m0

4.4 Gauge Symmetries: Noether’s Second and Third Theorems

In mechanics, we follow Deﬁnition2.4 of a gauge symmetry [62]. It takes the form (2.15): ⎛ ⎞

= ∂ + ⎝ iΛ( , j )χ a ⎠ ∂ , u t ua t q Λ i 0≤|Λ|≤m where χ is a section of some vector bundle E → R. In accordance with Theorem 2.5, we can restrict our consideration to vertical gauge symmetries ⎛ ⎞

= ⎝ iΛ( , j )χ a ⎠ ∂ . u ua t q Λ i (4.49) 0≤|Λ|≤m

Noether’s direct second theorem associates to a gauge symmetry of a Lagrangian L the Noether identities (NI) of its Lagrange operator δL as follows (cf. Theorem2.6). 72 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

Theorem 4.6 Let u (4.49) be a gauge symmetry of a Lagrangian L, then its Lagrange operator δL obeys the NI (4.50).

Proof The density uδL = dt σdt is variationally trivial and, therefore, its variational derivatives with respect to variables χ a vanish, i.e.,

E = (− )|Λ| ( iΛE ) = . a 1 dΛ ua i 0 (4.50) 0≤|Λ|

For instance, if the gauge symmetry u (4.49) is of second jet order in gauge parameters, i.e.,

= ( i χ a + itχ a + ittχ a )∂ , u ua ua t ua tt i the corresponding NI (4.50) read

i E − ( itE ) + ( ittE ) = ua i dt ua i dtt ua i 0

(cf. the expression (2.19)). If a Lagrangian L admits the gauge symmetry u (4.49), the weak conservation law (4.31) of the corresponding symmetry current Ju (4.32) holds. Because gauge symmetries depend on derivatives of gauge parameters, all gauge conservation laws in ﬁrst order Lagrangian mechanics possess the following peculiarity.

Theorem 4.7 If u (4.49) is a gauge symmetry of a ﬁrst order Lagrangian L, the corresponding symmetry current Ju (4.32) vanishes on-shell, i.e., J ≈ 0.

Proof Let a gauge symmetry u be at most of jet order N in gauge parameters. Then the symmetry current Ju is decomposed into a sum

J = Λχ a + t χ a + χ a. u Ja Λ Ja t Ja (4.51) 1<|Λ|≤N

The ﬁrst variational formula (4.30) takes a form ⎡ ⎤ ⎛ ⎞ N N = ⎣ i Λχ a + i χ a⎦ E − ⎝ Λχ a + χ a⎠ . 0 u a Λ ua i dt Ja Λ Ja |Λ|=1 |Λ|=1

χ a χ a |Λ|= ,..., χ a It falls into a set of equalities for each tΛ, Λ, 1 N, and as follows: = Λ, |Λ|= , 0 Ja N (4.52) =− itΛE + Λ + tΛ, ≤|Λ| < , 0 ua i Ja dt Ja 1 N (4.53) =− itE + + t , 0 ua i Ja dt Ja (4.54) =− i E + . 0 ua i dt Ja (4.55) 4.4 Gauge Symmetries: Noether’s Second and Third Theorems 73

With the equalities (4.52)–(4.54), the decomposition (4.51) takes a form

J = [( itΛE − tΛ]χ a + ( ittE − tt)χ a + ( itE +− t )χ a. u ua i dt Ja Λ ua i dt Ja t ua i dt Ja 1<|Λ|

A direct computation leads to the expression ⎛ ⎞ ⎛ ⎞

J = ⎝ itΛχ a + itχ a⎠ E − ⎝ tΛχ a + t χ a⎠ . u ua Λ ua i dt Ja Λ dt Ja (4.56) 1≤|Λ|

The ﬁrst summand of this expression vanishes on-shell. Its second one contains Λ |Λ|= ,..., Λ the terms dt Ja , 1 N. By virtue of the equalities (4.53), every dt Ja , |Λ| < tΛ N, is expressed in the terms vanishing on-shell and the term dt dt Ja . Iterating the procedure and bearing in mind the equality (4.52), one can easily show that the second summand of the expression (4.56) also vanishes on-shell. Thus, a symmetry current Ju vanishes on-shell. Let us note that the statement of Theorem 4.7 is a particular case of Noether’s third Theorem 2.8 that symmetry currents of gauge symmetries in Lagrangian theory are reduced to a superpotential because the superpotential U (2.26) equals zero on X = R.

4.5 Non-autonomous Hamiltonian Mechanics

As was mentioned above, a Hamiltonian formulation of non-autonomous nonrelativistic mechanics is similar to covariant Hamiltonian field theory on fibre bundles (Sect. 3.5) in the particular case of fibre bundles over R [62, 121, 135, 136]. In accordance with the Legendre map (4.16) and the homogeneous Legendre map (4.19), a phase space and a homogeneous phase space of mechanics on a configuration bundle Q → R are the vertical cotangent bundle V ∗ Q and the cotangent bundle T ∗ Q of Q, respectively. A key point is that a non-autonomous Hamiltonian system of k degrees of freedom on a phase space V ∗ Q is equivalent both to some autonomous symplectic Hamil- tonian system of k + 1 degrees of freedom on a homogeneous phase space T ∗ Q (Theorem 4.8) and to a particular first order Lagrangian system with the characteristic Lagrangian (4.76) on a configuration space V ∗ Q. Remark 4.10 It should be emphasized that this is not the most general case of a phase space of non-autonomous nonrelativistic mechanics which is defined as a fibred manifold Π → R provided with a Poisson structure such that the corresponding symplectic foliation belongs to the fibration Π → R [70]. Putting Π = V ∗ Q,wein fact restrict our consideration to Hamiltonian systems which admit the Lagrangian counterparts on a configuration space Q. 74 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

∗ i The cotangent bundle T Q is endowed with holonomic coordinates (t, q , p0, pi ), possessing the transition functions (4.20). It admits the canonical Liouville form (4.101): i ΞT = p0dt + pi dq , (4.57) the canonical symplectic form (4.102):

i ΩT = dΞT = dp0 ∧ dt + dpi ∧ dq , (4.58) and the corresponding canonical Poisson bracket (4.103):

0 0 i i ∞ ∗ { f, g}T = ∂ f ∂t g − ∂ g∂t f + ∂ f ∂i g − ∂ g∂i f, f, g ∈ C (T Q). (4.59)

There is the canonical one-dimensional afﬁne bundle (4.21):

ζ : T ∗ Q → V ∗ Q. (4.60)

A glance at the transformation law (4.20) shows that it is a trivial afﬁne bundle. Indeed, given a global section h of ζ , one can equip T ∗ Q with a global ﬁbre coordinate

I0 = p0 − h, I0 ◦ h = 0, possessing the identity transition functions. With respect to coordinates

i (t, q , I0, pi ), i = 1,...,m, (4.61) the ﬁbration (4.60) reads

∗ i i ∗ ζ : R × V Q (t, q , I0, pi ) → (t, q , pi ) ∈ V Q,

i ∗ where (t, q , pi ) are holonomic coordinates on the vertical cotangent bundle V Q possessing transition functions (B.20). Let us consider a subring of C∞(T ∗ Q) which comprises the pull-back ζ ∗ f onto T ∗ Q of functions f on the vertical cotangent bundle V ∗ Q by the ﬁbration ζ (4.60). This subring is closed under the Poisson bracket (4.59). Then by virtue of the well known theorem [62, 148], there exists a degenerate coinduced Poisson bracket

i i ∞ ∗ { f, g}V = ∂ f ∂i g − ∂ g∂i f, f, g ∈ C (V Q), (4.62) on a phase space V ∗ Q such that

∗ ∗ ∗ ζ { f, g}V ={ζ f,ζ g}T .

Holonomic coordinates on V ∗ Q are canonical for the Poisson structure (4.62). 4.5 Non-autonomous Hamiltonian Mechanics 75

With respect to the Poisson bracket (4.62), the Hamiltonian vector ﬁelds of functions on V ∗ Q read

i i ∞ ∗ ϑ f = ∂ f ∂i − ∂i f ∂ , f ∈ C (V Q), (4.63) { , } = ϑ , [ϑ ,ϑ ]=ϑ . f f V f df f f { f, f }V (4.64)

They are vertical vector fields on V ∗ Q → R. Accordingly, the characteristic distribution of the Poisson structure (4.62) is the vertical tangent bundle VV∗ Q ⊂ TV∗ Q of a fibre bundle V ∗ Q → R. The corresponding symplectic foliation on a phase space V ∗ Q coincides with the fibration V ∗ Q → R. However, the Poisson structure (4.62) fails to provide any dynamic equation on a phase space V ∗ Q → R because Hamiltonian vector fields (4.63) of functions on V ∗ Q are vertical vector fields. Hamiltonian dynamics on V ∗ Q is described as a particular Hamiltonian dynamics on fibre bundles [62, 121, 136]. A Hamiltonian on a phase space V ∗ Q → R is defined as a global section

∗ ∗ j h : V Q → T Q, p0 ◦ h = H (t, q , p j ), (4.65) of the afﬁne bundle ζ (4.60) (Deﬁnition 3.8). Given the Liouville form ΞT (4.57)on T ∗ Q, this section yields the pull-back Hamiltonian form

∗ k H = (−h) ΞT = pk dq − H dt (4.66) on V ∗ Q. This is the well-known invariant of Poincaré–Cartan [62]. It should be emphasized that, in contrast with a Hamiltonian in autonomous mechanics, the Hamiltonian H (4.65) is not a function on V ∗ Q, but it obeys the transformation law

H ( , i , ) = H ( , i , ) + ∂ i . t q pi t q pi pi t q

Remark 4.11 Any connection Γ (4.8) on a configuration bundle Q → R defines the i global section hΓ = pi Γ (4.65) of the affine bundle ζ (4.60) and the corresponding Hamiltonian form

k k i HΓ = pk dq − HΓ dt = pk dq − pi Γ dt. (4.67)

Furthermore, given a connection Γ , any Hamiltonian form (4.66) admits a splitting

H = HΓ − EΓ dt, (4.68) where i EΓ = H − HΓ = H − pi Γ (4.69) 76 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics is a function on V ∗ Q. It is called the Hamiltonian function relative to a reference frame Γ . With respect to the coordinates adapted to a reference frame Γ ,wehave EΓ = H . Given different reference frames Γ and Γ , the decomposition (4.68) leads at once to a relation

i i EΓ = EΓ + HΓ − HΓ = EΓ + (Γ − Γ )pi

(cf. (4.44)) between the Hamiltonian functions with respect to different reference frames.

Given the Hamiltonian form H (4.66), there exists a unique Hamiltonian connection k k γH = ∂t + ∂ H ∂k − ∂k H ∂ (4.70)

∗ on V Q → R such that γH dH = 0. It yields a ﬁrst order dynamic Hamilton equation k = ∂k H , =−∂ H qt ptk k (4.71)

∗ → R ( , k , , k , ) (cf. (3.61)) on V Q , where t q pk qt ptk are adapted coordinates on the first order jet manifold J 1V ∗ Q of V ∗ Q → R. A classical solution of the Hamilton equation (4.71)isanintegralsectionr for the connection γH (4.70). We agree to call (V ∗ Q, H) the Hamiltonian system of k = dim Q − 1 degrees of freedom. In order to describe evolution of a Hamiltonian system at any instant, the Hamilton connection γH (4.70) is assumed to be complete, i.e., it is an Ehresmann connection (Remark4.2). In this case, the Hamilton equation (4.71) admits a unique global classical solution through each point of a phase space V ∗ Q. By virtue of Theorem4.2, there exists a trivialization of a fibre bundle V ∗ Q → R (not necessarily compatible with its fibration V ∗ Q → Q) such that

i γH = ∂t , H = pi dq

i with respect to the associated bundle coordinates (t,q , pi ). A direct computation shows that the Hamilton vector ﬁeld γH (4.70) is an inﬁnitesimal generator of a ∗ one-parameter group of automorphisms of a Poisson manifold (V Q, {, }V ). Then i one can show that (t,q , pi ) are canonical coordinates for the Poisson bracket {, }V [98]. Since H = 0, the Hamilton equation (4.71) in these coordinates takes a form

i = , = , qt 0 pti 0

i i.e., (t,q , pi ) are the initial data coordinates.

Remark 4.12 In applications, the condition of the Hamilton connection γH (4.70) to be complete need not holds on the entire phase space (Chap. 5). In this case, 4.5 Non-autonomous Hamiltonian Mechanics 77 one consider its subsets, and sometimes we have different Hamiltonian systems on different subsets of V ∗ Q. As was mentioned above, one can associate to any non-autonomous Hamiltonian system on a phase space V ∗ Q an equivalent autonomous symplectic Hamiltonian system on the cotangent bundle T ∗ Q as follows (Theorem4.8). Given a Hamiltonian system (V ∗ Q, H), its Hamiltonian H (4.65) deﬁnes a function ∗ ∗ ∗ H = ∂t (ΞT − ζ (−h) ΞT )) = p0 + h = p0 + H (4.72) on T ∗ Q. Let us regard H ∗ (4.72) as a Hamiltonian of an autonomous Hamil- ∗ tonian system on a symplectic manifold (T Q,ΩT ). The corresponding autonomous Hamilton equation on T ∗ Q takes a form

˙ i i t = 1, p˙0 =−∂t H , q˙ = ∂ H , p˙i =−∂i H . (4.73)

∗ Remark 4.13 Let us note that the splitting H = p0 + H (4.72) is ill deﬁned. At the same time, any reference frame Γ yields the decomposition

∗ ∗ H = (p0 + HΓ ) + (H − HΓ ) = HΓ + EΓ , where HΓ is the Hamiltonian (4.67) and EΓ (4.69) is the Hamiltonian function relative to a reference frame Γ .

∗ ∗ A Hamiltonian vector ﬁeld ϑH ∗ of the function H (4.72)onT Q is

0 i i ∗ ϑH ∗ = ∂t − ∂t H ∂ + ∂ H ∂i − ∂i H ∂ ,ϑH ∗ ΩT =−dH .

Written relative to the coordinates (4.61), this vector ﬁeld reads

i i ϑH ∗ = ∂t + ∂ H ∂i − ∂i H ∂ . (4.74)

∗ It is identically projected onto the Hamiltonian connection γH (4.70)onV Q such that ζ ∗( ) ={H ∗,ζ∗ } , ∈ ∞( ∗ ). LγH f f T f C V Q (4.75)

Therefore, the Hamilton equation (4.71) is equivalent to the autonomous Hamilton equation (4.73). Obviously, the Hamiltonian vector ﬁeld ϑH ∗ (4.74) is complete if the Hamilton vector ﬁeld γH (4.70)isso. Thus, the following has been proved [32, 62, 99]. Theorem 4.8 A non-autonomous Hamiltonian system (V ∗ Q, H) of k degrees of freedom is equivalent to an autonomous Hamiltonian system (T ∗ Q, H ∗) of k + 1 ∗ degrees of freedom on a symplectic manifold (T Q,ΩT ) whose Hamiltonian is the function H ∗ (4.72). 78 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

We agree to call (T ∗ Q, H ∗) the homogeneous Hamiltonian system and H ∗ (4.72)thehomogeneous Hamiltonian. It is readily observed that the Hamiltonian form H (4.66) also is the Poincaré– Cartan form (4.18) of the characteristic Lagrangian

= ( ) = ( i − H ) L H h0 H pi qt dt (4.76) on the jet manifold J 1V ∗ Q of V ∗ Q → R.

Remark 4.14 In fact, the Lagrangian (4.76) is the pull-back onto J 1V ∗ Q of an ∗ 1 exterior form L H on a product V Q ×Q J Q.

The Lagrange operator (4.13) associated to the characteristic Lagrangian L H (4.76) reads

E = δ =[( i − ∂i H ) − ( + ∂ H ) i ]∧ . H L H qt dpi pti i dq dt

The corresponding Lagrange equation (4.15) is of first order, and it coincides with the Hamilton equation (4.71)onV ∗ Q. Due to this fact, Hamiltonian mechanics as like as covariant Hamiltonian field theory (Sect. 3.5) can be formulated as a specific Lagrangian mechanics on a configuration space V ∗ Q. In particular, let

t i j t u = u ∂t + u (t, q )∂i , u = 0, 1, (4.77) be a vector ﬁeld on a conﬁguration space Q. Its functorial lift (B.30) onto the cotangent bundle T ∗ Q is t i j i u = u ∂t + u ∂i − p j ∂i u ∂ . (4.78)

This vector field is identically projected onto a vector field, also given by the expression (4.78), on a phase space V ∗ Q as a base of the trivial fibre bundle (4.60). Then we have the equality

t i i i j Lu H = LJ 1u L H = (−u ∂t H + pi ∂t u − u ∂i H + pi ∂ j u ∂ H )dt for any Hamiltonian form H (4.66). This equality enables us to study conservation laws in Hamiltonian mechanics similarly to those in Lagrangian mechanics (Sect. 4.6). Lagrangian and Hamiltonian formulations of mechanics as like as those of field theory fail to be equivalent, unless Lagrangians are hyperregular (Definition 3.1). The comprehensive relations between Lagrangian and Hamiltonian systems can be established in the case of almost regular Lagrangians [62, 99, 121]. This is a particular case of the relations between Lagrangian and covariant Hamiltonian theories on fibre bundles (Sect.3.6). 4.5 Non-autonomous Hamiltonian Mechanics 79

If the ﬁrst order Lagrangian L (4.12) is hyperregular, it admits a unique weakly associated Hamiltonian form

i −1i j −1 j H = pi dq − (pi L − L (t, q , L ))dt (4.79)

(cf. (3.65)) which also is L-associated. Let s be a classical solution of the Lagrange equation (4.15) for a Lagrangian L. A direct computation shows that L ◦ J 1s is a classical solution of the Hamilton equation (4.71) for the Hamiltonian form H (4.79). Conversely, if r is a classical solution of the Hamilton equation (4.71)forthe Hamiltonian form H (4.79), then s = πΠ ◦ r is a solution of the Lagrange equation (4.15)forL. In the case of a regular Lagrangian L, the Lagrangian constraint space NL (4.17) ∗ ∗ is an open subbundle of the vertical cotangent bundle V Q → Q.IfNL = V Q, a weakly associated Hamiltonian form fails to be deﬁned everywhere on V ∗ Q in general. Let us restrict our consideration to almost regular Lagrangians L (Deﬁnition3.1). Then the following are reformulations of Theorems 3.9 and 3.10.

Theorem 4.9 Let a section r of V ∗ Q → R be a classical solution of the Hamilton equation (4.71) for a Hamiltonian form H weakly associated to an almost regular Lagrangian L. If r lives in the Lagrangian constraint space NL , a section s = π ◦ r of π : Q → R satisﬁes the Lagrange equation (4.15), while s = H ◦ r, where

: ∗ −→ 1 , i ◦ = ∂i H H V Q J Q qt H Q is the Hamiltonian map (3.62), obeys the Cartan equation (4.23).

Theorem 4.10 Given an almost regular Lagrangian L, let a section s of the jet bundle J 1 Q → R be a solution of the Cartan equation (4.23). Let H be a Hamiltonian form weakly associated to L, and let H satisfy a relation

H ◦ L ◦ s = J 1s, (4.80) where s is the projection of s onto Q. Then a section r = L ◦ s of a ﬁbre bundle V ∗ Q → R is a classical solution of the Hamilton equation (4.71)forH.

A set of Hamiltonian forms H weakly associated to an almost regular Lagrangian L is said to be complete if, for each classical solution s of a Lagrange equation, there exists a classical solution r of a Hamilton equation for a Hamiltonian form H from this set such that s = πΠ ◦ r (Deﬁnition3.13). By virtue of Theorem 4.10,asetof weakly associated Hamiltonian forms is complete if, for every classical solution s of a Lagrange equation for L, there exists a Hamiltonian form H from this set which fulﬁlls the relation (4.80) where s = J 1s, i.e.,

H ◦ L ◦ J 1s = J 1s. (4.81) 80 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

4.6 Hamiltonian Conservation Laws: Noether’s Inverse First Theorem

As was mentioned above, integrals of motion in Lagrangian mechanics can come from Lagrangian symmetries (Theorem 4.4), but not any integral of motion is of this type. In Hamiltonian mechanics, all integrals of motion are conserved symmetry currents (Theorem 4.15). One can think of this fact as being Noether’s inverse ﬁrst theorem.

Definition 4.6 An integral of motion of a Hamiltonian system (V ∗ Q, H) is defined as a smooth real function Φ on V ∗ Q which is an integral of motion of the Hamilton equation (4.71) in accordance with Definition 4.3, i.e., it satisfies the relation (4.26).

Since the Hamilton equation (4.71) is the kernel of the covariant differential DγH , this relation dt Φ ≈ 0 is equivalent to the equality

Φ = (∂ + γ i ∂ + γ ∂i )Φ = ∂ Φ +{H ,Φ} = , LγH t H i Hi t V 0 (4.82) i.e., the Lie derivative of Φ along the Hamilton connection γH (4.70) vanishes. At the same time, it follows from Theorem 4.3 that a vector field υ on V ∗ Q is a symmetry of the Hamilton equation (4.71) in accordance with Definition4.4 if and only if [γH ,υ]=0. Given the Hamiltonian vector field ϑΦ (4.63)ofΦ with respect to the Poisson bracket (4.62), it is easily justified that

[γ ,ϑ ]=ϑ . Φ γ Φ H L H

Thus, we can conclude the following.

Theorem 4.11 The Hamiltonian vector ﬁeld of an integral of motion is a symmetry of the Hamilton equation (4.71).

Given a Hamiltonian system (V ∗ Q, H),let(T ∗ Q, H ∗) be an equivalent homogeneous Hamiltonian system. It follows from the equality (4.75) that

ζ ∗( Φ) ={H ∗,ζ∗Φ} = ζ ∗(∂ Φ +{H ,Φ} ) LγH T t V (4.83) for any function Φ ∈ C∞(V ∗ Q). This formula is equivalent to that (4.82).

Theorem 4.12 A function Φ ∈ C∞(V ∗ Q) is an integral of motion of a Hamiltonian system (V ∗ Q, H) if and only if its pull-back ζ ∗Φ onto T ∗ Q is an integral of motion of a homogeneous Hamiltonian system (T ∗ Q, H ∗).

Proof The result follows from the equality (4.83):

{H ∗,ζ∗Φ} = ζ ∗( Φ) = . T LγH 0 (4.84) 4.6 Hamiltonian Conservation Laws: Noether’s Inverse First Theorem 81

Theorem 4.13 If Φ and Φ are integrals of motion of a Hamiltonian system, their Poisson bracket {Φ, Φ }V also is an integral of motion. Proof This fact results from the equalities (4.64) and (4.84).

Consequently, integrals of motion of a Hamiltonian system (V ∗ Q, H) constitute a real Lie subalgebra of a Poisson algebra C∞(V ∗ Q). Let us turn to Hamiltonian conservation laws. We are based on the fact that the Hamilton equation (4.71) also is a Lagrange equation of the characteristic Lagrangian L H (4.76). Therefore, by analogy with field theory (Sect. 3.7), one can study conservation laws in Hamiltonian mechanics on a phase space V ∗ Q similarly to those in Lagrangian mechanics on a configuration space V ∗ Q [62, 101]. Since the Hamilton equation (4.71) is of first order, we restrict our consideration to classical symmetries, i.e., vector fields on V ∗ Q.

Deﬁnition 4.7 A vector ﬁeld on a phase space V ∗ Q of a Hamiltonian system (V ∗ Q, H) is said to be its Hamiltonian symmetry if it is a Lagrangian symmetry of the characteristic Lagrangian L H . Let t i i t υ = υ ∂t + υ ∂i + υi ∂ ,υ= 0, 1, (4.85)

∗ ∗ 1 be a vector ﬁeld on a phase space V Q. Its prolongation onto V Q ×Q J Q (Remark4.14) reads

1υ = υt ∂ + υi ∂ + υ ∂i + υi ∂t . J t i i dt i

Then the ﬁrst variational formula (4.30) for the characteristic Lagrangian L H (4.76) takes a form

−υt ∂ H − υi ∂ H + υ ( i − ∂i H ) + υi = t i i qt pi dt (4.86) ( i υt − υi )( + ∂ H ) + (υ − υt )( i − ∂i H ) + ( υi − υt H ). qt pti i i pti qt dt pi

If υ (4.85) is a symmetry of L H , i.e.,

LJ 1υ L H = dt σdt, we obtain the weak Hamiltonian conservation law 0 ≈−dt J (4.31)oftheHamil- tonian symmetry current (4.32):

i t Jυ =−pi υ + υ H + σ. (4.87)

In accordance with item (iv) of Theorem 2.4, the vector ﬁeld υ (4.85)onV ∗ Q is a symmetry of the characteristic Lagrangian L H (4.76) if and only if

υi ( + ∂ H ) − υ ( i − ∂i H ) + υt ∂ H = (−J + υt H ). pti i i qt t dt υ (4.88) 82 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

A glance at this equality shows the following.

Theorem 4.14 The vector ﬁeld υ (4.85) is a Hamiltonian symmetry in accordance with Deﬁnition4.7 only if i i ∂ υi =−∂i υ . (4.89)

Remark 4.15 It is readily observed that the Hamiltonian connection γH (4.70)isa symmetry of the characteristic Lagrangian L H whose conserved Hamiltonian symmetry current (4.87) equals zero. It follows that, given a nonvertical Hamiltonian t symmetry υ, υ = 1, there exists a vertical Hamiltonian symmetry υ − γH with the same conserved Hamiltonian symmetry current as υ.

In view of Remark 2.4, any Hamiltonian symmetry, being classical symmetry of the characteristic Lagrangian L H (4.76), also is symmetry of the Hamilton equation (4.71). In accordance with Theorem4.4, the corresponding conserved Hamiltonian symmetry current (4.87) is an integral of motion of a Hamiltonian system which, thus, comes from its Hamiltonian symmetry. The converse also is true.

Theorem 4.15 AnyintegralofmotionΦ of a Hamiltonian system (V ∗ Q, H) is the J conserved Hamiltonian symmetry current −ϑΦ (4.87) along the Hamiltonian vector ﬁeld −ϑΦ (4.63)of−Φ.

Proof It follows from the relations (4.82) and (4.86) that

= (Φ − ∂i Φ). L−J 1ϑΦ dt pi

Φ = J Then the equality (4.87) results in a desired relation −ϑΦ .

This assertion can be regarded as above mentioned Noether’s inverse ﬁrst theorem. For instance, if the Hamiltonian symmetry υ (4.85) is projectable onto Q (i.e., i i j its components υ = u are independent of momenta pi ), then we υi =−p j ∂i u in accordance with the equality (4.89). Consequently, υ is the canonical lift u (4.78) onto V ∗ Q of the vector ﬁeld u (4.77)onQ.Ifu is a symmetry of the characteristic Lagrangian L H , it follows at once from the equality (4.88) thatu is an exact symmetry of L H . The corresponding conserved Hamiltonian symmetry current (4.87) reads

i t Ju = Ju =−pi u + u H (4.90)

(cf. (3.87)).

Definition 4.8 The vector field u (4.77) on a configuration space Q is said to be the basic Hamiltonian symmetry if its canonical lift u (4.78) onto V ∗ Q is a Hamiltonian symmetry. 4.6 Hamiltonian Conservation Laws: Noether’s Inverse First Theorem 83

If a basic Hamiltonian symmetry u is vertical, the corresponding conserved Hamil- tonian symmetry current (4.90):

i Ju =−pi u , (4.91) is a Noether current. Now let Γ be the connection (4.8)onQ. The corresponding symmetry current (4.90) is the Hamiltonian function (4.69):

i JΓ = JΓ = EΓ = H − pi Γ , (4.92) relative to a reference frame Γ . Given bundle coordinates adapted to Γ , we obtain the Lie derivative

=−∂ H . LJ 1ΓL H t

It follows that a connection Γ is a basic Hamiltonian symmetry if and only if the Hamiltonian H (4.65), written with respect to the coordinates adapted to Γ , is time- independent. In this case, the Hamiltonian function (4.92)isanintegralofmotionof a Hamiltonian system. As a consequence of Theorem 3.17, there is the following relation between Lagrangian symmetries and basic Hamiltonian symmetries if they are the same vector ﬁelds on a conﬁguration space Q.

Theorem 4.16 Let a Hamiltonian form H be associated to an almost regular Lagrangian L. Let r be a solution of the Hamilton equation (4.71) for H which lives in the Lagrangian constraint space NL (4.17). Let s = πΠ ◦ r be the corresponding solution of a Lagrange equation for L so that the relation (4.81) holds. Then, for any vector ﬁeld u (4.77) on a ﬁbre bundle Q → R, we have

1 Ju(r) = Ju(πΠ ◦ r), Ju(L ◦ J s) = Ju(s), (4.93)

1 where Ju is the symmetry current (4.35)onJ Y and Ju = Ju is the symmetry current (4.90)onV∗ Q.

By virtue of Theorems 4.9–4.10, it follows that:

• if Ju in Theorem4.16 is a conserved symmetry current, then the symmetry current Ju (4.93) is conserved on solutions of a Hamilton equation which live in the Lagrangian constraint space; • if Ju in Theorem4.16 is a conserved symmetry current, then the symmetry current Ju (4.93) is conserved on solutions s of a Lagrange equation which obey the condition (4.81).

In particular, let u = Γ be a connection and EΓ the energy function (4.41). Then the relations (4.93): 84 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

EΓ (r) = JΓ (r) = JΓ (πΠ ◦ r) = EΓ (πΠ ◦ r), 1 1 EΓ (L ◦ J s) = JΓ (L ◦ J s) = JΓ (s) = EΓ (s), show that the Hamiltonian function EΓ (4.92) can be treated as a Hamiltonian energy function relative to a reference frame Γ .

Remark 4.16 There exist Hamiltonian symmetries which are not basic in accordance with Deﬁnition 4.8. Therefore, they fail to be associated to Lagrangian symmetries. For instance, these are Hamiltonian symmetries (5.11) whose symmetry currents are components (5.9) of the Rung–Lenz vector in the Hamiltonian Kepler problem (Chap. 5).

4.7 Completely Integrable Hamiltonian Systems

It may happen that symmetries and the corresponding integrals of motion deﬁne a Hamiltonian system in full. This is the case of commutative and noncommutative completely integrable systems (henceforth, CISs) (Deﬁnition4.9). In view of Remark4.15, we can restrict our consideration to vertical symmetries υ (4.85) where υt = 0.

Deﬁnition 4.9 A non-autonomous Hamiltonian system (V ∗ Q, H) of n = dim Q−1 degrees of freedom is said to be completely integrable if it admits n ≤ k < 2n vertical classical symmetries υα which obey the following conditions.

(i) Symmetries υα everywhere are linearly independent. (ii) They form a k-dimensional real Lie algebra g of corank m = 2n − k with commutation relations ν [υα,υβ ]=cαβ υν . (4.94)

If k = n, then a Lie algebra g is commutative, and we are in the case of a commutative CIS.Ifn < k, the Lie algebra (4.94) is noncommutative, and a CIS is called noncommutative or superintegrable. The conditions of Definition4.9 can be reformulated in terms of integrals of Φ =−J υ motion α υα corresponding to symmetries α. By virtue of Noether’s inverse υ = ϑ first Theorem4.15, α Φα are the Hamiltonian vector fields (4.63)ofintegrals of motion Φα. In accordance with the relation (4.64), integrals of motion obey the commutation relations ν {Φα,Φβ }V = cαβ Φν . (4.95)

Then we come to an equivalent deﬁnition of a CISs [62, 132, 136].

Deﬁnition 4.10 A non-autonomous Hamiltonian system (V ∗ Q, H) of n = dim Q − 1 degrees of freedom is a CIS if it possesses n ≤ k < 2n integrals of motion Φ1,...,Φk , obeying the following conditions. 4.7 Completely Integrable Hamiltonian Systems 85

(i) All the functions Φα are independent, i.e., a k-form dΦ1 ∧···∧dΦk nowhere vanishes on V ∗ Q. It follows that a map

∗ ∗ ∗ k Φ : V Q → N = (Φ1(V Q), . . . , Φk (V Q)) ⊂ R (4.96)

is a fibred manifold (Definition B.1) over a connected open subset N ⊂ Rk . (ii) The commutation relations (4.95) are satisfied.

Given a non-autonomous CIS in accordance with Deﬁnition 4.10, the equivalent autonomous Hamiltonian system on a homogeneous phase space T ∗ Q (Theorem 4.8) possesses k + 1 integrals of motion

∗ ∗ ∗ (H ,ζ Φ1,...,ζ Φk ) (4.97) with the following properties (Theorem 4.12). (i) The integrals of motion (4.97) are mutually independent, and a map

∗ ∗ ∗ ∗ ∗ ∗ ∗ Φ : T Q → (H (T Q), ζ Φ1(T Q),...,ζ Φk (T Q)) = (4.98) ∗ ∗ (I0,Φ1(V Q),...,Φk (V Q)) = R × N = N

is a ﬁbred manifold. (ii) The integrals of motion (4.97) obey the commutation relations

∗ ∗ ν ∗ ∗ ∗ {ζ Φα,ζ Φβ }=cαβ ζ Φν , {H ,ζ Φα}=0.

They generate a real (k + 1) dimensional Lie algebra of corank 2n + 1 − k. As a result, integrals of motion (4.97) form an autonomous CIS on a symplectic ∗ manifold (T Q,ΩT ) in accordance with Deﬁnition 4.11. In order to describe it, one then can follow the Mishchenko–Fomenko theorem [18, 38, 106] extended to the case of noncompact invariant submanifolds [43, 129, 136]. Therefore, we turn to CISs (superintegrable systems) on a symplectic manifold.

Remark 4.17 Let Z be a manifold. Any exterior two-form Ω on Z yields a linear bundle morphism

∗ Ω : TZ→ T Z,Ω: v →−vΩ(z), v ∈ Tz Z, z ∈ Z. (4.99) Z

One says that a two-form Ω is nondegenerate if Ker Ω = 0. A closed nondegenerate form is called the symplectic form. Accordingly, a manifold Z equipped with a symplectic form is said to be the symplectic manifold. A symplectic manifold necessarily is even-dimensional. A closed two-form on Z is called presymplectic if it is not necessary degenerate. A vector ﬁeld u on a symplectic manifold (Z,Ω)is said to be Hamiltonian if a one-form uΩ is exact. Any smooth function f ∈ C∞(Z) on 86 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

Z defines a unique Hamiltonian vector field ϑ f , called the Hamiltonian vector field of a function f , such that

ϑ f Ω =−df,ϑf = Ω (df), (4.100) where Ω is the inverse isomorphism to Ω (4.99). Given an m-dimensional manifold M coordinated by (qi ),letT ∗ M be its cotangent bundle equipped with the holonomic i coordinates (q , q˙i ). It is endowed with the canonical Liouville form

i ΞT =˙qi dq (4.101) and the canonical symplectic form

i ΩT = dΞT = dq˙i ∧ dq . (4.102)

The Hamiltonian vector ﬁeld ϑ f (4.100) with respect to the canonical symplectic form (4.102) reads

i i ϑ f = ∂ f ∂i − ∂i f ∂ .

A symplectic form Ω on a manifold Z deﬁnes a Poisson bracket

∞ { f, g}=ϑgϑ f Ω, f, g ∈ C (Z).

∗ The canonical symplectic form ΩT (4.102)onT M yields the canonical Poisson bracket ∂ f ∂g ∂ f ∂g { f, g} = − . T i i (4.103) ∂q˙i ∂q ∂q ∂q˙i

Deﬁnition 4.11 Let (Z,Ω) bea2n-dimensional connected symplectic manifold, and let (C∞(Z), {,}) be a Poisson algebra of smooth real functions on Z. A subset

F = (F1,...,Fk ), n ≤ k < 2n, (4.104) of a Poisson algebra C∞(Z) is called the CIS or the superintegrable system if the following conditions hold.

(i) All the functions Fi (called the generating functions of a CIS) are independent, k k i.e., a k-form ∧ dFi nowhere vanishes on Z. It follows that a map F : Z → R is a submersion, i.e., F : Z → N = F(Z) (4.105)

k is a ﬁbred manifold over a domain N ⊂ R endowed with the coordinates (xi ) such that xi ◦ F = Fi . 4.7 Completely Integrable Hamiltonian Systems 87

(ii) There exist smooth real functions sij on N such that

{Fi , Fj }=sij ◦ F, i, j = 1,...,k. (4.106)

(iii) The (k × k)-matrix function s with the entries sij (4.106) is of constant corank m = 2n − k at all points of N.

Remark 4.18 If k = n, then s = 0, and we are in the case of commutative CISs when F1, ..., F − n are independent functions in involution. If k > n, the matrix s is necessarily nonzero. If k = 2n − 1, a CIS is called maximally integrable. The following two assertions clarify a structure of CISs [38, 43, 136]. Theorem 4.17 Given a symplectic manifold (Z,Ω),letF : Z → N be a fibred manifold such that, for any two functions f , f constant on fibres of F, their Poisson bracket { f, f } is so. By virtue of the well known theorem [62, 148], N is provided with an unique coinduced Poisson structure {,}N such that F is a Poisson morphism. Since any function constant on fibres of F is the pull-back of some function on N,theCIS(4.104) satisfies the condition of Theorem4.17 due to item (ii) of Defini- tion4.11. Thus, a base N of the fibration (4.105) is endowed with a coinduced Poisson structure of corank m. With respect to coordinates xi in item (i) of Definition4.11 its bivector field reads i j w = sij(xk )∂ ∧ ∂ . (4.107)

Theorem 4.18 Given a ﬁbred manifold F : Z → N in Theorem 4.17, the following conditions are equivalent [38]:

(i) a rank of the coinduced Poisson structure {, }N on N equals 2dim N − dim Z, (ii) the fibres of F are isotropic, (iii) the fibres of F are maximal integral manifolds of the involutive distribution spanned by the Hamiltonian vector fields of the pull-back F ∗C of Casimir functions C of the coinduced Poisson structure (4.107)onN.

It is readily observed that the fibred manifold F (4.105) obeys condition (i) of Theorem 4.18 due to item (iii) of Definition4.11, namely, k − m = 2(k − n). Fibres of the fibred manifold F (4.105) are called the invariant submanifolds.

Remark 4.19 In practice, condition (i) of Definition 4.11 fails to hold everywhere. k It can be replaced with that a subset Z R ⊂ Z of regular points (where ∧ dFi = 0) is open and dense. Let M be an invariant submanifold through a regular point z ∈ Z R ⊂ Z. Then it is regular, i.e., M ⊂ Z R.LetM admit a regular open saturated neighborhood UM (i.e., a fibre of F through a point of UM belongs to UM ). For instance, any compact invariant submanifold M has such a neighborhood UM .The restriction of functions Fi to UM defines a CIS on UM which obeys Definition4.11. In this case, one says that a CIS is considered around its invariant submanifold M. 88 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

Let (Z,Ω)be a 2n-dimensional connected symplectic manifold. Given the CIS (Fi ) (4.104)on(Z,Ω), the well known Mishchenko–Fomenko theorem (Theo- rem 4.20) states the existence of action-angle coordinates around its connected compact invariant submanifold [18, 38, 106]. This theorem has been extended to CISs with noncompact invariant submanifolds (Theorem 4.19) [43, 129, 136]. These submanifolds are diffeomorphic to a toroidal cylinder

Rm−r × T r , m = 2n − k, 0 ≤ r ≤ m. (4.108)

Theorem 4.19 Let the Hamiltonian vector fields ϑi of the functions Fi be complete, and let the fibres of the fibred manifold F (4.105) be connected and mutually diffeomorphic. Then the following hold. (I) The fibres of F (4.105) are diffeomorphic to the toroidal cylinder (4.108). (II) Given a fibre M of F (4.105), there exists its open saturated neighborhood UM which is a trivial principal bundle

m−r r F UM = NM × R × T −→ NM (4.109)

with the structure group (4.108). (III) A neighborhood UM is provided with the bundle action-angle coordinates s λ (Iλ, ps , q , y ), λ = 1,...,m, s = 1,...,n − m, such that: (i) the angle coordinates (yλ) are those on a toroidal cylinder, i.e., ﬁbre coordinates on the s ﬁbre bundle (4.109), (ii) (Iλ, ps , q ) are coordinates on its base NM where the action coordinates (Iλ) are values of Casimir functions of the coinduced Poisson structure {, }N on NM , and (iii) a symplectic form Ω on UM reads

λ s Ω = dIλ ∧ dy + dps ∧ dq .

Remark 4.20 The condition of the completeness of Hamiltonian vector fields of the generating functions Fi in Theorem4.19 is rather restrictive. One can replace this condition with that the Hamiltonian vector fields of the pull-back onto Z of Casimir functions on N are complete. If the conditions of Theorem4.19 are replaced with that fibres of the fibred manifold F (4.105) are compact and connected, this theorem restarts the Mishchenko– Fomenko theorem as follows. Theorem 4.20 Let the fibres of the fibred manifold F (4.105) be connected and compact. Then they are diffeomorphic to a torus T m, and statements (II)–(III) of Theorem4.19 hold.

Remark 4.21 In Theorem4.20, the Hamiltonian vector fields υλ are complete because fibres of the fibred manifold F (4.105) are compact. As well known, any vector field on a compact manifold is complete. To study a CIS, one conventionally considers it with respect to action-angle coordinates. A problem is that an action-angle coordinate chart on an open subbundle 4.7 Completely Integrable Hamiltonian Systems 89

U of the ﬁbred manifold Z → N (4.105) in Theorem 4.19 is local. The following generalizes this theorem to the case of global action-angle coordinates.

Deﬁnition 4.12 The CIS F (4.104) on a symplectic manifold (Z,Ω) in Deﬁni- tion4.11 is called globally integrable (or, shortly, global) if there exist global action- angle coordinates

A λ (Iλ, x , y ), λ = 1,...,m, A = 1,...,2(n − m), (4.110) such that: (i) the action coordinates (Iλ) are expressed in values of some Casimir λ functions Cλ on a Poisson manifold (N, {,}N ), (ii) the angle coordinates (y ) are coordinates on the toroidal cylinder Rm−r × T r ,0≤ r ≤ m, and (iii) a symplectic form Ω on Z reads

λ C A B Ω = dIλ ∧ dy + ΩAB(Iμ, x )dx ∧ dx .

It is readily observed that the action-angle coordinates on U in Theorem4.19 are global on U in accordance with Deﬁnition4.12. Forthcoming Theorem4.21 provides the sufﬁcient conditions of the existence of global action-angle coordinates of a CIS on a symplectic manifold (Z,Ω)[62, 101, 129, 136]. It generalizes the well-known result for the case of compact invariant submanifolds [31, 34, 38].

Theorem 4.21 A CIS F on a symplectic manifold (Z,Ω)is globally integrable if the following conditions hold.

(i) Hamiltonian vector fields ϑi of the generating functions Fi are complete. (ii) The fibred manifold F (4.105) is a fibre bundle with connected fibres. (iii) Its base N is simply connected and the cohomology H 2(N; Z) with coefficients in the constant sheaf Z is trivial. (iv) The coinduced Poisson structure {,}N on a base N admits m independent Casimir functions Cλ.

Theorem 4.21 restarts Theorem 4.19 if one considers an open subset V of N admitting the Darboux coordinates x A on symplectic leaves of U. If invariant submanifolds of a CIS are assumed to be compact, condition (i) of Theorem 4.21 is unnecessary since vector fields ϑλ on compact fibres of F are complete. Condition (ii) also holds by virtue of Theorem B.6. In this case, Theorem 4.21 reproduces the well known result in [31]. Furthermore, one can show that conditions (iii) of Theorem 4.21 guarantee that fibre bundles F in conditions (ii) of these theorems are trivial [136]. Therefore, Theorem 4.21 can be reformulated as follows.

Theorem 4.22 A CIS F on a symplectic manifold (Z,Ω)is global if and only if the following conditions hold. (i) The ﬁbred manifold F (4.105) is a trivial ﬁbre bundle. 90 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

(ii) The coinduced Poissonstructure {,}N on a base N admits m independent Casimir ∗ functions Cλ such that Hamiltonian vector ﬁelds of their pull-back F Cλ are complete.

Bearing in mind the autonomous CIS (4.97), let us turn to autonomous CISs whose generating functions are integrals of motion, i.e., they are in involution with a Hamiltonian H , and the functions (H , F1,...,Fk ) are nowhere independent, i.e.,

{H , Fi }=0, (4.111) k dH ∧ (∧ dFi ) = 0. (4.112)

Let us note that, in accordance with item (ii) of Theorem 4.22 and 4.23 below, the Hamiltonian vector ﬁeld of a Hamiltonian H of a CIS always is complete.

Theorem 4.23 It follows from the equality (4.112) that a Hamiltonian H is constant on invariant submanifolds. Hence, it is the pull-back of a function on N which is a Casimir function of the Poisson structure (4.107) because of conditions (4.111).

Theorem 4.23 leads to the following.

Theorem 4.24 Let H be a Hamiltonian of an autonomous global CIS provided with A λ the action-angle coordinates (Iλ, x , y ) (4.110). Then a Hamiltonian H depends only on the action coordinates Iλ. Consequently, the Hamilton equation of a global CIS takes a form ∂H λ A y˙ = , Iλ = const., x = const. ∂ Iλ

Remark 4.22 Given a Hamiltonian H of a Hamiltonian system on a symplectic manifold Z, it may happen that we have different CISs on different open subsets of Z. For instance, this is the case of the global Kepler problem (Chap.5).

Remark 4.23 Bearing in mind again the autonomous CIS (4.97), let us also consider CISs whose generating functions {F1,...,Fk } form a k-dimensional real Lie algebra g of corank m with commutation relations

{ , }= h , h = . Fi Fj cijFh cij const (4.113)

Then F (4.105) is a momentum mapping of Z to the Lie coalgebra g∗ provided with the coordinates xi in item (i) of Definition4.11 [58, 68]. In this case, the coinduced ∗ Poisson structure {,}N coincides with the canonical Lie–Poisson structure on g . Let V be an open subset of g∗ such that conditions (i) and (ii) of Theorem 4.22 are satisfied. Then an open subset F −1(V ) ⊂ Z is provided with the action-angle coordinates. Let Hamiltonian vector fields ϑi of the generating functions Fi which form a Lie algebra g be complete. Then they define a locally free Hamiltonian action on Z of some simply connected Lie group G whose Lie algebra is isomorphic to 4.7 Completely Integrable Hamiltonian Systems 91 g [109]. Orbits of G coincide with k-dimensional maximal integral manifolds of the regular distribution V on Z spanned by Hamiltonian vector fields ϑi [142]. Furthermore, Casimir functions of the Lie–Poisson structure on g∗ are exactly the coadjoint invariant functions on g∗. They are constant on orbits of the coadjoint action of G on g∗ which coincide with leaves of the symplectic foliation of g∗.

Now, let us return to the autonomous CIS (4.97) on homogeneous phase space of non-autonomous mechanics. There is the commutative diagram

ζ T ∗ Q −→ V ∗ Q Φ Φ ? ? ξ N −→ N where ζ (3.16) and ξ : N = R × N → N are trivial bundles. Thus, the fibred manifold (4.98) is the pull-back Φ = ξ ∗Φ of the fibred manifold Φ (4.96) onto N . Let the conditions of Theorem 4.19 hold. If the Hamiltonian vector fields

(γ ,ϑ ,...,ϑ ), ϑ = ∂i Φ ∂ − ∂ Φ ∂i , H Φ1 Φk Φα α i i α

∗ of integrals of motion Φα on V Q are complete, the Hamiltonian vector ﬁelds

( ∗ , ∗ ,..., ∗ ), ∗ = ∂i Φ ∂ − ∂ Φ ∂i , uH uζ Φ1 uζ Φk uζ Φα α i i α on T ∗ Q are complete. If fibres of the fibred manifold Φ (4.96) are connected and mutually diffeomorphic, the fibres of the fibred manifold Φ (4.98) also are well. Let M be a fibre of Φ (4.96) and h(M) the corresponding fibre of Φ (4.98). In accordance with Theorem4.19, there exists an open neighborhood U of h(M) which is a trivial principal bundle with the structure group

R1+m−r × T r (4.114) whose bundle coordinates are the action-angle coordinates

λ A (I0, Iλ, t, y , pA, q ), A = 1,...,n − m,λ= 1,...,k, (4.115) such that:

(i) (t, yλ) are coordinates on the toroidal cylinder (4.114), (ii) the symplectic form ΩT on U reads

α A ΩT = dI0 ∧ dt + dIα ∧ dy + dpA ∧ dq ,

(iii) the action coordinates (I0, Iα) are expressed in values of the Casimir functions A C0 = I0, Cα of the coinduced Poisson structure w = ∂ ∧ ∂A on N , 92 4 Lagrangian and Hamiltonian Nonrelativistic Mechanics

∗ (iv) a homogeneous Hamiltonian depends on action coordinates just as H = I0, ∗ ∗ λ (v) the integrals of motion ζ Φ1,...ζ Φk are independent of coordinates (t, y ). Endowed with the action-angle coordinates (4.115), the above mentioned neighborhood U is a trivial bundle U = R × UM where UM = ζ(U ) is an open neighborhood of a ﬁbre M of the ﬁbre bundle Φ (4.96). Then we come to the following.

Theorem 4.25 Let symmetries υα in Definition4.9 be complete, and let fibres of the fibred manifold Φ (4.96) defined by the corresponding conserved integrals of motion be connected and mutually diffeomorphic. Then there exists an open neighborhood UM of a fibre M of Φ (4.96) which is a trivial principal bundle with a structure group (4.114) whose bundle coordinates are the action-angle coordinates

A λ (pA, q , Iλ, t, y ), A = 1,...,k − n,λ= 1,...,m, such that: (i) (t, yλ) are coordinates on the toroidal cylinder (4.114), (ii) the Poisson bracket {, }V on UM reads

A A λ λ { f, g}V = ∂ f ∂Ag − ∂ g∂A f + ∂ f ∂λg − ∂ g∂λ f,

(iii) a Hamiltonian H depends only on action coordinates Iλ, λ (iv) the integrals of motion Φ1,...Φk are independent of coordinates (t, y ). Chapter 5 Global Kepler Problem

We provide a global analysis of the Kepler problem as an example of a mechanical system which is characterized by its symmetries in full. It falls into two distinct global CISs on different open subsets of a phase space. Their integrals of motion form the Lie algebras so(3) and so(2, 1) with compact and noncompact invariant submanifolds, respectively [62, 129, 136]. Let us consider a mechanical system of a point mass in the presence of a central potential. Its conﬁguration space is

Q = R × R3 → R (5.1) endowed with the Cartesian coordinates (t, qi ), i = 1, 2, 3. A Lagrangian of this mechanical system reads

1 L = (qi )2 − V (r), r 2 = (qi )2. (5.2) 2 t i i

The vertical vector ﬁelds va = b∂ − a∂ b q a q b (5.3) on Q (5.1) are inﬁnitesimal generators of the natural representation of a group SO(3) acting on R3. Their jet prolongations (4.28) read

1va = b∂ − a∂ + b∂t − a∂t . J b q a q b qt a qt b

It is easily justified that the vector fields (5.3) are exact symmetries of the Lagrangian (5.2). In accordance with Noether’s first theorem, the corresponding conserved Noether currents (4.39)areorbital momenta

a a b a b b a M = Jva = (q π − q π ) = q q − q q . b b b a t t (5.4)

They are integrals of motion, which however fail to be independent. Let us consider the Lagrangian system (5.2) where

V (r) =−r −1 (5.5) is a Kepler potential. This Lagrangian system admits additional integrals of motion

qa Aa = (qaqb − qbqa)qb − , (5.6) t t t r b besides the orbital momenta (5.4). They are components of the Rung–Lenz vector. However, there are no symmetries on Q (5.1) whose symmetry currents are Aa (5.6). Let us consider a Hamiltonian Kepler system on the conﬁguration space Q (5.1). ∗ 6 i Its phase space is V Q = R × R coordinated by (t, q , pi ). It is readily observed that the Lagrangian (5.2) with the Kepler potential (5.5)of a Kepler system is hyperregular. The associated Hamiltonian form reads 1 H = p dqi − (p )2 − r −1 dt. (5.7) i 2 i i

The corresponding characteristic Lagrangian L H (4.76)is 1 L = p qi − (p )2 + r −1 dt. H i t 2 i i

Then a Hamiltonian Kepler system possesses the following integrals of motion: • an energy function E = H ; • orbital momenta a = a − b ; Mb q pb q pa (5.8)

• components of the Rung–Lenz vector

qa Aa = (qa p − qb p )p − . (5.9) b a b r b

By virtue of the Noether’s inverse ﬁrst Theorem 4.15, these integrals of motion are conserved symmetry currents of the following Hamiltonian symmetries: • the exact symmetry ∂t , • the exact vertical symmetries

υa = b∂ − a∂ − ∂b + ∂a, b q a q b pa pb (5.10) 5 Global Kepler Problem 95

• the vertical symmetries qa υa = [p υa + (qb p − qa p )∂ ]−∂ ∂b. (5.11) b b a b b b r b

υa Let us note that the Hamiltonian symmetries b (5.10) are the canonical lift ∗ va (4.78) onto V Q of the vector fields b (5.3)onQ, which thus are basic Hamiltonian a symmetries, and integrals of motion Mb (5.8) are the Noether currents (4.91). At the same time, the Hamiltonian symmetries (5.11) do not come from any vector fields on a configuration space Q. Therefore, in contrast with the Rung–Lenz vector (5.11) in Hamiltonian mechanics, the Rung–Lenz vector (5.6) in Lagrangian mechanics fails to be a conserved symmetry current of a Lagrangian symmetry. As was mentioned above, the Hamiltonian symmetries of the Kepler problem make up CISs. Without a loss of generality, we further consider the Kepler problem on a configuration space R2. Its phase space is T ∗R2 = R4 provided with the Cartesian coordinates (qi , pi ), i = 1, 2, and the canonical symplectic form ΩT = dpi ∧ dqi . (5.12) i

Let us denote 2 2 2 i 2 p = (pi ) , r = (q ) ,(p, q) = pi qi . i i i

An autonomous Hamiltonian of a Kepler system reads

1 H = p2 − r −1 (5.13) 2 (cf. (5.7)). A Kepler system is a Hamiltonian system on a symplectic manifold Z = 4 R \{0} endowed with the symplectic form ΩT (5.12). Let us consider functions

M12 =−M21 = q1 p2 − q2 p1, (5.14) q q A = M p − i = q p2 − p (p, q) − i , i = 1, 2, (5.15) i ij j r i i r j on Z. They are integrals of motion of the Hamiltonian H (5.13) where M12 is an angular momentum and (Ai ) is the Rung–Lenz vector. Let us denote

2 2 2 2 2 2 M = (M12) , A = (A1) + (Aa) = 2M H + 1. (5.16)

Let Z0 ⊂ Z be a closed subset of points where M12 = 0. A direct computation shows that the functions (M12, Ai ) (5.14) and (5.15) are independent on an open 96 5 Global Kepler Problem submanifold U = Z \ Z0 of Z. At the same time, the functions (H , M12, Ai ) are independent nowhere on U because it follows from the expression (5.16) that

A2 − 1 H = (5.17) 2M2 on U. The well known dynamics of a Kepler system shows that the Hamiltonian vector ﬁeld of its Hamiltonian is complete on U (but not on Z). Poisson brackets of integrals of motion M12 (5.14) and Ai (5.15) read

A2 − 1 {M12, Ai }=η2i A1 − η1i A2, {A1, A2}=2H M12 = , (5.18) M12

2 where ηij is an Euclidean metric on R . It is readily observed that these relations take the form (4.106). However, the matrix function s of the relations (5.18) fails to be of constant rank at points where H = 0. Therefore, let us consider open submanifolds U− ⊂ U where H < 0 and U+ where H > 0. Then we observe that both a Kepler system with the Hamiltonian H (5.13) and the integrals of motion (Mij, Ai ) (5.14) and (5.15)onU− and a Kepler system with the Hamiltonian H (5.13) and the integrals of motion (Mij, Ai ) (5.14) and (5.15)onU+ are noncommutative CISs. Moreover, these CISs can be brought into the form (4.113)asfollows. Let us replace the integrals of motions Ai with the integrals of motion

−1/2 Li = Ai (−2H ) (5.19) on U−, and with the integrals of motion

−1/2 Ki = Ai (2H ) (5.20) on U+.TheCIS(M12, Li ) on U− obeys relations

{M12, Li }=η2i L1 − η1i L2, {L1, L2}=−M12. (5.21)

Let us denote Mi3 =−Li and put the indexes μ, ν, α, β = 1, 2, 3. Then the relations (5.21) are brought into a form

{Mμν , Mαβ }=ημβ Mνα + ηνα Mμβ − ημα Mνβ − ηνβ Mμα (5.22)

3 where ημν is an Euclidean metric on R . A glance at the expression (5.22) shows that the integrals of motion M12 (5.14) and Li (5.19) constitute a Lie algebra g = so(3). Its corank equals 1. Therefore the CIS (M12, Li ) on U− is maximally integrable. The equality (5.17) takes a form

M2 + L2 =−(2H )−1. (5.23) 5 Global Kepler Problem 97

The CIS (M12, Ki ) on U+ obeys relations

{M12, Ki }=η2i K1 − η1i K2, {K1, K2}=M12. (5.24)

Let us denote Mi3 =−Ki and put the indexes μ, ν, α, β = 1, 2, 3. Then the relations (5.24) are brought into a form

{Mμν, Mαβ }=ρμβ Mνα + ρνα Mμβ − ρμα Mνβ − ρνβ Mμα (5.25)

3 where ρμν is a pseudo-Euclidean metric of signature (+, +, −) on R . A glance at the expression (5.25) shows that the integrals of motion M12 (5.14) and Ki (5.20) constitute a Lie algebra so(2, 1). Its corank equals 1. Therefore the CIS (M12, Ki ) on U+ is maximally integrable. The equality (5.17) takes a form

K 2 − M2 = (2H )−1. (5.26)

Thus, the Kepler problem on a phase space R4 falls into two different maximally 4 integrable systems on open submanifolds U− and U+ of R . We agree to call them the Kepler CISs on U− and U+, respectively. Let us study the ﬁrst one, and let us put

F1 =−L1, F2 =−L2, F3 =−M12, (5.27)

{F1, F2}=F3, {F2, F3}=F1, {F3, F1}=F2.

We have a ﬁbred manifold ∗ F : U− → N ⊂ g , (5.28) which is the momentum mapping to a Lie coalgebra g∗ = so(3)∗, endowed with ∗ coordinates (xi ) such that integrals of motion Fi on g read Fi = xi (Remark 4.23). A base N of the ﬁbred manifold (5.28) is an open submanifold of g∗ given by a coordinate condition x3 = 0. It is a union of two contractible components with x3 > 0 and x3 < 0. A coinduced Lie–Poisson structure on N is given by the bivector

3 1 1 2 2 3 w = x2∂ ∧ ∂ + x3∂ ∧ ∂ + x1∂ ∧ ∂ . (5.29)

The coadjoint action of so(3) on N reads

2 3 3 1 1 2 ε1 = x3∂ − x2∂ ,ε2 = x1∂ − x3∂ ,ε3 = x2∂ − x1∂ .

2 + 2 + 2 = . Orbits of this action are given by an equation x1 x2 x3 const They are level = 2 + 2 + 2 surfaces of a Casimir function C x1 x2 x3 and, hence, a Casimir function

1 h =− (x2 + x2 + x2)−1. (5.30) 2 1 2 3 98 5 Global Kepler Problem

A glance at the expression (5.23) shows that the pull-back F ∗h of this Casimir function (5.30) onto U− is the Hamiltonian H (5.13) of a Kepler system on U−. As was mentioned above, the Hamiltonian vector field of F ∗h is complete. Fur- thermore, it is known that invariant submanifolds of the Kepler CIS on U− are compact. Therefore, the fibred manifold F (5.28) is a fibre bundle in accordance with Theorem B.6. Moreover, this fibre bundle is trivial because N is a disjoint union of two contractible manifolds. Consequently, it follows from Theorem 4.22 that the Kepler CIS on U− is global, i.e., it admits global action-angle coordinates as follows. The Poisson manifold N (5.28) can be endowed with the coordinates

(I, x1,γ), I < 0,γ= π/2, 3π/2, (5.31) deﬁned by the equalities

1 I =− (x2 + x2 + x2)−1, (5.32) 2 1 2 3 = (−( )−1 − 2)1/2 γ, = (−( )−1 − 2)1/2 γ. x2 2I x1 sin x3 2I x1 cos

It is readily observed that the coordinates (5.31) are Darboux coordinates of the w = ∂ ∧ ∂ Lie–Poisson structure (5.29)onU−, namely, x1 γ . Let ϑI be the Hamiltonian vector field of the Casimir function I (5.32). Its flows are invariant submanifolds of the Kepler CIS on U− (Remark 4.23). Let α be a parameter along the flow of this vector field, i.e.,

ϑI = ∂α. (5.33)

Then U− is provided with the action-angle coordinates (I, x1,γ,α) such that the Poisson bivector associated to the symplectic form ΩT on U− reads ∂ ∂ ∂ ∂ w = ∧ + ∧ . ∂ I ∂α ∂x1 ∂γ

Accordingly, Hamiltonian vector ﬁelds of integrals of motion Fi (5.27) take a form ∂ ϑ = , 1 ∂γ − / − / 1 1 1 2 ∂ 1 1 2 ∂ ϑ = − − x2 sin γ − x − − x2 sin γ − 2 4I 2 2I 1 ∂α 1 2I 1 ∂γ / 1 1 2 ∂ − − 2 γ , x1 cos 2I ∂x1 5 Global Kepler Problem 99

− / − / 1 1 1 2 ∂ 1 1 2 ∂ ϑ = − − x2 cos γ − x − − x2 cos γ + 3 4I 2 2I 1 ∂α 1 2I 1 ∂γ / 1 1 2 ∂ − − 2 γ . x1 sin 2I ∂x1

A glance at these expressions shows that the vector ﬁelds ϑ1 and ϑ2 fail to be complete on U− (Remark 4.20). One can say something more about the angle coordinate α. The vector ﬁeld ϑI (5.33) reads ∂ ∂H ∂ ∂H ∂ = − . ∂α ∂p ∂q ∂q ∂p i i i i i

This equality leads to relations

∂ ∂H ∂ ∂H qi = , pi =− , ∂α ∂pi ∂α ∂qi which take a form of the Hamilton equation. Therefore, the coordinate α is a cyclic time α = t mod 2π given by the well-known expression

α = φ − a3/2e sin(a−3/2φ), r = a(1 − e cos(a−3/2φ)), a = (2I )−1, e = (1 + 2IM2)1/2.

Now let us turn to the Kepler CIS on U+. It is a globally integrable system with noncompact invariant submanifolds as follows. Let us put

S1 =−K1, S2 =−K2, S3 =−M12, (5.34)

{S1, S2}=−S3, {S2, S3}=S1, {S3, S1}=S2.

We have a fibred manifold ∗ S : U+ → N ⊂ g , (5.35) which is the momentum mapping to a Lie coalgebra g∗ = so(2, 1)∗, endowed with ∗ the coordinates (xi ) such that integrals of motion Si on g read Si = xi . A base N of the fibred manifold (5.35) is an open submanifold of g∗ given by a coordinate condition x3 = 0. It is a union of two contractible components defined by conditions x3 > 0 and x3 < 0. A coinduced Lie–Poisson structure on N takes a form

3 1 1 2 2 3 w = x2∂ ∧ ∂ − x3∂ ∧ ∂ + x1∂ ∧ ∂ . (5.36)

The coadjoint action of so(2, 1) on N reads

2 3 3 1 1 2 ε1 =−x3∂ − x2∂ ,ε2 = x1∂ + x3∂ ,ε3 = x2∂ − x1∂ . 100 5 Global Kepler Problem

2 + 2− 2 = . The orbits of this coadjoint action are given by an equation x1 x2 x3 const They = 2 + 2 − 2 are the level surfaces of the Casimir function C x1 x2 x3 and, consequently, the Casimir function 1 h = (x2 + x2 − x2)−1. (5.37) 2 1 2 3 A glance at the expression (5.26) shows that the pull-back S∗h of this Casimir function (5.37) onto U+ is the Hamiltonian H (5.13) of a Kepler system on U+. As was mentioned above, the Hamiltonian vector field of S∗h is complete. Fur- thermore, it is known that invariant submanifolds of the Kepler CIS on U+ are diffeomorphic to R. Therefore, the fibred manifold S (5.35) is a fibre bundle in accordance with Theorem B.6. Moreover, this fibre bundle is trivial because N is a disjoint union of two contractible manifolds. Consequently, it follows from Theorem 4.22 that the Kepler CIS on U+ is globally integrable, i.e., it admits global action-angle coordinates as follows. The Poisson manifold N (5.35) can be endowed with the coordinates (I, x1,λ), I > 0, λ = 0, defined by the equalities

1 I = (x2 + x2 − x2)−1, 2 1 2 3 = (( )−1 − 2)1/2 λ, = (( )−1 − 2)1/2 λ. x2 2I x1 cosh x3 2I x1 sinh

These coordinates are Darboux coordinates of the Lie–Poisson structure (5.36)on w = ∂ ∧ ∂ N, namely, λ x1 . Let ϑI be the Hamiltonian vector field of the Casimir function I (5.32). Its flows are invariant submanifolds of the Kepler CIS on U+ (Remark 4.23). Let τ be a parameter along the flows of this vector field, i.e.,

ϑI = ∂τ . (5.38)

Then U+ (5.35) is provided with the action-angle coordinates (I, x1,λ,τ)such that the Poisson bivector associated to the symplectic form ΩT on U+ reads ∂ ∂ ∂ ∂ W = ∧ + ∧ . ∂ I ∂τ ∂λ ∂x1

Accordingly, Hamiltonian vector ﬁelds of integrals of motion Si (5.34) take a form ∂ ϑ =− , 1 ∂λ − / − / 1 1 1 2 ∂ 1 1 2 ∂ ϑ = − x2 cosh λ + x − x2 cosh λ + 2 4I 2 2I 1 ∂τ 1 2I 1 ∂λ / 1 1 2 ∂ − 2 λ , x1 sinh 2I ∂x1 5 Global Kepler Problem 101

− / − / 1 1 1 2 ∂ 1 1 2 ∂ ϑ = − x2 sinh λ + x − x2 sinh λ + 3 4I 2 2I 1 ∂τ 1 2I 1 ∂λ / 1 1 2 ∂ − 2 λ . x1 cosh 2I ∂x1

Similarly to the angle coordinate α (5.33), the angle coordinate τ (5.38) obeys the Hamilton equation ∂ ∂H ∂ ∂H qi = , pi =− . ∂τ ∂pi ∂τ ∂qi

Therefore, it is the time τ = t given by the well-known expression

τ = s − a3/2e sinh(a−3/2s), r = a(e cosh(a−3/2s) − 1), a = (2I )−1, e = (1 + 2IM2)1/2.

Chapter 6 Calculus of Variations on Graded Bundles

Throughout the book, by the Grassmann gradation is meant the Z2-gradation. It shortly is called the graded structure if there is no danger of confusion. The symbol [.] stands for the Grassmann parity. From the mathematical viewpoint, we restrict our consideration to simple graded manifolds (Definition6.4) and graded bundles over smooth manifolds (Defini- tion 6.5). A key point is that vector fields and exterior one-forms on a simple graded manifold with a body manifold Z are represented by sections of the vector bundles (6.20) and (6.22) over Z, respectively. We follow Definition6.7 of graded jet manifolds of graded bundles which is compatible with the conventional Definition B.11 of jets of fibre bundles. It differs from the definition of jets of modules over graded commutative rings [58, 131] and from that of jets of fibred-graded manifolds [73, 107], but reproduces the heuristic notion of jets of odd ghosts in BRST field theory [6, 22].

6.1 Grassmann-Graded Algebraic Calculus

Let us summarize the relevant notions of the Grassmann-graded algebraic calculus [7, 29, 61, 131]. Let K be a commutative ring. A K -module Q is termed graded if it is endowed with a grading automorphism γ such that γ 2 = Id. A graded module falls into a direct sum Q = Q0 ⊕ Q1 of K -modules Q0 and Q1 of even and odd elements such [q] that γ(q) = (−1) q, q ∈ Q[q]. One calls Q0 and Q1 the even and odd parts of Q, respectively. In particular, by a real graded vector space B = B0 ⊕ B1 is meant a graded n R-module. A real graded vector space is said to be (n, m)-dimensional if B0 = R m and B1 = R . A K -algebra algebra A is called graded if it a graded K -module such that

[aa ]=([a]+[a ])mod 2, a ∈ A[a], a ∈ A[a]. © Atlantis Press and the author(s) 2016 103 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_6 104 6 Calculus of Variations on Graded Bundles

Its even part A0 is a subalgebra of A and the odd one A1 is an A0-module. If A is a graded ring, then [1]=0.

Deﬁnition 6.1 A graded algebra A is called graded commutative if

aa = (−1)[a][a ]aa, where a and a are graded-homogeneous elements of A , i.e., they are either even or odd.

Given a graded algebra A , a left graded A -module Q is deﬁned as a left A - module where [aq]=([a]+[q])mod 2. Similarly, right graded A -modules are treated.

Deﬁnition 6.2 Let V be a real vector space, and let Λ =∧V be its exterior algebra endowed with the Grassmann gradation 2k 2k−1 Λ = Λ0 ⊕ Λ1,Λ0 = R ∧ V,Λ1 = ∧ V. (6.1) k=1 k=1

It is a real graded commutative ring, termed the Grassmann algebra.

A Grassmann algebra, seen as an additive group, admits the decomposition

2 Λ = R ⊕ R = R ⊕ R0 ⊕ R1 = R ⊕ (Λ1) ⊕ Λ1, where R is the ideal of nilpotents of Λ. The corresponding projections σ : Λ → R and s : Λ → R are called the body and soul maps, respectively. Let us note that there is a different definition of a Grassmann algebra [78] which is equivalent to the above mentioned one only in the case of an infinite-dimensional vector space V [29]. Hereafter, we restrict our consideration to Grassmann algebras of finite rank. Given a basis {ci } for a vector space V , elements of a Grassmann algebra Λ (6.1) take a form = i1 ··· ik , a ai1···ik c c

k=0,1,... (i1···ik ) where the second sum runs through all the tuples (i1 ···ik ) such that no two of them are permutations of each other.

Deﬁnition 6.3 A graded (non-associative) algebra g is termed a Lie superalgebra if its product [., .], called the graded Lie bracket,orsuperbracket obeys the relations

[ε, ε]=−(−1)[ε][ε ][ε,ε], (−1)[ε][ε ][ε, [ε,ε]] + (−1)[ε ][ε][ε, [ε,ε]] + (−1)[ε ][ε ][ε, [ε, ε]] = 0. 6.1 Grassmann-Graded Algebraic Calculus 105

Being decomposed in even and odd parts g = g0 ⊕ g1, a Lie superalgebra g obeys the relations

[g0, g0]⊂g0, [g0, g1]⊂g1, [g1, g1]⊂g1.

In particular, an even part g0 of a Lie superalgebra g is a Lie algebra. A graded K -module P is called the g-module if it is provided with an R-bilinear map

g × P (ε, p) → εp ∈ P, [εp]=([ε]+[p])mod 2, [ε, ε]p = (ε ◦ ε − (−1)[ε][ε ]ε ◦ ε)p.

If A is graded commutative, a graded K -module can be provided with a graded A -bimodule structure by letting

qa = (−1)[a][q]aq, a ∈ K , q ∈ Q.

Given a graded commutative ring A , the following are standard constructions of new graded A -modules from old ones. • A direct sum of graded modules and a graded quotient module are deﬁned just as those of modules over a commutative ring. • A tensor product P ⊗ Q of graded A -modules P and Q is an additive group generated by elements p ⊗ q, p ∈ P, q ∈ Q, obeying the relations

(p + p) ⊗ q = p ⊗ q + p ⊗ q, p ⊗ (q + q) = p ⊗ q + p ⊗ q, ap ⊗ q = (−1)[p][a] pa ⊗ q = (−1)[p][a] p ⊗ aq, a ∈ A .

In particular, the tensor algebra ⊗P of a graded K -module P is deﬁned as that (A.4) of a module over a commutative ring. Its quotient ∧P with respect to the ideal generated by elements

p ⊗ p − (−1)[p][p ] p ⊗ p, p, p ∈ P, is the bigraded exterior algebra of a graded module P with respect to the graded exterior product

p ∧ p = (−1)[p][p ] p ∧ p.

• A morphism Φ : P → Q of graded A -modules seen as additive groups is said to be even morphism (resp. odd morphism)ifΦ preserves (resp. change) the Grassmann parity of all graded-homogeneous elements of P and obeys the relations

Φ(ap) = (−1)[Φ][a]aΦ(p), p ∈ P, a ∈ A . 106 6 Calculus of Variations on Graded Bundles

A morphism Φ : P → Q of graded A -modules as additive groups is termed a graded A -module morphism if it is represented by a sum of even and odd morphisms. The set HomK (P, Q) of graded morphisms of a graded A -module P to a graded A -module ∗ Q is naturally a graded A -module. A graded A -module P = HomA (P, K ) is called the dual of a graded K -module P. Let B be a graded real vector space. Given a Grassmann algebra Λ, it can be brought into a graded Λ-module

ΛB = (ΛB)0 ⊕ (ΛB)1 = (Λ0 ⊗ B0 ⊕ Λ1 ⊗ B1) ⊕ (Λ1 ⊗ B0 ⊕ Λ0 ⊗ B1), called the superspace. A superspace

n m n m n|m B =[(⊕ Λ0) ⊕ (⊕ Λ1)]⊕[(⊕ Λ1) ⊕ (⊕ Λ0)] is said to be (n, m)-dimensional. A graded Λ0-module

n m n,m B = (⊕ Λ0) ⊕ (⊕ Λ1) is termed an (n, m)-dimensional supervector space.

6.2 Grassmann-Graded Differential Calculus

The differential calculus over graded commutative rings (Deﬁnition6.1) is developed similarly to that over commutative rings (Sects. A.1 and A.2) [61, 131, 134]. Let K be a commutative ring and A a graded commutative K -ring (Deﬁn- ition6.1). Let P and Q be graded A -modules. A K -module HomK (P, Q) of graded K -module homomorphisms Φ : P → Q can be endowed with the two graded A -module structures

(aΦ)(p) = aΦ(p), (Φ • a)(p) = Φ(ap), a ∈ A , p ∈ P.

Let us put

[a][Φ] δaΦ = aΦ − (−1) Φ • a, a ∈ A .

An element Δ ∈ HomK (P, Q) is said to be a Q-valued graded differential operator δ ◦···◦δ Δ = + ,..., of order s on P if a0 as 0 for any tuple of s 1 elements a0 as of A . In particular, zero order graded differential operators obey a condition

[a][Δ] δaΔ(p) = aΔ(p) − (−1) Δ(ap) = 0, a ∈ A , p ∈ P, 6.2 Grassmann-Graded Differential Calculus 107 i.e., they coincide with graded A -module morphisms P → Q. A ﬁrst order graded differential operator Δ satisﬁes a relation

([b]+[Δ])[a] [b][Δ] δa ◦ δb Δ(p) = abΔ(p) − (−1) bΔ(ap) − (−1) aΔ(bp) + (−1)[b][Δ]+([Δ]+[b])[a] = 0, a, b ∈ A , p ∈ P.

For instance, let P = A . Any zero order Q-valued graded differential operator Δ on A is deﬁned by its value Δ(1). Then there is a graded A -module isomorphism

Diff0(A , Q) = Q, Q q → Δq ∈ Diff0(A , Q), where Δq is given by the equality Δq (1) = q. A ﬁrst order Q-valued graded differential operator Δ on A fulﬁls the condition

Δ(ab) = Δ(a)b + (−1)[a][Δ]aΔ(b) − (−1)([b]+[a])[Δ]abΔ(1), a, b ∈ A .

It is called a Q-valued graded derivation of A if Δ(1) = 0, i.e., the Grassmann- graded Leibniz rule

Δ(ab) = Δ(a)b + (−1)[a][Δ]aΔ(b), a, b ∈ A , (6.2) holds. One obtains at once that any ﬁrst order graded differential operator on A falls into a sum

Δ(a) = Δ(1)a +[Δ(a) − Δ(1)a] of a zero order graded differential operator Δ(1)a and a graded derivation Δ(a) − Δ(1)a.If∂ is a graded derivation of A , then a∂ is so for any a ∈ A . Hence, graded derivations of A constitute a graded A -module d(A , Q), termed the graded derivation module. If Q = A , the graded derivation module dA also is a Lie superalgebra (Deﬁni- tion6.3) over a commutative ring K with respect to a superbracket

[u, u]=u ◦ u − (−1)[u][u ]u ◦ u, u, u ∈ A . (6.3)

Since dA is a Lie K -superalgebra, let us consider the Chevalley–Eilenberg complex C∗[dA ; A ] where a graded commutative ring A is a regarded as a dA -module [47, 131]. It is a complex

d d d 0 → A −→ C1[dA ; A ] −→···Ck [dA ; A ] −→··· (6.4) where

k k C [dA ; A ]=HomK (∧ dA , A ) 108 6 Calculus of Variations on Graded Bundles are dA -modules of K -linear graded morphisms of the graded exterior products k ∧ dA of a graded K -module dA to A . Let us bring homogeneous elements of k ∧ dA into a form

ε1 ∧···εr ∧ εr+1 ∧···∧εk ,εi ∈ dA0,εj ∈ dA1.

Then the Chevalley–Eilenberg coboundary operator d of the complex (6.4)isgiven by the expression

dc(ε1 ∧···∧εr ∧ ε1 ∧···∧εs ) = (6.5) r i−1 (−1) εi c(ε1 ∧···εi ···∧εr ∧ ε1 ∧···εs ) + i=1 s r (−1) εi c(ε1 ∧···∧εr ∧ ε1 ∧···ε j ···∧εs ) + = j 1 i+ j (−1) c([εi ,εj ]∧ε1 ∧···εi ···ε j ···∧εr ∧ ε1 ∧···∧εs ) + ≤ < ≤ 1 i j r c([εi ,εj ]∧ε1 ∧···∧εr ∧ ε1 ∧···εi ···ε j ···∧εs ) + ≤ < ≤ 1 i j s i+r+1 (−1) c([εi ,εj ]∧ε1 ∧···εi ···∧εr ∧ ε1 ∧···ε j ···∧εs ), 1≤i

in 0 → K −→ C∗[dA ; A ].

It is easily justiﬁed that this complex contains a subcomplex O∗[dA ] of A -linear graded morphisms. The N-graded module O∗[dA ] is provided with the structure of a bigraded A -algebra with respect to the graded exterior product

φ ∧ φ (u , ..., u + ) = 1 r s (6.6) ··· ··· i1 ir j1 js φ( ,..., )φ( ,..., ), Sgn1···r+s ui1 uir u j1 u js i1<···

··· ··· ∧···∧ = i1 ir j1 js ∧···∧ ∧ ∧···∧ . u1 ur+s Sgn1···r+s ui1 uir u j1 u js 6.2 Grassmann-Graded Differential Calculus 109

The graded Chevalley–Eilenberg coboundary operator d (6.5) and the graded exterior product ∧ (6.6)bringO∗[dA ] into a differential bigraded algebra (henceforth, DBGA) whose elements obey relations

φ ∧ φ = (−1)|φ||φ |+[φ][φ ]φ ∧ φ, (6.7) d(φ ∧ φ) = dφ ∧ φ + (−1)|φ|φ ∧ dφ. (6.8)

It is called the graded Chevalley–Eilenberg differential calculus over a graded commutative K -ring A . In particular, we have

1 ∗ O [dA ]=HomA (dA , A ) = dA . (6.9)

One can extend this duality relation to the graded interior product of u ∈ dA with any element φ ∈ O∗[dA ] by the rules

u (bda) = (−1)[u][b]u(a), a, b ∈ A , u (φ ∧ φ) = (u φ) ∧ φ + (−1)|φ|+[φ][u]φ ∧ (u φ). (6.10)

As a consequence, any graded derivation u ∈ dA of A yields a derivation

∗ Luφ = u dφ + d(u φ), φ ∈ O , u ∈ dA , (6.11) [u][φ] Lu(φ ∧ φ ) = Lu(φ) ∧ φ + (−1) φ ∧ Lu(φ ), termed the graded Lie derivative of a DBGA O∗[dA ]. Let us note that, if A is a commutative ring, the graded Chevalley–Eilenberg differential calculus comes to the familiar one (Sect. A.3). The minimal graded Chevalley–Eilenberg differential calculus O∗A ⊂ O∗[dA ] over a graded commutative ring A consists of monomials a0da1 ∧···∧dak , ai ∈ A . The corresponding complex

d d d 0 → K −→ A −→ O1A −→···Ok A −→··· (6.12) is called the bigraded de Rham complex of a graded commutative K -ring A .

6.3 Differential Calculus on Graded Bundles

In accordance with Serre–Swan Theorem 6.2 below, if a real graded commutative algebra A is generated by a projective module of finite rank over the ring C∞(Z) of smooth functions on some manifold Z, then A is isomorphic to the algebra of graded functions on a graded manifold with a body Z, and vice versa. Then the 110 6 Calculus of Variations on Graded Bundles minimal graded Chevalley–Eilenberg differential calculus O∗A over A is a DBGA of graded exterior forms on this graded manifold [61, 131, 134]. As was mentioned above, we restrict our consideration to simple graded manifolds modelled over vector bundles (Definition 6.4) and graded bundles over smooth manifolds (Definition6.5). Graded Manifolds A graded manifold of dimension (n, m) is defined as a local-ringed space (Z, A) (Definition C.1) where Z is an n-dimensional smooth manifold, and A = A0 ⊕ A1 is a sheaf of Grassmann algebras of rank m (Definition6.2) such that [7, 134]: • there is the exact sequence of sheaves

σ → R → A → ∞ → , R = A + (A )2, 0 CZ 0 1 1 (6.13)

∞ where CZ is the sheaf of smooth real functions on Z; • R/R2 ∞ is a locally free sheaf of CZ -modules of finite rank (with respect to point- wise operations), and the sheaf A is locally isomorphic to the exterior product ∧ ∞ (R/R2) CZ . A sheaf A is called the structure sheaf of a graded manifold (Z, A), and a manifold Z is said to be the body of (Z, A). Sections of the sheaf A are termed graded functions on a graded manifold (Z, A). They make up a graded commutative C∞(Z)-ring A(Z) called the structure ring of (Z, A). By virtue of the well-known Batchelor theorem [7, 14, 134], graded manifolds possess the following structure. Theorem 6.1 Let (Z, A) be a graded manifold. There exists a vector bundle E → Z with an m-dimensional typical fibre V such that the structure sheaf A of (Z, A) is isomorphic to the structure sheaf AE = S∧E∗ of germs of sections of the exterior bundle ∧E∗ (B.13), whose typical fibre is the Grassmann algebra ∧V ∗.

It should be emphasized that Batchelor’s isomorphism in Theorem6.1 fails to be canonical. In applications, it however is ﬁxed from the beginning. Therefore, we restrict our consideration to graded manifolds (Z, AE ) whose structure sheaf is the sheaf of germs of sections of some exterior bundle ∧E∗.

Deﬁnition 6.4 We agree to call (Z, AE ) the simple graded manifold modelled over a vector bundle E → Z, called its characteristic vector bundle.

Accordingly, the structure ring AE of a simple graded manifold (Z, AE ) is the structure module ∗ AE = AE (Z) =∧E (Z) (6.14) of sections of the exterior bundle ∧E∗. Automorphisms of a simple graded manifold (Z, AE ) are restricted to those induced by automorphisms of its characteristic vector bundles E → Z (Remark6.3). 6.3 Differential Calculus on Graded Bundles 111

Combining Batchelor Theorem6.1 and classical Serre–Swan Theorem A.10, we come to the following Serre–Swan theorem for graded manifolds [11, 61]. Theorem 6.2 Let Z be a smooth manifold. A graded commutative C∞(Z)-algebra A is isomorphic to the structure ring of a graded manifold with a body Z if and only if it is the exterior algebra of some projective C∞(Z)-module of ﬁnite rank. Proof By virtue of the Batchelor theorem, any graded manifold is isomorphic to a simple graded manifold (Z, AE ) modelled over some vector bundle E → Z.Its structure ring AE (6.14) of graded functions consists of sections of the exterior bundle ∧E∗. The classical Serre–Swan theorem states that a C∞(Z)-module is isomorphic to the module of sections of a smooth vector bundle over Z if and only if it is a projective module of ﬁnite rank. ( , A = ∞) Remark 6.1 One can treat a local-ringed space Z 0 CZ as a trivial graded manifold. It is a simple graded manifold whose characteristic bundle is E = Z ×{0}. Its structure module is a ring C∞(Z) of smooth real functions on Z.

A a Given a graded manifold (Z, AE ), every trivialization chart (U; z , y ) of the A a vector bundle E → Z yields a splitting domain (U; z , c ) of (Z, AE ). Graded functions on such a chart are Λ-valued functions

m 1 a a f = f ... (z)c 1 ···c k , (6.15) k! a1 ak k=0

( ) { a} ∗ where fa1···ak z are smooth functions on U and c is the ﬁbre basis for E .In particular, the sheaf epimorphism σ in (6.13) is induced by the body map of Λ. A a One calls {z , c } the local basis for the graded manifold (Z, AE ) [7]. Transition a = ρa( A) b → functions y b z y of bundle coordinates on E Z induce the corresponding transformation a = ρa( A) b c b z c (6.16) of the associated local basis for a graded manifold (Z, AE ) and the according coordinate transformation law of graded functions (6.15). Remark 6.2 Strictly speaking, elements ca of the local basis for a graded manifold a ∗ → ◦ a = δa are locally constant sections c of E X such that yb c b . Therefore, graded functions are locally represented by Λ-valued functions (6.15), but they are not Λ-valued functions on a manifold Z because of the transformation law (6.16). Remark 6.3 In general, automorphisms of a graded manifold take the form

ca = ρa(z A, cb), (6.17) where ρa(z A, cb) are local graded functions. Considering a simple graded manifold (Z, AE ), we restrict the class of graded manifold transformations (6.17) to the linear ones (6.16), compatible with given Batchelor’s isomorphism. 112 6 Calculus of Variations on Graded Bundles

Given a graded manifold (Z, A),bythesheaf dA of graded derivations of A is meant a subsheaf of endomorphisms of the structure sheaf A such that any section u ∈ dA(U) of dA over an open subset U ⊂ Z is a graded derivation of the real graded commutative algebra A(U), i.e., u ∈ d(A(U)). Conversely, one can show that, given open sets U ⊂ U, there is a surjection of the graded derivation modules

d(A(U)) → d(A(U )).

It follows that any graded derivation of a local graded algebra A(U) also is a local section over U of a sheaf dA. Global sections of dA are called graded vector fields on the graded manifold (Z, A). They make up a graded derivation module dA(Z) of a real graded commutative ring A(Z). This module is a real Lie superalgebra with respect to the superbracket (6.3). A key point is that graded vector fields u ∈ dAE on a simple graded manifold (Z, AE ) can be represented by sections of some vector bundle as follows [58, 100, 131]. Due to the canonical splitting VE = E × E, the vertical tangent bundle VE of E → Z can be provided with the fibre bases {∂/∂ca}, which are the duals of the a A a bases {c }. Then graded vector fields on a splitting domain (U; z , c ) of (Z, AE ) read ∂ u = u A∂ + ua , (6.18) A ∂ca where uλ, ua are local graded functions on U. In particular, ∂ ∂ ∂ ∂ ∂ ∂ ◦ =− ◦ ,∂◦ = ◦ ∂ . ∂ca ∂cb ∂cb ∂ca A ∂ca ∂ca A

The graded derivations (6.18) act on graded functions f ∈ AE (U) (6.15)bytherule ∂ a b A a b k a b u( f ... c ···c ) = u ∂ ( f ... )c ···c + u f ... (c ···c ). (6.19) a b A a b a b ∂ck This rule implies the corresponding coordinate transformation law

A = A, a = ρa j + A∂ (ρa) j u u u j u u A j c of graded vector ﬁelds. It follows that graded vector ﬁelds (6.18) can be represented by sections of the following vector bundle VE → Z. This vector bundle is locally isomorphic to a vector bundle

∗ VE ≈∧E ⊗(E ⊕ TZ), (6.20) Z Z and is characterized by an atlas of bundle coordinates (z A, z A ,vi ), k = a1...ak b1...bk 0,...,m, possessing the transition functions 6.3 Differential Calculus on Graded Bundles 113

A −1a1 −1ak A z ... = ρ ···ρ z ... , i1 ik i1 ik a1 ak ! i −1b1 −1bk i j k A i v ... = ρ ···ρ ρ v ... + z ... ∂Aρ , j1 jk j1 jk j b1 bk (k − 1)! b1 bk−1 bk which fulﬁl the cocycle condition (B.4). Thus, the graded derivation module dAE is isomorphic to the structure module VE (Z) of global sections of a vector bundle VE → Z. Given the structure ring AE of graded functions on a simple graded manifold (Z, AE ) and the real Lie superalgebra dAE of its graded derivations, let us consider the graded Chevalley–Eilenberg differential calculus

∗ ∗ S [E; Z]=O [dAE ] (6.21)

0 over AE where S [E; Z]=AE .

Theorem 6.3 Since a graded derivation module dAE is isomorphic to the struc- ∗ ture module of sections of a vector bundle VE → Z, elements of S [E; Z] are represented by sections of the exterior bundle ∧V E of the AE -dual V E → ZofVE .

The bundle V E is locally isomorphic to a vector bundle

∗ ∗ ∗ V E ≈∧E ⊗(E ⊕ T Z). (6.22) Z Z

A ∗ b ∗ With respect to the dual ﬁbre bases {dz } for T Z and {dc } for E , sections of V E take a coordinate form

A a φ = φAdz + φadc , together with transition functions

φ = ρ−1bφ ,φ = φ + ρ−1b∂ (ρa)φ j . a a b A A a A j bc

S 1[ ; ]=dA ∗ The duality isomorphism E Z E (6.9) is given by the graded interior product

A [φa ] a u φ = u φA + (−1) u φa.

Elements of S ∗[E; Z] are called graded exterior forms on a graded manifold (Z, AE ). ∗ A a Seen as an AE -algebra, the DBGA S [E; Z] (6.21) on a splitting domain (z , c ) is locally generated by the graded one-forms dzA, dci such that

dzA ∧ dci =−dci ∧ dzA, dci ∧ dcj = dcj ∧ dci . 114 6 Calculus of Variations on Graded Bundles

Accordingly, the graded Chevalley–Eilenberg coboundary operator d (6.5), termed the graded exterior differential, reads ∂ dφ = dzA ∧ ∂ φ + dca ∧ φ, A ∂ca

a where derivatives ∂λ, ∂/∂c act on coefﬁcients of graded exterior forms by the formula (6.19), and they are graded commutative with the graded exterior forms dzA and dca. The formulas (6.7)–(6.11) hold.

Theorem 6.4 The DBGA S ∗[E; Z] (6.21) is a minimal differential calculus over AE , i.e., it is generated by elements d f , f ∈ AE .

Proof The proof follows that of Theorem 1.8. Since dAE = VE (Z), it is a projective ∞ 1 C (Z)- and AE -module of ﬁnite rank, and so is its AE -dual S [E; Z]. Hence, dAE 1 1 is the AE -dual of S [E; Z] and, consequently, S [E; Z] is generated by elements df, f ∈ AE .

The bigraded de Rham complex (6.12) of a minimal graded Chevalley–Eilenberg differential calculus S ∗[E; Z] reads

d 1 d k d 0 → R → AE −→ S [E; Z] −→···S [E; Z] −→··· . (6.23)

∗ Its cohomology H (AE ) is called the de Rham cohomology of a simple graded manifold (Z, AE ). In particular, given a DGA O∗(Z) of exterior forms on Z, there exist a canonical monomorphism O∗(Z) → S ∗[E; Z] (6.24) and the body epimorphism S ∗[E; Z]→O∗(Z) which are cochain morphisms of the de Rham complexes (6.23) and (A.18).

Theorem 6.5 The de Rham cohomology of a simple graded manifold (Z, AE ) equals the de Rham cohomology of its body Z.

Ak ( , A ) Proof Let E denote the sheaf of germs of graded k-forms on Z E . Its structure module is S k [E; Z]. These sheaves constitute the complex

→ R −→ A −→d A1 −→···d Ak −→···d . 0 E E E (6.25)

Ak ∞ Its members E are sheaves of CZ -modules on Z and, consequently, are ﬁne and acyclic. Furthermore, the Poincaré lemma for graded exterior forms holds [7]. It follows that the complex (6.25) is a ﬁne resolution of the constant sheaf R on a manifold Z. Then, by virtue of Theorem C.5, there is an isomorphism

∗(A ) = ∗( ; R) = ∗ ( ) H E H Z HDR Z (6.26) 6.3 Differential Calculus on Graded Bundles 115

∗(A ) ∗ ( ) of the cohomology H E to the de Rham cohomology HDR Z of a smooth manifold Z.

Theorem 6.6 The cohomology isomorphism (6.26) is accompanied by the cochain monomorphism (6.24). Hence, any closed graded exterior form is decomposed into asumφ = σ + dξ where σ is a closed exterior form on Z.

Graded Bundles A morphism of graded manifolds (Z, A) → (Z , A) is defined as that of local- ringed spaces φ : Z → Z , Φ : A → φ∗A, (6.27) where φ is a manifold morphism and Φ is a sheaf morphism of A to the direct image φ∗A of A onto Z (Sect. C.3). The morphism (6.27) of graded manifolds is said to be: • a monomorphism if φ is an injection and Φ is an epimorphism; • an epimorphism if φ is a surjection and Φ is a monomorphism. An epimorphism of graded manifolds (Z, A) → (Z , A) where Z → Z is a fibre bundle is called the graded bundle [73, 140]. In this case, a sheaf monomorphism Φ induces a monomorphism of canonical presheaves A → A, which associates to each open subset U ⊂ Z the ring of sections of A over φ(U). Accordingly, there is a pull-back monomorphism of the structure rings A(Z ) → A(Z) of graded functions on graded manifolds (Z , A) and (Z, A). In particular, let (Y, A) be a graded manifold whose body Z = Y is a fibre bundle π : → ( , A = ∞) Y X. Let us consider a trivial graded manifold X 0 CX (Remark6.1). Then we have a graded bundle

( , A) → ( , ∞). Y X CX (6.28)

Let us denote it by (X, Y, A). Given a graded bundle (X, Y, A), the local basis for a graded manifold (Y, A) can be brought into a form (xλ, yi , ca) where (xλ, yi ) are bundle coordinates of Y → X.

Definition 6.5 We agree to call the graded bundle (6.28) over a trivial graded man- ( , ∞) ifold X CX the graded bundle over a smooth manifold. If Y → X is a vector bundle, the graded bundle (6.28) is a particular case of graded fibre bundles in [73, 107] when their base is a trivial graded manifold. → ( , ∞) Remark 6.4 Let Y X be a fibre bundle. Then a trivial graded manifold Y CY together with a real ring monomorphism C∞(X) → C∞(Y ) is a graded bundle ( , , ∞) X Y CY of trivial graded manifolds ( , ∞) → ( , ∞). Y CY X CX (6.29) 116 6 Calculus of Variations on Graded Bundles

Remark 6.5 A graded manifold (X, A) itself can be treated as the graded bundle (X, X, A) (6.28) associated to the identity smooth bundle X → X.

Let E → Z and E → Z be vector bundles and Φ : E → E their bundle morphism over a morphism φ : Z → Z . Then every section s∗ of the dual bundle E∗ → Z deﬁnes the pull-back section Φ∗s∗ of the dual bundle E∗ → Z by the law

∗ ∗ ∗ vz Φ s (z) = Φ(vz) s (ϕ(z)), vz ∈ Ez.

It follows that a bundle morphism (Φ, φ) yields a morphism of simple graded manifolds (Z, AE ) → (Z , AE ). (6.30)

∗ This is a pair (φ, Φ = φ∗ ◦ Φ ) of a morphism φ of body manifolds and the compo- ∗ ∗ sition φ∗ ◦ Φ of the pull-back AE f → Φ f ∈ AE of graded functions and the A a A a direct image φ∗ of a sheaf AE onto Z . Relative to local bases (z , c ) and (z , c ) for (Z, AE ) and (Z , AE ), the morphism (6.30) of simple graded manifolds reads = φ( ) Φ( a) = Φa( ) b z z , c b z c . The graded manifold morphism (6.30) is a monomorphism (resp. epimorphism) if Φ is a bundle injection (resp. surjection). In particular, the graded manifold morphism (6.30) is a graded bundle if Φ is a ﬁbre bundle. Let AE → AE be the corresponding pull-back monomorphism of the structure rings. By virtue of Theorem6.4 it yields a monomorphism of the DBGAs

S ∗[E; Z ]→S ∗[E; Z]. (6.31)

Let (Y, AF ) be a simple graded manifold modelled over a vector bundle F → Y . This is a graded bundle (X, Y, AF ):

( , A ) → ( , ∞) Y F X CX (6.32) modelled over a composite bundle

F → Y → X. (6.33)

The structure ring of graded functions on a simple graded manifold (Y, AF ) is the ∞ ∗ graded commutative C (X)-ring AF =∧F (Y ) (6.14). Let the composite bundle (6.33) be provided with adapted bundle coordinates (xλ, yi , qa) possessing transition functions

λ( μ), i ( μ, j ), a = ρa( μ, j ) b. x x y x y q b x y q

λ i a The corresponding local basis for a simple graded manifold (Y, AF ) is (x , y , c ) together with transition functions 6.3 Differential Calculus on Graded Bundles 117

λ( μ), i ( μ, j ), a = ρa( μ, j ) b. x x y x y c b x j c

We call it the local basis for a graded bundle (X, Y, AF ). Graded Jet Manifolds As was shown above, Lagrangian theory on a smooth fibre bundle Y → X is formulated in terms of the variational bicomplex on jet manifolds J ∗Y of Y . These are fibre bundles over X and, therefore, they can be regarded as trivial graded bundles ( , k , ∞ ) X J Y C J k Y . Then let us describe their partners in the case of graded bundles as follows. Let us note that, given a graded manifold (X, A) and its structure ring A , one 1 ∞ can define the jet module J A of a C (X)-ring A [58, 131]. If (X, AE ) is a simple 1 graded manifold modelled over a vector bundle E → X, the jet module J AE is a 1 ∗ 1 module of global sections of the jet bundle J (∧E ). A problem is that J AE fails to be a structure ring of some graded manifold. By this reason, we have suggested a different construction of jets of graded manifolds (Definition 6.6), though it is applied only to simple graded manifolds [61, 131, 134]. Let (X, AE ) be a simple graded manifold modelled over a vector bundle E → X. Let us consider a k-order jet manifold J k E of E. It is a vector bundle over X. Then k let (X, AJ k E ) be a simple graded manifold modelled over J E → X.

Deﬁnition 6.6 We agree to call (X, AJ k E ) the graded k-order jet manifold of a simple graded manifold (X, AE ).

λ a Given a splitting domain (U; x , c ) of a graded manifold (Z, AE ),wehavea splitting domain

(U; xλ, ca, ca, ca ,...ca ) λ λ1λ2 λ1...λk of a graded jet manifold (X, AJ k E ). As was mentioned above, a graded manifold is a particular graded bundle over its body (Remark 6.5). Then Deﬁnition 6.6 of graded jet manifolds is generalized to graded bundles over smooth manifolds as follows. Let (X, Y, AF ) be the graded bundle (6.32) modelled over the composite bundle (6.33). It is readily observed that the jet manifold J r F of F → X is a vector bundle r r λ i a r J F → J Y coordinated by (x , yΛ, qΛ),0≤|Λ|≤r.Let(J Y, AJ r F ) be a simple graded manifold modelled over this vector bundle. Its local generating basis λ i a is (x , yΛ, cΛ),0≤|Λ|≤r.

r Deﬁnition 6.7 We call (J Y, AJ r F ) the graded r-order jet manifold of a graded bundle (X, Y, AF ).

In particular, let Y → X be a smooth bundle seen as a trivial graded bundle ( , , ∞) ×{ }→ → X Y CY modelled over a composite bundle Y 0 Y X. Then its graded ( , r , ∞ ) r jet manifold is a trivial graded bundle X J Y C J r Y , i.e., the jet manifold J Y of Y . 118 6 Calculus of Variations on Graded Bundles

Thus, Definition6.7 of graded jet manifolds of graded bundles is compatible with the conventional Definition B.11 of jets of fibre bundles. The affine bundles J r+1Y → J r Y (1.1) and the corresponding fibre bundles J r+1 F → J r F also yield the graded bundles

r+1 r (J Y, AJ r+1 F ) → (J Y, AJ r F ), including the sheaf monomorphisms

r+ ∗ π 1 A → A + , r J r F J r 1 F (6.34)

πr+1∗A r+1 A → where r r is the pull-back onto J Y of the continuous ﬁbre bundle J r F J r Y . The sheaf monomorphism (6.34) induces a monomorphism of the canonical presheaves AJ r F → AJ r+1 F , (6.35)

r+1 which associates to each open subset U ⊂ J Y the ring of sections of AJ r+1 F over πr+1( ) r U . Accordingly, there is a pull-back monomorphism of the structure rings

S 0[ ; ]→S 0 [ ; ], S 0[ ; ]=S 0[ k ; k ], r F Y r+1 F Y k F Y J F J Y (6.36)

r r+1 of graded functions on graded manifolds (J Y, AJ r F ) and (J Y, AJ r+1 F ).Asa consequence, we have the inverse sequence of graded manifolds

1 r−1 r (Y, AF ) ←− (J Y, AJ 1 F ) ←−···(J Y, AJ r−1 F ) ←− (J Y, AJ r F ) ←−··· .

∞ One can think on its inverse limit (J Y, AJ ∞ F ) as the graded inﬁnite order jet manifold whose body is an inﬁnite order jet manifold J ∞Y and whose structure sheaf ∗ AJ ∞ F is a sheaf of germs of graded functions on graded manifolds (J Y, AJ ∗ F ) [61, ∞ 134]. However (J Y, AJ ∞ F ) fails to be a graded manifold in a strict sense because the inverse limit J ∞Y of the sequence (1.1) is a Fréche manifold, but not the smooth one. S ∗[ ; ] By virtue of Theorem 6.4, the differential calculuses r F Y of graded exterior r forms on graded manifolds (J Y, AJ r F ) are minimal. Therefore, the monomorphisms of structure rings (6.36) yields the pull-back monomorphisms (6.31)ofDBGAs

πr+1∗ : S ∗[ ; ]→S ∗ [ ; ], S ∗[ ; ]=S ∗[ k ; k ]. r r F Y r+1 F Y k F Y J F J Y (6.37)

As a consequence, we have a direct system of DBGAs

π ∗ πr∗ S ∗[ ; ] −→ S ∗[ ; ]−→···S ∗ [ ; ] −→r−1 S ∗[ ; ]−→··· . F Y 1 F Y r−1 F Y r F Y (6.38)

∗ The DBGA S∞[F; Y ] that we associate to a graded bundle (X, Y, AF ) is deﬁned as the direct limit 6.3 Differential Calculus on Graded Bundles 119

→ S ∗ [ ; ]= S ∗[ ; ] ∞ F Y lim r F Y (6.39) φ ∈ S ∗[ ; ] of the direct system (6.38). It consists of all graded exterior forms r F Y r on graded manifolds (J Y, AJ r F ) modulo the monomorphisms (6.37). Its elements obey the relations (6.7)–(6.8). O∗ = O∗( r ) → S ∗[ ; ] The cochain monomorphisms r Y J Y r F Y (6.24) provide a monomorphism of the direct system (1.6) to the direct system (6.38) and, consequently, a monomorphism ∗ ∗ O∞Y → S∞[F; Y ] (6.40)

∗ 0 of their direct limits. In particular, S∞[F; Y ] is an O∞Y -algebra. Accordingly, the S ∗[ ; ]→O∗ O0 body epimorphisms r F Y r Y yield an epimorphism of ∞Y -algebras

∗ ∗ S∞[F; Y ]→O∞Y. (6.41)

It is readily observed that the morphisms (6.40) and (6.41) are cochain morphisms ∗ between the de Rham complex (1.8)ofO∞Y and the de Rham complex

0 d 1 d k 0 → R −→ S∞[F; Y ] −→ S∞[F; Y ]···−→ S∞[F; Y ]−→··· (6.42)

∗ of a DBGA S∞[F; Y ]. Moreover, the corresponding homomorphisms of cohomology groups of these complexes are isomorphisms as follows [61]. Theorem 6.7 There is an isomorphism

∗(S ∗ [ ; ]) = ∗ ( ) H ∞ F Y HDR Y (6.43)

∗ ∗ of the cohomology H (S∞[F; Y ]) of the de Rham complex (6.42) to the de Rham ∗ ( ) cohomology HDR Y of Y . Proof The complex (6.42) is the direct limit of the de Rham complexes of the DBGAs S ∗[ ; ] r F Y . Therefore, the direct limit of cohomology groups of these complexes is the cohomology of the de Rham complex (6.42) (Theorem A.8). By virtue of the S ∗[ ; ] cohomology isomorphism (6.26), cohomology of the de Rham complex of r F Y equals the de Rham cohomology of J r Y and, consequently, that of Y , which is the strong deformation retract of any jet manifold J r Y . Hence, the isomorphism (6.43) holds. ∗ Theorem 6.8 Any closed graded exterior form φ ∈ S∞[F; Y ] is decomposed into asumφ = σ + dξ where σ is a closed exterior form on Y . ∗ One can think of elements of S∞[F; Y ] as being graded differential forms on an inﬁnite order jet manifold J ∞Y [61, 134]. Indeed, the presheaf monomorphisms (6.35) deﬁne a direct system of presheaves

∗ ∗ ∗ A −→ A −→···A −→··· , F J 1 F J r F 120 6 Calculus of Variations on Graded Bundles

∗ whose direct limit Q∞[F; Y ] is a presheaf of DBGAs of differential forms on an ∞ ∗ infinite order jet manifold J Y .LetQ∞[F; Y ] be the sheaf of DBGAs of germs ∞ 0 of the presheaf Q∞[F; Y ]. One can think of the pair (J Y, Q∞[F; Y ]) as being a graded Fréchet manifold, whose body is an infinite order jet manifold J ∞Y and 0 the structure sheaf Q∞[F; Y ] is a sheaf of germs of graded functions on graded r ∗ ∗ manifolds (J Y, AJ r F ). The structure module Q∞[F; Y ] of sections of Q∞[F; Y ] is ∗ ∞ a DBGA such that, given an element φ ∈ Q∞[F; Y ] and a point z ∈ J Y , there exist an open neighborhood U of z and a graded exterior form φ(k) on some finite order k φ| = π ∞∗φ(k)| jet manifold J Y so that U k U . In particular, there is a monomorphism

∗ ∗ S∞[F; Y ]→Q∞[F; Y ]. (6.44)

∗ Due to this monomorphism, one can restrict a DBGA S∞[F; Y ] to the coordinate chart (1.3)ofJ ∞Y with the local basis

a λ a a a λ i i i λ {cΛ, dx ,θΛ = dcΛ − cλ+Λdx ,θΛ = dyΛ − yλ+Λdx }, 0 ≤|Λ|,

a a λ i A where cΛ, θΛ are odd and dx , θΛ are even. Let the collective symbol s stand for A its elements. Accordingly, the notation sΛ for their jets and the notation

A A A λ θΛ = dsΛ − sλ+Λdx (6.45) for the contact forms are introduced. For the sake of simplicity, we further denote [A]=[s A].

Remark 6.6 Let (X, Y, AF ) and (X, Y , AF ) be graded bundles modelled over composite bundles F → Y → X and F → Y → X, respectively. Let F → F be a ﬁbre bundle over a ﬁbre bundle Y → Y over X. Then we have a graded bundle

(X, Y, AF ) → (X, Y , AF ) together with the pull-back monomorphism (6.31)ofDBGAs

S ∗[F ; Y ]→S ∗[F; Y ]. (6.46)

r r Let (X, J Y, AJ r F ) and (X, J Y , AJ r F ) be graded bundles modelled over composite bundles J r F → J r Y → X and J r F → J r Y → X, respectively. Since J r F → J r F is a ﬁbre bundle over a ﬁbre bundle J r Y → J r Y over X [53], we also get a graded bundle

r r (X, J Y, AJ r F ) → (X, J Y , AJ r F ) together with the pull-back monomorphism of DBGAs

S ∗[ ; ]→S ∗[ ; ]. r F Y r F Y (6.47) 6.3 Differential Calculus on Graded Bundles 121

The monomorphisms (6.46) and (6.47), r = 1, 2,..., provide a monomorphism of their direct limits (6.39):

∗ ∗ S∞[F ; Y ]→S∞[F; Y ]. (6.48)

Remark 6.7 Let (X, Y, AF ) and (X, Y , AF ) be graded bundles modelled over composite bundles F → Y → X and F → Y → X, respectively. We deﬁne their product over X as the graded bundle

(X, Y, AF ) ×(X, Y , AF ) = (X, Y × Y , AF × F ) (6.49) X X X modelled over a composite bundle

F × F = F × F → Y × Y → X. (6.50) X Y ×Y X

Let us consider the corresponding DBGA (6.39):

∗ S∞[F × F ; Y × Y ]. (6.51) X X

Then in accordance with Remark6.6, there are the monomorphisms (6.48):

∗ ∗ ∗ ∗ S∞[F; Y ]→S∞[F × F; Y × Y ], S∞[F ; Y ]→S∞[F × F; Y × Y ]. X X X X (6.52)

6.4 Grassmann-Graded Variational Bicomplex

Let (X, Y, AF ) be the graded bundle (6.32) modelled over the composite bundle ∗ (6.33) over an n-dimensional smooth manifold X, and let S∞[F; Y ] be the associated DBGA (6.39) of graded exterior forms on graded jet manifolds of (X, Y, AF ).Aswas mentioned above Grassmann-graded Lagrangian theory of even and odd variables on a graded bundle is formulated in terms of the variational bicomplex which the ∗ DBGA S∞[F; Y ] is split in [12, 59, 61, 134]. ∗ 0 k,r ADBGAS∞[F; Y ] is decomposed into S∞[F; Y ]-modules S∞ [F; Y ] of k- contact and r-horizontal graded forms together with the corresponding projections

∗ k,∗ m ∗ ∗,m hk : S∞[F; Y ]→S∞ [F; Y ], h : S∞[F; Y ]→S∞ [F; Y ].

In particular, the horizontal projection h0 takes a form

∗ 0,∗ λ λ i h0 : S∞[F; Y ]→S∞ [F, Y ], h0(dx ) = dx , h0(θΛ) = 0. 122 6 Calculus of Variations on Graded Bundles

∗ Accordingly, the graded exterior differential d on S∞[F; Y ] falls into a sum d = dV + dH of the vertical graded differential

◦ m = m ◦ ◦ m, (φ) = θ A ∧ ∂Λφ, φ ∈ S ∗ [ ; ], dV h h d h dV Λ A ∞ F Y and the total graded differential

λ dH ◦ hk = hk ◦ d ◦ hk , dH ◦ h0 = h0 ◦ d, dH (φ) = dx ∧ dλ(φ), where = ∂ + A ∂Λ dλ λ sλ+Λ A 0≤|Λ| are the graded total derivatives. These differentials obey the nilpotent relations (1.15). ∗ ∗ Similarly to the DGA O∞Y (1.7), a DBGA S∞[F; Y ] is provided with the graded projection endomorphism

1 ρ = ρ ◦ h ◦ hn : S ∗>0,n[F; Y ]→S ∗>0,n[F; Y ], (6.53) k k ∞ ∞ k>0 ρ(φ) = (− )|Λ|θ A ∧[ (∂Λ φ)],φ∈ S >0,n[ ; ], 1 dΛ A ∞ F Y 0≤|Λ| such that ρ ◦ dH = 0, and with the nilpotent graded variational operator

∗,n ∗+1,n δ = ρ ◦ dS∞ [F; Y ]→S∞ [F; Y ].

∗, With these operators a DBGA S∞ [F; Y ] is decomposed into the Grassmann-graded variational bicomplex ......

dV 6 dV 6 dV 6 −δ 6 ρ 1,0 dH 1,1 dH 1,n 0 → S∞ → S∞ → ··· S∞ → E1 → 0

dV 6 dV 6 dV 6 −δ 6 0 dH 0,1 dH 0,n 0,n (6.54) 0 → R → S∞ → S∞ → ··· S∞ ≡ S∞ 6 6 6 d d d 0 → R → O0(X) → O1(X) → ··· On(X) → 0 6 6 6 00 0

∗ ∗ k,n where S∞ = S∞[F; Y ] and Ek = ρ(S∞ [F; Y ]) (cf. (1.21)). 6.4 Grassmann-Graded Variational Bicomplex 123

We restrict our consideration to its short variational subcomplex

δ 0 dH 0,1 dH 0,n 0 → R → S∞[F; Y ] −→ S∞ [F; Y ]···−→ S∞ [F; Y ] −→ E1, (6.55) and the subcomplex of one-contact graded forms

ρ 1,0 dH 1,1 dH 1,n 0 → S∞ [F; Y ] −→ S∞ [F; Y ]···−→ S∞ [F; Y ] −→ E1 → 0. (6.56)

They possess the following cohomology [59, 61, 126, 134]. Theorem 6.9 Cohomology of the complex (6.55) equals the de Rham cohomology ∗ ( ) HDR Y of Y . Proof See the proof below. Theorem 6.10 The complex (6.56) is exact. Proof See the proof below.

Cohomology of the Grassmann-Graded Variational Bicomplex The proof of Theorem 6.9 follows the scheme of the proof of Theorem 1.12 [59, 61, 126, 134]. It falls into the three steps. (I) We start with showing that the complexes (6.55) and (6.56) are locally exact. Lemma 6.1 If Y = Rn+k → Rn, the complex (6.55) is acyclic. Proof Referring to [6] for the proof, we summarize a few formulas. Any horizontal 0,∗ graded form φ ∈ S∞ [F; Y ] admits a decomposition

1 dλ φ = φ + φ, φ = s A∂Λφ, 0 λ Λ A (6.57) ≤|Λ| 0 0

n+k 0,m

1 dλ α μ ...μ +ν ν α1 k−1 A 1 k μ A μ D φ = kδ(μ δμ ···δμ ) λs(α ...α )∂ φ(x ,λsΛ, dx ). λ 1 2 k 1 k−1 A ≤ 0 0 k

+ν ν The relation [D , dμ]φ = δμφ holds, and it leads to the desired expression

( − − )! n m 1 +ν ξ = D P ∂ν φ, (6.58) (n − m + k)! k k=0 +ν +ν = , = ··· 1 ··· k . P0 1 Pk dν1 dνk D D 124 6 Calculus of Variations on Graded Bundles

0,n Now let φ ∈ S∞ [F; Y ] be a graded density such that δφ = 0. Then its component n+k φ0 (6.57) is an exact n-form on R and φ = dH ξ, where ξ is given by the expression ξ = (− )|Σ| A ∂μ+Λφω . 1 sΞ dΣ A μ (6.59) |Λ|≥0 Σ+Ξ=Λ

We also quote the homotopy operator (5.107) in [108] which leads to the expression

1 dλ ξ = I (φ)(xμ,λs A, dxμ) , Λ λ (6.60) 0 Λμ + 1 I (φ) = · − +|Λ|+ ≤|Λ| μ n m 1 0⎡ ⎤ (μ + Λ + Ξ)! ⎣ Ξ A μ+Λ+Ξ ⎦ dΛ (−1) s dΞ ∂ (∂μ φ) , (μ + Λ)!Ξ! A 0≤|Ξ|

Λ!=Λ !···Λ ! Λ where μ1 μn , and μ denotes the number of occurrences of the index μ in Λ [108]. The graded forms (6.59) and (6.60)differinadH -exact graded form.

Lemma 6.2 If Y = Rn+k → Rn, the complex (6.56) is exact.

1,m

φΛ ∈ S 0,m [ ; ] where A ∞ F Y are horizontal graded m-forms. Let us introduce additional A A variables sΛ of the same Grassmann parity as sΛ. Then one can associate to each graded (1, m)-form φ (6.61) a unique horizontal graded m-form φ = φΛ A , A sΛ (6.62)

A whose coefﬁcients are linear in the variables sΛ, and vice versa. Let us put a modiﬁed total differential

Λ = + λ ∧ A ∂ , d H dH dx sλ+Λ A 0<|Λ|

Λ ∂ A acting on graded forms (6.62), where A is the dual of dsΛ. Comparing the equalities

A λ A A λ A d H sΛ = dx sλ+Λ, dH θΛ = dx ∧ θλ+Λ, 6.4 Grassmann-Graded Variational Bicomplex 125 one can easily justify that dH φ = d H φ. Let the graded (1, m)-form φ (6.61)be dH -closed. Then the associated horizontal graded m-form φ (6.62)isd H -closed and, by virtue of Lemma 6.1,itisd H -exact, i.e., φ = d H ξ, where ξ is a horizontal graded A (m − 1)-form given by the expression (6.58) depending on additional variables sΛ. A A glance at this expression shows that, since φ is linear in the variables sΛ,sois ξ = ξ Λ A . A sΛ

It follows that φ = dH ξ where ξ = ξ Λ ∧ θ A. A Λ

It remains to prove the exactness of the complex (6.56) at the last term E1.If ρ(σ) = (− )|Λ|θ A ∧[ (∂Λ σ)]= 1 dΛ A ≤|Λ| 0 (− )|Λ|θ A ∧[ σ Λ]ω = ,σ∈ S 1,n[ ; ], 1 dΛ A 0 ∞ F Y 0≤|Λ| a direct computation gives |Σ| A μ+Λ σ = d ξ, ξ =− (−1) θ ∧ dΣ σ ωμ. H Ξ A 0≤|Λ| Σ+Ξ=Λ

Remark 6.8 The proof of Lemma 6.2 fails to be extended to complexes of higher A B A B contact forms because the products θΛ ∧ θΣ and sΛsΣ obey different commutation rules.

∗ ∗ (II) Let us now prove Theorem 6.9 for a DBGA Q∞[F; Y ]. Similarly to S∞[F; Y ], ∗ ∗ the sheaf Q∞[F; Y ] and the DBGA Q∞[F; Y ] are decomposed into the Grassmann- graded variational bicomplexes. We consider their subcomplexes

δ 0 dH 0,1 dH 0,n 0 → R → Q∞[F; Y ] −→ Q∞ [F; Y ]···−→ Q∞ [F; Y ] −→ E1, (6.63) ρ 1,0 dH 1,1 dH 1,n 0 → Q∞ [F; Y ] −→ Q∞ [F; Y ]···−→ Q∞ [F; Y ] −→ E1 → 0, (6.64) δ 0 dH 0,1 dH 0,n 0 → R → Q∞[F; Y ] −→ Q∞ [F; Y ]···−→ Q∞ [F; Y ] −→ Γ(E1), (6.65) ρ 1,0 dH 1,1 dH 1,n 0 → Q∞ [F; Y ] −→ Q∞ [F; Y ]···−→ Q∞ [F; Y ] −→ Γ(E1) → 0, (6.66)

1,n where E1 = ρ(Q∞ [F; Y ]). By virtue of Lemmas 6.1 and 6.2, the complexes (6.63) ∗,∗ and (6.64) are acyclic. The terms Q∞ [F; Y ] of the complexes (6.63) and (6.64) 0 ∞ are sheaves of Q∞-modules (see (1.10)). Since J Y admits the partition of unity 0 just by elements of Q∞, these sheaves are ﬁne and, consequently, acyclic. By virtue of abstract de Rham Theorem C.5, cohomology of the complex (6.65) equals the 126 6 Calculus of Variations on Graded Bundles cohomology of J ∞Y with coefﬁcients in the constant sheaf R and, consequently, the de Rham cohomology of Y in accordance with the isomorphisms (B.79). Similarly, the complex (6.66) is proved to be exact. ∗ ∗ ∗ Due to monomorphisms O∞Y → S∞[F; Y ]→Q∞[F; Y ] (6.40) and (6.44), this proof gives something more.

0,m

0,m−1 φ = h0σ + dH ξ, ξ ∈ Q∞ [F; Y ], (6.67)

0,n where σ is a closed m-form on Y . Any δ-closed φ ∈ Q∞ [F; Y ] is a sum

0,n−1 φ = h0σ + dH ξ, ξ ∈ Q∞ [F; Y ], (6.68) where σ is a closed n-form on Y .

(III) It remains to prove that cohomology of the complexes (6.55) and (6.56) equals that of the complexes (6.65) and (6.66). The proof of this fact straightforwardly follows the proof of Theorem 1.12 [134]. Let the common symbol D stand for dH and δ. Bearing in mind the decompositions ∗ (6.67) and (6.68), it suffices to show that, if an element φ ∈ S∞[F; Y ] is D-exact ∗ ∗ in an algebra Q∞[F; Y ], then it is so in an algebra S∞[F; Y ]. Lemma 6.1 states that, if Y is a contractible bundle and a D-exact graded form φ ∞ ∗ on J Y is of finite jet order [φ] (i.e., φ ∈ S∞[F; Y ]), there exists a graded form ∗ ∞ ϕ ∈ S∞[F; Y ] on J Y such that φ = Dϕ. Moreover, a glance at the expressions (6.58) and (6.59) shows that a jet order [ϕ] of ϕ is bounded by an integer N([φ]), depending only on a jet order of φ. Let us call this fact the finite exactness of an ∞ operator D. Lemma6.1 shows that the finite exactness takes place on J Y |U over any domain U ⊂ Y . Let us prove the following.

Lemma 6.4 Given a family {Uα} of disjoint open subsets of Y , let us suppose that ∞ | the ﬁnite exactness takes place on J Y Uα over every subset Uα from this family. ∞ Then it is true on J Y over the union ∪ Uα of these subsets. α

∗ ∞ Proof Let φ ∈ S∞[F; Y ] be a D-exact graded form on J Y . The ﬁnite exact- (π ∞)−1(∪ ) φ = ϕ (π ∞)−1( ) [ϕ ] < ness on 0 Uα holds since D α on every 0 Uα and α N([φ]).

Lemma 6.5 If the ﬁnite exactness of an operator D takes place on J ∞Y over open subsets U, V ⊂ Y and their non-empty overlap U ∩ V , then it also is true on ∞ J Y |U∪V .

∗ ∞ Proof Let φ = Dϕ ∈ S∞[F; Y ] be a D-exact form on J Y . By assumption, it ϕ (π ∞)−1( ) ϕ (π ∞)−1( ) can be brought into the form D U on 0 U and D V on 0 V , where ϕU and ϕV are graded forms of bounded jet order. Let us consider their difference ϕ − ϕ (π ∞)−1( ∩ ) U V on 0 U V .ItisaD-exact graded form of bounded jet order [ϕU − ϕV ] < N([φ]) which, by assumption, can be written as ϕU − ϕV = Dσ 6.4 Grassmann-Graded Variational Bicomplex 127 where σ also is of bounded jet order [σ] < N(N([φ])). Lemma 6.6 below shows that σ = σU + σV where σU and σV are graded forms of bounded jet order on (π ∞)−1( ) (π ∞)−1( ) 0 U and 0 V , respectively. Then, putting

ϕ |U = ϕU − DσU ,ϕ|V = ϕV + DσV ,

φ ϕ (π ∞)−1( ) ϕ (π ∞)−1( ) we have a graded form , equal to D U on 0 U and D V on 0 V , ϕ − ϕ (π ∞)−1( ∩ ) respectively. Since the difference U V on 0 U V vanishes, we obtain φ = ϕ (π ∞)−1( ∪ ) D on 0 U V where

ϕ| = ϕ ϕ = U U ϕ| = ϕ V V is of bounded jet order [ϕ] < N(N([φ])).

∗ Lemma 6.6 Let U and V be open subsets of a bundle Y and σ ∈ G∞ a graded (π ∞)−1( ∩ ) ⊂ ∞ σ form of bounded jet order on 0 U V J Y . Then is decomposed into σ + σ σ σ (π ∞)−1( ) asum U V of graded forms U and V of bounded jet order on 0 U and (π ∞)−1( ) 0 V , respectively. Proof By taking a smooth partition of unity on U ∪ V subordinate to a cover {U, V } and passing to a function with support in V , one gets a smooth real function f on U ∪ V which equals 0 on a neighborhood of U \ V and 1 on a neighborhood of \ ∪ (π ∞)∗ (π ∞)−1( ∪ ) V U in U V .Let 0 f be the pull-back of f onto 0 U V . A graded ((π ∞)∗ )σ (π ∞)−1( ) form 0 f equals 0 on a neighborhood of 0 U and, therefore, can (π ∞)−1( ) σ be extended by 0 to 0 U . Let us denote it U . Accordingly, a graded form ( − (π ∞)∗ )σ σ (π ∞)−1( ) σ = σ + σ 1 0 f has an extension V by0to 0 V . Then U V is a desired decomposition because σU and σV are of jet order which does not exceed that of σ .

To prove the finite exactness of D on J ∞Y , it remains to choose an appropriate cover of Y . A smooth manifold Y admits a countable cover {Uξ } by domains Uξ , ξ ∈ N, and its refinement {Uij}, where j ∈ N and i runs through a finite set, such that Uij ∩ Uik =∅, j = k [67]. Then Y has a finite cover {Ui =∪j Uij}. Since the finite exactness of an operator D takes place over any domain Uξ , it also holds over any member Uij of the refinement {Uij} of {Uξ } and, in accordance with Lemma6.4, over any member of a finite cover {Ui } of Y . Then by virtue of Lemma6.5, the finite exactness of D takes place on J ∞Y over Y . k,n Similarly, one can show that, restricted to S∞ [F; Y ], an operator ρ is exact.

6.5 Grassmann-Graded Lagrangian Theory

Decomposed into the Grassmann-graded variational bicomplex (6.54), the DBGA ∗ S∞[F; Y ] (6.39) describes Grassmann-graded Lagrangian theory on a graded bundle (X, Y, AF ) modelled over the composite bundle (6.33) [61, 134]. 128 6 Calculus of Variations on Graded Bundles

Its graded Lagrangian is deﬁned as an element

0,n L = L ω ∈ S∞ [F; Y ] (6.69) of the graded variational complex (6.55). Accordingly, a graded exterior form δ = θ A ∧ E ω = (− )|Λ|θ A ∧ (∂Λ )ω ∈ L A 1 dΛ A L E1 (6.70) 0≤|Λ| is said to be its graded Euler–Lagrange operator. Its kernel deﬁnes a graded Euler– Lagrange equation δ = , E = (− )|Λ| (∂Λ ) = . L 0 A 1 dΛ A L 0 (6.71) 0≤|Λ|

The following is a corollary of Theorem6.9.

0,m

0,m−1 φ = h0σ + dH ξ, ξ ∈ S∞ [F; Y ], where σ is a closed m-form on Y . 0,n (ii) Any δ-closed graded Lagrangian L ∈ S∞ [F; Y ] is a sum

0,n−1 L = h0σ + dH ξ, ξ ∈ S∞ [F; Y ], where σ is a closed n-form on Y .

Proof The complex (6.55) possesses the same cohomology as the short variational subcomplex δ 0 dH 0,1 dH 0,n 0 → R → O∞Y −→ O∞ Y ···−→ O∞ Y −→ E1 (6.72)

∗ of the variational complex (1.22)oftheDGAO∞Y . The monomorphism (6.40) and the body epimorphism (6.41) yield the corresponding cochain morphisms of the complexes (6.55) and (6.72). Therefore, cohomology of the complex (6.55)isthe ∗ image of the cohomology of O∞Y .

Item (ii) of Theorem6.11 leads to the following.

Theorem 6.12 Any variationally trivial odd Lagrangian is dH -exact.

The exactness of the complex (6.56) by virtue of Theorem6.10 results in the following [59, 134]. 6.5 Grassmann-Graded Lagrangian Theory 129

Theorem 6.13 Given a graded Lagrangian L, there is the decomposition

dL = δL − d Ξ ,Ξ∈ S n−1[F; Y ], (6.73) H L ∞ A λνs ...ν Ξ = L + θ ∧ F 1 ωλ, (6.74) L νs ...ν1 A s=0 ν ...ν ν ...ν λν ...ν ν ...ν k 1 = ∂ k 1 L − k 1 + ψ k 1 , = , ,..., FA A dλ FA A k 1 2 where local graded functions σ obey the relations

(ν ν )...ν ψν = ,ψk k−1 1 = . A 0 A 0

Proof The decomposition (6.73) is a straightforward consequence of the exactness 1,n of the complex (6.56)atatermS∞ [F, Y ] and the fact that the endomorphism ρ (6.53) is a projector. The expression (6.74) results from a direct computation

λν ...ν −d Ξ =−d [θ A F λ + θ A F λν +···+θ A F s 1 H H A ν A νs ...ν1 A A λνs+1νs ...ν1 A λ A ν λν +θ ∧ F +···]∧ωλ =[θ dλ F + θ (F + dλ F ) +··· νs+1νs ...ν1 A A ν A A A νs+1νs ...ν1 λνs+1νs ...ν1 +θ (F + dλ F ) +···]∧ω = νs+1νs ...ν1 A A A λ A ν A νs+1νs ...ν1 [θ dλ F + θ (∂ L ) +···+θ (∂ L ) +···]∧ω = A ν A νs+1νs ...ν1 A θ A( λ − ∂ L ) ∧ ω + =−δ + . dλ FA A dL L dL

The graded exterior form ΞL (6.74) provides a global graded Lepage equivalent of a graded Lagrangian L. In particular, one can locally choose ΞL (6.74) where all graded functions ψ vanish. By analogy with the variational formula (1.36), the decomposition (6.73) is called the graded variational formula.

∗ Deﬁnition 6.8 We agree to call a pair (S∞[F; Y ], L) the Grassmann-graded Lagrangian system on a graded bundle (X, Y, AF ).

6.6 Noether’s First Theorem: Supersymmetries

∗ Given a Grassmann-graded Lagrangian system (S∞[F; Y ], L), by its inﬁnitesimal transformations are meant contact graded derivations of a real graded commutative 0 0 0 ring S∞[F; Y ]. They constitute a S∞[F; Y ]-module dS∞[F; Y ] which is a real Lie superalgebra relative to the Lie superbracket (6.3). The following holds [59, 61].

0 0 Theorem 6.14 The derivation module dS∞[F; Y ] is isomorphic to the S∞[F; Y ]- 1 ∗ 1 dual (S∞[F; Y ]) of the module of graded one-forms S∞[F; Y ]. 130 6 Calculus of Variations on Graded Bundles

∗ Proof At first, let us show that S∞[F; Y ] is generated by elements df, f ∈ 0 1 0 S∞[F; Y ]. It suffices to justify that any element of S∞[F; Y ] is a finite S∞[F; Y ]- 0 1 linear combination of elements df, f ∈ S∞[F; Y ]. Indeed, every φ ∈ S∞[F; Y ] is a graded exterior form on some finite order jet manifold J r Y , i.e., a section of r a vector bundle V J r F → J Y in accordance with Theorem6.3. Then by virtue of ∞( r ) S 1[ ; ] the classical Serre–Swan Theorem A.10, a C J Y -module r F Y of graded one-forms on J r Y is a projective module of finite rank, i.e., φ is represented by a ∞( r ) ∈ S 0[ ; ]⊂S 0 [ ; ] finite C J Y -linear combination of elements df, f r F Y ∞ F Y . 1 ∗ Any element Φ ∈ (S∞[F; Y ]) yields a derivation ϑΦ ( f ) = Φ(df) of a real ring 0 1 0 S∞[F; Y ]. Since a module S∞[F; Y ] is generated by elements df, f ∈ S∞[F; Y ], 1 ∗ 0 different elements of (S∞[F; Y ]) provide different derivations of S∞[F; Y ], i.e., 0 ∗ 0 there is a monomorphism (S∞[F; Y ]) → dS∞[F; Y ]. By the same formula, 0 0 any derivation ϑ ∈ dS∞[F; Y ] sends df → ϑ(f ) and, since S∞[F; Y ] is gen- 1 0 erated by elements df, it defines a morphism Φϑ : S∞[F; Y ]→S∞[F; Y ]. Moreover, different derivations ϑ provide different morphisms Φϑ .Thus,wehave 0 1 ∗ a monomorphism dS∞[F; Y ]→(S∞[F; Y ]) and, consequently, isomorphism 0 0 ∗ dS∞[F; Y ]=(S∞[F; Y ]) .

The proof of Theorem6.14 gives something more. It follows that a DBGA ∗ S∞[F; Y ] is the Chevalley–Eilenberg minimal differential calculus over a real 0 graded commutative ring S∞[F; Y ]. 0 1 Let ϑ φ, ϑ ∈ dS∞[F; Y ], φ ∈ S∞[F; Y ], denote the corresponding interior ∗ product. Extended to a DBGA S∞[F; Y ], it obeys the rule

|φ|+[φ][ϑ] ∗ ϑ (φ ∧ σ) = (ϑ φ) ∧ σ + (−1) φ ∧ (ϑ σ), φ,σ ∈ S∞[F; Y ].

∞ ∗ Restricted to a coordinate chart (1.3)ofJ Y , an algebra S∞[F; Y ] is a free 0 λ A A S∞[F; Y ]-module generated by graded one-forms dx and θΛ , [θΛ ]=[A].Dueto 0 the isomorphism stated in Theorem6.14, any graded derivation ϑ ∈ dS∞[F; Y ]reads ϑ = ϑλ∂ + ϑ A∂ + ϑ A∂Λ, λ A Λ A (6.75) 0<|Λ|

∂Λ [∂Λ]=[ ] where the graded derivations A , A A , obey the relations

∂Λ( B ) = ∂Λ B = δ B δΛ A sΣ A dsΣ A Σ

λ A A up to permutations of multi-indices Λ and Σ. Its coefﬁcients ϑ , ϑ , ϑΛ are local graded functions of ﬁnite jet order possessing the transformation law

∂xλ ∂sA ∂sA ϑλ = ϑμ,ϑA = ϑ B + ϑμ, ∂xμ ∂s B ∂xμ ∂sA ∂sA ϑA = Λ ϑ B + Λ ϑμ. Λ ∂ B Σ ∂ μ |Σ|≤|Λ| sΣ x 6.6 Noether’s First Theorem: Supersymmetries 131

0 Every graded derivation ϑ (6.75) of a graded commutative ring ring S∞[F; Y ] yields a graded derivation (called the graded Lie derivative) Lϑ of a DBGA ∗ S∞[F; Y ] given by the relations

∗ Lϑ φ = ϑ dφ + d(ϑ φ), φ ∈ S∞[F; Y ], [ϑ][φ] Lϑ (φ ∧ σ) = Lϑ (φ) ∧ σ + (−1) φ ∧ Lϑ (σ ).

The graded derivation ϑ (6.75) is called contact if the graded Lie derivative Lϑ ∗ preserves the ideal of contact graded forms of the DBGA S∞[F; Y ] generated by contact graded one-forms (6.45).

A ∗ Theorem 6.15 With respect to a local generating basis (s ) for a DBGA S∞[F; Y ], any its contact graded derivation takes a form ϑ = ϑ + ϑ = υλ +[υ A∂ + (υ A − Aυμ)∂Λ], H V dλ A dΛ sμ A (6.76) |Λ|>0 where ϑH and ϑV denotes the horizontal and vertical parts of ϑ.

Proof The expression (6.76) results from a direct computation similar to that of the expression (2.1).

A glance at the expression (6.76) shows that a contact graded derivation ϑ is the inﬁnite order jet prolongation ϑ = J ∞υ (6.77) of its restriction

υ = υλ∂ + υ A∂ = υ + υ = υλ + ( A∂ − A∂λ ) λ A H V dλ u A sλ A (6.78) to a graded commutative ring S 0[F; Y ]. By analogy with υ (2.3), we call υ (6.78)the generalized graded vector field on a simple graded manifold (Y, AF ). This fails to be a graded vector field on (Y, AF ) because its component depends on jets of elements of the local generating basis for (Y, AF ) in general. At the same tine, any graded vector field u on (Y, AF ) is the generalized graded vector field (6.78) generating the contact graded derivation J ∞u (6.77). In particular, the vertical contact graded derivation (6.78) reads ϑ = υ A∂ + υ A∂Λ. A dΛ A (6.79) |Λ|>0

Theorem 6.16 Any vertical contact graded derivation (6.79) obeys the relations

ϑ dH φ =−dH (ϑ φ), (6.80) ∗ Lϑ (dH φ) = dH (Lϑ φ), φ ∈ S∞[F; Y ]. (6.81) 132 6 Calculus of Variations on Graded Bundles

Proof The proof is similar to that of the equalities (2.8) and (2.9).

The vertical contact graded derivation ϑ (6.79)issaidtobenilpotent if ( φ) = (υ B ∂Σ (υ A)∂Λ + (− )[s B ][υ A]υ B υ A∂Σ ∂Λ)φ = Lϑ Lϑ Σ B Λ A 1 Σ Λ B A 0 0≤|Σ|,0≤|Λ|

0,∗ for any horizontal graded form φ ∈ S∞ [F, Y ]. Theorem 6.17 The vertical contact graded derivation (6.79) is nilpotent only if it is odd.

Remark 6.9 If there is no danger of confusion, the common symbol υ further stands for a generalized graded vector ﬁeld υ (6.78), the contact graded derivation ϑ (6.77) determined by υ, and the Lie derivative Lϑ . We call all these operators, in brief, a graded derivation of the structure algebra of a graded Lagrangian system. ← A Remark 6.10 For the sake of convenience, right graded derivations υ = ∂Aυ also are considered. They act on graded differential forms φ on the right by the rules

← ← ← υ (φ) = φυ + (φυ ), θ ∂Σ A = δ AδΣ , d d ΛB B Λ ← ← ← υ (φ ∧ φ) = (−1)[φ ] υ (φ) ∧ φ + φ ∧ υ (φ).

∗ Definition 6.9 Let (S∞[F; Y ], L) be a Grassmann-graded Lagrangian system (Definition6.8). The generalized graded vector field υ (6.78) is called the supersymmetry (henceforth, SUSY) of a graded Lagrangian L if a graded Lie derivative Lϑ L of L along the contact graded derivation ϑ (6.77)isdH -exact, i.e., Lϑ L = dH σ . A corollary of the graded variational formula (6.73)isthegraded first variational formula for a graded Lagrangian [9, 59, 61].

Theorem 6.18 The graded Lie derivative of a graded Lagrangian along any contact graded derivation (6.76) fulﬁls the graded ﬁrst variational formula

Lϑ L = υV δL + dH (h0(ϑ ΞL )) + dV (υH ω)L , (6.82) where ΞL is the graded Lepage equivalent (6.74) of a graded Lagrangian L. Proof The formula (6.82) is derived from the decomposition (6.73) and the relations (6.80) and (6.81) similarly to the proof of Theorem 2.3.

A glance at the expression (6.82) shows the following.

Theorem 6.19 (i) A generalized graded vector ﬁeld υ is SUSY only if it is projected onto X. (ii) Any projectable generalized graded vector ﬁeld is SUSY of a variationally trivial graded Lagrangian. 6.6 Noether’s First Theorem: Supersymmetries 133

(iii) A generalized graded vector ﬁeld υ is SUSY if and only if its vertical part υV (6.78) is well. (iv) It is SUSY if and only if the graded density υV δLisdH -exact.

SUSY of a graded Lagrangian L constitute a real vector subspace SGL of the 0 graded derivation module dS∞[F; Y ]. By virtue of item (ii) of Theorem 6.19,the Lie superbracket

L[ϑ,ϑ ] =[Lϑ , Lϑ ] of SUSY is SUSY, and their vector space SGL is a real Lie superalgebra. An immediate corollary of the graded ﬁrst variational formula (6.82)isNoether’s ﬁrst theorem in a very general setting.

Theorem 6.20 If the generalized graded vector ﬁeld υ (6.78) is SUSY of a graded Lagrangian L, the ﬁrst variational formula (6.82) leads to a weak conservation law

0 ≈−dH (−h0(ϑ ΞL ) + σ) (6.83) of the SUSY current μ Jυ = Jϑ ωμ =−h0(ϑ ΞL ) + σ (6.84) on the shell (6.71).

Chapter 7 Noether’s Second Theorems

As was mentioned above, any Euler–Lagrange operator satisfies Noether identities (henceforth, NI) which, therefore, should be separated into the trivial and notrivial ones. These NI obey first-stage NI, which in turn are subject to the second-stage NI, and so on. We follow the general notion of NI of differential operators on fibre bundles (Definition D.1). They are represented by one-cycles of a certain chain complex (Theorem 7.1). Its boundaries are trivial NI, and nontrivial NI modulo the trivial ones are given by first homology of this complex. To describe (k + 1)-stage NI, let us assume that nontrivial k-stage NI are generated by a projective C∞(X)-module C(k) of finite rank and that a certain homology regularity condition (Definition 7.2) holds. In this case, (k + 1)-stage NI are represented by (k + 2)-cycles of some chain complex of modules of antifields isomorphic to C(i), i ≤ k, by virtue of Serre–Swan Theorem 6.2. Accordingly, trivial (k + 1)-stage NI are defined as its boundaries (Sect. 7.1). Iterating the arguments, we come to the exact Koszul–Tate (henceforth, KT) complex (7.28) with the boundary KT operator (7.26) whose nilpotentness is equivalent to all nontrivial NI (7.12) and higher-stage NI (7.29) (Theorem 7.5). Noether’s inverse second Theorem 7.9 associates to the KT complex (7.28)the cochain sequence (7.38) with the ascent operator (7.39), called the gauge operator. Its components (7.44) and (7.47) are nontrivial gauge and higher-stage gauge supersymmetries of Grassmann-graded Lagrangian theory which are indexed by Grassmann- graded ghosts, but not gauge parameters (Remark 7.7). Herewith, k-stage gauge supersymmetries act on (k − 1)-stage ghosts. Conversely, given these gauge and higher-stage supersymmetries, Noether’s direct second theorem extended to higher-gauge supersymmetries (Theorem 7.10) states that the corresponding NI and higher-stage NI hold. As was mentioned above, gauge symmetries fail to form an algebra in general. We say that gauge and higher-stage gauge symmetries are algebraically closed if the gauge operator (7.39) admits the nilpotent BRST extension (7.66) where k-stage gauge symmetries are generalized to k-stage BRST transformations acting both on (k − 1)-stage and k-stage ghosts (Sect.7.5). The BRST operator (7.66) brings the cochain sequence (7.38) into the BRST complex. © Atlantis Press and the author(s) 2016 135 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_7 136 7 Noether’s Second Theorems

The KT and BRST complexes provide the above mentioned BRST extension of original Lagrangian field theory. This extension exemplifies so called field-antifield theory whose Lagrangians are required to satisfy the particular condition (7.82) called the classical master equation. We show that an original Lagrangian is extended to a proper solution of the master equation if the gauge operator (7.39) admits a nilpotent BRST extension (Theorem 7.16) [12]. Given the BRST operator (7.66), a desired proper solution of the master equation is constructed by the formula (7.86).

7.1 Noether Identities: Reducible Degenerate Lagrangian Systems

∗ Let (S∞[F; Y ], L) be a Grassmann-graded Lagrangian system (Deﬁnition 6.8). Without a lose of generality, let its graded Lagrangian L (6.69) be even, and let the Euler–Lagrange operator δL (6.70) be at least of ﬁrst order.

Remark 7.1 Let us introduce the following notation. If E → X is a vector bundle, we call

n E = E∗ ⊗ ∧ T ∗ X X the density-dual of E. Given a ﬁbre bundle Y → X, by the density-dual of its vertical tangent bundle VY is meant a ﬁbre bundle

n VY = V ∗Y ⊗ ∧ T ∗ X. (7.1) Y

If Y = E is a vector bundle, then VE = E ×X E.Let

0 1 E = E ⊕X E (7.2) be a graded vector bundle over X possessing an even part E0 → X and the odd one E1 → X.Itsgraded density-dual is deﬁned to be

1 0 E = E ⊕ E , (7.3) X

1 0 with an even part E → X and the odd one E → X. Given a graded vector bundle ( , 0 × , A ) E, we consider the product X E X Y E×X F (6.49) of graded bundles over the product (6.50) of composite bundles F (6.33) and E → E 0 → X. The corresponding DBGA (6.51) reads ∗ 0 S∞[F × E; Y × E ]. (7.4) X X 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems 137

In particular, we treat the composite bundle F (6.33) as a graded vector bundle over Y only with an odd part. The density-dual (7.1) of the vertical tangent bundle VFof F → X is

n VF = V ∗ F ⊗ ∧ T ∗ X → F, F where V ∗ F is the vertical cotangent bundle of F → X. It however is not a vector bundle over Y . Therefore, we restrict our consideration to the case of a pull-back composite bundle F (6.33), that is,

F = Y × F 1 → Y → X, X where F 1 → X is a vector bundle. Then

1 n VF = F ⊕((V ∗Y ⊗ ∧ T ∗ X) ⊕ F 1) (7.5) Y Y Y is a graded vector bundle over Y . It can be seen as a product of composite bundles

1 1 VF1 = F ⊕ F 1 → F → X, VY → Y → X. X

Let we consider the corresponding graded bundle (6.49) and DBGA (6.51):

∗ ∗ 1 ∗ 1 1 P∞[VF; Y ]=S∞[VF; Y × F ]=S∞[VF × VY; Y × F ]. (7.6) X X X

Theorem 7.1 One can associate to any Grassmann-graded Lagrangian system ∗ (S∞[F; Y ], L) the chain complex (7.7) whose one-boundaries vanish on-shell.

Proof Let us consider the density-dual VF (7.5) of the vertical tangent bun- ∗ dle VF → F, and let us enlarge an original DBGA S∞[F; Y ] to the DBGA ∗ A P∞[VF; Y ] (7.6) with the local generating basis (s , s A), [s A]=([A]+1)mod 2. Following the terminology of Lagrangian BRST theory [6, 63], we agree to call its ∗ elements s A the antiﬁelds of antiﬁeld number Ant[s A]=1. A DBGA P∞[VF; Y ] A is endowed with the nilpotent right graded derivation δ = ∂ EA, where EA are the variational derivatives (6.70). Then we have the chain complex

δ δ 0,n 0,n 0←Im δ ←− P∞ [VF; Y ]1 ←− P∞ [VF; Y ]2 (7.7)

0,n of densities of antiﬁeld number ≤ 2. Its one-boundaries δΦ, Φ ∈ P∞ [VF; Y ]2, by very deﬁnition, vanish on-shell.

Any one-cycle A,Λ 0,n Φ = Φ sΛAω ∈ P∞ [VF; Y ]1 (7.8) 0≤|Λ| 138 7 Noether’s Second Theorems of the complex (7.7) is a differential operator on a bundle VF such that it is linear on ﬁbres of VF → F and its kernel contains a graded Euler–Lagrange operator δL (6.70), i.e., A,Λ δΦ = 0, Φ dΛEAω = 0. (7.9) 0≤|Λ|

In accordance with Deﬁnition D.1, the one-cycles (7.8) deﬁne the NI (7.9)of an Euler–Lagrange operator δL, which we agree to call Noether identities of a ∗ Grassmann-graded Lagrangian system (S∞[F; Y ], L). In particular, one-chains Φ (7.8) are necessarily NI if they are boundaries. There- fore, these NI are called trivial. They are of the form (AΛ)(BΣ) (AΛ)(BΣ) [A][B] (BΣ)(AΛ) Φ = T dΣ EB sΛAω, T =−(−1) T . 0≤|Λ|,|Σ|

Accordingly, nontrivial NI modulo the trivial ones are associated to elements of the first homology H1(δ) of the complex (7.7). A graded Lagrangian L is called degenerate if there exist nontrivial NI. Nontrivial NI can obey first-stage NI. In order to describe them, let us assume that the module H1(δ) is finitely generated. Namely, there exists a graded projective ∞ C (X)-module C(0) ⊂ H1(δ) of finite rank possessing a local basis {Δr ω}: Δ ω = ΔA,Λ ω, ΔA,Λ ∈ S 0 [ ; ], r r sΛA r ∞ F Y (7.10) 0≤|Λ| such that any element Φ ∈ H1(δ) factorizes as r,Ξ r,Ξ 0 Φ = Φ dΞ Δr ω, Φ ∈ S∞[F; Y ], (7.11) 0≤|Ξ| through elements (7.10)ofC(0). Thus, all nontrivial NI (7.9) result from NI δΔ = ΔA,Λ E = , r r dΛ A 0 (7.12) 0≤|Λ| called the complete NI. Certainly, the factorization (7.11) is independent of specification of a local basis {Δr ω}. Note that, being representatives of H1(δ), the graded densities Δr ω (7.10) are not δ-exact. A Lagrangian system whose nontrivial NI are finitely generated is called finitely degenerate. Hereafter, degenerate Lagrangian systems only of this type are considered. 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems 139

Theorem 7.2 If the homology H1(δ) of the complex (7.7) is ﬁnitely generated in the above mentioned sense, this complex can be extended to the one-exact chain complex (7.15) with a boundary operator whose nilpotency conditions are equivalent to the complete NI (7.12).

Proof By virtue of Serre–Swan Theorem 6.2, the graded module C(0) is isomorphic to a module of sections of the density-dual E0 (7.3) of some graded vector bundle ∗ E0 → X. Let us enlarge a DBGA P∞[VF; Y ] toaDBGA

∗ P { }=P∗ [ ⊕ ; ]=S ∗ [ ⊕ ; ×( 1 ⊕ 1)] ∞ 0 ∞ VF E0 Y ∞ VF E0 Y F E0 (7.13) Y Y X X

A possessing the local generating basis (s , s A, cr ) where cr are Noether antiﬁelds of Grassmann parity [cr ]=([Δr ]+1)mod 2 and antiﬁeld number Ant[cr ]=2. The DBGA (7.13) is provided with an odd right derivation

r δ0 = δ + ∂ Δr (7.14) which is nilpotent if and only if the complete NI (7.12) hold. Then δ0 (7.14)isa boundary operator of the chain complex

δ δ , δ , 0,n 0 0 n 0 0 n 0←Im δ ← P∞ [VF; Y ]1 ← P∞ {0}2 ← P∞ {0}3 (7.15) of graded densities of antiﬁeld number ≤ 3. Let H∗(δ0) denote its homology. We have

H0(δ0) = H0(δ) = 0.

Furthermore, any one-cycle Φ up to a boundary takes the form (7.11) and, therefore, it is a δ0-boundary ⎛ ⎞ r,Ξ ⎝ r,Ξ ⎠ Φ = Φ dΞ Δr ω = δ0 Φ cΞr ω . 0≤|Σ| 0≤|Σ|

Hence, H1(δ0) = 0, i.e., the complex (7.15) is one-exact.

Let us consider the second homology H2(δ0) of the complex (7.15). Its two-chains read r,Λ (A,Λ)(B,Σ) Φ = G + H = G cΛr ω + H sΛAsΣ B ω. (7.16) 0≤|Λ| 0≤|Λ|,|Σ| 140 7 Noether’s Second Theorems

Its two-cycles deﬁne the ﬁrst-stage NI r,Λ δ0Φ = 0, G dΛΔr ω =−δH. (7.17) 0≤|Λ|

Conversely, let the equality (7.17) hold. Then it is a cycle condition of the two-chain (7.16).

Remark 7.2 It should be emphasized that this definition of first-stage NI is independent on specification of a generating module C(0). Given a different one, there exists a chain isomorphism between the corresponding complexes (7.15).

The ﬁrst-stage NI (7.17)aretrivial either if the two-cycle Φ (7.16)isaδ0-boundary or its summand G vanishes on-shell. Therefore, nontrivial ﬁrst-stage NI fail to exhaust the second homology H2(δ0) of the complex (7.15) in general.

Theorem 7.3 Nontrivial ﬁrst-stage NI modulo the trivial ones are identiﬁed with 0,n elements of the homology H2(δ0) if and only if any δ-cycle φ ∈ P∞ {0}2 is a δ0- boundary.

Proof It sufﬁces to show that, if a summand G of the two-cycle Φ (7.16)isδ-exact, then Φ is a boundary. If G = δΨ , let us write

Φ = δ0Ψ + (δ − δ0)Ψ + H. (7.18)

Hence, the cycle condition (7.17) reads

δ0Φ = δ((δ − δ0)Ψ + H) = 0.

0,n Since any δ-cycle φ ∈ P∞ {0}2, by assumption, is δ0-exact, then (δ − δ0)Ψ + H is 0,n a δ0-boundary. Consequently, Φ (7.18)isδ0-exact. Conversely, let Φ ∈ P∞ {0}2 be a δ-cycle, i.e.,

(A,Λ)(B,Σ) (A,Λ)(B,Σ) δΦ = 2Φ sΛAδsΣ B ω = 2Φ sΛAdΣ EB ω = 0.

(A,Λ)(B,Σ) It follows that Φ δsΣ B = 0 for all indices (A,Λ). Omitting a δ-boundary term, we obtain

(A,Λ)(B,Σ) (A,Λ)(r,Ξ) Φ sΣ B = G dΞ Δr .

Hence, Φ takes a form

(A,Λ)(r,Ξ) Φ = G dΞ Δr sΛAω. 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems 141

Then there exists a three-chain

(A,Λ)(r,Ξ) Ψ = G cΞr sΛAω such that (A,Λ)(r,Ξ) δ0Ψ = Φ + σ = Φ + G dΛEAcΞr ω. (7.19)

Owing to an equality δΦ = 0, we have δ0σ = 0. Thus, σ in the expression (7.19) is δ-exact δ0-cycle. By assumption, it is a δ0-exact, i.e., σ = δ0ψ. Consequently, a δ-cycle Φ is a δ0-boundary Φ = δ0Ψ − δ0ψ.

Remark 7.3 It is easily justiﬁed that the two-cycle Φ (7.16)isδ0-exact if and only if Φ up to a δ-boundary takes a form (r,Σ)(r ,Λ) Φ = G dΣ Δr dΛΔr ω. 0≤|Λ|,|Σ|

NI of a Lagrangian system are called irreducible if there are no nontrivial first- stage NI. Accordingly, a degenerate Lagrangian system is said to be reducible (resp. irreducible) if it admits (resp. does not admit) nontrivial first-stage NI. If the condition of Theorem 7.3 is satisfied, let us assume that nontrivial first-stage NI are finitely generated as follows. There exists a graded projective C∞(X)-module C ⊂ (δ ) {Δ ω} (1) H2 0 of finite rank possessing a local basis r1 : r,Λ Δ ω = Δ cΛ ω + h ω, (7.20) r1 r1 r r1 0≤|Λ| such that any element Φ ∈ H2(δ0) factorizes as ,Ξ ,Ξ Φ = Φr1 Δ ω, Φr1 ∈ S 0 [ ; ], dΞ r1 ∞ F Y (7.21) 0≤|Ξ| through elements (7.20)ofC(1). Thus, all nontrivial first-stage NI (7.17) result from the equalities r,Λ Δ dΛΔ + δh = 0, (7.22) r1 r r1 0≤|Λ| called the complete first-stage NI. Note that, by virtue of the condition of Theorem 7.3, Δ ω δ first summands of the graded densities r1 (7.20) are not -exact. A degenerate Lagrangian system is called finitely reducible if admits finitely generated nontrivial first-stage NI. Theorem 7.4 The one-exact complex (7.15) of a finitely reducible Lagrangian system is extended to the two-exact one (7.24) with a boundary operator whose nilpotency conditions are equivalent to the complete NI (7.12) and the complete first-stage NI (7.22). 142 7 Noether’s Second Theorems

Proof By virtue of Serre–Swan Theorem 6.2, a graded module C(1) is isomorphic to a module of sections of the density-dual E (7.3) of some graded vector bundle ∗ 1 E1 → X. Let us enlarge the DBGA P∞{0} (7.13)toaDBGA

∗ ∗ P∞{1}=P∞[VF⊕ E0 ⊕ E1; Y ] Y Y

{ A, , , } possessing the local generating basis s s A cr cr1 where cr1 are first-stage Noether [ ]=([Δ ]+ ) antifields of Grassmann parity cr1 r1 1 mod 2 and of antifield number [ ]= Ant cr1 3. This DBGA is provided with an odd right derivation

δ = δ + ∂r1 Δ 1 0 r1 (7.23) which is nilpotent if and only if the complete NI (7.12) and the complete ﬁrst-stage NI (D.13) hold. Then δ1 (7.23) is a boundary operator of the chain complex

δ δ , δ , δ , 0,n 0 0 n 1 0 n 1 0 n 0←Im δ ← P∞ [VF; Y ]1 ← P∞ {0}2 ← P∞ {1}3 ← P∞ {1}4 (7.24) of graded densities of antiﬁeld number ≤ 4. Let H∗(δ1) denote its homology. It is readily observed that

H0(δ1) = H0(δ), H1(δ1) = H1(δ0) = 0.

By virtue of the expression (7.21), any two-cycle of the complex (7.24) is a boundary ⎛ ⎞ ,Ξ ,Ξ Φ = Φr1 Δ ω = δ ⎝ Φr1 ω⎠ . dΞ r1 1 cΞr1 0≤|Ξ| 0≤|Ξ|

It follows that H2(δ1) = 0, i.e., the complex (7.24) is two-exact.

If the third homology H3(δ1) of the complex (7.24) is not trivial, its elements correspond to second-stage NI which the complete ﬁrst-stage ones satisfy, and so on. Iterating the arguments, one comes to the following.

∗ Deﬁnition 7.1 A degenerate Grassmann-graded Lagrangian system (S∞[F; Y ], L) is called N-stage reducible if it admits ﬁnitely generated nontrivial N-stage NI, but no nontrivial (N + 1)-stage ones.

It is characterized as follows [11].

Theorem 7.5 One can associate to degenerate N-stage reducible Lagrangian theory the exact KT complex (7.28) with the boundary operator (7.26) whose nilpotency property restarts all NI and higher-stage NI (7.12) and (7.29) if these identities are ﬁnitely generated and if and only if this complex obeys the homology regularity condition (Deﬁnition 7.2). 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems 143

Namely, one can show the following.

∗ • There are graded vector bundles E0,...,EN over X, and a DBGA P∞[VF; Y ] is enlarged to a DBGA

∗ ∗ P∞{N}=P∞[VF⊕ E0 ⊕ ···⊕E N ; Y ] (7.25) Y Y Y

( A, , , ,..., ) with the local generating basis s s A cr cr1 crN , where crk are Noether [ ]= + k-stage antiﬁelds of antiﬁeld number Ant crk k 2. • The DBGA (7.25) is provided with the nilpotent right graded derivation ,Λ δ = δ = δ + ∂r ΔA + ∂rk Δ , KT N r sΛA rk (7.26) ≤|Λ| ≤ ≤ 0 1 k N r − ,Λ Δ ω = Δ k 1 cΛ ω + (7.27) rk rk rk−1 0≤|Λ| , (r − ,Σ)(A,Ξ) 0 n (h k 2 cΣ sΞ +···)ω ∈ P {k − 1} + , rk rk−2 A ∞ k 1 0≤|Σ|,|Ξ|

of antiﬁeld number −1. The index k =−1 here stands for s A. The nilpotent derivation δKT (7.26) is called the KT operator. 0,n • With this graded derivation, a module P∞ {N}≤N+3 of densities of antiﬁeld number ≤ (N + 3) is decomposed into an exact KT chain complex

δ δ , δ , 0,n 0 0 n 1 0 n 0←Im δ ←− P∞ [VF; Y ]1 ←− P∞ {0}2 ←− P∞ {1}3 ··· (7.28) δ N−1 0,n δKT 0,n δKT 0,n ←− P∞ {N − 1}N+1 ←− P∞ {N}N+2 ←− P∞ {N}N+3

which satisﬁes the following condition.

Deﬁnition 7.2 One says that the homology regularity condition holds if any δk

0,n 0,n φ ∈ P∞ {k}k+3 ⊂ P∞ {k + 1}k+3 is a δk+1-boundary.

Remark 7.4 The exactness of the complex (7.28) means that any δk

This item means the following.

0,n Theorem 7.6 Any δk -cocycle Φ ∈ P∞ {k}k+2 is a k-stage NI, and vice versa.

0,n Proof Any (k + 2)-chain Φ ∈ P∞ {k}k+2 takes a form ,Λ ( ,Ξ)( ,Σ) Φ = + = rk ω + ( A rk−1 +···)ω. G H G cΛrk H sΞ AcΣrk−1 (7.30) 0≤|Λ| 0≤Σ,Ξ

If it is a δk -cycle, then the equality ⎛ ⎞ rk ,Λ ⎝ rk− ,Σ ⎠ G dΛ Δ 1 cΣ + (7.31) rk rk−1 ≤|Λ| ≤|Σ| 0 0 ( ,Ξ)( ,Σ) δ A rk−1 = H sΞ AcΣrk−1 0 0≤Σ,Ξ is a k-stage NI. Conversely, let the conditions (7.31) hold. Then it can be extended to a cycle condition as follows. It is brought into a form ⎛ ⎞ ,Λ ( ,Ξ)( ,Σ) δ ⎝ rk + A rk−1 ⎠ = k G cΛrk H sΞ AcΣrk−1 0≤|Λ| 0≤Σ,0≤Ξ ,Λ ( ,Ξ)( ,Σ) − rk + A rk−1 Δ . G dΛhrk H sΞ AdΣ rk−1 0≤|Λ| 0≤Σ,Ξ

A glance at the expression (7.27) shows that the term in the right-hand side of 0,n this equality belongs to P∞ {k − 2}k+1.Itisaδk−2-cycle and, consequently, a δk−1-boundary δk−1Ψ in accordance with the homology regularity condition

(Deﬁnition 7.2). Then the equality (7.31)isacΣrk−1 -dependent part of a cycle condition ⎛ ⎞ ,Λ ( ,Ξ)( ,Σ) δ ⎝ rk + A rk−1 − Ψ ⎠ = , k G cΛrk H sΞ AcΣrk−1 0 0≤|Λ| 0≤Σ,Ξ but δk Ψ does not make a contribution to this condition. 7.1 Noether Identities: Reducible Degenerate Lagrangian Systems 145

0,n Theorem 7.7 Any trivial k-stage NI is a δk -boundary Φ ∈ P∞ {k}k+2.

Proof The k-stage NI (7.31) are trivial either if a δk -cycle Φ (7.30)isaδk -boundary or its summand G vanishes on-shell. Let us show that, if a summand G of Φ (7.30) is δ-exact, then Φ is a δk -boundary. If G = δΨ , one can write

Φ = δk Ψ + (δ − δk )Ψ + H.

Hence, the δk -cycle condition reads

δk Φ = δk−1((δ − δk )Ψ + H) = 0.

0,n By virtue of the homology regularity condition, any δk−1-cycle φ ∈ P∞ {k − 1}k+2 is δk -exact. Then (δ−δk )Ψ + H is a δk -boundary. Consequently, Φ (7.30)isδk -exact.

Theorem 7.8 All nontrivial k-stage NI (7.31), by assumption, factorize as ,Ξ ,Ξ Φ = Φrk Δ ω, Φr1 ∈ S 0 [ ; ], dΞ rk ∞ F Y 0≤|Ξ| through the complete ones (7.29).

It may happen that a Grassmann-graded Lagrangian field system possesses nontrivial NI of any stage. However, we restrict our consideration to N-reducible Lagrangian systems for a finite integer N. In this case, the KT operator (7.26) and the gauge operator (7.39) below contain finite terms.

7.2 Noether’s Inverse Second Theorem

Different variants of the second Noether theorem have been suggested in order to relate reducible NI and gauge symmetries [6, 9, 10, 49]. The inverse second Noether theorem (Theorem 7.9), that we formulate in homology terms, associates to the KT complex (7.28) of nontrivial NI the cochain sequence (7.38) with the ascent operator u (7.39) whose components are nontrivial complete gauge and higher-stage gauge SUSY of a Grassmann-graded Lagrangian system (Deﬁnitions 7.4–7.5). Let us start with the following notation. ∗ Remark 7.5 Given the DBGA P∞{N} (7.25), we consider a DBGA

∗ ∗ P∞{N}=P∞[F ⊕ E0 ⊕ ···⊕EN ; Y ], (7.32) Y Y Y 146 7 Noether’s Second Theorems

( A, r , r1 ,..., rN ) [ rk ]=([ ]+ ) possessing the local generating basis s c c c , c crk 1 mod 2, and a DBGA

∗ ∗ P∞{N}=P∞[VF⊕ E0 ⊕···⊕EN ⊕ E0 ⊕ ···⊕E N ; Y ] (7.33) Y Y Y Y Y

( A, , r , r1 ,..., rN , , ,..., ) with the local generating basis s s A c c c cr cr1 crN . Their elements crk are called k-stage ghosts of ghost number gh[crk ]=k + 1 and antifield ∞ ( ) number Ant[crk ]=−(k+1).AC (X)-module C k of k-stage ghosts is the density- ∗ ∗ dual of the module Ck of k-stage antifields. The DBGAs P∞{N} (7.25) and P∞{N} ∗ (7.32) are subalgebras of P∞{N} (7.33). The KT operator δKT (7.26) naturally is ∗ extended to a graded derivation of a DBGA P∞{N}. ∗ Λ Remark 7.6 Any graded differential form φ ∈ S∞[F; Y ] and any finite tuple ( f ), Λ 0 0 ≤|Λ|≤k, of local graded functions f ∈ S∞[F; Y ] obey the following relations [10]: Λ |Λ| Λ f dΛφ ∧ ω = (−1) dΛ( f )φ ∧ ω + dH σ, (7.34) ≤|Λ|≤ ≤|Λ| 0 k 0 |Λ| Λ Λ (−1) dΛ( f φ) = η( f ) dΛφ, (7.35) 0≤|Λ|≤k 0≤|Λ|≤k Λ |Σ+Λ| (|Σ + Λ|)! Σ+Λ η( f ) = (− ) dΣ f , 1 |Σ|!|Λ|! (7.36) 0≤|Σ|≤k−|Λ| η(η( f ))Λ = f Λ. (7.37)

0,n Theorem 7.9 Given the KT complex (7.28), a module of graded densities P∞ {N} is decomposed into a cochain sequence

0,n u 0,n 1 u 0,n 2 u 0 → S∞ [F; Y ] −→ P∞ {N} −→ P∞ {N} −→··· , (7.38) ( ) ( ) ∂ ∂ ∂ u = u + u 1 +···+u N = u A + ur +···+urN−1 , (7.39) ∂s A ∂cr ∂crN−1 graded in ghost number. Its ascent operator u (7.39) is an odd derivation of ghost number 1 where u (7.44) is a SUSY of a graded Lagrangian L and the graded derivations u(k) (7.47), k = 1,...,N, obey the relations (7.46). Proof Given the KT operator (7.26), let us extend an original graded Lagrangian L to a graded Lagrangian = + = + rk Δ ω = + δ ( rk ω) Le L L1 L c rk L KT c crk (7.40) 0≤k≤N 0≤k≤N of zero antiﬁeld number. It is readily observed that the KT operator δKT (7.26)isan 0,n exact symmetry of the extended Lagrangian Le ∈ P∞ {N} (7.40). Since a graded derivation δKT is vertical, it follows from the ﬁrst variational formula (6.82) that 7.2 Noether’s Inverse Second Theorem 147

← ← δ Le δ Le δLe E + Δ ω = υ AE + υrk ω = σ, A rk A dH (7.41) δs δc δcrk A 0≤k≤N rk 0≤k≤N ← δ L ← υ A = e = u A + w A = cr η(ΔA)Λ + cri η(∂ A(h ))Λ, δs Λ r Λ ri A 0≤|Λ| 1≤i≤N 0≤|Λ| ← δ L ← rk e rk rk rk+1 rk Λ ri rk Λ υ = = u + w = cΛ η(Δ ) + cΛη(∂ (hr )) . δc rk+1 i rk 0≤|Λ| k+1

The equality (7.41) is split into a set of equalities

← δ (cr Δ ) r E ω = u AE ω = d σ , (7.42) δs A A H 0 A ← ← δ (crk Δ ) δ (crk Δ ) rk E + rk Δ ω = d σ , (7.43) δs A δc ri H k A 0≤i

0 of P∞{0} is SUSY of a graded Lagrangian L. Every equality (7.43) falls into a set of equalities graded by the polynomial degree in antiﬁelds. Let us consider that of them which is linear in antiﬁelds crk−2 .Wehave ⎛ ⎞ ← δ ( ,Σ)( ,Ξ) ⎝ rk rk−2 A ⎠ E ω + c h cΣr − sΞ A A δ rk k 2 s A ≤|Σ|,|Ξ| 0⎛ ⎞ ← δ r − ,Σ ,Ξ ⎝ rk Δ k 1 ⎠ Δrk−2 ω = σ . c r cΣr cΞr − dH k δc k k−1 rk−1 k 2 rk−1 0≤|Σ| 0≤|Ξ|

This equality is brought into a form ⎛ ⎞ |Ξ| ⎝ rk (rk− ,Σ)(A,Ξ) ⎠ (−1) dΞ c h 2 cΣ E ω + rk rk−2 A 0≤|Ξ| 0≤|Σ| r − r − ,Ξ u k 1 Δ k 2 cΞ ω = d σ . rk−1 rk−2 H k 0≤|Ξ| 148 7 Noether’s Second Theorems

Using the relation (7.34), we obtain an equality r (r − ,Σ)(A,Ξ) c k h k 2 cΣ dΞ E ω + (7.45) rk rk−2 A ≤|Ξ| ≤|Σ| 0 0 r − r − ,Ξ u k 1 Δ k 2 cΞ ω = d σ . rk−1 rk−2 H k 0≤|Ξ|

The variational derivative of both its sides with respect to crk−2 leads to a relation ∂ r − r − r − dΣ u k 1 u k 2 = δ(α k 2 ), (7.46) ∂ rk−1 ≤|Σ| cΣ 0 r − (r − )(A,Ξ) Σ r α k 2 =− η(h k 2 ) dΣ (c k sΞ ), rk A 0≤|Σ| which the odd derivation ( ) ∂ Λ ∂ k = rk−1 = rk η(Δrk−1 ) , = ,..., , u u cΛ r k 1 N (7.47) ∂crk−1 k ∂crk−1 0≤|Λ| satisﬁes. Graded derivations u (7.44) and u(k) (7.47) are assembled into the ascent operator u (7.39) of the cochain sequence (7.38).

7.3 Gauge Supersymmetries: Noether’s Direct Second Theorem

A glance at the expression (7.44) shows that SUSY u is a linear differential operator ∞ (0) on the C (X)-module C of ghosts with values into a real vector space SGL of SUSY of L. Bearing in mind Deﬁnition 2.4 of gauge symmetries, we come to the following generalization of this notion to Grassmann-graded Lagrangian theory on graded bundles in a general setting (Deﬁnition 7.3). ∗ Let (S∞[F; Y ], L) be a Grassmann-graded Lagrangian system on a graded bun- A dle (X, Y, AF ) with the local generating basis (s ).LetE be the graded vector ∗ 0 bundle (7.2). Let us consider the corresponding DBGA S∞[F ×X E; Y ×X E ] (7.4) together with the monomorphisms (6.52)ofDBGAs

∗ ∗ 0 ∗ 0 ∗ 0 S∞[F; Y ]→S∞[F × E; Y × E ], S∞[E; E ]→S∞[F × E; Y × E ]. X X X X

0,n Given a graded Lagrangian L ∈ S∞ [F; Y ], let us deﬁne its pull-back

0,n 0,n 0 L ∈ S∞ [F; Y ]⊂S∞ [F × E; Y × E ], (7.48) X X 7.3 Gauge Supersymmetries: Noether’s Direct Second Theorem 149 and consider an extended Grassmann-graded Lagrangian system

∗ 0 (S∞[F × E; Y × E ], L) (7.49) X X provided with the local generating basis (s A, cr ).

Definition 7.3 Gauge SUSY of a graded Lagrangian L is defined to be SUSY u of its pull-back (7.48) (Definition 6.9) such that the contact graded derivation J ∗u 0 0 (6.77)oftheringS∞[F ×X E; Y ×X E ] equals zero on a subring

0 0 0 0 S∞[E; E ]⊂S∞[F × E; Y × E ]. X X

In view of the second condition in Deﬁnition 7.3, the graded variables cr of the extended Lagrangian system (7.49) can be treated as graded gauge parameters of gauge SUSY u. Bearing in mind Remark 2.8, we further assume that gauge SUSY u is linear in r r graded gauge parameters c and their jets cΛ. Then it reads ⎛ ⎞ ⎛ ⎞ = ⎝ λΛ( μ) r ⎠ ∂ + ⎝ AΛ( μ, B ) r ⎠ ∂ . u ur x cΛ λ ur x sΣ cΛ A (7.50) 0≤|Λ|≤m 0≤|Λ|≤m

By virtue of item (iii) of Theorem 6.19, the generalized graded vector ﬁeld υ (7.50) is gauge SUSY if and only if its vertical part is so. Therefore, we can restrict our consideration to vertical gauge SUSY. In accordance with Deﬁnition 7.3, the graded derivation u (7.44) in Noether’s inverse second Theorem 7.9) is gauge SUSY of a graded Lagrangian L. Therefore, graded gauge parameters cr of gauge SUSY (7.50) are called the ghosts.

Remark 7.7 In contrast with Definitions 2.3 and 2.4, if a graded Lagrangian L depends only on even variables, gauge SUSY in Definition 7.3 are parameterized by odd ghosts, but not even gauge parameters. Given the gauge symmetry u (2.15) 0 defined as a derivation of the real ring O∞[Y × E], one can associate to it gauge SUSY ⎛ ⎞ ⎛ ⎞ = ⎝ λΛ( μ) a ⎠ ∂ + ⎝ iΛ( μ, j ) a ⎠ ∂ , u ua x cΛ λ ua x yΣ cΛ i 0≤|Λ|≤m 0≤|Λ|≤m

0 which is an odd derivation of a real ring S∞[E; Y ], and vice versa.

The gauge SUSY u (7.44) is complete in the following sense. Let R r,Ξ Δ ω C G R dΞ r 0≤|Ξ| 150 7 Noether’s Second Theorems be some projective C∞(X)-module of ﬁnite rank of the nontrivial NI (7.11) parameterized by the corresponding ghosts C R. We have the equalities ⎛ ⎞ = R r,Ξ ⎝ ΔA,Λ E ⎠ ω = 0 C G R dΞ r dΛ A ≤|Ξ| ≤|Λ| 0 ⎛ 0 ⎞ ⎝ η( r )Ξ R ⎠ ΔA,Λ E ω + (σ ) = G R CΞ r dΛ A dH ≤|Λ| ≤|Ξ| 0 0 ⎛ ⎞ (− )|Λ| ⎝ΔA,Λ η( r )Ξ R ⎠ E ω + σ = 1 dΛ r G R CΞ A dH ≤|Λ| ≤|Ξ| 0 ⎛ 0 ⎞ η(ΔA)Λ ⎝ η( r )Ξ R ⎠ E ω + σ = r dΛ G R CΞ A dH ≤|Λ| ≤|Ξ| 0 ⎛ 0 ⎞ A,Λ ⎝ η( r )Ξ R ⎠ E ω + σ. ur dΛ G R CΞ A dH 0≤|Λ| 0≤|Ξ|

It follows that the graded derivation ⎛ ⎞ ∂ ⎝ r Ξ R ⎠ A,Λ dΛ η(G ) C u R Ξ r ∂s A 0≤|Ξ| is SUSY of a graded Lagrangian L and, consequently, its gauge SUSY parameterized by ghosts C R. It factorizes through the gauge SUSY (7.44) by putting ghosts r = η( r )Ξ R . c G R CΞ 0≤|Ξ|

Definition 7.4 The odd derivation u (7.44)issaidtobeacomplete nontrivial gauge SUSY of a graded Lagrangian L associated to the complete NI (7.12). Turn now to the relation (7.46). For k = 1, it takes a form ∂ r A = δ(αA) dΣ u r u ∂cΣ 0≤|Σ| of a first-stage gauge SUSY condition on-shell which the nontrivial gauge SUSY u (7.44) satisfies. Therefore, one can treat the odd derivation ∂ (1) r r r1 r Λ u = u , u = cΛη(Δ ) , ∂cr r1 0≤|Λ| 7.3 Gauge Supersymmetries: Noether’s Direct Second Theorem 151 as first-stage gauge SUSY associated to the complete first-stage NI ⎛ ⎞ ⎛ ⎞ r,Λ ⎝ A,Σ ⎠ ⎝ (B,Σ)(A,Ξ) ⎠ Δ dΛ Δ sΣ =−δ h sΣ sΞ . r1 r A r1 B A 0≤|Λ| 0≤|Σ| 0≤|Σ|,|Ξ|

Iterating the arguments, one comes to the relation (7.46) which provides a k-stage gauge SUSY condition which is associated to the complete k-stage NI (7.29). The odd derivation u(k) (7.47) is called the k-stage gauge SUSY. It is complete as follows. Let R rk ,Ξ C k G dΞ Δ ω Rk rk 0≤|Ξ| be a projective C∞(X)-module of ﬁnite rank of nontrivial k-stage NI (7.11) factorizing through the complete ones (7.27) and parameterized by the corresponding ghosts C Rk . One can show that it deﬁnes k-stage gauge SUSY factorizing through u(k) (7.47) by putting k-stage ghosts Ξ crk = η(Grk ) C Rk . Rk Ξ 0≤|Ξ|

Deﬁnition 7.5 The odd derivation u(k) (7.47)issaidtobeacomplete nontrivial k-stage gauge SUSY of a graded Lagrangian L.

Gauge SUSY of a graded Lagrangian are called irreducible if they do not admit the nontrivial ﬁrst-stage ones. In accordance with Deﬁnitions 7.4–7.5, components of the ascent operator u (7.39) are complete nontrivial gauge and higher-stage gauge SUSY. Therefore, we agree to call this operator the gauge operator.

Remark 7.8 With the gauge operator (7.39), the extended Lagrangian Le (7.40) takes aform ∗ = + ( rk−1 )ω + + σ, Le L u c crk−1 L1 dH (7.51) 0≤k≤N

∗ where L1 is a term of polynomial degree in antiﬁelds exceeding 1. The correspondence of complete nontrivial gauge and higher-stage gauge SUSY to complete nontrivial Noether and higher-stage Noether identities is unique due to forthcoming Noether’s direct second theorem.

Theorem 7.10 (i) If u (7.44) is gauge SUSY, the variational derivative of a dH -exact A r density u EAω (7.42) with respect to ghosts c leads to the equality δ ( AE ω) = (− )|Λ| [ AΛE ]= r u A 1 dΛ ur A (7.52) 0≤|Λ| 152 7 Noether’s Second Theorems (− )|Λ| (η(ΔA)ΛE ) = (− )|Λ|η(η(ΔA))Λ E = , 1 dΛ r A 1 r dΛ A 0 0≤|Λ| 0≤|Λ|

which reproduces the complete Noether identities (7.12) by means of the relation (7.37). (ii) Given the k-stage gauge symmetry condition (7.46), the variational derivative of the equality (7.45) with respect to ghosts crk leads to the equality, reproducing the k-stage Noether identities (7.29) by means of the relations (7.35)–(7.37).

Remark 7.9 If the gauge SUSY u (4.49) is of second jet order in ghosts, i.e.,

= ( A r + Aμ r + Aνμ r )∂ , u ur c ur cμ ur cνμ A (7.53) the corresponding NI (7.52) take a form

AE − ( AμE ) + ( AνμE ) = , ur A dμ ur A dνμ ur A 0 (7.54) and vice versa (cf. Remark 2.6).

Remark 7.10 A glance at the expression (7.54) shows that, if the gauge SUSY u (7.53) is independent of jets of ghosts, then all variational derivatives of a graded Lagrangian equal zero, i.e., this Lagrangian is variationally trivial. Therefore, such gauge SUSY usually are not considered. At the same time, let a graded Lagrangian L be variationally trivial. Its variational derivatives EA ≡ 0 obey irreducible complete NI δΔA = 0,ΔA = s A. (7.55)

By the formula (7.39), the associated irreducible gauge SUSY is given by the gauge operator ∂ u = cA . (7.56) ∂s A

7.4 Noether’s Third Theorem: Superpotential

Being SUSY, the gauge SUSY u (7.50) defines the weak conservation law (6.83)in accordance with Noether’s first Theorem 6.20. The peculiarity of this conservation law is that the SUSY current Ju (6.84) is the graded total differential (7.57), i.e., it is reduced to a superpotential (Definition 2.6).

Theorem 7.11 If u (7.50) is gauge SUSY of a graded Lagrangian L, the corresponding SUSY current Ju (6.84) along u is linear in ghosts and their jets (up to a dH -closed term), and it takes a form

μ νμ Ju = W + dH U = (W + dνU )ωμ, (7.57) 7.4 Noether’s Third Theorem: Superpotential 153 where a term W vanishes on-shell and U νμ =−U μν is a superpotential which takes the form (7.64).

Proof Let the gauge SUSY u (7.50) be at most of jet order N in ghosts. Then the SUSY current Ju (6.84), being linear in ghosts and their jets, is decomposed into a sum μ μμ ...μ μμ ...μ μμ μ J = J 1 M cr + J k M cr + J M cr + J cr , (7.58) u r μ1...μM r μk ...μM r μM r 1

N M A μ ...μ r A r μμ ...μ r μ r 0 =[ u k N c + u c ]E − dμ( J k M c + J c ). V r μk ...μN V r A r μk ...μM r k=1 k=1

This equality provides the following set of equalities for each cr , cr (k = μμ1...μM μk ...μM 1,...,M − N − 1), cr (k = 1,...,N − 1), cr and cr : μk ...μN μ

(μμ )...μ = 1 M , 0 Jr (7.59) (μ μ )...μ νμ ...μ = k k+1 M + k M , ≤ < − , 0 Jr dν Jr 1 k M N (7.60) μ ...μ (μ μ )...μ νμ ...μ = A k N E − k k+1 N − k N , ≤ < , 0 uV r A Jr dν Jr 1 k N (7.61) = A μE − μ − νμ, 0 uV r A Jr dν Jr (7.62) where (μν) means symmetrization of indices in accordance with the splitting

μ μ ...μ (μ μ )...μ [μ μ ]...μ k k+1 N = k k+1 N + k k+1 N . Jr Jr Jr

We also have the equality

= A E − μ. 0 uV r A dμ Jr

With the equalities (7.59)–(7.62), the decomposition (7.58) takes a form μ [μμ ]...μ r [μμ ]...μ νμμ ...μ r J = J 1 M c + [(J k M − dν J k M )c ]+ u r μ1...μM r r μk ...μM < ≤ − 1 k M N i μμ ...μ νμμ ...μ [μμ ]...μ r [(u A k N E − dν J k N + J k N )c ]+ V r A r r μk ...μN 1

A direct computation

μ [μν]μ2...μM r [μν]μ2...μM r J = dν (J cμ ...μ ) − dν J cμ ...μ + u r 2 M r 2 M [μν]μ + ...μ r [dν (J k 1 M c ) − r μk+1...μM 1

[μν]μk+1...μM r νμμk ...μM r dν J cμ ...μ − dν J cμ ...μ ]+ r k+1 M r k M A μμ ...μ νμμ ...μ r [(u k N E − dν J k N )c + V r A r μk ...μN 1

[μν]μ2...μM r = dν (J cμ ...μ ) + r 2 M [μν]μ + ...μ r (νμ)μ ...μ r [dν (J k 1 M c ) − dν J k M c ]+ r μk+1...μM r μk ...μM < ≤ − 1 k M N A μμ ...μ (νμ)μ ...μ r [μν]μ + ...μ r [(u k N E − dν J k N )c + dν (J k 1 N c )]+ V r A r μk ...μN r μk+1...μN 1

The ﬁrst summand of this expression vanishes on-shell. Its second one contains (νμ )μ ...μ terms dν J k k+1 M , k = 1,...,M. By virtue of the equalities (7.60)–(7.61), (νμ )μ ...μ every dν J k k+1 M is expressed into the terms vanishing on-shell and the term (νμ )μ ...μ dν J k−1 k M . Iterating the procedure and bearing in mind the equality (7.59), one can easily show that a second summand of the expression (7.63) also vanishes on-shell. Thus a SUSY current takes the form (7.57) where νμ [νμ]μ ...μ [νμ] U =− J k M cr − J cr . (7.64) μk ...μM r 1

Remark 7.11 If an exact gauge SUSY

= ( λ r + λμ r )∂ + ( A r + Aμ r )∂ u ur c ur cμ λ ur c ur cμ A 7.4 Noether’s Third Theorem: Superpotential 155 of a graded Lagrangian L depends at most on ﬁrst jets of ghosts, then the decomposition (7.63) takes a form

J μ = A μE r − ( [νμ] r ) = u uV r Ac dν Jr c (7.65) ( iμ − A λμ) r E + [( A[μ − A λ[μ) r ∂ν]L + [νμ] r L ]. ua r sλ ur c A dν ur sλ ur c A ur c

7.5 Lagrangian BRST Theory

In contrast with the KT operator (7.26), the gauge operator u (7.38) need not be nilpotent. let us study its extension to a nilpotent graded derivation ( ) ∂ b = u + γ = u + γ k = u + γ rk−1 (7.66) ∂ rk−1 ≤ ≤ + ≤ ≤ + c 1 k N 1 1 k N 1 ∂ ∂ ∂ ∂ = u A + γ r + urk + γ rk+1 ∂s A ∂cr ∂crk ∂crk+1 0≤k≤N−1 of ghost number 1 by means of antiﬁeld-free terms γ (k) of higher polynomial degree r ri in ghosts c i and their jets cΛ,0≤ i < k. We call b (7.66)theBRST operator, where k-stage gauge SUSY are extended to k-stage BRST transformations acting both on (k − 1)-stage and k-stage ghosts [12, 60, 134]. If the BRST operator exists, the cochain sequence (7.38) is brought into a BRST complex

0,n 1 b 0,n 1 b 0,n 2 b 0 → P∞ {N} −→ P∞ {N} −→ P∞ {N} −→··· .

There is following necessary condition of the existence of such a BRST extension.

Theorem 7.12 The gauge operator (7.38) admits the nilpotent extension (7.66) only if the higher-stage gauge SUSY conditions (7.46) and the higher-stage NI (7.29)are satisﬁed off-shell.

Proof It is easily justiﬁed that, if the graded derivation b (7.66) is nilpotent, then the right hand sides of the equalities (7.46) equal zero, i.e.,

u(k+1)(u(k)) = 0, 0 ≤ k ≤ N − 1, u(0) = u. (7.67)

Using the relations (7.34)–(7.37), one can show that, in this case, the right hand sides of the higher-stage NI (7.29) also equal zero [9]. It follows that the summand Grk Δ δ δ of each cocycle rk (7.27)is k−1-closed. Then its summand hrk also is k−1-closed and, consequently, δk−2-closed. Hence it is δk−1-exact by virtue of the homology Δ regularity condition (Deﬁnition 7.2). Therefore, rk contains only a term Grk linear in antiﬁelds. 156 7 Noether’s Second Theorems

It follows at once from the equalities (7.67) that the higher-stage gauge operator

(1) (N) uHS = u − u = u +···+u is nilpotent, and u(u) = u(u). (7.68)

Therefore, the nilpotency condition for the BRST operator b (7.66) takes a form

b(b) = (u + γ)(u) + (u + uHS + γ )(γ ) = 0. (7.69)

Let us denote

γ (0) = 0, ( ) ( ) γ (k) = γ k +···+γ k , k = 1,...,N + 1, (2) (⎛k+1) ⎞ r − ,Λ ,...,Λ r rk−1 ⎝ k 1 k1 ki rk1 ki ⎠ γ( ) = γ( ) ,..., cΛ ···cΛ , i i rk1 rki k1 ki +···+ = + − ≤|Λ | k1 ki k 1 i 0 k j γ (N+2) = 0,

γ (k) ≤ ≤ + where (i) are terms of polynomial degree 2 i k 1 in ghosts. Then the nilpotent property (7.69)ofb falls into a set of equalities

u(k+1)(u(k)) = 0, 0 ≤ k ≤ N − 1, (7.70) ( + γ (k+1))( (k)) + (γ (k)) = , ≤ ≤ + , u (2) u uHS (2) 0 0 k N 1 (7.71) (k+1) (k) (k) k γ (u ) + u(γ ) + u (γ( )) + (7.72) (i) (i−1) HS i γ (γ (k) ) = , − ≤ ≤ + , (m) (i−m+1) 0 i 2 k N 1 2≤m≤i−1 of ghost polynomial degree 1, 2 and 3 ≤ i ≤ N + 3, respectively. The equalities (7.70) are exactly the conditions (7.67) in Theorem 7.12. The equality (7.71)fork = 0 reads ( + γ (1))( ) = , ( ( A)∂Λ B + (γ r ) B,Λ) = . u u 0 dΛ u A u dΛ ur 0 (7.73) 0≤|Λ|

It takes a form of the Lie antibracket [ , ]=− γ (1)( ) =− (γ r ) B,Λ∂ u u 2 u 2 dΛ ur B (7.74) 0≤|Λ| 7.5 Lagrangian BRST Theory 157 of the odd gauge SUSY u. Its right-hand side is a nonlinear differential operator on (0) a module C of ghosts taking values into a real space SGL of SUSY. Following Remark 2.8, we treat it as a generalized gauge SUSY factorizing through gauge SUSY u. Thus, we come to the following. Theorem 7.13 The gauge operator (7.38) admits the nilpotent extension (7.66) only if the Lie antibracket of the odd gauge SUSY u (7.44) is generalized gauge SUSY factorizing through u. The equalities (7.71)–(7.72)fork = 1 take a form

( + γ (2))( (1)) + (1)(γ (1)) = , u (2) u u 0 (7.75) γ (2)( (1)) + ( + γ (1))(γ (1)) = . (3) u u 0 (7.76)

In particular, if a Lagrangian system is irreducible, i.e., u(k) = 0 and u = u,the BRST operator reads = + γ (1) = A∂ + γ r ∂ = A,Λ r ∂ + γ r,Λ,Ξ p q ∂ . b u u A r ur cΛ A pq cΛcΞ r 0≤|Λ| 0≤|Λ|,|Ξ|

In this case, the nilpotency conditions (7.75)–(7.76) are reduced to the equality

(u + γ (1))(γ (1)) = 0. (7.77)

Furthermore, let gauge SUSY u be affine in fields s A and their jets. It follows from the nilpotency condition (7.73) that the BRST term γ (1) is independent of original fields and their jets. Then the relation (7.77) takes a form of the Jacobi identity

γ (1))(γ (1)) = 0 (7.78)

γ r,Λ,Ξ ( ) for coefﬁcient functions pq x in the Lie antibracket (7.74). The relations (7.74) and (7.78) motivate us to think of the equalities (7.71)–(7.72) in a general case of reducible gauge symmetries as being sui generis generalized commutation relations and Jacobi identities of gauge SUSY,respectively [60]. Based on Theorem 7.13, we therefore say that nontrivial gauge symmetries are algebraically closed (in the terminology of [63]) if the gauge operator u (7.39) admits the nilpotent BRST extension b (7.66). Remark 7.12 A Grassmann-graded Lagrangian system is called Abelian if its gauge SUSY u is Abelian and the higher-stage gauge SUSY are independent of original ﬁelds, i.e., if u(u) = 0. It follows from the relation (7.68) that, in this case, the gauge operator itself is the BRST operator u = b. For instance, let a graded Lagrangian L be variationally trivial (Remark 7.10). Its variational derivatives Ei ≡ 0 obey irreducible complete NI (7.55):

δΔi = 0,Δi = si . 158 7 Noether’s Second Theorems

The associated irreducible gauge SUSY is given by the gauge operator (7.56). Thus, a Lagrangian system with a variationally trivial Lagrangian is Abelian, and u (7.56) is the BRST operator. The topological BF theory exempliﬁes a reducible Abelian Lagrangian system (Chap. 12).

∗ The DBGA P∞{N} (7.33) is a particular ﬁeld-antiﬁeld theory of the following type [6, 12, 63]. Let us consider a pull-back composite bundle

W = Z × Z → Z → X X where Z → X is a vector bundle. Let us regard it as a graded vector bundle over Z possessing only odd part. The density-dual VW of the vertical tangent bundle VW of W → X is a graded vector bundle

n VW = ((Z ⊕ V ∗ Z) ⊗ ∧ T ∗ X) ⊕ Z Z Z Y

∗ over Z (cf. (7.5)). Let us consider the DBGA P∞[VW; Z] (7.6) with the local a a a generating basis (z , za), [za]=([z ]+1)mod 2. Its elements z and za are called ﬁelds and antiﬁelds, respectively. Graded densities of this DBGA are endowed with the antibracket

← ←

δ L δL δ L δL {Lω,L ω}= + (− )[L ]([L ]+1) ω. a 1 a δza δz δza δz

With this antibracket, one associates to any (even) Lagrangian Lω the odd vertical graded derivations

← ← δ L ∂ a υL = E ∂ = , a a (7.79) δza ∂z ← ∂ δL a υL = ∂ E = , (7.80) a ∂ δ a za z δL ∂ δL ∂ l [a]+1 ϑL = υL + υ = (− ) + , L 1 a a (7.81) δz ∂za δz ∂za such that

ϑL(L ω) ={Lω,L ω}. 7.5 Lagrangian BRST Theory 159

Theorem 7.14 The following conditions are equivalent. (i) The antibracket of a Lagrangian Lω is dH -exact, i.e.,

← δ L δL {Lω,Lω}= ω = d σ. 2 a H (7.82) δza δz

(ii) The graded derivation υ (7.79) is a symmetry of a Lagrangian Lω. (iii) The graded derivation υ (7.80) is a symmetry of Lω. (iv) The graded derivation ϑL (7.81) is nilpotent.

Proof By virtue of the ﬁrst variational formula (6.82), conditions (ii) and (iii) are equivalent to condition (i). The equality (7.82) is equivalent to that the odd den- ← ← a a sity E Eaω is variationally trivial. Replacing right variational derivatives E with (−1)[a]+1E a, we obtain [a] a 2 (−1) E Eaω = dH σ. a

The variational operator acting on this relation results in the equalities (− )[a]+|Λ| (∂Λ(E aE )) = 1 dΛ b a ≤|Λ| 0 [a] a Λ Λ a (−1) [η(∂bE ) EΛa + η(∂bEa) EΛ)]=0, ≤|Λ| 0 [a]+|Λ| Λb a (−1) dΛ(∂ (E Ea)) = ≤|Λ| 0 [a] b a Λ b a (−1) [η(∂ E ) EΛa + η(∂ Ea)EΛ]=0. 0≤|Λ|

Due to the identity

(δ ◦ δ)( ) = ,η(∂E )Λ = (− )[A][B]∂ΛE , L 0 B A 1 A B we obtain (− )[a][(− )[b]([a]+1)∂ΛaE E + (− )[b][a]∂ΛE E a]= , 1 1 b Λa 1 a b Λ 0 ≤|Λ| 0 (− )[a]+1[(− )([b]+1)([a]+1)∂ΛaE bE + (− )([b]+1)[a]∂ΛE bE a]= 1 1 Λa 1 a Λ 0 0≤|Λ|

b for all Eb and E . This is exactly condition (iv). 160 7 Noether’s Second Theorems

The equality (7.82) is called the classical master equation. For instance, any variationally trivial Lagrangian satisﬁes the master equation. A solution of the master equation (7.82) is called nontrivial if both the derivations (7.79) and (7.80) do not vanish. ∗ Being an element of the DBGA P∞{N} (7.33), an original Lagrangian L obeys the master equation (7.82) and yields the graded derivations υL = 0(7.79) and υ L = δ (7.80), i.e., it is a trivial solution of the master equation. The graded derivations (7.79)–(7.80) associated to the extended Lagrangian Le (7.51) are extensions

← ← δ L ∗ ∂ δ L ∗ ∂ 1 1 υe = u + + , δs ∂s A δc ∂crk A 0≤k≤N rk ∂ δL υ = δ + 1 e KT A ∂s A δs of the gauge and KT operators, respectively. However, the Lagrangian Le need not satisfy the master equation. Therefore, let us consider its extension

L E = Le + L = L + L1 + L2 +··· (7.83) by means of even densities Li , i ≥ 2, of zero antiﬁeld number and polynomial degree i in ghosts. The corresponding graded derivations (7.79)–(7.80) read

← ← δ L ∂ δ L ∂ υE = υe + + , (7.84) δs ∂s A δc ∂crk A 0≤k≤N rk ∂ δL ∂ δL υ E = υe + + . ∂s δs A ∂c δcrk A 0≤k≤N rk

The Lagrangian L E (7.83) where L + L1 = Le is called a proper extension of an original Lagrangian L. The following is a corollary of Theorem 7.14.

Theorem 7.15 A Lagrangian L is extended to a proper solution L E (7.83)ofthe master equation only if the gauge operator u (7.38) admits a nilpotent extension.

By virtue of condition (iv) of Theorem 7.14, this nilpotent extension is the deriva- ϑ = υ + υl tion E E E (7.81), called the KT-BRST operator. With this operator, the 0,n module of densities P∞ {N} is split into the KT-BRST complex

0,n 0,n 0,n ···−→P∞ {N}2 −→ P∞ {N}1 −→ P∞ {N}0 −→ (7.85) 0,n 1 0,n 2 P∞ {N} −→ P∞ {N} −→··· . 7.5 Lagrangian BRST Theory 161

Putting all ghosts zero, we obtain a cochain morphism of this complex onto the 0,n KT complex, extended to P∞ {N} and reversed into the cochain one. Letting all antiﬁelds zero, we come to a cochain morphism of the KT-BRST complex (7.85) onto the cochain sequence (7.38), where the gauge operator is extended to the antiﬁeld-free part of the KT-BRST operator.

Theorem 7.16 If the gauge operator u (7.38) can be extended to the BRST operator b (7.66), then the master equation has a nontrivial proper solution = + γ rk−1 ω = L E Le crk−1 (7.86) ≤ ≤ 1 k N + rk−1 ω + σ. L b c crk−1 dH 0≤k≤N

Proof By virtue of Theorem 7.12, if the BRST operator b (7.66) exists, the densities Δ rk (7.27) contain only the terms Grk linear in antifields. It follows that the extended Lagrangian Le (7.40) and, consequently, the Lagrangian L E (7.86) are affine in antifields. In this case, we have

← ← A A rk rk u = δ (Le), u = δ (Le) for all indices A and rk and, consequently,

← ← A A rk rk b = δ (LE ), b = δ (LE ), i.e., b = υE is the graded derivation (7.84) deﬁned by the Lagrangian L E . Its nilpotency condition takes a form

← ← A rk b( δ (LE )) = 0, b( δ (LE )) = 0.

Hence, we obtain

← ← (L ) = ( δ A(L ) + δ rk (L ) ) = , b E b E s A E crk 0 i.e., b is a variational symmetry of L E . Consequently, L E obeys the master equation.

For instance, let a gauge symmetry u be Abelian, and let the higher-stage gauge symmetries be independent of original ﬁelds, i.e., u(u) = 0. Then u = b and L E = Le. The proper solution L E (7.86) of the master equation is called the BRST extension of an original Lagrangian L. As was mentioned above, it is a necessary step towards BV quantization of classical Lagrangian ﬁeld theory in terms of functional integrals [6, 13, 44, 50, 63].

Chapter 8 Yang–Mills Gauge Theory on Principal Bundles

In classical gauge field theory, gauge fields conventionally are described as principal connections on principal bundles [37, 61, 100, 102, 127]. We consider their first order Yang–Mills Lagrangian theory. This is irreducible degenerate Lagrangian theory (Sect.8.6) where conserved Noether currents are reduced to a superpotential (Sect. 8.5). However, an energy-momentum current fails to be conserved because of a background world metric.

8.1 Geometry of Principal Bundles

We consider smooth principle bundles with a structure real Lie group of nonzero dimension [100, 134]. Principal Bundles Let G,dimG > 0, be a real Lie group which acts on a smooth manifold V on the left. A smooth ﬁbre bundle Y → X with a typical ﬁbre V is called a bundle with a structure group G or, in brief, the G-bundle if it possesses an atlas

Ψ ={(Uα,ψα), ραβ },ψα = ραβ ψβ , (8.1) whose transition functions ραβ (B.3) factorize as

Id ×ζ ραβ : Uα ∩ Uβ × V −→ Uα ∩ Uβ × (G × V ) −→ Uα ∩ Uβ × V (8.2)

G through G-valued local smooth functions ραβ : Uα ∩ Uβ → G on X. This means that transition morphisms ραβ (x) (B.6) are elements of G acting on V . Transition functions (8.2) are called G-valued. Let πP : P → X (8.3)

© Atlantis Press and the author(s) 2016 163 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_8 164 8 Yang–Mills Gauge Theory on Principal Bundles be a G-bundle whose typical fibre is the group space of G, which a group G acts on by left multiplications. It is called the principal bundle with a structure group G. Equivalently, a principal G-bundle is defined as a fibre bundle P (8.3) which admits an action of G on P on the right by a fibrewise morphism

RGP : G × P −→ P, Rg : p → pg,πP (p) = πP (pg), p ∈ P, (8.4) X X which is free and transitive on each ﬁbre of P. As a consequence, the quotient of P with respect to the action (8.4)ofG is diffeomorphic to a base X, i.e., P/G = X.

Remark 8.1 Given the principal bundle P (8.3), a ﬁbre bundle Y → X with a structure group G and a typical ﬁbre V is said to be associated to P (or, in brief, P-associated) if it is isomorphic to the quotient

Y = (P × V )/G (8.5) of the product P × V by identiﬁcation of elements (p, v) and (pg, g−1v) for all g ∈ G.AnyG-bundle is associated to some principal G-bundle.

A principal G-bundle P is equipped with the bundle atlas (8.1):

P ΨP ={(Uα,ψα ), ραβ } (8.6)

P whose trivialization morphisms ψα obey the condition

P P ψα (pg) = gψα (p), g ∈ G.

P Due to this property, every trivialization morphism ψα determines a unique local P section zα : Uα → P such that (ψα ◦ zα)(x) = 1. The transformation law for zα reads zβ (x) = zα(x)ραβ (x), x ∈ Uα ∩ Uβ . (8.7)

Conversely, a family ΨP ={(Uα, zα), ραβ } (8.8) of local sections of P which obey the transformation law (8.7) uniquely deﬁnes a bundle atlas ΨP of a principal bundle P. In particular, a principal bundle admits a global section if and only if it is trivial.

Remark 8.2 Every bundle atlas ΨP (8.8) of a principal G-bundle P deﬁnes a unique associated bundle atlas

−1 Ψ ={(Uα,ψα(x) =[zα(x)] )} of the P-associated bundle Y (8.5). 8.1 Geometry of Principal Bundles 165

Let us consider the tangent morphism to the right action RGP (8.4)ofG on P. Since TG = G ×gl where gl is the left Lie algebra of G, this morphism takes a form

TRGP : (G × gl ) × TP−→ TP, (8.9) X X and its restriction to T1G ×X TPprovides a homomorphism

gl ε → ξε ∈ T (P) (8.10) of the left Lie algebra gl of G to the Lie algebra T (P) of vector fields on P. Vector fields ξε (8.10) are obviously vertical. They are called fundamental vector fields {ε } g ξ = ξ [82]. Given a basis r for l , the corresponding fundamental vector fields r εr form a family of m = dim gl nowhere vanishing and linearly independent sections of the vertical tangent bundle VP of P → X. Consequently, this bundle is trivial VP= P × gl by virtue of Theorem B.12. Restricting the tangent morphism TRGP (8.9)to TRGP : 0(G) × TP−→ TP, (8.11) X X we obtain the tangent prolongation of the structure group action RGP (8.4). In brief, it is called the action of G on T P. Since the action of G (8.4)onP is fibrewise, its action (8.11) is restricted to the vertical tangent bundle VPof P. A vector field ξ on P is called equivariant if ξ(pg) = TRg ◦ ξ(p). Obviously, it is projectable. Taking the quotient of the tangent bundle TP → P and the vertical tangent bundle VPof P by G (8.11), we obtain the vector bundles

TG P = TP/G, VG P = VP/G (8.12) over X. Sections of TG P → X are equivariant vector fields on P. Accordingly, sections of VG P → X are equivariant vertical vector fields on P. Hence, a typical fibre of VG P → X is the right Lie algebra gr of G subject to the adjoint representation of a structure group G. Therefore, VG P (8.12) is called the Lie algebra bundle. Given the bundle atlas ΨP (8.6)ofP, there is the corresponding atlas (8.1):

Ψ ={( ,ψ ), } Uα α Adραβ (8.13) of a Lie algebra bundle VG P, which is provided with the corresponding bundle μ m −1 coordinates (Uα; x ,χ ) with respect to the ﬁbre frames {em = ψα (x)(εm )}, where {εm} is a basis for the right Lie algebra gr . These coordinates obey a transformation rule

m m ρ(χ )εm = χ Adρ−1 (εm ).

A glance at this transformation rule shows that VG P is a G-bundle P. 166 8 Yang–Mills Gauge Theory on Principal Bundles

Accordingly, the vector bundle TG P (8.12) is endowed with bundle coordinates μ μ m (x , x˙ ,χ ) with respect to the ﬁbre frames {∂μ, em }. Their transformation rule is

m m μ m ρ(χ )εm = χ Adρ−1 (εm) +˙x Rμ εm . (8.14)

For instance, if G is a matrix group, this transformation rule reads

m m −1 μ −1 ρ(χ )εm = χ ρ εm ρ −˙x ∂μ(ρ )ρ. (8.15)

Since the second term in the right-hand sides of expressions (8.14) and (8.15) depend on derivatives of a G-valued function ρ on X, the vector bundle TG P (8.12) fails to be a G-bundle. The Lie bracket of equivariant vector ﬁelds on P is equivariant, and it induces the Lie bracket of sections of the vector bundle TG P → X. This bracket reads

λ p μ q ξ = ξ ∂λ + ξ ep,η= η ∂μ + η eq , (8.16) μ λ μ λ [ξ,η]=(ξ ∂μη − η ∂μξ )∂λ + (8.17) (ξ λ∂ ηr − ηλ∂ ξ r + r ξ pηq ) . λ λ cpq er

Putting ξ λ = 0 and ημ = 0 in the formulas (8.16) and (8.17), we obtain the Lie bracket [ξ,η]= r ξ pηq cpq er (8.18) of sections of the Lie algebra bundle VG P. Principal Connections In gauge field theory, gauge fields are conventionally described as principal connections on principal bundles. Principal connections on a principal bundle P (8.3)are connections on P which are equivariant with respect to the right action (8.4)ofa structure group G on P. Let J 1 P be the first order jet manifold of a principal G-bundle P → X (8.3). Connections on a principal bundle P → X (Definition B.10) are global sections

A : P → J 1 P (8.19) of the afﬁne jet bundle J 1 P → P modelled over a vector bundle

∗ ∗ T X ⊗ VP= (T X ⊗ gl ). P P

Let us consider the jet prolongation

1 1 1 1 J RG : J (X × G) × J P → J P X 8.1 Geometry of Principal Bundles 167 of the morphism RGP (8.4). Restricting this morphism to

1 1 1 J RG : 0(G) × J P → J P, X we obtain the jet prolongation of the structure group action RGP (8.4). In brief, it is called the action of G on J 1 P. It reads

1 : 1 → ( 1 ) = 1( ). J Rg jx p jx p g jx pg (8.20)

The quotient of the afﬁne jet bundle J 1 P by G (8.20) is an afﬁne bundle

C = J 1 P/G → X (8.21)

∗ modelled over a vector bundle C = T X ⊗X VG P → X. Hence, there is the vertical splitting VC = C ⊗X C (8.22) of the vertical tangent bundle VC of C → X. Taking the quotient with respect to the action of a structure group G, one can reduce the canonical imbedding (B.48) (where Y = P) to a bundle monomorphism

∗ μ m λC : C −→ T X ⊗ TG P,λC : dx ⊗ (∂μ + χμ em ). (8.23) X X

It follows that, given atlases ΨP (8.6)ofP and Ψ (8.13)ofTG P, the afﬁne bundle C λ m (8.21) is provided with bundle coordinates (x , aμ ) possessing transition functions ∂ ν m m m x ρ(a )ε = (a Adρ−1 (ε ) + R ε ) . (8.24) μ m ν m ν m ∂x μ If G is a matrix group, these transition functions read ∂ ν m m −1 −1 x ρ(a )ε = (a ρ (ε )ρ − ∂μ(ρ )ρ) . μ m ν m ∂x μ

A glance at this expression shows that C (8.21) as like as the vector bundle TG P (8.12) fails to be a bundle with a structure group G. As was mentioned above, a connection A (8.19) on a principal bundle P → X is called a principal connection if it is equivariant under the action (8.20) of a structure group G, i.e.,

1 A(pg) = (J Rg ◦ A)(p) g ∈ G.

There is obvious one-to-one correspondence between the principal connections on a principal G-bundle P and the global sections

A : X → C (8.25) 168 8 Yang–Mills Gauge Theory on Principal Bundles of the quotient bundle C → X (8.21), called the bundle of principal connections.

Theorem 8.1 Since a bundle of principal connections C → X is afﬁne, principal connections on a principal bundle always exist (Theorem B.4).

Due to the bundle monomorphism (8.23), any principal connection A (8.25)is represented by a TG P-valued principal connection form

λ q A = dx ⊗ (∂λ + Aλeq ) (8.26)

(cf. (B.55)). Taking the quotient with respect to the action of a structure group G, one can reduce the exact sequence (B.21) (where Y = P) to the exact sequence

0 → VG P −→ TG P −→ TX → 0. (8.27) X

The principal connection A (8.26) yields a splitting of this exact sequence.

Remark 8.3 Any principal connection A (8.26) on a principal bundle P → X deﬁnes a unique connection on the P-associated ﬁbre bundle Y (8.5) which reads

= λ ⊗ (∂ + p i ∂ ), A dx λ Aλ Ip i where {Ip} is a representation of the Lie algebra gr of G in V [83].

In gauge ﬁeld theory, coefﬁcients of the principal connection form (8.26)are treated as gauge potentials. We use this term in order to refer to sections A (8.25)of the bundle C → X of principal connections. Let P → X be a principal G-bundle. The Frölicher–Nijenhuis bracket (B.37) on the space O∗(P) ⊗ T (P) of tangent-valued forms on P is compatible with the right action RGP (8.4). Therefore, it induces the Frölicher–Nijenhuis bracket on a space ∗ O (X) ⊗ TG P(X) of TG P-valued forms on X, where TG P(X) is a vector space of sections of a vector bundle TG P → X. Note that, as it follows from the exact sequence 1 (8.27), there is an epimorphism TG P(X) → T (X).LetA ∈ O (X) ⊗ TG P(X) be the principal connection (8.26). The associated Nijenhuis differential (B.40) is

r r+1 dA : O (X) ⊗ TG P(X) → O (X) ⊗ VG P(X), r dAφ =[A,φ]FN,φ∈ O (X) ⊗ TG P(X).

The strength of a principal connection A is deﬁned as a VG P-valued two-form

1 1 F = d A = [A, A] ∈ O2(X) ⊗ V P(X), (8.28) A 2 A 2 FN G 1 r λ μ F = Fλμdx ∧ dx ⊗ e , (8.29) A 2 r r =[∂ + p ,∂ + q ]r = ∂ r − ∂ r + r p q . Fλμ λ Aλ ep μ Aμeq λ Aμ μ Aλ cpq Aλ Aμ 8.1 Geometry of Principal Bundles 169

Remark 8.4 The strength FA (8.28) is not the standard curvature (B.63) of a principal connection because A (8.26) is not a tangent-valued form.

Canonical Principal Connection Since gauge potentials are represented by sections of the bundle of principal connections C → X (8.21), gauge field theory is formulated as first order Lagrangian theory on a configuration space C. A glance at the expression (8.20) shows that the fibre bundle J 1 P → C is a trivial principal G-bundle. It is canonically isomorphic to the pull-back bundle

1 J P = PC = C × P → C. (8.30) X

Given a principal G-bundle P → X and its jet manifold J 1 P, let us consider the canonical morphism θ(1) (B.48) where Y = P. By virtue of Remark B.3, this 1 morphism deﬁnes a map θ : J P ×P TP → VP. Taking its quotient with respect to G, we obtain a morphism

θ p C × TG P −→ VG P,θ(∂λ) =−aλ ep,θ(ep) = ep. (8.31) X

This means that the exact sequence (8.27) admits the canonical splitting over C [51, 100]. In view of this fact, let us consider the pull-back bundle PC (8.30). Since

VG (C × P) = C × VG P, TG (C × P) = TC× TG P, (8.32) X X X X the exact sequence (8.27) for the principal bundle PC reads

0 → C × VG P −→ TC× TG P −→ TC → 0. (8.33) X C X

It is readily observed that the morphism (8.31) yields the horizontal splitting

TC× TG P −→ C × TG P −→ C × VG P, X X X of the exact sequence (8.33) and, consequently, it deﬁnes the principal connection

A : TC → TC× TG P, X A = λ ⊗ (∂ + p ) + r ⊗ ∂λ ∈ O1( ) ⊗ ( × )( ) dx λ aλ ep daλ r C TG C P X (8.34) X on the principal bundle PC → C (8.30). It follows that the principal bundle PC admits the canonical principal connection (8.34). 170 8 Yang–Mills Gauge Theory on Principal Bundles

Following the expression (8.28), let us deﬁne the strength

1 1 2 FA = dA A = [A , A ]∈O (C) ⊗ V P(X), 2 2 G r μ 1 r p q λ μ FA = (daμ ∧ dx + c a aμdx ∧ dx ) ⊗ e , (8.35) 2 pq λ r of the canonical principal connection A (8.34). It is called the canonical strength ∗ because, given the principal connection A (8.25)onP → X, the pull-back A FA = FA is the strength (8.29)ofA. With the VG P-valued two-form FA (8.35)onC, let us deﬁne the VG P-valued horizontal two-form F = h0(FA ):

1 r λ μ r r r r p q F = Fλμdx ∧ dx ⊗ ε , Fλμ = aλμ − aμλ + c a aμ, (8.36) 2 r pq λ on J 1C. It is called the strength form. For each principal connection A (8.25)onP, 1 ∗ the pull-back J A F = FA is the strength (8.29)ofA. The strength form (8.36) yields an afﬁne surjection

1 2 ∗ F /2 : J C −→ C ×(∧ T X ⊗ VG P) (8.37) C X over C of the afﬁne jet bundle J 1C → C to the vector one

2 ∗ C ×(∧ T X ⊗ VG P) → C. X

By virtue of Theorem B.10, its kernel C+ = Ker F /2 is an afﬁne subbundle of J 1C → C. Thus, we have the canonical splitting of the afﬁne jet bundle

1 2 ∗ J C = C+ ⊕ C− = C+ ⊕(C × ∧ T X ⊗ VG P), (8.38) C C X

r 1 r r 1 r r r p q aλμ = (Sλμ + Fλμ) = (aλμ + aμλ − c a aμ) (8.39) 2 2 pq λ 1 r r r p q + (aλμ − aμλ + c a aμ). 2 pq λ = F The corresponding canonical projections are pr 2 (8.37) and

= S : 1 → . pr 1 J C C+ (8.40)

The splitting (8.38) exempliﬁes that (3.100), but it is not related to a Lagrangian. 8.2 Principal Gauge Symmetries 171

8.2 Principal Gauge Symmetries

In gauge ﬁeld theory, principal gauge transformations are deﬁned as principal automorphisms of a principal bundle P. An automorphism ΦP of a principal G-bundle P is called principal if it is equivariant under the right action (8.4) of a structure group G on P, i.e.,

ΦP (pg) = ΦP (p)g, g ∈ G, p ∈ P. (8.41)

In particular, every vertical principal automorphism of a principal bundle P looks like ΦP (p) = pf(p), p ∈ P, where f is a G-valued equivariant function on P, i.e.,

f (pg) = g−1 f (p)g, g ∈ G. (8.42)

There is one-to-one correspondence

s(πP (p))p = pf(p), p ∈ P, between the equivariant functions f (8.42) and the global sections s of the associated group bundle G πP G : P → X (8.43) whose typical fibre is a group G which acts on itself by the adjoint representation. In order to describe gauge symmetries of gauge field theory on a principal bundle P, let us restrict our consideration to local one-parameter groups of local principal automorphisms of P. Their infinitesimal generators are equivariant vector fields ξ on P, and vice versa. We call ξ the principal vector fields or the infinitesimal gauge transformation. They are represented by sections ξ (8.16) of the vector bundle TG P (8.12). Principal vector fields constitute a real Lie algebra TG P(X) with respect to the Lie bracket (8.17). Vertical principal vector fields are sections

p ξ = ξ ep (8.44) of the Lie algebra bundle VG P → X (8.12). They form a ﬁnite-dimensional Lie ∞ C (X)-algebra G (X) = VG P(X) with respect to the bracket (8.18).

Remark 8.5 Any principal automorphism ΦP (8.41)ofP yields a unique principal automorphism

ΦY : (p, v)/G → (ΦP (p), v)/G, p ∈ P, v ∈ V, of the P-associated bundle Y (8.5). As a consequence, any principal vector field ξ (8.44) yields an associated principal vector field on a P-associated bundle Y which is an infinitesimal generator of local one-parameter group of local principal automorphisms of Y . 172 8 Yang–Mills Gauge Theory on Principal Bundles

As was mentioned above, a bundle of principal connections C fails to be a G- bundle, but principal automorphisms of P yield automorphisms of C as follows. Any local one-parameter group of local principal automorphism ΦP (8.41) of a principal 1 bundle P admits the jet prolongation J ΦP (B.51) to a local one-parameter group of local equivariant automorphism of the jet manifold J 1 P which, in turn, yields a one-parameter group of principal automorphisms ΦC of the bundle of principal connections C (8.21) [80, 100]. Its infinitesimal generator is a vector field on C, called the principal vector field on C and regarded as an infinitesimal gauge transformation of C. Thus, any principal vector field ξ (8.16)onP yields a principal vector field uξ on C, which can be obtained as follows. Using the morphism (8.31), we get a morphism ξ θ : C → VG P, which is a section of of the Lie algebra bundle VG (C ×X P) → C in accordance with the first formula (8.32). Then the equation

uξ FA = dA (ξ θ) uniquely determines a principal vector ﬁeld uξ on C. A direct computation leads to

= ξ μ∂ + (∂ ξ r + r pξ q − r ∂ ξ ν )∂μ. uξ μ μ cpqaμ aν μ r (8.45)

In particular, if ξ is the vertical principal vector ﬁeld (8.44), we obtain the vertical principal vector ﬁeld (8.45):

= (∂ ξ r + r pξ q )∂μ. uξ μ cpqaμ r (8.46)

1 Remark 8.6 The jet prolongation (B.52) of the vector ﬁeld uξ (8.45) onto J C reads

1 = + (∂ ξ r + r p∂ ξ q + r p ξ q J uξ uξ λμ cpqaμ λ cpqaλμ (8.47) − r ∂ ξ ν − r ∂ ξ ν − r ∂ ξ ν )∂λμ. aν λμ aλν μ aνμ λ r

It is readily justiﬁed that the monomorphism

TG P(X) ξ → uξ ∈ T (C) (8.48) obeys the equality u[ξ,η] =[uξ , uη], (8.49) i.e., it is a monomorphism of the real Lie algebra TG P(X) of sections of a vector bundle TG X → to the real Lie algebra T (C) of vector ﬁelds on C. In particular, the image of the Lie algebra G (X) ⊂ TG P(X) in T (C) also is a real Lie algebra, but ∞ ∞ not the C (X)-one because u f ξ = fuξ , f ∈ C (X).

Remark 8.7 A glance at the expression (8.45) shows that the monomorphism (8.48) is a linear first order differential operator which sends sections of the pull-back bundle C ×X TG P → C onto sections of the tangent bundle TC → C. Refereing 8.2 Principal Gauge Symmetries 173 to Definition 2.3, we therefore can treat principal vector fields (8.45) as infinitesimal gauge transformations depending on gauge parameters ξ ∈ TG P(X).

8.3 Noether’s Direct Second Theorem: Yang–Mills Lagrangian

Let us consider first order Lagrangian gauge theory on a principal bundle P.Its configuration space is the bundle of principal connections C (8.21), endowed with μ m bundle coordinates (x , aμ ) possessing transition functions (8.24). Given a first order Lagrangian n L = L ω : J 1C → ∧ T ∗ X (8.50) on the jet manifold J 1C, the corresponding Euler–Lagrange operator (3.2) reads

E = E μθr ∧ ω = (∂μ − ∂λμ)L θr ∧ ω. L r μ r dλ r μ (8.51)

Its kernel deﬁnes the Euler–Lagrange equation

E μ = (∂μ − ∂λμ)L = . r r dλ r 0

Let us assume that the gauge theory Lagrangian L (8.50)onJ 1C is invariant under vertical principal gauge transformations (or, in short, gauge invariant). This means that vertical principal vector ﬁelds uξ (8.46) are exact symmetries of L, i.e.,

= , 1 = + (∂ ξ r + r p∂ ξ q + r p ξ q )∂λμ, LJ 1uξ L 0 J uξ uξ λμ cpqaμ λ cpqaλμ r (8.52)

(see the formula (8.47)). Then it follows from Remark 8.7 that vertical principal vector fields uξ (8.46) are gauge symmetries of L (Definition 2.3) whose gauge parameters are sections ξ (8.44) of the Lie algebra bundle VG P. We call them principal gauge symmetries. Let us apply Noether’s direct second theorem to these gauge symmetries. In this case, the first variational formula (3.26) for the Lie derivative (8.52) takes a form

= (∂ ξ r + r pξ q )E μ + [(∂ ξ r + r pξ q )∂λμL )]. 0 μ cpqaμ r dλ μ cpqaμ r (8.53)

It leads to the gauge invariance conditions

∂μλL + ∂λμL = , p p 0 (8.54) E μ + ∂λμL + q p∂μν L = , r dλ r cpr aν q 0 (8.55) r ( pE μ + ( p∂λμL )) = . cpq aμ r dλ aμ r 0 (8.56) 174 8 Yang–Mills Gauge Theory on Principal Bundles

One can regard the equalities (8.54) and (8.56) as the conditions of a Lagrangian L to be gauge invariant. They are brought into a form

∂μλL + ∂λμL = , p p 0 (8.57) ∂μL + r p∂μν L = , q cpqaν r 0 (8.58) r ( p∂μL + p ∂λμL ) = . cpq aμ r aλμ r 0 (8.59)

q r r Let us utilize the coordinates (aμ, Fλμ, Sλμ) (8.39) which correspond to the canonical splitting (8.38) of the afﬁne jet bundle J 1C → C. With respect to these coordinates, the Eq. (8.57) reads ∂L = . p 0 (8.60) ∂Sμλ

Then the Eq. (8.58) takes a form ∂L = . q 0 (8.61) ∂aμ

A glance at the equalities (8.60) and (8.61) shows that a gauge invariant Lagrangian factorizes through the strength F (8.36). Then the Eq. (8.59), written as ∂L r F p = , cpq λμ r 0 ∂Fλμ shows that a gauge symmetry uξ of L is exact. The following thus has been proved. Theorem 8.2 A gauge theory Lagrangian (8.50) possesses the exact gauge symmetry uξ (8.46) only if it factorizes through the strength F (8.36). A corollary of this result is the well-known Utiyama theorem [24]. Theorem 8.3 There is a unique gauge invariant quadratic ﬁrst order Lagrangian which is the Yang–Mills Lagrangian 1 G λμ βν p q L = a g g F Fμν |g| ω, g = det(gμν), (8.62) YM 4 pq λβ

G where a is a G-invariant bilinear form on the right Lie algebra gr and g is a world metric on X (Remark B.12).

The Euler–Lagrange operator (8.51) of the Yang–Mills Lagrangian LYM reads E = E μθ μ ∧ ω = (δn + n p)( G μα λβ F q | |)θr ∧ ω. YM r r r dλ crpaλ anq g g αβ g μ (8.63)

Its kernel deﬁnes the Yang–Mills equation E μ = (δn + n p)( G μα λβ F q | |) = . r r dλ crpaλ anq g g αβ g 0 (8.64) 8.3 Noether’s Direct Second Theorem: Yang–Mills Lagrangian 175

∗ We call a Lagrangian system (S∞[C], LYM) the Yang–Mills gauge theory. Remark 8.8 In gauge ﬁeld theory, there are Lagrangians, e.g., the Chern–Simons ones (Chap. 11) which do not factorize through the strength of a gauge ﬁeld, and whose gauge symmetries uξ (8.46) are not exact.

8.4 Noether’s First Theorem: Conservation Laws

Since the gauge symmetry uξ (8.46) of the Yang–Mills Lagrangian (8.62) is exact, the ﬁrst variational formula (8.53) leads to the gauge conservation law

≈ (− μ∂λμL ) 0 dλ uξ r r YM of a Noether current J λ =−(∂ ξ r + r pξ q )( G μα λβ F q | |). ξ μ cpqaμ arqg g αβ g (8.65)

In accordance with Theorem 2.8, the Noether current (8.65) is brought into the superpotential form (7.65) which reads

λ r μ r [νμ] J = ξ E + dν (ξ ∂ L ), (8.66) ξ r r YM νμ = ξ r G να μβ F q | |. U arqg g αβ g

Let us study energy-momentum conservation laws in Yang–Mills gauge theory. If a background world metric g is specified, one can find a particular vector field τ on X and its lift γτ (B.27) onto C which is an exact symmetry of the Yang– Mills Lagrangian (8.62). Then the energy-momentum current (3.33) along such a symmetry is conserved. We here treat the energy-momentum conservation law as the gauge one in the case of an arbitrary world metric g and any vector field τ on X. Let A (8.26) be a principal connection on P. For any vector field τ on X,this connection yields a section

λ p λ τ A = τ ∂λ + Aλ τ ep of the vector bundle TG P → X. It, in turn, deﬁnes the principal vector ﬁeld (8.45) on the bundle of principal connection C which reads

τ = τ λ∂ + (∂ ( r τ ν ) + r p q τ ν − r ∂ τ ν )∂μ. A λ μ Aν cpqaμ Aν aν μ r (8.67)

Let us consider an energy-momentum current along this vector ﬁeld [100, 119]. Since the Yang–Mills Lagrangian (8.62) depends on a background world metric g, the vector ﬁeld Aτ (8.67) is not its exact symmetry in general. Following the procedure in Sect. 2.4.6, let us consider the total Lagrangian 176 8 Yang–Mills Gauge Theory on Principal Bundles 1 G λμ βν p q L = a σ σ F Fμν |σ|ω, σ = det(σμν ), (8.68) 4 pq λβ

2 2 1 ∗ on the jet manifold J (C ×X ∨ TX) of the total configuration space C ×X ∨ T X, 2 where a tensor bundle ∨ TX is provided with holonomic fibre coordinates (σ μν ). Given a vector field τ on X, there exists its canonical lift (B.29):

λ α νβ β να τ = τ ∂λ + (∂ν τ σ + ∂ν τ σ )∂αβ ,

2 onto a tensor bundle ∨ T ∗ X. It is an inﬁnitesimal general covariant transformations 2 of ∨ T ∗ X (Sect. 10.1). Thus, we have the lift

τ = τ + τ = τ λ∂ + (∂ ( r τ ν ) + r p q τ ν − r ∂ τ ν )∂μ A A λ μ Aν cpqaμ Aν aν μ r (8.69) α νβ β να + (∂ν τ σ + ∂ν τ σ )∂αβ

2 ∗ of a vector ﬁeld τ on X onto a product C ×X ∨ T X. Its is readily observed that the vector ﬁeld τA (8.69) is an exact symmetry of the total Lagrangian (8.68). One the shell (8.64), we then obtain the weak transformation law (3.43):

≈ (∂ τ α νβ + ∂ τ β να − ∂ αβ τ λ)∂ L − J λ 0 ν g ν g λg αβ dλ A (8.70) of the energy-momentum current

J λ = ∂λμL [τ ν r − ∂ ( r τ ν ) − r p q τ ν + r ∂ τ ν ]−τ λL A r YM aνμ μ Aν cpqaμ Aν aν μ YM (8.71) along the vector ﬁeld Aτ (8.67). The weak identity (8.70) can be written in a form ≈ ∂ τ μ λ | |−τ μ{ β } λ | |− J λ, 0 λ tμ g μ λ tβ g dλ A (8.72)

β where {μ λ} are the Christoffel symbols (B.74) of a world metric g and μ | |= μα∂ L = (F q ∂μν − δμ)L tβ g 2g αβ YM βν q β YM is the metric energy-momentum tensor of a gauge ﬁeld. In particular, let a principal connection B be a classical solution of the Yang–Mills equation (8.64). Let us consider the lift (8.67) of a vector ﬁeld τ on X onto C by means of a principal connection A = B. In this case, the energy-momentum current (8.71) reads J λ ◦ = τ μ( λ ◦ ) | |. B B tμ B g (8.73) 8.4 Noether’s First Theorem: Conservation Laws 177

Then the identity (8.72)onasolutionB comes to a covariant conservation law Γ λ ∇λ ((tμ ◦ B) |g|) = 0, (8.74) where ∇ is the covariant differential (B.62) with respect to the Levi–Civita connection β {μ λ} (B.74) of a background metric g. Note that, considering a different lift γτ of a vector ﬁeld τ on X to a principal vector ﬁeld on C, we obtain an energy-momentum current along γτ which differs J λ from B (8.73) in a Noether current (8.66). Since such a current is reduced to a superpotential, one can always bring the energy-momentum transformation law (8.70) into the covariant conservation law (8.74).

8.5 Hamiltonian Gauge Theory

The Yang–Mills Lagrangian (8.62) exempliﬁes almost regular quadratic Lagrangians analyzed in Sect.3.8. It admits a complete set of weakly associated Hamiltonian forms (8.78). A key point that neither these Hamiltonian forms nor the corresponding characteristic Lagrangians LH (8.79) inherit principal gauge symmetries of a Yang– Mills Lagrangian so that the part (8.82) of the covariant Hamilton equation plays the role of gauge-type conditions. Restricted to the Lagrangian constraint space, they however lead to the gauge invariant constrained Lagrangian L N (8.81). The phase space of gauge ﬁeld theory is the Legendre bundle (3.6) which due to the vertical splitting (8.22)ofVC is

n ∗ ∗ πΠC : Π → C,Π= ∧ T X ⊗ TX⊗[C × C] . (8.75) C C

λ p μλ It is endowed with holonomic coordinates (x , aλ , pm ). The Legendre bundle Π (8.75) admits the canonical decomposition (3.103):

Π = Π+ ⊕ Π−, C 1 1 pμλ = R(μλ) + P[μλ] = p(μλ) + p[μλ] = (pμλ + pλμ) + (pμλ − pλμ). m m m m m 2 m m 2 m m The Legendre map (3.5) induced by the Yang–Mills Lagrangian (8.62) takes a form

p(μλ) ◦ L = 0, (8.76) m YM [μλ] ◦ = G μα λβ F n | |. pm LYM amng g αβ g (8.77) It follows that Ker LYM = C+, and the Lagrangian constraint space (3.9)is

1 NL = LYM(J C) = Π−. 178 8 Yang–Mills Gauge Theory on Principal Bundles

It is deﬁned by the Eq. (8.76). Obviously, NL is an imbedded submanifold of Π, and the Yang–Mills Lagrangian LYM is almost regular. Following the procedure in Sect.3.8, let us consider connections Γ on a ﬁbre bundle C → X which take their values into Ker L, i.e.,

Γ : → ,Γr − Γ r + r p q = . C C+ λμ μλ cpqaλ aμ 0

Given a symmetric world connection K (B.71) on T ∗ X, every principal connection B on a principal bundle P → X gives rise to the connection ΓB : C → C+ such 1 that ΓB ◦ B = S ◦ J B where S is the projection (8.40) [61]. It reads

r 1 r r r p q r p q p q β r r Γ λμ = [∂μ Bλ + ∂λ Bμ − c a aμ + c (a Bμ + aμ B )]−Kλ μ(aβ − Bβ ). B 2 pq λ pq λ λ Given this connection, the corresponding Hamiltonian form (3.109):

= λμ r ∧ ω − λμΓ r ω − H ω, HB pr daμ λ pr B λμ YM (8.78) 1 mn [μλ] [νβ] H = a gμν gλβ p p |g|, YM 4 G m n is weakly associated to the Yang–Mills Lagrangian LYM. It is the Poincaré–Cartan form of the Lagrangian

=[ λμ( r − Γ r ) − H ]ω L H pr aλμ B λμ YM (8.79)

1 on Π ×C J C. The pull-back of any Hamiltonian form HB (8.78) onto the Lagrangian constraint space NL is the constrained Hamiltonian form (3.76):

∗ [λμ] r 1 r p q H = i H = p (daμ ∧ ωλ + c a aμω) − H ω. (8.80) N N B r 2 pq λ YM

1 The corresponding constrained Lagrangian L N on NL ×C J C reads

= ( [λμ]F r − H )ω. L N pr λμ YM (8.81)

Note that, in contrast with the Lagrangian (8.79), the constrained Lagrangian L N (8.81) possesses gauge symmetries as follows. Gauge symmetries uξ (8.45)ofthe Yang–Mills Lagrangian give rise to vector ﬁelds

= (∂ ξ r + r q ξ p)∂μ − r ξ p λμ∂q uξ μ cqpaμ r cqp pr λμ on Π. Their projection onto NL provides gauge symmetries of the Lagrangian L N in accordance with Theorem 3.21. The Hamiltonian form HB (8.78) yields the covariant Hamilton equation which consist of the equation (8.77) and the equations 8.5 Hamiltonian Gauge Theory 179

m m m aλμ + aμλ = 2ΓB (λμ), (8.82) λμ = q p [λμ] − q p (λμ) + μ (λν). pλr cpr rλ pq crpBλ pq Kλ ν pr (8.83)

The Hamilton equations (8.82)–(8.77) are similar to the equations (3.111) and (3.112), respectively. The Hamilton equations (8.77) and (8.83) restricted to the Lagrangian constraint space (8.76) are precisely the constrained Hamilton equations (3.77) for the constrained Hamiltonian form HN (8.80), and they are equivalent to the Yang-Mills equation (8.64) for gauge potentials A = πΠC ◦ r. Different Hamiltonian forms HB lead to different equations (8.82). This equation is independent of momenta and, thus, it exempliﬁes the gauge-type condition ΓB ◦ A = S ◦ J 1 A (3.111). A glance at this condition shows that, given a solution A of the Yang-Mills equations, there always exists a Hamiltonian form HB (e.g., HB=A) which obeys the condition (3.74), i.e.,

1 1 HB ◦ LYM ◦ J A = J A.

Consequently, the Hamiltonian forms HB (8.78) parameterized by principal connections B constitute a complete set.

8.6 Noether’s Inverse Second Theorem: BRST Extension

Gauge invariance conditions (8.54)–(8.56) lead to NI which the Euler–Lagrange operator EYM (8.63) of a Yang–Mills Lagrangian satisﬁes. These NI are associated to the gauge symmetry uξ (8.46). By virtue of the formula (2.19), they read

p q E μ + E μ = . crqaμ p dμ r 0 (8.84)

Theorem 8.4 The Noether identities (8.84) are nontrivial.

Proof Following the procedure in Sect.7.1, let us consider the density dual

∗ n ∗ ∗ ∗ n ∗ VC = V C ⊗ ∧ T X = (T X ⊗ VG P) ⊗ ∧ T X (8.85) C X C

∗ of the vertical tangent bundle VC of C → X, and let us enlarge a DGA S∞[C] to the DBGA (7.6):

∗ ∗ P∞[VC; C]=S∞[VC; C],

( r , μ) μ possessing the local generating basis aμ ar where ar are odd antiﬁelds. Providing this DBGA with the nilpotent right graded derivation ∂ δ = E μ, ∂ μ r ar 180 8 Yang–Mills Gauge Theory on Principal Bundles let us consider the chain complex (7.7). Its one-chains

Δ = p q μ + μ r crqaμa p dμar (8.86) are δ-cycles which deﬁne the NI (8.84). Clearly, they are not δ-boundaries. Therefore, the NI (8.84) are nontrivial.

Theorem 8.5 The Noether identities (8.84) are complete.

Proof The second order Euler–Lagrange operator EYM (8.63) takes its values into a space of sections of a vector bundle

∗ ∗ n ∗ (T X ⊗ VG P) ⊗ ∧ T X → X. X X

Let Φ be a ﬁrst order differential operator on this vector bundle such that Φ◦EYM = 0. Φ1 ◦ E 2 This condition holds only if the highest derivative term of the composition YM Φ1 Φ E 2 of the ﬁrst order derivative term of and the second order derivative term YM E Φ1 = Δ1 = μ of YM vanishes. This is the case only of r dμar . ∞ The graded densities Δr ω (8.86) constitute a local basis for a C (X)-module C(0) isomorphic to the module VG P(X) of sections of the density dual VG P of the ∗ Lie algebra bundle VG P → X. Let us enlarge a DBGA P∞[VC; C] to a DBGA

∗ ∗ P∞{0}=S∞[VC; C × VG P] X

( r , μ, ) with the local generating basis aμ ar cr where cr are even Noether antifields. Theorem 8.6 NI (8.84) are irreducible. ∗ Proof Providing a DBGA P∞{0} with a nilpotent odd derivation ∂ δ0 = δ + Δr , ∂cr let us consider the chain complex (7.15). Let us assume that Φ (7.16)isatwo- cycle of this complex, i.e., the relation (7.17) holds. It is readily observed that Φ obeys this relation only if its first term G is δ-exact, i.e., the first-stage NI (7.17)are trivial.

By virtue of Theorems 8.4–8.6, Yang–Mills gauge theory is an irreducible degenerate Lagrangian theory characterized by complete NI (8.84). Following Noether’s inverse second Theorem 7.9, let us consider a DBGA

∗ ∗ P∞{0}=S∞[VC⊕ VG P; C × VG P] (8.87) C X 8.6 Noether’s Inverse Second Theorem: BRST Extension 181

( r , μ, r , ) with the local generating basis aμ ar c cr where cr are odd ghosts. The gauge operator u (7.39) associated to NI (8.84) reads

= = ( r + r p q )∂μ. u u cμ cpqaμc r (8.88)

It is an odd gauge symmetry of the Yang–Mills Lagrangian LYM which can be r obtained from the gauge symmetry uξ (8.46) by replacement of gauge parameters ξ with odd ghosts cr (Remark 7.7). Since gauge symmetries uξ form the Lie algebra (8.49), the gauge operator u (8.88) admits the nilpotent BRST extension

∂ 1 ∂ b = (cr + cr a pcq ) − cr c pcq , μ pq μ r pq r ∂aμ 2 ∂c which is the well-known BRST operator in Yang–Mills gauge theory [63]. Then, by virtue of Theorem 7.16, the Yang–Mills Lagrangian LYM is extended to a proper solution of the master equation

r r p q μ 1 r p q L = L + (cμ + c aμc )a ω − c c c c ω. E YM pq r 2 pq r

Chapter 9 SUSY Gauge Theory on Principal Graded Bundles

By analogy to gauge field theory on principal bundles (Chap. 8), we develop SUSY gauge theory as theory of graded principal connections on principal graded bundles. A key point is that we consider simple principal graded bundles (Definition 9.2) subject to an action of an even Lie group, but not the whole graded Lie one. In this case, odd gauge potentials in comparison with the even ones are linear, but not affine objects (Remark 9.2). However, they admit affine gauge transformations (9.8) parameterized by ghosts. Our goal is Yang–Mills theory of graded gauge fields and its BRST extension. Graded principal bundles and connections on these bundles [100, 140] can be studied similarly to principal superbundles and principal superconnections [7, 100]. Since the definition of a graded Lie group and its Lie superalgebra involves the notion of a Hopf algebra, we start with the following construction [58, 140]. Let (Z, A) be a graded manifold. One considers the finite dual A(Z)◦ of its structure ring A(Z) which consists of elements a of the dual A(Z)∗ of A(Z) vanishing on an ideal of A(Z) of finite codimension. This is brought into a graded commutative coalgebra with a comultiplication

Δ◦(a)( f ⊗ f ) = a( ff), f, f ∈ A(Z), a ∈ A(Z)◦,

◦ ◦ and a counit (a) = a(1A(Z)). In particular, A(Z) includes the evaluation elements ◦ δz such that δz( f ) = (σ ( f ))(z). Given an evaluation element δz, elements u ∈ A(Z) aresaidtobeprimitive relative to δz if they obey a relation

◦ Δ (u) = u ⊗ δz + δz ⊗ u.

These elements are derivations of A(Z) at z, i.e.,

[u][ f ] u( ff ) = u( f )δz( f ) + (−1) δz( f )u( f ).

© Atlantis Press and the author(s) 2016 183 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_9 184 9 SUSY Gauge Theory on Principal Graded Bundles

Deﬁnition 9.1 A graded Lie group (G, G ) is deﬁned as a graded manifold whose body G is a real Lie group, and the structure ring G (G) is a graded Hopf algebra (Δ, , S) such that a body epimorphism σ : G (G) → C∞(G) is a Hopf algebra morphism where a ring C∞(G) of smooth real functions on a Lie group G possesses a real Hopf algebra structure with the co-operations:

Δ( f )(g, g) = f (gg), ( f ) = f (1), S( f )(g) = f (g−1), f ∈ C∞(G).

One can show that the ﬁnite dual G (G)◦ of G (G) is equipped with the structure of a real Hopf algebra with the multiplication law

a b = (a ⊗ b) ◦ Δ, a, b ∈ G (G)◦. (9.1)

With respect to this multiplication, the evaluation elements δg, g ∈ G, provided with the product δg δg = δgg (9.1), constitute a group isomorphic to G. They are called the group-like elements. It is readily observed that the set of primitive elements of ◦ G (G) relative to δ1 is a real Lie superalgebra g with respect to the bracket

[u, u]=u u − (−1)[u][u ]u u.

It is called the Lie superalgebra of a graded Lie group (G, G ). Its even part g0 is a Lie algebra of a Lie group G. One says that a graded Lie group (G, G ) acts on a graded manifold (Z, A) on the right if there exists a morphism of graded manifolds

(ϕ, Φ) : (Z, A) × (G, G ) → (Z, A) such that the corresponding ring morphism

Φ : A(Z) → A(Z) ⊗ G (G) deﬁnes a structure of a right G (G)-comodule on A(Z), i.e.,

(Id ⊗ Δ) ◦ Φ = (Φ ⊗ Id ) ◦ Φ, (Id ⊗ ) ◦ Φ = Id .

For the right action (ϕ, Φ) and for each element a ∈ G (G)◦, one can introduce a linear map Φa = (Id ⊗ a) ◦ Φ : A(Z) → A(Z). (9.2)

In particular, if a is a primitive element with respect to δe, then Φa ∈ dA(Z). Let us consider the right action of a graded Lie group (G, G ) on itself. If Φ = Δ and a = δg is a group-like element, then

rg = (Id ⊗ δg) ◦ Δ (9.3) 9 SUSY Gauge Theory on Principal Graded Bundles 185 is an even graded algebra isomorphism which corresponds to the right translation G → Gg. Similarly, the left action

lg = (δg ⊗ Id ) ◦ Δ (9.4) is deﬁned. If a ∈ g, then la is a derivation of G (G). Given a basis {ui } for g,the Φ dG ( ) dG ( ) G ( ) derivations ui constitute the global basis for G , i.e., G is a free left G - module. In particular, there is a decomposition

G (G) = G (G) ⊕R G (G), G (G) ={f ∈ G (G) : Φu( f ) = 0, u ∈ g0}, G (G) ={f ∈ G (G) : Φu( f ) = 0, u ∈ g1}.

Since G (G) =∼ C∞(G), one can show the following [1, 21].

Theorem 9.1 A graded Lie group (G, G ) is a simple graded manifold modelled over a trivial bundle ×∧g∗ → , G 1 G (9.5) endowed with the right action rg,g∈ G, (9.3) of a Lie group G.

Let us turn now to the notion of a principal graded bundle. Following Sect. 6.3, we however restrict our consideration to graded bundles over a smooth manifold X. The right action (ϕ, Φ) of a graded Lie group (G, G ) on a graded manifold (Z, A) is called free if, for each z ∈ Z, the morphism Φz : A(Z) → G (G) is such that the ◦ ◦ dual morphism Φz∗ : G (G) → A(Z) is injective. A right action (ϕ, Φ) of (G, G ) on (Z, A) is said to be regular if the morphism

(ϕ × ) ◦ Δ : ( , A) × ( , G ) → ( , A) × ( , A) pr 1 Z G Z Z deﬁnes a closed graded submanifold of (Z, A) × (Z, A).

Remark 9.1 Let us note that (Z , A) is said to be a graded submanifold of (Z, A) if there exists a morphism (Z , A) → (Z, A) such that the corresponding morphism A(Z )◦ → A(Z)◦ is an inclusion. A graded submanifold is called closed if dim (Z , A)

Then we come to the following variant of the well-known theorem on the quotient of a graded manifold [1, 140].

Theorem 9.2 A right action (ϕ, Φ) of (G, G ) on (Z, A) is regular if and only if the quotient (Z/G, A/G ) is a graded manifold, i.e., there exists an epimorphism of graded manifolds (Z, A) → (Z/G, A/G ) compatible with a surjection Z → Z/G.

In view of this theorem, a principal graded bundle (P, A) can be deﬁned as a locally trivial submersion (P, A) → (P/G, A/G ) with respect to the right regular 186 9 SUSY Gauge Theory on Principal Graded Bundles free action of (G, G ) on (P, A). In an equivalent way, one can say that a principal graded bundle is a graded manifold (P, A) together with a free right action of a graded Lie group (G, G ) on (P, A) such that the quotient (P/G, A/G ) is a graded manifold and the natural surjection (P, A) → (P/G, A/G ) is a submersion. Obviously, P → P/G is a familiar principal bundle with a structure group G. As was mentioned above, we restrict our consideration to the case of a principal ( = / , A/G = ∞) graded bundle whose base a trivial graded manifold X P G CX . i.e., principal graded bundles over a smooth manifold X.

Deﬁnition 9.2 A principal graded bundle is deﬁned to be as a simple graded manifold (P, AF ) whose body is a principal bundle P → X with a structure group G, and which is modelled over a composite bundle

F = P × W → P → X (9.6) X where W → X is a P-associated bundle (Remark 8.1) with a typical ﬁbre g1 provided with the left action (9.4) of a group G. A principal graded bundle (P, AF ) is subject to the right ﬁbrewise action (9.2) of a group G.

Remark 9.2 Being kept as a simple graded manifold, a graded bundle (P, AF ) fails to admit an action of a graded Lie group (G, G ). Therefore, it is not a principal graded bundle in a strict sense.

Turn now to Yang–Mills SUSY gauge theory. Let g be a ﬁnite-dimensional real Lie superalgebra with a basis {εr }, r = 1,...,m, r and real structure constants cij such that

r =−(− )[i][ j] r , [ ]=[]+[ ], cij 1 c ji r i j (− )[i][b] r j + (− )[a][i] r j + (− )[b][a] r j = , 1 cijcab 1 cajcbi 1 cbjcia 0 where [r] denotes the Grassmann parity of εr . Its even part g0 is a Lie algebra of some Lie group G. Let the adjoint representation of g0 in g is extended to the corresponding action of G on g. Let P be a principal bundle with a structure group G. Given Yang–Mills gauge theory of principal connections on P (Sect.8.3), we aim to extend it to Yang–Mills SUSY gauge theory associated to a Lie superalgebra g. Let us consider a simple graded manifold (G, G ) modelled over the trivial vector bundle (9.5). In accordance with Theorem 9.1, this is a graded Lie group whose Lie superalgebra is g. Following Definition 9.2, let us define a principal graded bundle (P, AF ) modelled over the composite bundle (9.6). Let J 1 F be a jet manifold of the fibre bundle F → X (9.6). It is a composite bundle J 1 F → J 1 P → X, (9.7) 9 SUSY Gauge Theory on Principal Graded Bundles 187

1 1 1 where J F → J P is a vector bundle. Then a simple graded manifold (J P, AJ 1 F ) modelled over J 1 F → J 1 P is a graded jet manifold of a principal graded bundle (P, AF ) in accordance with Deﬁnition 6.7. The composite bundle (9.7) is provided with the jet prolongation of the right action (9.2)ofG onto (P, AF ). By analogy with the bundle of principal connections C (8.21), let us consider the quotient

C = J 1 F/G → C → X.

Since C → C is a vector bundle, we have a simple graded manifold (C, AC ) modelled over a vector bundle C → C. It is a graded bundle over a manifold X. We develop SUSY gauge theory as a Grassmann-graded Lagrangian theory on a graded bundle (C, AC ) (see Sect. 6.5). Given a basis {εr } for g, we have a local basis λ r r (x , aλ) of Grassmann parity [aλ]=[r] for a graded bundle (C, AC ). Similarly to r the splitting (8.39), jets of the elements aλ admit the decomposition

r 1 r r aλμ = (Fλμ + Sλμ) = 2 1 r r r i j 1 r r r i j (aλμ − aμλ + c aλaμ) + (aλμ + aμλ − c aλaμ). 2 ij 2 ij

∗ Let us consider the DBGA S∞[C ; C] (6.39). Given the universal enveloping ij algebra g of g, we assume that there exists an even quadratic Casimir element h εi ε j of g such that the matrix hij is nondegenerate. Let g be a world metric on X. Then a graded Yang–Mills Lagrangian is deﬁned as

1 λμ βν i j L = h g g Fλβ Fμν ω. YM 4 ij E λ Its variational derivatives r obey the irreducible complete NI

r i E λ + E λ = . c jiaλ r dλ j 0

∗ Therefore, let us enlarge the differential bigraded algebra S∞[C ; C] to the differential bigraded algebra

∗ ∗ P∞{0}=S∞[C ⊕ C ⊕ VF/G ⊕ VF/G; P], X X X where E denotes the density dual (7.1) of a fibre bundle E → X. Its local basis ( λ, r , λ, r , ) r λ x aλ ar c cr contains gauge fields aλ, their antifields ar of Grassmann parity [ λ]=([ ]+ ) r [ r ]=([ ]+ ) ar r 1 mod 2, the ghosts c of Grassmann parity c r 1 mod 2, and the Noether antifields cr of Grassmann parity [cr ]=[r]. Then the gauge operator (7.44) reads ∂ = ( r − r j i ) u cλ c jic aλ r (9.8) ∂aλ 188 9 SUSY Gauge Theory on Principal Graded Bundles

(cf. (8.88)). It admits the nilpotent BRST extension

∂ 1 ∂ b = (cr − cr c j ai ) − (− )[i]cr ci c j . λ ji λ r 1 ij r ∂aλ 2 ∂c

The corresponding proper solution (7.86) of the master equation takes a form

r r j i λ 1 [i] r i j L = L + (cλ − c c aλ)a ω − (−1) c c c c ω. E YM ji r 2 ij r Chapter 10 Gauge Gravitation Theory on Natural Bundles

Gravitation theory in the absence of matter fields can be formulated as gauge field theory on natural bundles over an oriented four-dimensional manifold X [61, 124, 130]. It is metric-affine gravitation theory whose dynamic variables are linear world connections and pseudo-Riemannian world metrics on X (Sect.10.3). Its first order Lagrangians are invariant under general covariant transformations. Infinitesimal generators of local one-parameter groups of these transformations are the functorial lift (Definition B.6) of vector fields on X onto a natural bundle. They are exact gauge symmetries whose gauge parameters are vector fields on X. By virtue of Noether’s first Theorem2.7, the corresponding conserved symmetry current is the energy-momentum current (10.32) which is reduced to the generalized Komar superpotential (10.37) in accordance with Noether’s third Theorem 2.8. Throughout this chapter, by X is meant an oriented simply connected four- dimensional manifold, called a world manifold.

10.1 Relativity Principle: Natural Bundles

Gauge gravitation theory has been vigorously developed since 1960s in different variants which in common lead to above-mentioned metric-affine gravitation theory (Sect. 10.3). We refer the reader to [17, 71, 77, 124] for the history and the references on the subject. It seems reasonable to believe that gauge gravitation theory must incorporates Einstein’s General Relativity and, therefore, it should be based on Relativity and Equivalence Principles reformulated in fibre bundle terms [77, 116]. Relativity Prin- ciple states that gauge symmetries of classical gravitation theory are general covariant transformations. Fibre bundles possessing general covariant transformations constitute the category of so called natural bundles [83, 145]. Let π : Y → X be a smooth fibre bundle. Any automorphism (Φ, f ) of Y ,by definition, is projected as π ◦ Φ = f ◦ π onto a diffeomorphism f of its base X. The converse need not be true. A diffeomorphism of X need not give rise to an © Atlantis Press and the author(s) 2016 189 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_10 190 10 Gauge Gravitation Theory on Natural Bundles automorphism of Y , unless Y → X is a trivial bundle. Given a local one-parameter group (Φt , ft ) of local automorphisms of Y , its infinitesimal generator is a projectable vector field u on Y . This vector field is projected as τ ◦ π = T π ◦ u onto a vector field τ on X. Its flow is the one-parameter group ( ft ) of diffeomorphisms of X which are projections of automorphisms (Φt , ft ) of Y . Conversely, let τ be a vector field on X. There is exists its lift γτ onto Y (Definition B.5), e.g., by means of a connection on Y → X, but this lift need not be functorial (Definition B.6). A fibre bundle Y → X is called the natural bundle if there exists the functorial lift of vector fields on X onto Y (Definition B.7). One treats this lift as an infinitesimal general covariant transformation of Y . Its flow is a local one-parameter group of local general covariant transformations. Following Relativity Principle, one thus should develop gauge gravitation theory as a field theory on natural bundles [61, 130]. Remark 10.1 In general, there exist diffeomorphisms of X which do not belong to any one-parameter group of diffeomorphisms of X. In a general setting, one therefore considers a monomorphism f → f of the group of diffeomorphisms of X to the group of bundle automorphisms of a natural bundle T → X. Automorphisms f are called general covariant transformations of T . No vertical automorphism of T , unless it is the identity morphism, is a general covariant transformation. The group of automorphisms of a natural bundle is a semi-direct product of its subgroup of vertical automorphisms and the subgroup of general covariant transformations. Tensor bundles, including the tangent and cotangent bundles of X, admit the functorial lifts τ (B.29) and (B.30) of vector fields τ on X and, consequently, they exemplify natural bundles. Remark 10.2 Any diffeomorphism f of X gives rise to the tangent automorphisms f = Tf of TX which is a general covariant transformation of TX as a natural bundle. Tensor bundles over a world manifold X have the structure group

+ GL4 = GL (4, R). (10.1)

The associated principal bundle is a ﬁbre bundle

πLX : LX → X (10.2) of oriented linear frames in tangent spaces to a world manifold X. It is called the frame bundle. Its sections are termed the frame ﬁelds. Given holonomic frames {∂μ} in the tangent bundle TX associated to the holonomic μ atlas ΨT (B.15), every element {Ha} of a frame bundle LX takes a form Ha = Ha ∂μ, μ 4 where Ha is a matrix of the natural representation of a group GL4 in R . These matrices constitute bundle coordinates

∂xμ (xλ, H μ), H μ = H λ, a a ∂xλ a 10.1 Relativity Principle: Natural Bundles 191 on LX associated to its holonomic atlas

ΨT ={(Uι, zι ={∂μ})} (10.3) given by the local frame ﬁelds zι ={∂μ}. With respect to these coordinates, the right action (8.4)ofGL4 on LX reads

: μ → μ b , ∈ . Rg Ha Hb g a g GL4

A frame bundle LX is equipped with the canonical R4-valued one-form

a μ θLX = Hμdx ⊗ ta, (10.4)

4 a μ where {ta} is a ﬁxed basis for R and Hμ is the inverse matrix of Ha . λ A frame bundle LX → X is a natural bundles. Any vector ﬁeld τ = τ ∂λ on X admits the functorial lift

μ α ν ∂ τ = τ ∂μ + ∂ν τ H a ∂ α Ha onto LX deﬁned by the condition Lτ θLX = 0. Moreover, all ﬁbre bundles associated to the frame bundle LX (10.2) are natural bundles. However, there are natural bundles which are not associated to LX.

10.2 Equivalence Principle: Lorentz Reduced Structure

Gauge field theory deals with the three types of classical fields [61, 127]. These are gauge potentials, matter fields and Higgs fields. Higgs fields are responsible for spontaneous symmetry breaking. In classical gauge theory on a principal bundle P → X, spontaneous symmetry breaking is characterized by the reduction of a structure Lie group G of this principal bundle to a closed subgroup H of exact symmetries [77, 80, 115, 146]. Let P (8.3) be a principal G-bundle, and let H,dimH > 0, be a closed (and, consequently, Lie) subgroup of G. Then there is a composite bundle

P → P/H → X, where

πPΣ PΣ = P −→ P/H (10.5) is a principal bundle with a structure group H and

π Σ = P/H = (P × G/H)/G −→Σ X X (10.6) 192 10 Gauge Gravitation Theory on Natural Bundles is a P-associated bundle with a structure group G and a typical ﬁbre G/H on which G acts on the left. One says that a structure Lie group G of the principal bundle P → X (8.3)is reduced to its closed subgroup H if the following equivalent conditions hold [61, 82, 124, 133].

• A principal bundle P admits the bundle atlas ΨP (8.6) with H-valued transition functions ραβ . • There exists a reduced principal subbundle PH of P with a structure group H.

Remark 10.3 It is easily justiﬁed that these conditions are equivalent. If PH ⊂ P is a reduced subbundle, its atlas (8.8) given by local sections zα of PH → X is a desired atlas of P. Conversely, let ΨP ={(Uα, zα), ραβ },beanatlasofP with H-valued transition functions ραβ . For any x ∈ Uα ⊂ X, let us deﬁne a submanifold zα(x)H ⊂ Px . These submanifolds form a desired H-subbundle of P because zα(x)H = zβ (x)Hρβα(x) on the overlaps Uα ∩ Uβ .

Theorem 10.1 There is one-to-one correspondence

h = π −1 ( ( )) P PΣ h X (10.7)

h between the reduced principal H-subbundles ih : P → P of P and the global sections h of the quotient bundle P/H → X(10.6) [82].

A reduced principal H-subbundle of a principal G-bundle is called the reduced H-structure. A glance at the formula (10.7) shows that a reduced principal H-bundle h ∗ P is the restriction h PΣ (B.12) of the principal H-bundle PΣ (10.5)toh(X) ⊂ Σ. In classical field theory, global sections of the quotient bundle P/H → X (10.6) are interpreted as Higgs fields [61, 115, 125]. In gravitation theory on natural bundles, Equivalence Principle reformulated in geometric terms requires that the structure group GL4 (10.1) of the frame bundle LX (10.2) is reducible to a Lorentz subgroup SO(1, 3) [77, 116, 124]. It means that these fibre bundles admit atlases with SO(1, 3)-valued transition functions or, equivalently, that there exist reduced principal subbundles of LX with a Lorentz structure group. A reduced principle SO(1, 3)-subbundle (or, shortly,a Lorentz subbundle)ofa frame bundle LX is called the reduced Lorentz structure.

Remark 10.4 There is the well-known topological obstruction to the existence of a Lorentz reduced structure on a world manifold X. All noncompact manifolds and compact manifolds whose Euler characteristic equals zero admit a reduced Lorentz structure [33].

By virtue of Theorem10.1, there is one-to-one correspondence between the reduced Lorentz subbundles L g X of a frame bundle LX and the global sections g of the quotient bundle ΣPR = LX/SO(1, 3). (10.8) 10.2 Equivalence Principle: Lorentz Reduced Structure 193

Its typical ﬁbre is Gl4/SO(1, 3), and its global sections are pseudo-Riemannian world metrics g on X. Let us call ΣPR (10.8)themetric bundle. For the sake of convenience, one usually identiﬁes the metric bundle (10.8) with an open subbundle of the tensor bundle 2 ΣPR ⊂ ∨ TX. (10.9)

Therefore, the metric bundle ΣPR (10.8) can be equipped with the bundle coordinates (xλ,σμν ). In General Relativity, a pseudo-Riemannian world metric g describes a gravitational ﬁeld. Hereafter, by a metric g is meant just a global section of the metric bundle (10.8). Every metric g deﬁnes an associated Lorentz bundle atlas

g g Ψ ={(Uι, zι ={ha})} (10.10)

g of a frame bundle LX such that the corresponding local sections zι of LX take their values into the Lorentz subbundle L g X and the transition functions of Ψ g (10.10) between the frames {ha} are SO(1, 3)-valued. The frames (10.10):

{ = μ( )∂ }, μ = μ ◦ g, ∈ , ha ha x μ ha Ha zι x Uι (10.11) are called the tetrad frames. Certainly, a Lorentz bundle atlas Ψ g is not unique. Given a Lorentz bundle atlas Ψ g, the pull-back

a h∗ a λ h = h ⊗ ta = zι θLX = hλ(x)dx ⊗ ta (10.12)

g of the canonical form θLX (10.4) by a local section zι is called the (local) tetrad form. The tetrad form (10.12) determines the tetrad coframes

a a μ {h = hμ(x)dx }, x ∈ Uι, (10.13) in the cotangent bundle T ∗ X. They are the dual of the tetrad frames (10.11). The μ a coefﬁcients ha and hμ of the tetrad frames (10.11) and coframes (10.13) are called the tetrad functions. They are transition functions between the holonomic atlas ΨT (10.3) and the Lorentz atlas Ψ g (10.10) of a frame bundle LX. With respect to the Lorentz atlas Ψ h (10.10), a tetrad ﬁeld h can be represented by the R4-valued tetrad form (10.12). Relative to this atlas, a pseudo-Riemannian world metric g takes the well-known form

a b a b g = η(h ⊗ h) = ηabh ⊗ h , gμν = hμhν ηab, where η is the Minkowski metric of signature (+, −−−) in R4. 194 10 Gauge Gravitation Theory on Natural Bundles

10.3 Metric-Afﬁne Gauge Gravitation Theory

Based on Relativity and Equivalence Principles formulated in geometric terms (Sects.10.1 and 10.2), gravitation theory can be developed as gauge field theory on natural bundles provided with reduced Lorentz structures [61, 77, 116, 130]. It is metric-affine gravitation theory whose Lagrangian LMA is invariant under general covariant transformations. In the absence of matter fields, its dynamic variables are world connections Γ and pseudo-Riemannian metrics g on a world manifold X. Since the tangent bundle TX is associated to a frame bundle LX, every world connection Γ (B.70): λ μ ν ˙ Γ = dx ⊗ (∂λ + Γλ ν x˙ ∂μ), (10.14) on TX is associated to a principal connection on LX. It follows that world connections are represented by sections of the quotient bundle

1 CW = J LX/GL4, (10.15) called the bundle of world connections. With respect to the holonomic atlas ΨT (10.3), the bundle of world connections CW (10.15) is provided with coordinates ν β β 2 ν μ λ ν ν ∂x ∂x γ ∂x ∂ x ∂x (x , kλ α), k α = kμ β + , λ ∂xγ ∂xα ∂xα ∂xμ∂xβ ∂xλ

ν ν so that, for any section Γ of CW → X, its coordinates kλ α ◦ Γ = Γλ α are components of the linear connection Γ (10.14). Though the bundle of world connections CW → X (10.15) is not LX-associated, it is a natural bundle. It admits the functorial lift ∂ τ = τ μ∂ +[∂ τ α ν − ∂ τ ν α − ∂ τ ν α + ∂ τ α] C μ ν kμ β β kμ ν μ kν β μβ α (10.16) ∂kμ β of vector ﬁelds τ on X [100]. 1 The ﬁrst order jet manifold J CW of the bundle of world connections admits the canonical splitting (8.38). In order to obtain its coordinate expression, let us consider the strength (8.29) of the linear world connection Γ (10.14). It reads

1 b a λ μ 1 α λ μ FΓ = Fλμ I dx ∧ dx = Rμν β dx ∧ dx , 2 a b 2 ( a)α = α a where Ib β Hb Hβ are generators of the group GL4 (10.1) in ﬁbres of TX with respect to the holonomic frames, and

α α α γ α γ α Rλμ β = ∂λΓμ β − ∂μΓλ β + Γλ β Γμ γ − Γμ β Γλ γ 10.3 Metric-Afﬁne Gauge Gravitation Theory 195 are components of the curvature (B.72) of a world connection Γ . Accordingly, the 1 above mentioned canonical splitting (8.38)ofJ CW can be written in a form

α 1 α α kλμ β = (Rλμ β + Sλμ β ) = (10.17) 2 1 α α γ α γ α (kλμ β − kμλ β + kλ β kμ γ − kμ β kλ γ ) + 2 1 α α γ α γ α (kλμ β + kμλ β − kλ β kμ γ + kμ β kλ γ ). 2

It is readily observed that, if Γ is a section of CW → X, then

α 1 α Rλμ β ◦ J Γ = Rλμ β .

Due the canonical vertical splitting (B.41) of the vertical tangent bundle VTX of TX, the curvature (B.72) of a world connection Γ can be represented by a tangent- valued two-form

1 α β λ μ R = Rλμ β x˙ dx ∧ dx ⊗ ∂α 2 on TX. Then the Ricci tensor

1 λ μ β R = Rλμ β dx ⊗ dx c 2 of a world connection Γ is deﬁned. Owing to the above mentioned vertical splitting (B.41) of VTX, the torsion form T (B.73) of Γ deﬁnes the tangent-valued torsion two-form

1 ν λ μ ν ν ν T = Tμ λdx ∧ dx ⊗ ∂ν , Tμ λ = Γμ λ − Γλ μ. (10.18) T 2 Remark 10.5 Given a pseudo-Riemannian metric g, every world connection Γ (10.14) admits a decomposition

1 Γμνα ={μνα}+Sμνα + Cμνα 2 in the Christoffel symbols {μνα} (B.74), the nonmetricity tensor

Γ Cμνα = Cμαν =∇μ gνα = ∂μgνα + Γμνα + Γμαν and the contorsion 1 Sμνα =−Sμαν = (Tνμα + Tναμ + Tμνα + Cανμ − Cναμ), 2 196 10 Gauge Gravitation Theory on Natural Bundles where Tμνα =−Tανμ are coefﬁcients of the torsion form (10.18)ofΓ .Aworld connection Γ is called the metric connection for a pseudo-Riemannian world metric g if the covariant differential ∇Γ g (B.62) of a metric g with respect to a connection Γ Γ vanishes, i.e., metricity condition ∇μ gνα = 0 holds. A metric connection reads

1 Γμνα ={μνα}+ (Tνμα + Tναμ + Tμνα). 2 For instance, the Levi–Civita connection (B.74) is a symmetric metric connection.

As was mentioned above, world connections are represented by sections of the bundle of world connections CW (10.15), whereas pseudo-Riemannian world metrics are described by sections of the open subbundle (10.9). Therefore, a conﬁguration space of metric-afﬁne gauge theory is a bundle product

Y = ΣPR × CW (10.19) X

λ μν α coordinated by (x ,σ , kμ β ). Its ﬁrst order jet manifold is

1 1 1 J Y = J ΣPR × J CW, (10.20) X

1 where J CW admits the canonical splitting (8.38) given by the coordinate expression (10.17). Let us consider the DGA (1.7):

∗ ∗ S∞[Y ]=O∞Y, (10.21)

αβ α possessing the local generating basis (σ , kμ β ). A Lagrangian of metric-afﬁne gauge gravitation theory is a ﬁrst order Lagrangian

αβ α αβ α LMA = LMA(σ , kμ β ,σλ , kλμ β )ω (10.22) on the jet manifold (10.20). Its Euler–Lagrange operator reads

αβ μ β α δLMA = (Eαβ dσ + E α dkμ β ) ∧ ω. (10.23)

The fibre bundle (10.19) is a natural bundle admitting the functorial lift ∂ τ = τ μ∂ + (σ νβ ∂ τ α + σ αν∂ τ β ) + ΣC μ ν ν ∂σαβ (10.24) ∂ (∂ τ α ν − ∂ τ ν α − ∂ τ ν α + ∂ τ α) ν kμ β β kμ ν μ kν β μβ α ∂kμ β 10.3 Metric-Affine Gauge Gravitation Theory 197 of vector fields τ on X (cf. (10.16)). Following Definition2.3, one can treat vector fields τΣC (10.24) as infinitesimal gauge transformations whose gauge parameters are vector fields τ on X. Lagrangians LMA of metric-affine gauge gravitation theory are assumed to be invariant under general covariant transformations. This means that infinitesimal gauge transformations τΣC (10.24) are exact symmetries of a Lagrangian LMA.By analogy with Theorem8.2, one can show that, if the first order Lagrangian LMA αβ (10.22) does not depend on jet coordinates σλ and if it possesses exact gauge symmetries (10.24), it factorizes as

αβ α α LMA = LMA(σ , tν σ , Rλμ β )ω (10.25)

α through the terms Rλμ β (10.17) and the torsion terms

α α α tν σ = kν σ − kσ ν . (10.26)

. In particular, a Hilbert–Einstein Lagrangian in metric-afﬁne gauge gravitation theory (see Remark10.7) reads √ 1 λν α L = R −σω, R = σ Rαλ ν ,σ= det(σαβ ). (10.27) HE 2κ

10.4 Energy-Momentum Gauge Conservation Law

Since infinitesimal general covariant transformations τΣC (10.24) are assumed to be exact gauge symmetries of a metric-affine gravitation Lagrangian, let us study the corresponding conservation law. This is an energy-momentum conservation law (Definition3.3) because the vector fields τΣC are the lift (B.27) of vector fields on X [52, 120, 130]. There are several approaches to discover an energy-momentum conservation law in gravitation theory. Here we treat this conservation law as a particular gauge conservation law . Accordingly, the energy-momentum of gravity is seen as a particular symmetry current (see, e.g., [5, 19, 71, 74]). Since infinitesimal general covariant transformations τΣC (10.24) are gauge symmetries, the corresponding energy-momentum current reduces to a superpotential (Theorem2.8). Let us assume that the metric-affine gravitation Lagrangian LMA (10.25) is independent of torsion terms (10.26). Then the following relations take place: ∂L π λν β =−π νλ β ,πλν β = MA , α α α α (10.28) ∂kλν β ∂L MA = π λν σ β − π λν β σ . α α kλ σ σ kλ α (10.29) ∂kν β 198 10 Gauge Gravitation Theory on Natural Bundles

Let us introduce the compact notation

A α α εσ ε σ α α ε ε α α ε α ε y = kμ β , uμ β γ = δμδβ δγ , uμ β γ = kμ β δγ − kμ γ δβ − kγ β δμ.

Then the vector ﬁelds (10.24) take a form

λ νβ α αν β Aβ α Aβμ α τΣC = τ ∂λ + (σ ∂ν τ + σ ∂ν τ )∂αβ + (u α ∂β τ + u α ∂βμτ )∂A.

We also have the equalities

π λ Aβμ = π λμ β ,πε Aβ =−∂ε β L − π εβ γ σ . Au α α Au α α MA σ kα γ

1 = Since LJ τΣC LMA 0, the ﬁrst variational formula (3.26) takes a form

νβ α αν β λ αβ 0 = (σ ∂ν τ + σ ∂ν τ − τ σλ )δαβ LMA + (10.30) Aβ α Aβμ α λ A (u α ∂β τ + u α ∂βμτ − τ yλ )δALMA − [π λ( Aτ α − Aβ ∂ τ α − Aεβ ∂ τ α) − τ λL ]. dλ A yα u α β u α εβ MA

The ﬁrst variational formula (10.30) on-shell leads to a weak conservation law

≈− [π λ( Aτ α − Aβ ∂ τ α − Aεβ ∂ τ α) − τ λL ] 0 dλ A yα u α β u α εβ MA (10.31) of the energy-momentum current of metric-afﬁne gravity

J λ = π λ( Aτ α − Aβ ∂ τ α − Aεβ ∂ τ α) − τ λL . MA A yα u α β u α εβ MA (10.32)

Due to the arbitrariness of gauge parameters τ λ, the ﬁrst variational formula (10.30) falls into the set of equalities (4.52)–(4.55) which read

(λε σ) π γ = 0, (10.33) ( Aεσ ∂ + Aε ∂σ )L = , u γ A u γ A MA 0 (10.34) δβ L + ( σ βμδ + Aβ δ )L + (π μ Aβ ) − Aπ β = , α MA 2 αμ u α A MA dμ A u α yα A 0 (10.35) ∂λLMA = 0.

It is readily observed that the equalities (10.33) and (10.34) hold due to the relations Aπ β (10.28) and (10.29), respectively. Substituting the term yα A from the expression (10.35) in the energy-momentum conservation law (10.31), one brings this conservation law into a form

≈− [ σ λμτ αδ L + Aλ τ αδ L − π λ Aβ ∂ τ α + 0 dλ 2 αμ MA u α A MA Au α β (10.36) (π λμ β )∂ τ α + (π μ Aλ )τ α − (π λμ β ∂ τ α)]. dμ α β dμ A u α dμ α β 10.4 Energy-Momentum Gauge Conservation Law 199

After separating a variational derivatives, the energy-momentum conservation law (10.36) of metric-afﬁne gravity takes the following superpotential form [52, 120]:

λμ α 0 ≈−dλ[2σ τ δαμLMA + λ μ γ σ μ λ σ λ γ α (kμ γ δ α LMA − kμ αδ σ LMA − kα γ δ σ LMA)τ + λ μ α μ λ α μλ ν α α σ δ α LMA∂μτ − dμ(δ α LMA)τ + dμ(π α (∂ν τ − kσ ντ ))], where the energy-momentum current on-shell reduces to the generalized Komar superpotential μλ μλ ν α α σ UMA = π α (∂ν τ − kσ ν τ ). (10.37)

We can rewrite this superpotential as

∂L μλ = MA ( τ α + α τ σ ), UMA 2 α Dν tν σ ∂Rμλ ν

α α where Dν is the covariant derivative relative to the connection kν σ and tν σ is its torsion (10.26).

Remark 10.6 Let us consider the Hilbert–Einstein Lagrangian (10.27) of metric- afﬁne gauge gravitation theory. Then the generalized Komar superpotential (10.37) comes to the well known Komar superpotential if we substitute the Levi–Civita α α connection kν σ ={ν σ } (10.40).

10.5 BRST Gravitation Theory

TX × Y = TX × ΣPR × CW, X X X

∗ ∗ and let us enlarge the DGA S∞[Y ] (10.21)toaDBGAP∞[TX; Y ] possessing the αβ α μ αβ α μ local generating basis (σ , kμ β , c ) of even ﬁelds (σ , kμ β ) and odd ghosts (c ). Taking the vertical part of vector ﬁelds τΣC (10.24) and replacing gauge parameters τ λ with ghosts cλ, we obtain the odd vertical graded derivation ∂ ∂ = αβ + α = u u αβ uμ β α (10.38) ∂σ ∂kμ β αβ ∂ (σ νβ cα + σ ανcβ − cλσ ) + ν ν λ ∂σαβ 200 10 Gauge Gravitation Theory on Natural Bundles ∂ ( α ν − ν α − ν α + α − λ α ) . cν kμ β cβ kμ ν cμkν β cμβ c kλμ β α ∂kμ β

Since it is SUSY of a metric-afﬁne Lagrangian LMA, its Euler–Lagrange operator δLMA (10.23) obeys complete NI

αβ μβ α μ β −σλ Eαβ − 2dμ(σ Eλβ − kλμ β E α − (10.39) μ α α μ α μ ν β μ β dμ[(kν β δλ − kν λδβ − kλ β δν )E α ]+dμβ E λ = 0.

Remark 10.7 Let us note that the Hilbert–Einstein Lagrangian LHE of General Rela- tivity depends only on metric variables σ αβ . It is a reduced second order Lagrangian which differs from the first order one LHE in a variationally trivial term. The infinitesimal gauge covariant transformations τΣC (10.24) are variational (but not exact) symmetries of the first order Lagrangian LHE, and the graded derivation u (10.38)is so. It reads

αβ ∂ u = (σ νβ cα + σ ανcβ − cλσ ) . ν ν λ ∂σαβ

Then the corresponding NI (10.39) take a familiar form

μ β μ ∇μEλ = (dμ +{μ λ})Eβ = 0,

μ μα where Eλ = σ Eαλ and

β 1 βν {μ λ}=− σ (dμσνλ + dλσμν − dν σμλ) (10.40) 2 μν are the Christoffel symbols expressed into function σαβ of σ given by the relations μα μ σ σαβ = δβ .

The NI (10.39) are irreducible. Therefore, the graded derivation (10.38) is a gauge operator of gauge gravitation theory. It possesses a nilpotent BRST extension ∂ b = u + cλ cμ . μ ∂cλ

Accordingly, an original gravitation Lagrangian LMA admits a BRST extension to a proper solution of the master equation which reads

αβ α μ β λ μ L E = LMA + u σ αβ ω + uμ β k α ω + cμc cλω,

μ β where σ αβ , k α and cλ are the corresponding antiﬁelds [8, 61]. Chapter 11 Chern–Simons Topological Field Theory

In classical field theory, there are Lagrangian field models whose Lagrangians are independent of a world metric on a base X. These are topological field theories of Schwartz type [16]. Here we address Chern–Simons topological field theory on a principal bundle. In comparison with Yang–Mills gauge theory in Chap. 8, it possesses principal gauge symmetries which are neither vertical nor exact (Remark 8.8). The corresponding conservation laws (11.11) and (11.12) involve Noether currents and the energy-momentum ones. Note that one usually considers the local Chern–Simons Lagrangian which is the local Chern–Simons form derived from the local transgression formula for the Chern characteristic form. The global Chern–Simons Lagrangian is well defined, but depends on a background gauge potential [20, 41, 57, 61]. Given a principal bundle P → X with a structure Lie group G,letC → X be the bundle of principal connections (8.21), A the canonical principal connection (8.34) on the principal G-bundle PC (8.30), and FA (8.35) its strength. Let

(χ) = χr1 ...χrk Ik br1...rk (11.1) be a G-invariant polynomial of degree k > 1 on the right Lie algebra gr of G. With FA (8.35), one can associate to this polynomial Ik the closed 2k-form

( ) = r1 ∧···∧ rk , ≤ , P2k FA br1...rk FA FA 2k n (11.2) on C which is invariant under automorphisms of C induced by vertical principal automorphisms of P. Given a section A of C → X, the pull-back

∗ P2k (FA) = A P2k (FA ) (11.3)

© Atlantis Press and the author(s) 2016 201 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0_11 202 11 Chern–Simons Topological Field Theory of the form P2k (FA ) (11.2) is a closed 2k-form on X where FA is the strength (8.29) of a principal connection A. It is called the characteristic form. The characteristic forms (11.3) possess the following important properties [37, 82, 102].

• Every characteristic form P2k (FA) (11.3) is a closed form, i.e., dP2k (FA) = 0; • The difference P2k (FA)− P2k (FA ) of characteristic forms is an exact form, when- ever A and A are different principal connections on a principal bundle P.

It follows that characteristic forms P2k (FA) possess the same de Rham cohomology class [P2k (FA)] for all principal connections A on P. The association

(χ) →[ ( )]∈ ∗ ( ) Ik P2k FA HDR X is the well-known Weil homomorphism. The de Rham cohomology class [P2k (FA)] is a topological invariant. Choosing a certain family of characteristic forms (11.3), one therefore can obtain characteristic classes of a principal bundle P. Let Ik (11.1)beaG-invariant polynomial of degree k > 1 on the right Lie algebra gr of G.LetP2k (FA ) (11.2) be the corresponding closed 2k-form on C and P2k (FA) (11.3) its pullback onto X by means of a section A of C → X.Letthe same symbol P2k (FA) stand for its pull-back onto C. Since C → X is an afﬁne bundle and, consequently, the de Rham cohomology of C equals that of X,the exterior forms P2k (FA ) and P2k (FA) possess the same de Rham cohomology class [P2k (FA )]=[P2k (FA)] for any principal connection A. Consequently, the exterior forms P2k (FA ) and P2k (FA) on C differ from each other in an exact form

P2k (FA ) − P2k (FA) = dS2k−1(a, A). (11.4)

This relation is called the transgression formula on C [57, 61]. Its pull-back by means of a section B of C → X gives the transgression formula on a base X:

P2k (FB ) − P2k (FA) = dS2k−1(B, A).

For instance, let i c(FA ) = det 1 + FA = 1 + c (FA ) + c (FA ) +··· 2π 1 2 be the total Chern form on a bundle of principal connections C. Its components ck (FA ) are Chern characteristic forms on C.IfP2k (FA ) = ck (FA ) is the characteristic Chern 2k-form, then S2k−1(a, A) (11.4)isaChern–Simons (2k − 1)-form. In particular, one can choose a local section A = 0. In this case, S2k−1(a, 0) is called the local Chern–Simons form.LetS2k−1(A, 0) be its pull-back onto X by means of a section A of C → X. Then the Chern–Simons form S2k−1(a, A) (11.4) admits the decomposition

S2k−1(a, A) = S2k−1(a, 0) − S2k−1(A, 0) + dK2k−1. (11.5) 11 Chern–Simons Topological Field Theory 203

The transgression formula (11.4) also yields the transgression formula

h0(P2k (FA ) − P2k (FA)) = dH (h0S2k−1(a, A)), 1

h0S2k−1(a, A) = k P2k (t, A)dt, (11.6) 0 r r μ r r P (t, A) = b ... (a 1 − A 1 )dx 1 ∧ F 2 (t, A) ∧···∧F k (t, A), 2k r1 rk μ1 μ1 1 r j r j r j r j F (t, A) = [taλ μ + (1 − t)∂λ Aμ − taμ λ 2 j j j j j j

r j 1 r j p p q − (1 − t)∂μ Aλ + cpq(taλ + (1 − t)Aλ )(taμ j j 2 j j j λ μ + (1 − t)Aq ]dx j ∧ dx j ⊗ e , μ j r

1 on J C (where br1...rk are coefﬁcients of the invariant polynomial (11.1)). If 2k − 1 = dim X, the density (11.6) is the global Chern–Simons Lagrangian

LCS(A) = h0S2k−1(a, A) (11.7) of Chern–Simons topological ﬁeld theory. It depends on a background gauge ﬁeld A. The decomposition (11.5) induces the decomposition

LCS(A) = h0S2k−1(a, 0) − h0S2k−1(A, 0) + dH h0 K2k−1, where

LCS = h0S2k−1(a, 0) is the local Chern–Simons Lagrangian. For instance, if dim X = 3, the global Chern–Simons Lagrangian (11.7) reads ( ) = 1 εαβγ m (F n − 1 n p q ) ω LCS A hmn aα βγ cpqaβ aγ (11.8) 2 3 1 αβγ m n 1 n p q αβγ m n − h ε Aα (F βγ − c A Aγ ) ω − dα(h ε aβ Aγ )ω, 2 mn A 3 pq β mn where εαβγ is the skew-symmetric Levi–Civita tensor. Since the density

−S2k−1(A, 0) + dH h0 K2k−1 is variationally trivial, the global Chern–Simons Lagrangian (11.7) possesses the same NI and gauge symmetries as the local one (11.7). They are the following. 204 11 Chern–Simons Topological Field Theory

Infinitesimal generators of local one-parameter groups of principal automorphisms of a principal bundle P are equivariant vector fields on P. They are identified with sections ξ (8.16) of the vector bundle TG P → X (8.12), and yield the principal vector fields uξ (8.45) on the bundle of principal connections C. Sections ξ (8.16) play a role of gauge parameters. Lemma 11.1 Principal vector fields (8.45) locally are symmetries of the global Chern–Simons Lagrangian LCS(B) (11.7). Proof Since dim X = 2k − 1, the transgression formula (11.4) takes a form

P2k (FA ) = dS2k−1(a, A).

The Lie derivative LJ 1υ acting on its sides results in an equality

= ( S ( , )) = ( S ( , )), 0 d uξ d 2k−1 a A d LJ 1uξ 2k−1 a A

S ( , ) i.e., the Lie derivative LJ 1uξ 2k−1 a A locally is d-exact. Consequently, the hor- S ( , ) izontal form h0LJ 1uξ 2k−1 a A locally is dH -exact. A direct computation shows that

S ( , ) = ( S ( , )) + . h0LJ 1uξ 2k−1 a A LJ 1uξ h0 2k−1 a A dH S

It follows that the Lie derivative LJ 1υ LCS(A) of the global Chern–Simons Lagrangian along any vector ﬁeld uξ (8.45) locally is dH -exact, i.e., this vector ﬁeld is locally a symmetry of LCS(A). By virtue of item (iii) of Theorem 2.4, a vertical part

= (− r ξ p q + ∂ ξ r − r ∂ ξ μ − ξ μ r )∂λ uV cpq aλ λ aμ λ aμλ r (11.9) of the vector ﬁeld uξ (8.45) locally is a symmetry of LCS(A), too. Given the ﬁbre bundle TG P → X (8.12), let the same symbol also stand for the pull-back of TG P onto C. Let us consider the DBGA (7.6):

∗ ∗ P∞[TG P; C]=S∞[TG P; C],

r λ r r possessing the local generating basis (aλ, c , c ) of even ﬁelds aλ and odd ghosts λ r c , c . Substituting these ghosts for gauge parameters in the vector ﬁeld uV (11.9) (Remark 7.7), we obtain the odd vertical graded derivation

= (− r p q + r − μ r − μ r )∂λ u cpqc aλ cλ cλ aμ c aμλ r (11.10)

∗ of a DBGA P∞[TG P; C]. This graded derivation as like as the vector ﬁelds υV (11.9) locally is a symmetry of the global Chern–Simons Lagrangian LCS(A) (11.7), i.e., an odd density LJ 1u(LCS(A)) locally is dH -exact. Hence, it is δ-closed and, 11 Chern–Simons Topological Field Theory 205 consequently, dH -exact in accordance with Theorem 6.12. Thus, we have proved the following.

Lemma 11.2 The graded derivation u (11.10) is a gauge symmetry of the global Chern–Simons Lagrangian LCS(A) (11.7), and the vector ﬁelds uV (11.9) and uξ (8.45) are well.

Therefore, let us turn to Noether’s first theorem. Let us consider the vertical principal vector field ξ (8.44). In accordance with Lemma 11.2, the corresponding vertical principal vector field uξ (8.46) is a gauge symmetry of the Chern–Simons Lagrangian (11.7), i.e.,

( ) = σ. LJ 1uξ LCS A dH

As a consequence, Chern–Simons topological ﬁeld theory with the Lagrangian (11.7) admits a weak conservation law

≈− (J + σ) 0 dH uξ (11.11)

J + σ of a symmetry current uξ where

J =−∂λμ[L ( )](∂ ξ r + r pξ q ) uξ r CS A μ cpqaμ is the Noether current (3.32) along a vertical principal vector field uξ . By virtue of J + σ Noether’s third Theorem 2.8, a conserved current uξ is reduced to a superpotential. Let us study energy-momentum conservation laws in Chern–Simons topological field theory. Given a gauge field A, lets us consider the lift Aτ (8.67) of vector field τ on X onto the bundle of principal connection C. It is a principal vector field on C and, thus, is a symmetry of the Chern–Simons Lagrangian (11.7) by virtue of Lemma 11.2. Then we come to a weak conservation law

0 ≈−dH (JA + σ ) (11.12)

of a symmetry current JA + σ where

J λ = ∂λμL [τ ν r − ∂ ( r τ ν ) − r p q τ ν + r ∂ τ ν ]−τ λL A r CS aνμ μ Aν cpqaμ Aν aν μ CS is an energy-momentum current along the vector ﬁeld Aτ (8.67). Turn now to Noether’s second theorem where the graded vector ﬁeld u (11.10) is a gauge symmetry. By virtue of the formulas (2.18)–(2.19), the corresponding NI read

δ =− r i E λ − E λ = , j c jiaλ r dλ j 0 (11.13) δ =− r E λ + ( r E λ) = . μ aμλ r dλ aμ r 0 (11.14) 206 11 Chern–Simons Topological Field Theory

They are irreducible and nontrivial, unless dim X = 3. Therefore, the gauge operator (7.39)isu = u. It admits a nilpotent BRST extension

μ ∂ 1 ∂ ∂ b = (−cr c j ai + cr − c ar − cμar ) − cr ci c j + cλ cμ . ji λ λ λ μ μλ r ij r μ λ (11.15) ∂aλ 2 ∂c ∂c

( λ, , ) P∗ [ ; ] In order to include antiﬁelds ar cr cμ , let us enlarge a DBGA ∞ TG P C toaDBGA

∗ ∗ P∞{0}=S∞[VC⊕ TG P; C × TG P] C X where VC is the density dual (8.85) of the vertical tangent bundle VC of C → X and TG P is the density dual of TG P → X (cf. (8.87)). By virtue of Theorem 7.16, given the BRST operator b (11.15), the global Chern–Simons Lagrangian LCS(A) (11.7) is extended to the proper solution (7.86) of the master equation which reads

r p q r μ r μ r λ 1 r i j λ μ L = L (A) +[(−c c a + cλ − c aμ − c aμλ)a − c c c c + cμc cλ]ω. E CS pq λ λ r 2 ij r If dim X = 3, the global Chern–Simons Lagrangian takes the form (11.8). Its Euler–Lagrange operator is

δ ( ) = E λθr ∧ ω, E λ = ελβγ F p . LCS B r λ r hrp βγ

A glance at the NI (11.13)–(11.14) shows that they are equivalent to NI

δ =− r i E λ − E λ = , j c jiaλ r dλ j 0 δ = δ + r δ = μF r E λ = . μ μ aμ r c λμ r 0 (11.16)

These NI deﬁne the gauge symmetry u (11.10) written in a form

= (− r p q + r + μF r )∂λ u cpqc aλ cλ c λμ r (11.17)

r r r μ where c = c − aμc . It is readily observed that, if dim X = 3, the NI δμ (11.16) μF r ∂λ are trivial. Then the corresponding part c λμ r of the gauge symmetry u (11.17) also is trivial. Consequently, the nontrivial gauge symmetry of the Chern–Simons Lagrangian (11.8)is

= (− r p q + r )∂λ. u cpqc aλ cλ r Chapter 12 Topological BF Theory

We consider topological BF theory of two exterior forms A and B of form degree |A|+|B|=dim X − 1 on a smooth manifold X [16]. It is reducible degenerate Lagrangian theory which satisfies the homology regularity condition (Definition 7.2) [11, 137]. Its dynamic variables A and B are sections of a fibre bundle

p q Y = ∧ T ∗ X ⊕ ∧ T ∗ X, p + q = n − 1 > 1,

( λ, , ) coordinated by x Aμ1...μp Bν1...νq . Without a loss of generality, let q be even and ∗ q ≥ p. The corresponding DGA is O∞Y (1.7). There are the canonical p- and q-forms

μ μ ν ν = 1 ∧···∧ p , = 1 ∧···∧ q A Aμ1...μp dx dx B Bν1...νq dx dx on Y . A Lagrangian of topological BF theory is

μ ...μ = ∧ = ε 1 n ω, LBF A dH B Aμ1...μp dμp+1 Bμp+2...μn (12.1) where ε is the Levi–Civita symbol. It is a reduced ﬁrst order Lagrangian whose Euler–Lagrange operator (3.2) is of ﬁrst order. It reads

μ ...μ ν ...ν δ = E 1 p ∧ ω + E p+2 n ∧ ω, L A dAμ1...μp B dBνp+2...νn (12.2) μ ...μ μ ...μ E 1 p = ε 1 n , A dμp+1 Bμp+2...μn (12.3) μ + ...μ μ ...μ E p 2 n =−ε 1 n . B dμp+1 Aμ1...μp (12.4)

The corresponding Euler–Lagrange equation can be written in a form

dH B = 0, dH A = 0. (12.5)

It obeys the NI dH dH B = 0, dH dH A = 0. (12.6)

μ ...μ μ ...μ E 1 p E p+2 n One can regard the components A (12.3) and B (12.4)oftheEuler– p n ∗ Lagrange operator (12.2)asa(∧ TX) ⊗X (∧ T X)-valued differential operator on q ∗ q n ∗ a ﬁbre bundle ∧ T X and a (∧ TX) ⊗X (∧ T X)-valued differential operator on a p n−1 ﬁbre bundle ∧ T ∗ X, respectively. They are of the same type as the ∧ TX-valued differential operator (D.19) in Example in Appendix D (cf. the equations (12.5)) and (D.18). Therefore, the analysis of NI of the differential operators (12.3) and (12.4) is a repetition of that of NI of the operator (D.19) (cf. NI (12.6) and (D.23)). Following Example in Appendix D, let us consider the family of vector bundles

− − − − p k 1 ∗ q k 1 ∗ Ek = ∧ T X × ∧ T X, 0 ≤ k < p − 1, X − q p ∗ Ek = R × ∧ T X, k = p − 1, X − − q k 1 ∗ Ek = ∧ T X, p − 1 < k < q − 1,

Eq−1 = X × R.

∗ ∗ Let us enlarge the DGA O∞Y to the DBGA P∞{q − 1} (7.33) which is

∗ ∗ P∞{q − 1}=P∞[VY ⊕ E0 ⊕···⊕Eq−1 ⊕ E0 ⊕ ···⊕Eq−1; Y ]. Y Y Y Y Y

It possesses the local generating basis

{ , ,ε ,...,ε ,ε,ξ ,...,ξ ,ξ, Aμ1...μp Bν1...νq μ2...μp μp ν2...νq νq μ1...μp ν1...νq μ ...μ μ ν ...ν ν A , B , ε 2 p ,...,ε p , ε, ξ 2 q ,...,ξ q , ξ} of Grassmann parity

[ε ]=[ξ ]=( + ) , [ε]= , [ξ]= , μk ...μp νk ...νq k 1 mod 2 p mod 2 0 μ ...μ ν ...ν [ε k p ]=[ξ k q ]=k mod 2, [ε]=(p + 1)mod 2, [ξ]=1, of ghost number

[ε ]= [ξ ]= , [ε]= + , [ξ]= + , gh μk ...μp gh νk ...νq k gh p 1 gh q 1 and of antiﬁeld number

μ ...μ ν ...ν Ant[A 1 p ]=Ant[B p+1 q ]=1, μ ...μ ν ...ν Ant[ε k p ]=Ant[ξ k q ]=k + 1, 12 Topological BF Theory 209

Ant[ε]=p, Ant[ε]=q.

One can show that the homology regularity condition holds (see Lemma D.5) and ∗ that the DBGA P∞{q − 1} is endowed with the KT operator ∂ ∂ μ1...μp ν1...νq δKT = μ ...μ E + ν ...ν E + ∂ A 1 p A ∂ B 1 q B ∂ μ ...μ ∂ ∂ ν ...ν ∂ ν k p μp k q q μ ...μ Δ + dμ ε + ν ...ν Δ + dν ξ , ∂ε k p A ∂ε p ∂ξ k q B ∂ξ q 2≤k≤p 2≤k≤q

μ ...μ μ1...μp μ + ...μ μ μ ...μ Δ 2 p = ,Δk 1 p = ε k k+1 p , ≤ < , A dμ1 A A dμk 2 k p ν ...ν ν ...ν ν ...ν ν ν ...ν Δ 2 q = 1 q ,Δk+1 q = ξ k k+1 q , ≤ < . B dν1 B B dνk 2 k q

Its nilpotentness provides the complete NI (12.5):

μ ...μ ν ...ν E 1 p = , E 1 q = , dμ1 A 0 dν1 B 0 and the (k − 1)-stage ones

μ ...μ Δ k p = , = ,..., , dμk A 0 k 2 p ν ...ν Δ k q = , = ,..., , dνk B 0 k 2 q

(cf. (D.25)). It follows that the topological BF theory is (q − 1)-reducible. Applying Noether’s inverse second Theorem 7.9, one obtains the gauge operator (7.39) which reads ∂ ∂ = ε + ξ + u dμ1 μ2...μp dν1 ν2...νq (12.7) ∂ Aμ μ ...μ ∂ Bν ν ...ν 1 2 p 1 2 q ∂ ∂ ε +···+ ε + dμ2 μ3...μp dμp ∂εμ μ ...μ ∂εμ 2 3 p p ∂ ∂ dν ξν ...ν +···+dν ξ . 2 3 q ∂ξ q ∂ξ ν2ν3...νq νq

In particular, a gauge symmetry of the Lagrangian LBF (12.1)is ∂ ∂ u = dμ εμ ...μ + dν ξν ...ν . 1 2 p ∂ 1 2 q ∂ Aμ1μ2...μp Bν1ν2...νq

This gauge symmetry is Abelian. It also is readily observed that higher-stage gauge symmetries are independent of original ﬁelds. Consequently, topological BF theory is Abelian, and its gauge operator u (12.7) is nilpotent. Thus, it is the BRST operator 210 12 Topological BF Theory b = u. As a result, the Lagrangian LBF is extended to the proper solution of the master equation L E = Le (7.40) which reads μ1...μp μ ...μ μ = + ε + ε ε k p + ε ε p + Le LBF μ2...μp dμ1 A μk+1...μp dμk dμp 1

Acronyms

BRST Becchi–Rouet–Stora–Tyitin CIS Completely integrable system DBGA Differential bigraded algebra DGA Differential graded algebra KT Koszul–Tate NI Noether identities SUSY Supersymmetry

Notation

C∞(Z) ring of smooth real functions on a smooth manifold Z ∂ B B A partial derivative with respect to coordinates with indices A dλ, dΛ total derivative and multiderivative d exterior differential dH total (horizontal) differential dV vertical differential DΓ covariant differential with respect to a connection Γ dA derivation module of a ring A λ {∂λ}, {dx } holonomic frame and coframe E∗ the dual of a vector bundle E EL Euler–Lagrange operator of a Lagrangian L f ∗Y pull-back bundle f ∗φ pull-back exterior form gl , gr left and right Lie algebras h0 horizontal projection J 1Φ jet prolongation of a bundle morphism Φ

J 1s jet prolongation of a section s J 1u jet prolongation of a vector filed u J k Yk-order jet manifold J ∞Y infinite order jet manifold Jυ symmetry current along a generalized vector field υ LX frame bundle O∗A de Rham complex of a commutative ring A O∗(Z) DGA of exterior forms on a smooth manifold Z ∗ O∞Y DGA of exterior forms on all jet manifolds of a fibre bundle Y TZ, T ∗ Z tangent and cotangent bundles of a manifold Z T (Z) real Lie algebra of vector fields on a manifold Z Tf tangent prolongation of a manifold morphism f VY, V ∗Y vertical tangent and cotangent bundles of a fibre bundle Y (Z, UE ) simple graded manifold modelled over a vector bundle E → Z ∗ S (E, Z) DBGA of exterior forms on a simple graded manifold (Z, UE ) Y (X) vector space of global sections of a vector bundle Y → X (Y, F, UF ) graded bundle modelled over a composite bundle F → Y → X ∗ S∞(F, Y ) DBGA of exterior forms on all jet manifolds of a graded bundle δ variational operator ω local volume form i θΛ local contact one-form τ functorial lift of a vector field τ Γτ horizontal lift of a vector field τ 1 unit element of algebras and groups 0 canonical zero-valued section of a vector bundle ⊗P, ∧P tensor and exterior algebras of a module P ∧E exterior bundle |.| form degree [.] Grassmann parity

Convention

Throughout the book, all algebras, except the Lie and graded Lie ones, are associative, and modules over commutative algebras are commutative bimodules. Unless otherwise stated, all morphisms are smooth (i.e., of class C∞), and manifolds are smooth real and ﬁnite-dimensional. They are Hausdorff second-countable and, consequently, locally compact and paracompact topological spaces (Remark B.1). Only proper covers throughout the book are considered (Remark C.4).

By the gradation is meant the N-gradation, unless it is the Grassmann Z2-gradation. The standard symbols ⊗, ∨, and ∧ stand for the tensor, symmetric, and exterior products, respectively. The interior product (contraction) is denoted by . Appendix A Differential Calculus over Commutative Rings

In this Appendix, the relevant basics on differential operators and the Chevalley– Eilenberg differential calculus over commutative rings are summarized [58, 85, 131]. The differential calculus over a ring C∞(X) of smooth real functions on a manifold X especially is considered (Sect.A.4). A key point is the well-known Serre–Swan theorem (Theorem A.10) which states the categorial equivalence between the vector bundles over a manifold X and the projective C∞(X)-modules of ﬁnite rank. In particular, the Chevalley–Eilenberg differential calculus over a real ring C∞(X) is just the DGA of exterior forms on a smooth manifold X.

A.1 Commutative Algebra

This section addresses the algebraic calculus over commutative algebras [93, 97].

Algebras and Modules

An algebra A is an additive group which is additionally provided with distributive multiplication. All algebras throughout the book are associative, unless they are the Lie and graded Lie ones. By a ring is meant a unital algebra, i.e., it contains the unit element 1 = 0. A ﬁeld is a commutative ring whose nonzero elements are invertible. A subset I of an algebra A is called the left (resp. right) ideal if it is a subgroup of an additive group A and ab ∈ I (resp. ba ∈ I ) for all a ∈ A , b ∈ I .IfI is both a left and right ideal, it is termed the two-sided ideal. A two-sided ideal is a subalgebra, but a proper ideal (i.e., I = A ) of a ring is not a subring. Let A be a commutative ring. Of course, its ideals are two-sided. Its proper ideal is said to be maximal if it does not belong to another proper ideal. A commutative ring A is called local if it has a unique maximal ideal. This ideal consists of all non-invertible elements of A . A proper two-sided ideal I of a commutative ring is © Atlantis Press and the author(s) 2016 213 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0 214 Appendix A: Differential Calculus over Commutative Rings termed prime if ab ∈ I implies either a ∈ I or b ∈ I . Any maximal two-sided ideal is prime. Given a two-sided ideal I ⊂ A , an additive quotient group A /I is an algebra, called the quotient algebra.IfA is a ring, then A /I is so. Given an algebra A , an additive group P is said to be the left (resp. right) A - module if it is provided with distributive multiplication A × P → P by elements of A such that (ab)p = a(bp) (resp. (ab)p = b(ap)) for all a, b ∈ A and p ∈ P. If A is a ring, one additionally assumes that 1p = p = p1 for all p ∈ P. Left and right module structures are usually written by means of left and right multiplications (a, p) → apand (a, p) → pa, respectively. If P is both a left module over an algebra A and a right module over an algebra A , it is called the (A − A )-bimodule (an A -bimodule if A = A ). If A is a commutative algebra, an (A − A )-bimodule P is said to be commutative if ap = pa for all a ∈ A and p ∈ P. Any left or right module over a commutative algebra A can be brought into a commutative bimodule. Therefore, unless otherwise stated, any module over a commutative algebra A is termed the A -module. A module over a ﬁeld is called the vector space. If an algebra A is a module over aringK ,itissaidtobetheK -algebra. Any algebra can be seen as a Z-algebra.

Remark A.1 Any K -algebra A can be extended to a ring Aby the adjunction of the unit element 1 to A . It is deﬁned as a direct sum of K -modules K ⊕A provided with the multiplication

(λ1, a1)(λ2, a2) = (λ1λ2,λ1a2 + λ2a1 + a1a2), λ1,λ2 ∈ K , a1, a2 ∈ A .

Elements of Acan be written as (λ, a) = λ1 + a, λ ∈ K , a ∈ A . Let us note that, if A is a ring, the unit element 1A in A fails to be that in A . In this case, a ring A is isomorphic to a product of A and an algebra K (1 − 1A ).

Hereafter, all associative algebras in this Appendix are assumed to be commutative, unless they are graded. The following are standard constructions of new modules from the old ones.

• A direct sum P1 ⊕ P2 of A -modules P1 and P2 is an additive group P1 × P2 provided with an A -module structure

a(p1, p2) = (ap1, ap2), p1,2 ∈ P1,2, a ∈ A .

{ } ⊕ (. . . , Let Pi i∈I be a set of modules. Their direct sum Pi contains elements pi ,...)of a Cartesian product Pi so that pi = 0 at most for a ﬁnite number of indices i ∈ I. • A tensor product P ⊗ Q of A -modules P and Q is an additive group which is generated by elements p ⊗ q, p ∈ P, q ∈ Q, obeying relations

(p + p ) ⊗ q = p ⊗ q + p ⊗ q, p ⊗ (q + q ) = p ⊗ q + p ⊗ q , pa ⊗ q = p ⊗ aq, p ∈ P, q ∈ Q, a ∈ A . Appendix A: Differential Calculus over Commutative Rings 215

It is provided with an A -module structure

a(p ⊗ q) = (ap) ⊗ q = p ⊗ (qa) = (p ⊗ q)a.

If a ring A is treated as an A -module, a tensor product A ⊗A Q is canonically isomorphic to Q via the assignment

A ⊗ Q a ⊗ q ↔ aq ∈ Q. A

• Given a submodule Q of an A -module P, the quotient P/Q of an additive group P with respect to its subgroup Q also is provided with an A -module structure. It is called the quotient module. • AsetHomA (P, Q) of A -linear morphisms of an A -module P to an A -module ∗ Q is naturally an A -module. An A -module P = Hom A (P, A ) is termed the dual of an A -module P. There is a natural monomorphism P → P∗∗. An A -module P is called free if it admits a basis, i.e., a linearly independent subset I ⊂ P spanning P such that each element of P has a unique expression as a linear combination of elements of I with a finite number of nonzero coefficients from an algebra A . Any vector space is free. Any module is isomorphic to a quotient of a free module. A module is said to be finitely generated (or of finite rank)ifitis a quotient of a free module with a finite basis. One says that a module P is projective if it is a direct summand of a free module, i.e., there exists a module Q such that P ⊕ Q is a free module. A module P is projective if and only if P = pS where S is a free module and p is a projector of S, i.e., p ◦ p = p.IfP is a projective module of finite rank over a ring, then its dual P∗ is so, and P∗∗ is isomorphic to P.

Sequences and Limits of Modules

Now we focus exposition on sequences, direct and inverse limits of modules [97, 104]. A composition of module morphisms

i j P −→ Q −→ T is said to be exact at Q if Ker j = Im i. A composition of module morphisms

i j 0 → P −→ Q −→ T → 0(A.1) 216 Appendix A: Differential Calculus over Commutative Rings is called the short exact sequence if it is exact at all the terms P, Q, and T .This condition implies that: (i) i is a monomorphism, (ii) Ker j = Im i, and (iii) j is an epimorphism onto a quotient T = Q/P. Theorem A.1 Given an exact sequence of modules (A.1) and another A -module R, the sequence of modules

j ∗ i ∗ 0 → Hom A (T, R) −→ Hom A (Q, R) −→ Hom (P, R) is exact at the ﬁrst and second terms, i.e., j ∗ is a monomorphism, but i ∗ need not be an epimorphism. One says that the exact sequence (A.1)issplit if there exists a monomorphism s : T → Q such that j ◦ s = Id T or, equivalently,

Q = i(P) ⊕ s(T ) =∼ P ⊕ T.

Theorem A.2 The exact sequence (A.1) is always split if T is a projective module. A directed set I is a set with an order relation < which satisﬁes the following three conditions: (i) i < i, for all i ∈ I ; (ii) if i < j and j < k, then i < k; (iii) for any i, j ∈ I, there exists k ∈ I such that i < k and j < k. It may happen that i = j, but i < j and j < i simultaneously. A family of A -modules {Pi }i∈I , indexed by a directed set I , is termed the direct < i : → system if, for any pair i j, there exists a morphism r j Pi Pj such that

i = , i ◦ j = i , < < . ri Id Pi r j rk rk i j k

A direct system of modules admits a direct limit. i Deﬁnition A.1 This is a module P∞ together with morphisms r∞ : Pi → P∞ such i = j ◦ i < that r∞ r∞ r j for all i j. A module P∞ consists of elements of a direct sum ⊕I Pi modulo the identiﬁcation of elements of Pi with their images in Pj for all i < j. An example of a direct system is a direct sequence

i ri+1 P0 −→ P1 −→···Pi −→··· , I = N.

It should be noted that direct limits also exist in the categories of commutative algebras and rings, but not in a category of noncommutative groups. Theorem A.3 Direct limits commute with direct sums and tensor products of modules. Namely, let {Pi } and {Qi } be two direct systems of modules over the same algebra which are indexed by the same directed set I , and let P∞ and Q∞ be their direct limits. Then direct limits of direct systems {Pi ⊕Qi } and {Pi ⊗Qi } are P∞⊕Q∞ and P∞ ⊗ Q∞, respectively. Appendix A: Differential Calculus over Commutative Rings 217

{ , i } { ,ρi } Theorem A.4 A morphism of a direct system Pi r j I to a direct system Qi j I consists of an order preserving map f : I → I and morphisms Fi : Pi → Q f (i) ρ f (i) ◦ = ◦ i which obey compatibility conditions f ( j) Fi Fj r j .IfP∞ and Q∞ are limits of these direct systems, there exists a unique morphism F∞ : P∞ → Q∞ such that f (i) i ρ∞ ◦ Fi = F∞ ◦ r∞.

Theorem A.5 Moreover, direct limits preserve monomorphisms and epimorphisms. Namely, if all Fi : Pi → Q f (i) are monomorphisms or epimorphisms, so is Φ∞ : P∞ → Q∞. Let short exact sequences

Fi Φi 0 → Pi −→ Qi −→ Ti → 0(A.2) for all i ∈ I deﬁne a short exact sequence of direct systems of modules {Pi }I , {Qi }I , and {Ti }I which are indexed by the same directed set I . Then there exists a short exact sequence of their direct limits

F∞ Φ∞ 0 → P∞ −→ Q∞ −→ T∞ → 0. (A.3)

In particular, a direct limit of quotient modules Qi /Pi is a quotient module Q∞/P∞. By virtue of Theorem A.3, if all the exact sequences (A.2) are split, the exact sequence (A.3) is well.

k Remark A.2 Let P be an A -module. We denote P⊗k = ⊗ P. Let us consider a direct system of A -modules with respect to monomorphisms

A −→ (A ⊕ P) −→···(A ⊕ P ⊕···⊕ P⊗k ) −→··· .

Its direct limit ⊗ P = A ⊕ P ⊕···⊕ P⊗k ⊕··· (A.4) is an N-graded A -algebra with respect to a tensor product ⊗. It is called the tensor algebra of a module P. Its quotient

k ∧P = A ⊕ P ⊕···⊕∧ P ⊕··· by an ideal generated by elements p ⊗ p + p ⊗ p, p, p ∈ P,isanN-graded commutative algebra with respect to the exterior product

1 p ∧ p = (p ⊗ p − p ⊗ p). 2 It is called the exterior algebra of a module P. 218 Appendix A: Differential Calculus over Commutative Rings

We restrict our consideration of inverse systems of modules to inverse sequences

π i+1 P0 ←− P1 ←−···Pi ←−···i . (A.5)

It admits an inverse limit. Deﬁnition A.2 The inverse limit of the inverse sequence of modules (A.5)isa ∞ π ∞ : ∞ → i π ∞ = π j ◦ π ∞ module P together with morphisms i P P so that i i j for i i i all i < j. It consists of elements (. . . , p ,...), p ∈ P , of the Cartesian product i i = π j ( j ) < P such that p i p for all i j. Theorem A.6 Inverse limits preserve monomorphisms, but not epimorphisms. If a sequence

Fi Φi 0 → Pi −→ Qi −→ T i , i ∈ N, of inverse systems of modules {Pi }, {Qi } and {T i } is exact, so is a sequence of inverse limits

F∞ Φ∞ 0 → P∞ −→ Q∞ −→ T ∞.

In contrast with direct limits (Deﬁnition A.1), the inverse ones (Deﬁnition A.2) exist in the category of groups which are not necessarily commutative.

Remark A.3 Let {Pi } be a direct sequence of modules. Given another module Q, modules Hom (Pi , Q) constitute an inverse system such that its inverse limit (Deﬁnition A.2) is isomorphic to Hom (P∞, Q).

Complexes

Turn now to complexes of modules over a commutative ring [97, 104]. Let K be a commutative ring. An inverse sequence

∂ ∂1 ∂2 p+1 0←B0 ←− B1 ←−···Bp ←−··· (A.6) of K -modules Bp and homomorphisms ∂p is said to be the chain complex if

∂p ◦ ∂p+1 = 0, p ∈ N, i.e., Im ∂p+1 ⊂ Ker ∂p. Homomorphisms ∂p are called the boundary operators. Elements of a module Bp, its submodules Ker ∂p ⊂ Bp and Im ∂p+1 ⊂ Ker ∂p are termed p-chains, p-cycles and p-boundaries, respectively. The p-homology group of the chain complex B∗ (A.6) is deﬁned as the K -quotient module Appendix A: Differential Calculus over Commutative Rings 219

Hp(B∗) = Ker ∂p/Im ∂p+1.

The chain complex (A.6) is exact at a term Bp if and only if Hp(B∗) = 0. This complex is said to be k-exact if its homology groups Hp≤k (B∗) are trivial. It is called exact if all its homology groups are trivial, i.e., it is an exact sequence. A direct sequence

δ0 δ1 δ p 0 → B0 −→ B1 −→···B p −→··· (A.7) of modules B p and their homomorphisms δ p is said to be the cochain complex (or, shortly, the complex)ifδ p+1 ◦ δ p = 0, p ∈ N, i.e., Im δ p ⊂ Ker δ p+1. Homomor- phisms δ p are termed the coboundary operators. For the sake of convenience, let us denote δ p=−1 : 0 → B0. Elements of a module B p, its submodules Ker δ p ⊂ B p and Im δ p−1 are called p-cochains, p-cocycles and p-coboundaries, respectively. A p-cohomology group of the complex B∗ (A.7) is the quotient K -module

H p(B∗) = Ker δ p/Im δ p−1.

The complex (A.7) is exact at a term B p if and only if H p(B∗) = 0. This complex is an exact sequence if all its cohomology groups are trivial. A complex (B∗,δ∗) is called acyclic if its cohomology groups H 0

hp+1 : B p+1 → B p, p ∈ N, p+1 p p−1 p p h ◦ δ + δ ◦ h = Id B , p ∈ N+.

Indeed, if δ pb p = 0, then b p = δ p−1(hpb p), and H p>0(B∗) = 0. A complex (B∗,δ∗) is said to be a resolution of a module B if it is acyclic and H 0(B∗) = Ker δ0 = B. The following are the standard constructions of new complexes from the old ones. • ( ∗,δ∗) ( ∗,δ∗) ∗ ⊕ ∗ Given complexes B1 1 and B2 2 , their direct sum B1 B2 is a complex of modules

( ∗ ⊕ ∗)p = p ⊕ p B1 B2 B1 B2

with respect to the coboundary operators

δ p ( p + p) = δ p p + δ p p. ⊕ b1 b2 1 b1 2 b2

• Given a subcomplex (C∗,δ∗) of a complex (B∗,δ∗),thequotient complex B∗/C∗ is deﬁned as a complex of quotient modules B p/C p provided with the coboundary operators δ p[b p]=[δ pb p], where [b p]∈B p/C p denotes the coset of an element b p. 220 Appendix A: Differential Calculus over Commutative Rings

• ( ∗,δ∗) ( ∗,δ∗) ∗ ⊗ ∗ Given complexes B1 1 and B2 2 , their tensor product B1 B2 is a complex of modules

( ∗ ⊗ ∗)p =⊕ k ⊗ r B1 B2 B1 B2 k+r=p

with respect to the coboundary operators

δ p ( k ⊗ r ) = (δk k ) ⊗ r + (− )k k ⊗ (δr r ). ⊗ b1 b2 1 b1 b2 1 b1 2b2 γ : ∗ → ∗ A cochain morphism of complexes B1 B2 is deﬁned as a family of degree-preserving homomorphisms

γ p : p → p,δp ◦ γ p = γ p+1 ◦ δ p, ∈ N. B1 B2 2 1 p

p ∈ p γ p( p) ∈ p It follows that if b B1 is a cocycle or a coboundary, then b B2 is so. Therefore, a cochain morphism of complexes yields an induced homomorphism of their cohomology groups

[γ ]∗ : ∗( ∗) → ∗( ∗). H B1 H B2

Let short exact sequences

γ ζ 0 → C p −→p B p −→p F p → 0 for all p ∈ N deﬁne a short exact sequence of complexes

γ ζ 0 → C∗ −→ B∗ −→ F ∗ → 0, (A.8) where γ is a cochain monomorphism and ζ is a cochain epimorphism onto the quotient F∗ = B∗/C∗.

Theorem A.7 The short exact sequence of complexes (A.8) yields the long exact sequence of their cohomology groups

[γ ]0 [ζ]0 τ 0 0 → H 0(C∗) −→ H 0(B∗) −→ H 0(F ∗) −→ H 1(C∗) −→··· [γ ]p [ζ]p τ p −→ H p(C∗) −→ H p(B∗) −→ H p(F ∗) −→ H p+1(C∗) −→··· .

Theorem A.8 A direct sequence of complexes

γ k ∗ −→ ∗ −→··· ∗ −→k+1 ∗ −→··· B0 B1 Bk Bk+1 Appendix A: Differential Calculus over Commutative Rings 221

∗ ∗ ∗ admits a direct limit B∞ which is a complex whose cohomology H (B∞) is a direct limit of the direct sequence of cohomology groups

[γ k ] ∗( ∗) −→ ∗( ∗) −→··· ∗( ∗) −→k+1 ∗( ∗ ) −→··· . H B0 H B1 H Bk H Bk+1

A.2 Differential Operators on Modules and Rings

This section addresses the notion of a (linear) differential operator on a module over a commutative ring [53, 85, 131]. Let K be a commutative ring and A a commutative K -ring. Let P and Q be A - modules. A K -module Hom K (P, Q) of K -module homomorphisms Φ : P → Q can be endowed with two different A -module structures

(aΦ)(p) = aΦ(p), (Φ • a)(p) = Φ(ap), a ∈ A , p ∈ P. (A.9)

For the sake of convenience, we will refer to the second one as an A •-module structure. Let us put δaΦ = aΦ − Φ • a, a ∈ A .

Deﬁnition A.3 An element Δ ∈ Hom K (P, Q) is called the (linear) Q-valued δ ◦···◦δ Δ = + differential operator of order s on P if a0 as 0 for any tuple of s 1 elements a0,...,as of A . • AsetDiffs (P, Q) of these operators inherits the A - and A -module structures (A.9). Of course, an s-order differential operator also is of (s + 1)-order. In particular, zero order differential operators obey a condition

δaΔ(p) = aΔ(p) − Δ(ap) = 0, a ∈ A , p ∈ P, and, consequently, they coincide with A -module morphisms P → Q. A ﬁrst order differential operator Δ satisﬁes a condition

δb ◦ δa Δ(p) = baΔ(p) − bΔ(ap) − aΔ(bp) + Δ(abp) = 0, a, b ∈ A .

Let P = A . Any zero order Q-valued differential operator Δ on A is deﬁned by its value Δ(1). Then there is an isomorphism Diff 0(A , Q) = Q via the association

Q q → Δq ∈ Diff 0(A , Q), where Δq is given by an equality Δq (1) = q. A ﬁrst order Q-valued differential operator Δ on A fulﬁls a condition

Δ(ab) = bΔ(a) + aΔ(b) − baΔ(1), a, b ∈ A . 222 Appendix A: Differential Calculus over Commutative Rings

Deﬁnition A.4 A ﬁrst order Q-valued differential operator Δ on A is termed the Q-valued derivation of A if Δ(1) = 0, i.e., it obeys the Leibniz rule

Δ(ab) = Δ(a)b + aΔ(b), a, b ∈ A .

One obtains at once that any ﬁrst order differential operator on A falls into a sum

Δ(a) = aΔ(1) +[Δ(a) − aΔ(1)] of a zero order differential operator aΔ(1) and a derivation Δ(a) − aΔ(1).If∂ is a derivation of A , then a∂ is well for any a ∈ A . Hence, derivations of A constitute an A -module d(A , Q), called the derivation module. There is an A - module decomposition

Diff 1(A , Q) = Q ⊕ d(A , Q). (A.10)

If Q = A , the derivation module dA of A also is a Lie algebra over a ring K with respect to a Lie bracket

[u, u ]=u ◦ u − u ◦ u, u, u ∈ A .

Accordingly, the decomposition (A.10) takes a form

Diff 1(A ) = A ⊕ dA .

A derivation u of A is termed inner if there exists an element q ∈ A such that u(a) = qa − aq.

Remark A.4 Let us note that, given a (noncommutative) K -algebra A , by a derivation of A usually is meant a K -module morphism u : A → A which obeys the Leibniz rule

u(ab) = u(a)b + au(b), a, b ∈ A .

However, this derivation rule differs from that (6.2) of derivations of a graded commutative ring and that in [93].

A.3 Chevalley–Eilenberg Differential Calculus

Any commutative K -ring A admits the differential calculus as follows. By a gradation throughout this section is meant the N-gradation. A graded algebra Ω∗ over a commutative ring K is deﬁned as a direct sum Appendix A: Differential Calculus over Commutative Rings 223

Ω∗ =⊕Ωk , k ∈ N, k of K -modules Ωk , provided with an associative multiplication law α ·β, α, β ∈ Ω∗, such that α · β ∈ Ω|α|+|β|, where |α| denotes the degree of an element α ∈ Ω|α|.In particular, it follows that Ω0 is a (noncommutative) K -algebra A , while Ωk>0 are A -bimodules and Ω∗ is an (A − A )-algebra. A graded algebra is said to be graded commutative if

α · β = (−1)|α||β|β · α, α, β ∈ Ω∗.

Deﬁnition A.5 A graded algebra Ω∗ is called the differential graded algebra (DGA) if it is a cochain complex of K -modules

δ δ δ 0 → K −→ A −→ Ω1 −→···Ωk −→··· (A.11) with respect to a coboundary operator δ which obeys the graded Leibniz rule

δ(α · β) = δα · β + (−1)|α|α · δβ. (A.12)

In particular, δ : A → Ω1 is a Ω1-valued derivation of a K -algebra A .

The cochain complex (A.11) is termed the de Rham complex of a DGA (Ω∗,δ). This algebra also is said to be the differential calculus over A . Cohomology H ∗(Ω∗) of the complex (A.11) is called the de Rham cohomology of a DGA (Ω∗,δ). A morphism γ between two DGAs (Ω∗,δ)and (Ω ∗,δ ) is deﬁned as a cochain morphism, i.e., γ ◦ δ = γ ◦ δ . It yields the corresponding morphism of the de Rham cohomology groups of these algebras. One considers a minimal differential graded subalgebra Ω∗A of a DGA Ω∗ which contains A . Seen as an (A − A )-algebra, it is generated by elements δa, a ∈ A , and consists of monomials α = a0δa1 ···δak , ai ∈ A , whose product obeys the juxtaposition rule

(a0δa1) · (b0δb1) = a0δ(a1b0) · δb1 − a0a1δb0 · δb1 in accordance with the equality (A.12). A DGA (Ω∗A ,δ) is termed the minimal differential calculus over A . Now, let us show that any commutative K -ring A deﬁnes the differential calculus. As was mentioned above, the derivation module dA of A also is a Lie K -algebra. Given its Chevalley–Eilenberg complex C∗[dA ; A ] [47, 131], let us consider the extended Chevalley–Eilenberg complex

in 0 → K −→ C∗[dA ; A ] 224 Appendix A: Differential Calculus over Commutative Rings of the Lie algebra dA with coefﬁcients in a ring A , regarded as a dA -module [58, 131]. This complex contains a subcomplex O∗[dA ] of A -multilinear skew- symmetric maps

k φk : × dA → A with respect to the Chevalley–Eilenberg coboundary operator k i dφ(u0,...,uk ) = (−1) ui (φ(u0,...,ui ,...,uk )) + (A.13) i=0 i+ j (−1) φ([ui , u j ], u0,...,ui ,...,u j ,...,uk ). i< j

Indeed, a direct veriﬁcation shows that if φ is an A -multilinear map, so is dφ.In particular, (da)(u) = u(a), a ∈ A , u ∈ dA , 1 (dφ)(u0, u1) = u0(φ(u1)) − u1(φ(u0)) − φ([u0, u1]), φ ∈ O [dA ], 0 1 O [dA ]=A , O [dA ]=Hom A (dA , A ).

It follows that d(1) = 0 and d is an O1[dA ]-valued derivation of A . A graded module O∗[dA ] is provided with the structure of a graded A -algebra with respect to the product

φ ∧ φ (u , ..., u + ) = 1 r s ··· ··· i1 ir j1 js φ( ,..., )φ ( ,..., ), sgn1···r+s ui1 uir u j1 u js i1<···

... where sgn... denotes the sign of a permutation. This product obeys the relations

d(φ ∧ φ ) = d(φ) ∧ φ + (−1)|φ|φ ∧ d(φ ), φ, φ ∈ O∗[dA ],

φ ∧ φ = (−1)|φ||φ |φ ∧ φ. (A.14)

By virtue of the first one, (O∗[dA ], d) is a K -DGA, whereas the relation (A.14) shows that O∗[dA ] is a graded commutative algebra. Definition A.6 A DGA (O∗[dA ], d) is called the Chevalley–Eilenberg differential calculus over a K -ring A [58, 131]. Definition A.7 The minimal Chevalley–Eilenberg differential calculus O∗A over aringA consists of the monomials a0da1 ∧···∧dak , ai ∈ A . Its complex

d d d 0 → K −→ A −→ O1A −→···Ok A −→··· (A.15) Appendix A: Differential Calculus over Commutative Rings 225 is termed the de Rham complex of a K -ring A , and its cohomology H ∗(A ) is said to be the de Rham cohomology of A .

A.4 Differential Calculus over C∞(X). Serre–Swan Theorem

Let X be a smooth manifold (Remark B.1) and C∞(X) a real ring of smooth real functions on X. ∞ Let CX be a sheaf of germs of local smooth real functions on a manifold X (Remark C.1). Its structure module of global sections is a ring C∞(X). Similarly to 0 ∞ ∞ a sheaf CX of germs of continuous real functions, a stalk Cx of a sheaf CX at a point x ∈ X has a unique maximal ideal of germs of smooth functions vanishing at ∞ ∞ x. Consequently, CX is a local-ringed space (Remark C.7). Though a sheaf CX is defined on a topological space X, it fixes a unique smooth manifold structure on X as follows. Theorem A.9 Let X be a paracompact topological space and (X, A) a local-ringed space. Let X admit an open cover {Ui } such that a sheaf A restricted to each Ui is (Rn, ∞ ) isomorphic to a local-ringed space CRn . Then X is an n-dimensional smooth manifold together with a natural isomorphism of local-ringed spaces (X, A) and ( , ∞) X CX . One can think of this result as being an alternative definition of smooth real manifolds in terms of local-ringed spaces. A smooth manifold X also is algebraically reproduced as a certain subspace of the spectrum of a real ring C∞(X) of smooth real functions on X [4, 58]. In particular, there is one-to-one correspondence between the manifold morphisms X → X and the R-ring morphisms C∞(X ) → C∞(X). Since a manifold admits the partition of unity by smooth functions (Remark B.1), ∞ it follows from Theorem C.8 that any sheaf of CX -modules on X, including a sheaf ∞ CX itself, is fine and, consequently, acyclic (Remark C.6). For instance, let Y → X be a vector bundle. The germs of its sections form a sheaf ∞ of C (X)-modules, called the structure sheaf SY of a vector bundle Y → X.The sheaf SY is fine. Its structure module of global sections coincides with the structure module Y (X) of global sections of a vector bundle Y → X. The forthcoming Serre– Swan theorem shows that these modules exhaust all projective modules of finite rank over C∞(X). This theorem originally has been proved in the case of a compact manifold X, but it is generalized to an arbitrary manifold [58, 111]. Theorem A.10 Let X be a smooth manifold. A C∞(X)-module P is isomorphic to the structure module of a smooth vector bundle over X if and only if it is a projective module of finite rank. Theorem A.10 states the categorial equivalence between the vector bundles over a manifold X and the projective modules of finite rank over a ring C∞(X) of smooth real functions on X. The following are corollaries of this equivalence 226 Appendix A: Differential Calculus over Commutative Rings

• The structure module Y ∗(X) of the dual Y ∗ → X of a vector bundle Y → X is the C∞(X)-dual Y (X)∗ of the structure module Y (X) of Y → X. • Any exact sequence of vector bundles

0 → Y −→ Y −→ Y → 0 (A.16)

over the same base X yields an exact sequence

0 → Y (X) −→ Y (X) −→ Y (X) → 0 (A.17)

of their structure modules, and vice versa. In accordance with Theorem A.2,the exact sequence (A.17) always is split. Every its splitting defines that of the exact sequence (A.16), and vice versa. • The derivation module of a real ring C∞(X) coincides with a C∞(X)-module T (X) of vector fields on X, i.e., with the structure module of the tangent bundle TXof X. Hence, it is a projective C∞(X)-module of finite rank. It is the C∞(X)- dual T (X) = O1(X)∗ of the structure module O1(X) of the cotangent bundle T ∗ X of X which is a module of one-forms on X and, conversely, O1(X) = T (X)∗. It follows that the Chevalley–Eilenberg differential calculus over a real ring C∞(X) (Definition A.6)isexactlyaDGA(O∗(X), d) of exterior forms on a manifold X, where the Chevalley–Eilenberg coboundary operator d (A.13) coincides with the exterior differential (B.31). Moreover, one can show that (O∗(X), d) is the minimal differential calculus (Definition A.7), i.e., the C∞(X)-module O1(X) is generated by elements df, f ∈ C∞(X) [58]. Accordingly, the de Rham complex (A.15)ofa real ring C∞(X) is the de Rham complex

d d d 0 → R −→ C∞(X) −→ O1(X) −→···Ok (X) −→··· (A.18) of exterior forms on a manifold X. Its cohomology is called the de Rham cohomology ∗ ( ) HDR X of X. It obeys de Rham Theorem C.10. Appendix B Differential Calculus on Fibre Bundles

Unless otherwise stated, ﬁbre bundles throughout the book are smooth manifolds. The differential calculus on ﬁbre bundles and their differential geometry are phrased in terms of jet manifolds [25, 53, 85].

B.1 Geometry of Fibre Bundles

We start with manifolds and fibred manifolds (Definition B.1). Fibre bundles are locally trivial fibred manifolds (Definition B.2).

Remark B.1 We follow the notion of a ﬁnite-dimensional real smooth manifold without boundary. It conventionally is assumed to be Hausdorff and second-countable (i.e., it has a countable base for topology). Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space (i.e., it has a countable dense subset) and a paracompact space (Remark C.5). Being paracompact, a smooth manifold admits the partition of unity by smooth real functions. One also can show that, given two disjoint closed subsets N and N of a smooth manifold X, there exists a smooth function f on X such that f |N = 0 and f |N = 1. Unless otherwise stated, manifolds are assumed to be connected and, consequently, arcwise connected.

Let Z be a manifold. We denote by

π : → ,π∗ : ∗ → Z TZ Z Z T Z Z its tangent and cotangent bundles, respectively. Given coordinates (zα) on Z,they are equipped with the holonomic bundle coordinates

© Atlantis Press and the author(s) 2016 227 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0 228 Appendix B: Differential Calculus on Fibre Bundles

∂z λ (zλ, z˙λ), z˙ λ = z˙μ, ∂zμ μ λ ∂z (z , z˙λ), z˙ = z˙μ, λ ∂zλ

λ with respect to the holonomic frames {∂λ} and coframes {dz } in the tangent and cotangent spaces to Z, respectively. Any manifold morphism f : Z → Z yields the tangent morphism

∂ f λ Tf : TZ → TZ , z˙ λ ◦ Tf = z˙μ. ∂xμ Let us consider manifold morphisms of maximal rank. They are immersions (in particular, imbeddings) and submersions. An injective immersion is a submanifold, and a surjective submersion is a ﬁbred manifold (in particular, a ﬁbre bundle). Given manifolds M and N,bytherank of a morphism f : M → N at a point p ∈ M is meant the rank of the linear morphism

Tp f : Tp M → T f (p) N.

For instance, if f is of maximal rank at p ∈ M, then Tp f is injective when dim M ≤ dim N and surjective when dim N ≤ dim M. In this case, f is called the immersion and the submersion at a point p ∈ M, respectively. Since p → rank p f is a lower semicontinuous function, then the morphism Tp f is of maximal rank on an open neighborhood of p, too. It follows from the inverse function theorem that, if f is an immersion (resp. submersion) at p, then it is locally injective (resp. surjective) around p.If f is both an immersion and a submersion, it is termed a local diffeomorphism at p. In this case, there exists an open neighborhood U of p such that f : U → f (U) is a diffeomorphism onto an open set f (U) ⊂ N. A manifold morphism f is called the immersion (resp. submersion) if it is an immersion (resp. submersion) at all points of M. A submersion is necessarily an open map, i.e., it sends open subsets of M onto open subsets of N. If an immersion f is open (i.e., f is a homeomorphism onto f (M) equipped with the relative topology from N), it is termed the imbedding. A pair (M, f ) is called a submanifold of N if f is an injective immersion. A submanifold (M, f ) is an imbedded submanifold if f is an imbedding. For the sake of simplicity, we usually identify (M, f ) with f (M).If M ⊂ N, its natural injection is denoted by iM : M → N. There are the following criteria for a submanifold to be imbedded.

Theorem B.1 Let (M, f ) be a submanifold of N.

(i) The map f is an imbedding if and only if, for each point p ∈ M, there exists a (cubic) coordinate chart (V,ψ) of N centered at f (p) so that f (M) ∩ V consists of all points of V with coordinates (x 1,...,xm , 0,...,0). Appendix B: Differential Calculus on Fibre Bundles 229

(ii) Suppose that f : M → N is a proper map, i.e., the pre-images of compact sets are compact. Then (M, f ) is a closed imbedded submanifold of N. In particular, this occurs if M is a compact manifold. (iii) If dim M = dim N, then (M, f ) is an open imbedded submanifold of N.

Fibred Manifolds and Fibre Bundles

Deﬁnition B.1 Atriple

π : Y → X, dim X = n > 0, (B.1) is called a fibred manifold if a manifold morphism π is a surjective submersion, i.e., the tangent morphism T π : TY → TXis a surjection. One says that Y is a total space of a fibred manifold (B.1), X is its base, π is a −1 fibration, and Yx = π (x) is a fibre over x ∈ X. Any fibre is an imbedded submanifold of Y of dimension dim Y − dim X. Unless otherwise stated, we assume that dim Y = dim X, i.e., fibred manifolds with discrete fibres are not considered. Theorem B.2 A surjection (B.1) is a fibred manifold if and only if a manifold λ i λ Y admits an atlas of coordinate charts (UY ; x , y ) such that (x ) are coordi- λ λ μ nates on π(UY ) ⊂ X and coordinate transition functions read x = f (x ), y i = f i (xμ, y j ). These coordinates are termed fibred coordinates compatible with a fibration π.

By a local section of the surjection (B.1) is meant an injection s : U → Y of an open subset U ⊂ X such that π ◦ s = Id U, i.e., a section sends any point x ∈ X into a fibre Yx over this point. A local section also is defined over any subset N ∈ X as the restriction to N of a local section over an open set containing N.If U = X, one calls s the global section. Hereafter, by a section is meant both a global section and a local section (over an open subset). Theorem B.3 The surjection π (B.1) is a fibred manifold if and only if, for each point y ∈ Y , there exists a local section s of π : Y → X passing through y.

The range s(U) of a local section s : U → Y of a fibred manifold Y → X is an imbedded submanifold of Y .Italsoisaclosed map, which sends closed subsets of U onto closed subsets of Y .Ifs is a global section, then s(X) is a closed imbedded submanifold of Y . Global sections of a fibred manifold need not exist. Theorem B.4 Let Y → X be a fibred manifold whose fibres are diffeomorphic to Rm . Any its section over a closed imbedded submanifold (e.g., a point) of X is extended to a global section [141]. In particular, such a fibred manifold always has a global section. 230 Appendix B: Differential Calculus on Fibre Bundles

λ i Given ﬁbred coordinates (UY ; x , y ), a section s of a ﬁbred manifold Y → X is i i represented by collections of local functions {s = y ◦ s} on π(UY ).

Definition B.2 A fibred manifold Y → X is called a fibre bundle if it admits a −1 λ i −1 fibred coordinate atlas {(π (Uξ ); x , y )} over a cover {π (Uι)} of Y which is the inverse image of a cover U ={Uξ } of X.

In this case, there exists a manifold V , called the typical ﬁbre, such that Y is locally diffeomorphic to the splittings

−1 ψξ : π (Uξ ) → Uξ × V, (B.2) glued together by means of transition functions

−1 ξζ = ψξ ◦ ψζ : Uξ ∩ Uζ × V → Uξ ∩ Uζ × V (B.3) on overlaps Uξ ∩ Uζ . Transition functions ξζ fulﬁl the cocycle condition

ξζ ◦ ζκ = ξκ (B.4) on all overlaps Uξ ∩ Uζ ∩ Uι. Restricted to a point x ∈ X, trivialization morphisms ψξ (B.2) and transition functions ξζ (B.3) deﬁne diffeomorphisms of ﬁbres

ψξ (x) : Yx → V, x ∈ Uξ , (B.5) ξζ(x) : V → V, x ∈ Uξ ∩ Uζ . (B.6)

Trivialization charts (Uξ ,ψξ ) together with transition functions ξζ (B.3) constitute a bundle atlas Ψ ={(Uξ ,ψξ ), ξζ} (B.7) of a fibre bundle Y → X. Two bundle atlases are said to be equivalent if their union also is a bundle atlas, i.e., there exist transition functions between trivialization charts of different atlases. A fibre bundle Y → X is uniquely defined by a bundle atlas. Given the atlas Ψ (B.7), there is a unique manifold structure on Y for which π : Y → X is a fibre bundle with a typical fibre V and a bundle atlas Ψ . All atlases of a fibre bundle are equivalent.

Remark B.2 The notion of a fibre bundle (Definition B.2) is the notion of a smooth locally trivial fibre bundle. In a general setting, a continuous fibre bundle is defined as a continuous surjective submersion of topological spaces Y → X. A continuous map π : Y → X is termed a submersion if, for any point y ∈ Y , there exists an open neighborhood U of the point π(y) and a right continuous inverse σ : U → Y of π through y so that π ◦ σ = Id U and σ ◦ π(y) = y, i.e., there exists a local section σ of π and π is open. The notion of a locally trivial continuous fibre bundle is a Appendix B: Differential Calculus on Fibre Bundles 231 repetition of that of a smooth fibre bundle, where trivialization morphisms ψξ and transition functions ξζ are continuous.

We have the following useful criteria for a ﬁbred manifold to be a ﬁbre bundle.

Theorem B.5 If a fibration π : Y → X is a proper map, then Y → X is a fibre bundle. In particular, a fibred manifold with a compact total space is a fibre bundle.

Theorem B.6 A fibred manifold whose fibres are diffeomorphic either to a compact manifold or Rr is a fibre bundle [105].

A comprehensive relation between fibred manifolds and fibre bundles is given in Remark B.11. It involves the notion of an Ehresmann connection. Unless otherwise stated, we restrict our consideration to fibre bundles. Without a loss of generality, we further assume that a cover U for a bundle atlas of Y → X also is a cover for a manifold atlas of the base X. Then, given the bundle atlas Ψ (B.7), a fibre bundle Y is provided with the associated bundle coordinates

λ λ i i −1 x (y) = (x ◦ π)(y), y (y) = (y ◦ ψξ )(y), y ∈ π (Uξ ),

λ i where x are coordinates on Uξ ⊂ X and y , called the fibre coordinates, are coordinates on a typical fibre V . The forthcoming Theorems B.7 and B.9 describe the particular covers which one can choose for a bundle atlas. A fibred manifold Y → X is termed trivial if it is diffeomorphic to a product X × V .

Theorem B.7 Any ﬁbre bundle over a contractible base is trivial.

However, a ﬁbred manifold over a contractible base need not be trivial, even its ﬁbres are mutually diffeomorphic [53]. It follows from Theorem B.7 that any cover of a base X consisting of domains (i.e., contractible open subsets) is a bundle cover.

Theorem B.8 Every ﬁbre bundle Y → X admits a bundle atlas over a countable cover U of X where each member Uξ of U is a domain whose closure is compact [67].

If a base X is compact, there is a bundle atlas of Y over a ﬁnite cover of X which obeys the condition of Theorem B.8.

Theorem B.9 Every ﬁbre bundle Y → X admits a bundle atlas over a ﬁnite cover of X, but its members need not be contractible and connected [61].

Morphisms of fibre bundles, by definition, are fibrewise morphisms, sending a fibre to a fibre. Namely, a bundle morphism of a fibre bundle π : Y → X to a fibre bundle π : Y → X is defined as a pair (Φ, f ) of manifold morphisms which form a commutative diagram 232 Appendix B: Differential Calculus on Fibre Bundles

Φ Y −→ Y π π ,π ◦ Φ = f ◦ π. (B.8) ? ? f X −→ X

Bundle injections and surjections are called bundle monomorphisms and epimorphisms, respectively. A bundle diffeomorphism is termed the bundle isomorphism, or the bundle automorphism if it is an isomorphism to itself. For the sake of brevity, a bundle morphism over f = Id X often is said to be the bundle morphism over X, and is denoted by Y −→ Y . In particular, a bundle automorphism over X is called X the vertical automorphism. A bundle monomorphism Φ : Y → Y over X is called the subbundle of a fibre bundle Y → X if Φ(Y ) is a submanifold of Y . There is the following useful criterion for an image and an inverse image of a bundle morphism to be subbundles. Theorem B.10 Let Φ : Y → Y be a bundle morphism over X. Given a global section s of the fibre bundle Y → X such that s(X) ⊂ Φ(Y ), by the kernel of a bundle morphism Φ with respect to a section s is meant the inverse image Ker s Φ = Φ−1(s(X)) of s(X) by Φ.IfΦ : Y → Y is a bundle morphism of constant rank over X, then Φ(Y ) and Ker s Φ are subbundles of Y and Y , respectively. There are the following standard constructions of new fibre bundles from the old ones. • Given a fibre bundle π : Y → X and a manifold morphism f : X → X,bythe pull-back of Y by f is termed the manifold

f ∗Y ={(x , y) ∈ X × Y : π(y) = f (x )}

together with the natural projection (x , y) → x . It is a ﬁbre bundle over X such that a ﬁbre of f ∗Y over a point x ∈ X is that of Y over the point f (x ) ∈ X. There is the canonical bundle morphism

∗ fY : f Y (x , y)|π(y)= f (x ) → y ∈ Y. f

Any section s of a ﬁbre bundle Y → X yields the pull-back section f ∗s(x ) = (x , s( f (x )) of f ∗Y → X .

• If X ⊂ X is a submanifold of X and i X is the corresponding natural injection, ∗ = | then the pull-back bundle iX Y Y X is called the restriction of a ﬁbre bundle Y to a submanifold X ⊂ X.IfX is an imbedded submanifold, any section of the

pull-back bundle Y |X → X is the restriction to X of some section of Y → X. • Let π : Y → X and π : Y → X be ﬁbre bundles over the same base X. Their

bundle product Y ×X Y over X is deﬁned as the pull-back

∗ Y × Y = π ∗Y or Y × Y = π Y X X Appendix B: Differential Calculus on Fibre Bundles 233

together with its natural surjection onto X. Fibres of the bundle product Y × Y × are the Cartesian products Yx Yx of ﬁbres of ﬁbre bundles Y and Y . Let us consider a composition

π : Y → Σ → X (B.9) of ﬁbre bundles

πY Σ : Y → Σ, (B.10)

πΣ X : Σ → X. (B.11)

One can show that it is a fibre bundle, called the composite bundle. It is provided with bundle coordinates (xλ,σm , yi ), where (xλ,σm ) are bundle coordinates on the fibre bundle (B.11), i.e., transition functions of coordinates σ m are independent of coordinates yi . For instance, the tangent bundle TY of a fibre bundle Y → X is a composite bundle TY → Y → X. The following fact makes composite bundles useful for applications.

Theorem B.11 Given the composite bundle (B.9), let h be a global section of the ﬁbre bundle Σ → X(B.11). Then the restriction

Y h = h∗Y (B.12) of the ﬁbre bundle Y → Σ (B.10)toh(X) ⊂ Σ is a subbundle of the ﬁbre bundle Y → X.

Vector and Afﬁne Bundles

A vector bundle is a ﬁbre bundle Y → X such that:

−1 • its typical fibre V and all the fibres Yx = π (x), x ∈ X, are real finite-dimensional vector spaces; • there is the bundle atlas Ψ (B.7)ofY → X whose trivialization morphisms ψξ (B.5) and transition functions ξζ (B.6) are linear isomorphisms. Accordingly, a vector bundle is provided with linear fibre coordinates (yi ) pos- i = i ( ) j sessing linear transition functions y A j x y .Wehave

i i −1 y = y ei (π(y)) = y ψξ (π(y)) (ei ), π(y) ∈ Uξ , where {ei } is a fixed basis for the typical fibre V of Y , and {ei (x)} are the fibre bases (or the frames) for the fibres Yx of Y associated to the bundle atlas Ψ . 234 Appendix B: Differential Calculus on Fibre Bundles

By virtue of Theorem B.4, any vector bundle has a global section, e.g., the canonical global zero-valued section0(x) = 0. Global sections of a vector bundle Y → X constitute a projective C∞(X)-module Y (X) of finite rank. It is called the structure module of a vector bundle. Theorem B.12 Let a vector bundle Y → X admit m = dim V nowhere vanishing m global sections si which are linearly independent, i.e., ∧ si = 0. Then Y is trivial. By a morphism of vector bundles is meant a linear bundle morphism, which is a linear fibrewise map whose restriction to each fibre is a linear map. Given a linear bundle morphism Φ : Y → Y of vector bundles over X, its kernel Ker Φ is defined as the inverse image Φ−1(0(X)) of the canonical zero-valued section 0(X) of Y . By virtue of Theorem B.10,ifΦ is of constant rank, its kernel and its range are vector subbundles of the vector bundles Y and Y , respectively. For instance, monomorphisms and epimorphisms of vector bundles fulfil this condition. There are the following particular constructions of new vector bundles from the old ones. • Let Y → X be a vector bundle with a typical fibre V .ByY ∗ → X is denoted the dual vector bundle with the typical fibre V ∗ dual of V .Aninterior product of Y and Y ∗ is defined as a fibred morphism

:Y ⊗ Y ∗ −→ X × R. X

• Let Y → X and Y → X be vector bundles with typical ﬁbres V and V , respec-

tively. Their Whitney sum Y ⊕X Y is a vector bundle over X with a typical ﬁbre V ⊕ V . • Let Y → X and Y → X be vector bundles with typical ﬁbres V and V , respec-

tively. Their tensor product Y ⊗X Y is a vector bundle over X with a typical ﬁbre V ⊗ V . Similarly, the exterior product of vector bundles Y ∧ Y is deﬁned. The X exterior product

2 k ∧ Y = X × R ⊕ Y ⊕ ∧ Y ⊕ ···∧ Y, k = dim Y − dim X, (B.13) X X X

is called the exterior bundle. Remark B.3 Given vector bundles Y and Y over the same base X, every linear bundle morphism Φ : { ( )}→{Φk ( ) ( )}∈ Yx ei x i x ek x Yx

over X deﬁnes a global section Φ : → Φk ( ) i ( ) ⊗ ( ) x i x e x ek x

of the tensor product Y ⊗ Y ∗, and vice versa. Appendix B: Differential Calculus on Fibre Bundles 235

i j A sequence Y −→ Y −→ Y of vector bundles over the same base X is called exact at Y if Ker j = Im i. A sequence of vector bundles

i j 0 → Y −→ Y −→ Y → 0(B.14) over X is said to be a short exact sequence if it is exact at all terms Y , Y , and Y . This means that i is a bundle monomorphism, j is a bundle epimorphism, and Ker j = Im i. Then Y is the quotient bundle Y/Y whose structure module is the quotient Y (X)/Y (X) of the structure modules of Y and Y . Given an exact sequence of vector bundles (B.14), there is the exact sequence of their duals

j ∗ i ∗ 0 → Y ∗ −→ Y ∗ −→ Y ∗ → 0.

One says that an exact sequence (B.14)issplit if there exists a bundle monomorphism s : Y → Y such that j ◦ s = Id Y or, equivalently,

Y = i(Y ) ⊕ s(Y ) = Y ⊕ Y .

The following is a corollary of Theorem A.2 and Serre–Swan Theorem A.10.

Theorem B.13 Every exact sequence of vector bundles (B.14) is split.

The tangent bundle TZand the cotangent bundle T ∗ Z of a manifold Z exemplify vector bundles.

Remark B.4 Given an atlas ΨZ ={(Uι,φι)} of a manifold Z, the tangent bundle is provided with the holonomic bundle atlas

ΨT ={(Uι,ψι = T φι)}, (B.15) where T φι is the tangent morphism to φι. The associated linear ﬁbre coordinates are λ holonomic (or induced) coordinates (z˙ ) with respect to the holonomic frames {∂λ} in tangent spaces Tz Z.

A tensor product of tangent and cotangent bundles

m k T = (⊗ TZ) ⊗ (⊗ T ∗ Z), m, k ∈ N, (B.16)

α ···α is called the tensor bundle, provided with holonomic ﬁbre coordinates x˙ 1 m pos- β1···βk sessing transition functions

α α ν ν α ···α ∂x 1 ∂x m ∂x 1 ∂x k μ ···μ ˙ 1 m = ··· ··· ˙ 1 m . xβ ···β μ μ β β xν ···ν 1 k ∂x 1 ∂x m ∂x 1 ∂x k 1 k

Its sections are termed tensor ﬁelds or (m, k)-tensor ﬁelds. 236 Appendix B: Differential Calculus on Fibre Bundles

Let πY : TY → Y be the tangent bundle of a ﬁbre bundle π : Y → X.Given bundle coordinates (xλ, yi ) on Y , it is equipped with the holonomic coordinates (xλ, yi , x˙λ, y˙i ) with transition functions

∂x λ ∂y i ∂y i x˙ λ = x˙μ, y˙ i = y˙ j + x˙μ. ∂xμ ∂y j ∂xμ

Definition B.3 The vertical tangent bundle of a fibre bundle Y → X is defined to be a subbundle VY = Ker (T π) of the tangent bundle TY → Y which consists of vectors tangent to fibres of Y .

A vertical tangent bundle is provided with the holonomic bundle coordinates λ i i i (x , y , y =˙y |x˙λ=0) with respect to the vertical frames {∂i }. They possess transition functions

∂y i y i = y j . ∂y j

Every bundle morphism Φ : Y → Y yields a linear bundle morphism

∂Φi V Φ : VY → VY , y i ◦ V Φ = y j (B.17) Φ ∂y j of the vertical tangent bundles. It is called the vertical tangent morphism. In many important cases, the vertical tangent bundle VY → Y of a ﬁbre bundle Y → X is trivial, and it is isomorphic to the bundle product

VY = Y × Y (B.18) X where Y → X is some vector bundle. It follows that VY can be provided with bundle coordinates (xλ, yi , yi ) such that transition functions of ﬁbre coordinates yi are independent of coordinates yi . One calls (B.18)thevertical splitting. For instance, every vector bundle Y → X admits the canonical vertical splitting V Y = Y ⊕X Y . Let T ∗Y → Y be the cotangent bundle of a ﬁbre bundle Y → X. It is provided λ i with the holonomic coordinates (x , y , x˙λ, y˙i ) with transition functions

μ j j ∂x ∂y ∂y x˙ = x˙μ + y˙ , y˙ = y˙ . λ ∂x λ ∂x λ j i ∂y i j

Definition B.4 The vertical cotangent bundle V ∗Y → Y of a fibre bundle Y → X is defined as the dual of the vertical tangent bundle VY → Y (Definition B.3). Appendix B: Differential Calculus on Fibre Bundles 237

It is not a subbundle of the cotangent bundle T ∗Y , but there is the canonical surjection ∗ λ i i ∗ ζ : T Y ˙xλdx +˙yi dy →˙yi dy ∈ V Y, (B.19) where {dyi }, possessing transition functions

∂y i dy i = dyj , ∂y j are the duals of the holonomic frames {∂i } of the vertical tangent bundle VY.The λ i vertical cotangent bundle is endowed with the holonomic coordinates (x , y , pi = y˙i ) possessing transition functions

∂y j p = p . (B.20) i ∂y i j

For any ﬁbre bundle Y , there exist the exact sequences of vector bundles

π 0 → VY −→ TY −→T Y × TX → 0, (B.21) X 0 → Y × T ∗ X → T ∗Y → V ∗Y → 0. (B.22) X

Their splitting, by definition, is a connection on Y → X (Sect.B.3). Let π : Y → X be a vector bundle with a typical fibre V .Anaffine bundle modelled over a vector bundle Y → X is a fibre bundle π : Y → X whose typical fibre V is an affine space modelled over V such that the following conditions hold.

• All the fibres Yx of Y are affine spaces modelled over the corresponding fibres Y x of the vector bundle Y . • There is an affine bundle atlas Ψ ={(Uα,ψχ ), χζ} of Y → X whose local trivializations morphisms ψχ (B.5) and transition functions χζ (B.6) are affine isomorphisms. Dealing with affine bundles, we use only affine fibre coordinates (yi ) associated to an affine bundle atlas Ψ . There are the bundle morphisms

Y × Y −→ Y,(yi , yi ) → yi + yi , X X Y × Y −→ Y ,(yi , y i ) → yi − y i , X X where (yi ) are linear fibre coordinates on a vector bundle Y . By virtue of Theorem B.4, affine bundles admit global sections, but in contrast with vector bundles, there is not their canonical global section. Every global section s of an affine bundle Y → X modelled over a vector bundle Y → X yields the bundle morphisms 238 Appendix B: Differential Calculus on Fibre Bundles

Y y → y − s(π(y)) ∈ Y , (B.23) Y y → s(π(y)) + y ∈ Y. (B.24)

In particular, every vector bundle Y has a natural structure of an affine bundle owing to the morphisms (B.24) where s = 0 is the canonical zero-valued section of Y . Due to the morphism (B.23), any affine bundle Y → X admits fibre coordinates yi = yi − si (π(y)), possessing linear transition functions. By a morphism of affine bundles is meant an affine bundle morphism Φ : Y → Y whose restriction to each fibre of Y is an affine map. Every affine bundle morphism Φ : Y → Y of an affine bundle Y modelled over a vector bundle Y to an affine bundle Y modelled over a vector bundle Y yields an unique linear bundle morphism

∂Φi Φ : Y → Y , y i ◦ Φ = y j , ∂y j called the linear derivative of Φ. Every affine bundle Y → X modelled over a vector bundle Y → X admits the canonical vertical splitting V Y = Y ×X Y . Let us note that Theorems B.8 and B.9 on a particular cover for bundle atlases remain true in the case of linear and affine atlases of vector and affine bundles.

Vector Fields

Vector ﬁelds on a manifold Z are global sections of the tangent bundle TZ → Z of Z. The set T (Z) of vector ﬁelds on Z is both a C∞(Z)-module and a real Lie algebra with respect to the Lie bracket

λ μ λ μ λ λ [v, u]=(v ∂λu − u ∂λv )∂μ, u = u ∂λ, v = v ∂λ.

Given a vector field u on X,acurve c : R ⊃ (, ) → Z in Z is said to be the integral curve of u if Tc = u(c). Every vector field u on a manifold Z can be seen as an infinitesimal generator of a local one-parameter group of local diffeomorphisms (a flow), and vice versa [82]. One-dimensional orbits of this group are integral curves of u. A vector field is termed complete if its flow is a one-parameter group of diffeomorphisms of Z. For instance, every vector field on a compact manifold is complete. A vector field u on a fibre bundle Y → X is called projectable if it projects onto a vector field on X, i.e., there exists a vector field τ on X such that τ ◦ π = T π ◦ u. A projectable vector field takes a coordinate form

λ μ i μ j λ u = u (x )∂λ + u (x , y )∂i ,τ= u ∂λ. (B.25) Appendix B: Differential Calculus on Fibre Bundles 239

Its flow is a local one-parameter group of automorphisms of Y → X over a local one-parameter group of diffeomorphisms of X whose infinitesimal generator is τ. A projectable vector field is termed vertical if its projection onto X vanishes, i.e., if it lives into the vertical tangent bundle VY. Let γ : T (X) → T (Y ) (B.26)

λ be an R-linear module morphism which sends a vector ﬁeld τ = τ ∂λ on a base X onto a projectable vector ﬁeld

λ i γτ = τ ∂λ + (γ τ) ∂i (B.27) on Y over τ.

Definition B.5 The module morphism (B.26) is called the lift of vector fields on a base X onto a fibre bundle Y .

In particular, a vector field τ on a base X of a fibre bundle Y → X gives rise to the horizontal vector field Γτ (B.55)onY by means of a connection Γ on this bundle.

Definition B.6 The lift γ (B.26) of vector fields is said to be functorial if it is a monomorphism of a real Lie algebra T (X) of vector fields on X toarealLie algebra T (Y ) of vector fields on Y , i.e.,

[γτ,γτ ]=γ [τ,τ ],τ,τ ∈ T (X). (B.28)

For instance, the above mentioned lift Γτof vector fields by means of a connection Γ is functorial if and only if this connection is flat (Theorem B.16). Every tensor bundle (B.16) admits the canonical functorial lift of vector fields

μ α να2···αm ν α1···αm ˙ β ···β τ = τ ∂μ +[∂ν τ 1 x˙ +···−∂β τ x˙ −···]∂ 1 k , (B.29) β1···βk 1 νβ2···βk α1···αm

˙ λ where we employ the compact notation ∂λ = ∂/∂x˙ . In particular, we have the canonical functorial lifts ∂ ∂ τ = τ μ∂ + ∂ τ α ˙ν , τ = τ μ∂ − ∂ τ ν ˙ μ ν x α μ β xν (B.30) ∂x˙ ∂x˙β of a vector ﬁeld τ on X onto the tangent and cotangent bundles TXand T ∗ X.

Definition B.7 A fibre bundle admitting functorial lift of vector fields on its base is called the natural bundle [83, 145]. 240 Appendix B: Differential Calculus on Fibre Bundles

Differential Forms

An exterior r-form on a manifold Z is deﬁned as a section

1 λ λ φ = φλ ...λ dz 1 ∧···∧dz r r! 1 r

r of the exterior product ∧ T ∗ Z → Z, where

λ λ 1 λ ...λ μ μ dz 1 ∧···∧dz r = ε 1 r μ ...μ dx 1 ⊗···⊗dx r , r! 1 r ...λ ...λ ...... λ ...λ ...... λ ...λ ... ε i j =−ε j i =−ε i j , ...μp...μk ...... μp...μk ...... μk ...μp... λ ...λ ε 1 r = . λ1...λr 1

Let Or (Z) denote a vector space of exterior r-forms on a manifold Z. By deﬁn- ition, O0(Z) = C∞(Z) is a ring of smooth real functions on Z. All exterior forms on Z constitute an N-graded commutative algebra O∗(Z) of global sections of the exterior bundle ∧T ∗ Z (B.13). It is endowed with an exterior product

1 λ λ 1 μ μ φ = φλ ...λ dz 1 ∧···∧dz r ,σ= σμ ...μ dz 1 ∧···∧dz s , r! 1 r s! 1 s

1 ν ν + φ ∧ σ = φν ...ν σν ...ν dz 1 ∧···∧dz r s = r!s! 1 r r+1 r+s

1 ν ...ν + α α + ε 1 r s α ...α φν ...ν σν ...ν dz 1 ∧···∧dz r s , r!s!(r + s)! 1 r+s 1 r r+1 r+s φ ∧ σ = (−1)|φ||σ |σ ∧ φ, where the symbol |φ| stands for the form degree. A graded algebra O∗(Z) also is provided with an exterior differential

μ 1 μ λ λ dφ = dz ∧ ∂μφ = ∂μφλ ...λ dz ∧ dz 1 ∧···∧dz r (B.31) r! 1 r which obeys the relations

d ◦ d = 0, d(φ ∧ σ) = d(φ) ∧ σ + (−1)|φ|φ ∧ d(σ ).

The exterior differential d (B.31) brings O∗(Z) into a DGA which is the minimal Chevalley–Eilenberg differential calculus O∗A over a real ring A = C∞(Z).Itsde Rham complex is (A.18). Given a manifold morphism f : Z → Z , any exterior k-form φ on Z yields the pull-back exterior form f ∗φ on Z given by a condition

f ∗φ(v1,...,vk )(z) = φ(Tf(v1),...,Tf(vk ))( f (z)) Appendix B: Differential Calculus on Fibre Bundles 241

1 k for an arbitrary collection of tangent vectors v ,...,v ∈ Tz Z. We have the relations

f ∗(φ ∧ σ) = f ∗φ ∧ f ∗σ, df∗φ = f ∗(dφ).

In particular, given a ﬁbre bundle π : Y → X, the pull-back onto Y of exterior forms on X by π provides a monomorphism of graded commutative algebras O∗(X) → O∗(Y ). Elements of its range π ∗O∗(X) are called basic forms. Exterior forms

r ∗ 1 λ λ φ : Y → ∧ T X,φ= φλ ...λ dx 1 ∧···∧dx r , r! 1 r on Y such that uφ = 0 for an arbitrary vertical vector ﬁeld u on Y aresaidtobe horizontal forms. Horizontal forms of degree n = dim X are termed densities.We use for them the compact notation

1 μ μ L = Lμ ...μ dx 1 ∧···∧dx n = L ω, L = L ... , n! 1 n 1 n 1 n 1 μ μ ω = dx ∧···∧dx = εμ ...μ dx 1 ∧···∧dx n , (B.32) n! 1 n ωλ = ∂λω, ωμλ = ∂μ∂λω,

ε ε = where is the skew-symmetric Levi–Civita symbol with a component μ1...μn 1. The interior product (or contraction) of a vector ﬁeld u and an exterior r-form φ on a manifold Z is given by a coordinate expression

r − (− )k 1 λ 1 λ λ k λ uφ = u k φλ ...λ ...λ dz 1 ∧···∧dz ∧···∧dz r = r! 1 k r k=1 1 μ α α u φμα ...α dz 2 ∧···∧dz r , (r − 1)! 2 r where the caretdenotes omission. It obeys relations

|φ| φ(u1,...,ur ) = ur ···u1φ, u(φ ∧ σ) = uφ ∧ σ + (−1) φ ∧ uσ.

The Lie derivative of an exterior form φ along a vector ﬁeld u is

Luφ = udφ + d(uφ), Lu(φ ∧ σ) = Luφ ∧ σ + φ ∧ Luσ.

It is a derivation of the graded algebra O∗(Z) such that

Lu ◦ Lu − Lu ◦ Lu = L[u,u ].

In particular, if f is a function, then Lu f = u( f ) = udf. 242 Appendix B: Differential Calculus on Fibre Bundles

Remark B.5 An exterior form φ is invariant under a local one-parameter group of diffeomorphisms G(t) of Z (i.e., G(t)∗φ = φ) if and only if its Lie derivative along the inﬁnitesimal generator u of this group vanishes, i.e., Luφ = 0.

A tangent-valued r-form on a manifold Z is deﬁned as a section

1 μ λ1 λr φ = φλ ...λ dz ∧···∧dz ⊗ ∂μ r! 1 r

r of a tensor bundle ∧ T ∗ Z ⊗ TZ → Z.

Remark B.6 There is one-to-one correspondence between the tangent-valued one- forms φ on a manifold Z and the linear bundle endomorphisms φ : TZ → TZ, φ : Tz Z v → vφ(z) ∈ Tz Z, (B.33) φ∗ : ∗ → ∗ , φ∗ : ∗ ∗ → φ( ) ∗ ∈ ∗ , T Z T Z Tz Z v z v Tz Z (B.34) over Z (Remark B.3). For instance, the canonical tangent-valued one-form

λ θZ = dz ⊗ ∂λ (B.35) on Z corresponds to the identity morphisms (B.33) and (B.34).

Remark B.7 Let Z = TX, and let TTX be the tangent bundle of TX. There is a bundle endomorphism ˙ ˙ J(∂λ) = ∂λ, J(∂λ) = 0 of TTX over X. It corresponds to the canonical tangent-valued form

λ ˙ θJ = dx ⊗ ∂λ (B.36) on the tangent bundle TX. It is readily observed that J ◦ J = 0.

A space O∗(Z) ⊗ T (Z) of tangent-valued forms is provided with the Frölicher– Nijenhuis bracket

r s r+s [ , ]FN : O (Z) ⊗ T (Z) × O (Z) ⊗ T (Z) → O (Z) ⊗ T (Z),

[α ⊗ u,β⊗ v]FN = (α ∧ β) ⊗[u, v]+(α ∧ Luβ) ⊗ v − (B.37) r r (Lvα ∧ β) ⊗ u + (−1) (dα ∧ uβ) ⊗ v + (−1) (vα ∧ dβ) ⊗ u, α ∈ Or (Z), β ∈ Os (Z), u, v ∈ T (Z). Appendix B: Differential Calculus on Fibre Bundles 243

Its coordinate expression is

1 ν μ ν μ [φ,σ]FN = (φλ ...λ ∂ν σλ ...λ − σλ ...λ ∂ν φλ ...λ − r!s! 1 r r+1 r+s r+1 r+s 1 r μ ν μ ν λ λ + rφ ∂λ σ + sσ ∂λ φ )dz 1 ∧···∧dz r s ⊗ ∂μ, λ1...λr−1ν r λr+1...λr+s νλr+2...λr+s r+1 λ1...λr φ ∈ Or (Z) ⊗ T (Z), σ ∈ Os (Z) ⊗ T (Z).

There are the relations

|φ||ψ|+1 [φ,σ]FN = (−1) [σ, φ]FN, |φ||σ | [φ,[σ, θ]FN]FN =[[φ,σ]FN,θ]FN + (−1) [σ, [φ,θ]FN]FN. (B.38)

Given a tangent-valued form θ,theNijenhuis differential on O∗(Z) ⊗ T (Z) is deﬁned as a morphism

∗ dθ : ψ → dθ ψ =[θ,ψ]FN,ψ∈ O (Z) ⊗ T (Z). (B.39)

By virtue of the relation (B.38), it has a property

|φ||ψ| dφ[ψ, θ]FN =[dφψ, θ]FN + (−1) [ψ, dφθ]FN.

In particular, if θ = u is a vector ﬁeld, the Nijenhuis differential is a Lie derivative of tangent-valued forms

1 ν μ ν μ Luσ = duσ =[u,σ]FN = (u ∂ν σλ ...λ − σλ ...λ ∂ν u + s! 1 s 1 s μ ν λ λ s sσ ∂λ u )dx 1 ∧···∧dx s ⊗ ∂μ,σ∈ O (Z) ⊗ T (Z). νλ2...λs 1

Let Y → X be a ﬁbre bundle. In the book, we deal with the following particular subspaces of the space O∗(Y ) ⊗ T (Y ) of tangent-valued forms on Y : • horizontal tangent-valued forms

r 1 μ ∗ λ1 λr i φ : Y → ∧ T X ⊗ TY,φ= dx ∧···∧dx ⊗ [φλ ...λ (y)∂μ + φλ ...λ (y)∂i ], Y r! 1 r 1 r

• vertical-valued forms

r 1 ∗ i λ1 λr φ : Y → ∧ T X ⊗ VY,φ= φλ ...λ (y)dx ∧···∧dx ⊗ ∂i , Y r! 1 r

i λ • vertical-valued one-forms, called the soldering forms, σ = σλ(y)dx ⊗ ∂i , i λ • basic soldering forms σ = σλ(x)dx ⊗ ∂i .

Remark B.8 The tangent bundle TXis provided with the canonical basic soldering form θJ (B.36). Due to the canonical vertical splitting 244 Appendix B: Differential Calculus on Fibre Bundles

VTX = TX× TX, (B.40) X the canonical soldering form (B.36)onTX deﬁnes the canonical tangent-valued form θX (B.35)onX. By this reason, tangent-valued one-forms on a manifold X also are called the soldering forms.

Let us also mention TX-valued forms

r 1 μ ∗ λ1 λr φ : Y → ∧ T X ⊗ TX,φ= φλ ...λ dx ∧···∧dx ⊗ ∂μ, (B.41) Y r! 1 r and V ∗Y -valued forms

r ∗ ∗ 1 λ λ φ : → ∧ ⊗ ,φ= φ 1 ∧···∧ r ⊗ i . Y T X V Y λ1...λr i dx dx dy (B.42) Y r!

Let us note that (B.41) are not tangent-valued forms, while (B.42) are not exterior forms. They exemplify vector-valued forms. Given a vector bundle E → X,bya k ∗ ∗ E-valued k-form on X is meant a section of a ﬁbre bundle (∧ T X) ⊗X E → X.

B.2 Jet Manifolds

This section addresses first and second order jet manifolds of sections of fibre bundles (see Sect.B.4) for higher order jets) [53, 83, 89, 133, 138]. Given a fibre bundle Y → X with bundle coordinates (xλ, yi ), let us consider 1 the equivalence classes jx s of its sections s, which are identified by their values i i s (x) and the values of their partial derivatives ∂μs (x) at a point x ∈ X.Theyare called the first order jets of sections at x. Let us note that the definition of jets is 1 1 ∈ coordinate-independent. A key point is that the set J Y of first order jets jx s, x X, λ i i is a smooth manifold with respect to the adapted coordinates (x , y , yλ) such that ∂ μ i 1 i i x j i yλ( j s) = ∂λs (x), y λ = (∂μ + yμ∂ j )y . (B.43) x ∂x λ

Remark B.9 Let us note that there are different notions of jets. Jets of sections are the particular jets of maps [83, 110] and the jets of submanifolds [53, 85]. Let us also mention the jets of modules over a commutative ring [85, 131] which are representative objects of differential operators on modules. In particular, given a manifold X, the jets of a projective C∞(X)-module P of ﬁnite rank are exactly the jets of sections of the vector bundle over X whose module of sections is P in accordance with the Serre–Swan Theorem A.10. The notion of jets is extended to modules over graded commutative rings, though the jets of modules over a noncommutative ring fails to be deﬁned [58, 131]. Appendix B: Differential Calculus on Fibre Bundles 245

Definition B.8 Endowed with a smooth manifold structure, a set of first order jets J 1Y is termed the first order jet manifold of a fibre bundle Y → X.

A jet manifold J 1Y admits the natural ﬁbrations

π 1 : 1 1 → ∈ , J Y jx s x X (B.44) π 1 : 1 1 → ( ) ∈ . 0 J Y jx s s x Y (B.45)

π 1 A glance at the transformation law (B.43) shows that 0 is an afﬁne bundle modelled over a vector bundle T ∗ X ⊗ VY → Y. (B.46) Y

π 1 π 1 It is convenient to call (B.44)thejet bundle, while 0 (B.45)issaidtobethe afﬁne jet bundle. Let us note that, if Y → X is a vector or an afﬁne bundle, the jet bundle π 1 (B.44)isso. Jets can be expressed in terms of familiar tangent-valued forms owing to the canonical imbeddings

1 ∗ λ i λ λ(1) : J Y → T X ⊗ TY,λ(1) = dx ⊗ (∂λ + yλ∂i ) = dx ⊗ dλ, (B.47) Y Y 1 ∗ i i λ i θ(1) : J Y → T Y ⊗ VY,θ(1) = (dy − yλdx ) ⊗ ∂i = θ ⊗ ∂i , (B.48) Y Y

i where dλ are said to be total derivatives, and θ are called local contact forms.

Remark B.10 We usually identify a jet manifold J 1Y with its images under the 1 = ( λ, i , i ) canonical morphisms (B.47) and (B.48), and represent the jets jx s x y yμ by the tangent-valued forms λ(1) (B.47) and θ(1) (B.48).

Sections and morphisms of ﬁbre bundles admit prolongations to jet manifolds as follows. Any section s of a ﬁbre bundle Y → X has the jet prolongation to a section

( 1 )( ) = 1 , i ◦ 1 = ∂ i ( ), J s x jx s yλ J s λs x (B.49) of a jet bundle J 1Y → X.Asection of a jet bundle J 1Y → X is termed integrable if it is the jet prolongation (B.49) of some section of a ﬁbre bundle Y → X. Any bundle morphism Φ : Y → Y (B.8) over a diffeomorphism f admits the jet prolongation to a bundle morphism of afﬁne jet bundles

∂( −1)μ 1 1 1 i 1 f i J Φ : J Y −→ J Y , y λ ◦ J Φ = dμΦ . (B.50) Φ ∂x λ

Any projectable vector field u (B.25) on a fibre bundle Y → X has the jet prolongation to a projectable vector field 246 Appendix B: Differential Calculus on Fibre Bundles

1 = λ∂ + i ∂ + ( i − i ∂ μ)∂λ, J u u λ u i dλu yμ λu i (B.51) on a jet manifold J 1Y . In order to obtain (B.51), the canonical bundle morphism

1 1 i i i μ r1 : J TY → TJ Y, y˙λ ◦ r1 = (y˙ )λ − yμx˙λ is used. In particular, there is the canonical isomorphism

1 1 i i VJ Y = J VY, y˙λ = (y˙ )λ.

Taking the ﬁrst order jet manifold of the jet bundle J 1Y → X, we obtain the repeated jet manifold J 1 J 1Y provided with the adapted coordinates λ i i i i (x , y , yλ,yμ, yμλ) possessing transition functions

∂ α ∂ α ∂ α i x i i x i i x i yλ = dα y , yλ = dα y , y μλ = μ dα y λ, ∂x λ ∂x λ ∂x = ∂ + j ∂ + j ∂ν , = ∂ + j ∂ + j ∂ν . dα α yα j yνα j dα α yα j yνα j

There exist the two following different affine fibrations of J 1 J 1Y over J 1Y : • the familiar affine jet bundle (B.45):

1 1 1 i i π11 : J J Y → J Y, yλ ◦ π11 = yλ,

• the afﬁne bundle

1π 1 : 1 1 → 1 , i ◦ 1π 1 = i . J 0 J J Y J Y yλ J 0 yλ

∈ 1 1 π ( ) = 1π 1( ) The points q J J Y , where 11 q J 0 q , form an affine subbundle J2Y → J 1Y of J 1 J 1Y called the sesquiholonomic jet manifold. It is given by i i λ i i i coordinate conditions yλ = yλ, and is coordinated by (x , y , yλ, yμλ).Asecond order jet manifold J 2Y of a fibre bundle Y → X can be defined as the affine subbundle 2 1 i i of the fibre bundle J Y → J Y by means of coordinate conditions yλμ = yμλ.It λ i i i i is endowed with adapted coordinates (x , y , yλ, yλμ = yμλ), possessing transition functions ∂ α ∂ α i x i i x i yλ = dα y , y μλ = μ dα y λ. ∂x λ ∂x

A second order jet manifold J 2Y also can be introduced as a set of equivalence 2 → classes jx s of sections s of a fibre bundle Y X, which are identified by their values and the values of their first and second order partial derivatives at points x ∈ X.The 2 equivalence classes jx s are called the second order jets of sections. Let s be a section of a fibre bundle Y → X, and let J 1s be its jet prolongation to a section of a jet bundle J 1Y → X. The latter gives rise to a section J 1 J 1s of a Appendix B: Differential Calculus on Fibre Bundles 247 repeated jet bundle J 1 J 1Y → X. This section lives in a second order jet manifold J 2Y . It is termed the second order jet prolongation of a section s, and is denoted by J 2s.

B.3 Connections on Fibre Bundles

There are several equivalent definitions of a connection on a fibre bundle. We start with the traditional notion of a connection as a splitting of the exact sequences (B.21) and (B.22) (Definition B.9), but then follow its Definition B.10 as a global section of the affine jet bundle (B.45)[83, 100, 133, 138].

Connections as Tangent-Valued Forms

Let Y → X be a ﬁbre bundle coordinated by (xλ, yi ) (Deﬁnition B.2).

Definition B.9 A connection on a fibre bundle Y → X is defined as a linear bundle monomorphism

λ λ i Γ : Y × TX → TY,Γ:˙x ∂λ →˙x (∂λ + Γλ ∂i ), (B.52) X over Y which splits the exact sequence (B.21), i.e., πT ◦ Γ = Id (Y ×X TX).

This also is a definition of connections on fibred manifolds (Remark B.11). i By virtue of Theorem B.13, a connection always exists. Local functions Γλ (y) in (B.52) are called the components of a connection Γ with respect to bundle coordinates (xλ, yi ) on Y → X. The image of Y ×X TX by a connection Γ defines the horizontal subbundle HY ⊂ TY which splits the tangent bundle TY as follows:

λ i λ i i λ i TY = HY ⊕ VY, x˙ ∂λ +˙y ∂i =˙x (∂λ + Γλ ∂i ) + (y˙ −˙x Γλ )∂i . (B.53) Y

i i λ Its annihilator is locally generated by one-forms dy − Γλ dx . The morphism Γ (B.52) uniquely yields a horizontal tangent-valued one-form

λ i Γ = dx ⊗ (∂λ + Γλ ∂i ) (B.54) on Y which projects onto the canonical tangent-valued form θX (B.35)onX. With this form, called the connection form, the morphism (B.52) reads

i Γ : ∂λ → ∂λΓ = ∂λ + Γλ ∂i . 248 Appendix B: Differential Calculus on Fibre Bundles

Given a connection Γ and the corresponding horizontal subbundle (B.53), a vector field u on a fibre bundle Y → X is termed horizontal if it lives in HY. A horizontal vector field takes a form

λ i u = u (y)(∂λ + Γλ ∂i ).

In particular, let τ be a vector ﬁeld on X. By means of the connection form Γ (B.54), we obtain a projectable horizontal vector ﬁeld

λ i Γτ = τΓ = τ (∂λ + Γλ ∂i ) (B.55) on Y , called the horizontal lift of τ. Conversely, any projectable horizontal vector ﬁeld u on Y is the horizontal lift Γτof its projection τ on X. Moreover, the horizontal distribution HY is generated by the horizontal lifts Γτ (B.55) of vector ﬁelds τ on X. The horizontal lift

T (X) τ → Γτ ∈ T (Y ) is a C∞(X)-linear module morphism. Given the splitting (B.52), the dual splitting of the exact sequence (B.22)is

∗ ∗ i i i λ Γ : V Y → T Y,Γ: dy → dy − Γλ dx . (B.56)

Hence, a connection Γ on Y → X also is represented by a vertical-valued form

i i λ Γ = (dy − Γλ dx ) ⊗ ∂i .

Remark B.11 Let π : Y → X be a fibred manifold. Any connection Γ on Y → X yields a horizontal lift of a vector field on X onto Y , but need not defines the similar lift of a path in X into Y .Let

R ⊃[, ]t → x(t) ∈ X, R t → y(t) ∈ Y, be paths in X and Y , respectively. Then t → y(t) is called a horizontal lift of x(t) if

π(y(t)) = x(t), y˙(t) ∈ Hy(t)Y, t ∈ R, where HY ⊂ TY is a horizontal subbundle associated to Γ . If, for each path x(t) −1 (t0 ≤ t ≤ t1) and any y0 ∈ π (x(t0)), there exists a horizontal lift y(t)(t0 ≤ t ≤ t1) so that y(t0) = y0, then Γ is termed the Ehresmann connection. A ﬁbred manifold is a ﬁbre bundle if and only if it admits an Ehresmann connection [67]. Appendix B: Differential Calculus on Fibre Bundles 249

Connections as Jet Bundle Sections

Throughout the book, we follow the equivalent Definition B.10 of connections on a fibre bundle Y → X as sections of the affine jet bundle J 1Y → Y (B.45). Let Y → X be a fibre bundle and J 1Y its first order jet manifold. Given the canonical morphisms (B.47) and (B.48), we have the corresponding morphisms

1 1 λ(1) : J Y × TX ∂λ → dλ = ∂λλ(1) ∈ J Y × TY, X Y 1 ∗ i i i 1 ∗ θ(1) : J Y × V Y dy → θ = θ(1)dy ∈ J Y × T Y Y Y

(Remark B.3). They yield the canonical horizontal splittings of pull-back bundles

1 J Y × TY = λ(1)(TX) ⊕ VY, (B.57) Y J 1Y λ i λ i i λ i x˙ ∂λ +˙y ∂i =˙x (∂λ + yλ∂i ) + (y˙ −˙x yλ)∂i , 1 ∗ ∗ ∗ J Y × T Y = T X ⊕ θ(1)(V Y ), (B.58) Y J 1Y λ i i λ i i λ x˙λdx +˙yi dy = (x˙λ +˙yi yλ)dx +˙yi (dy − yλdx ).

Let Γ be a global section of J 1Y → Y . Substituting the tangent-valued form

λ i λ(1) ◦ Γ = dx ⊗ (∂λ + Γλ ∂i ) in the canonical splitting (B.57), we obtain the familiar horizontal splitting (B.53) of TY by means of a connection Γ on Y → X. Accordingly, substitution of the tangent-valued form

i i λ θ(1) ◦ Γ = (dy − Γλ dx ) ⊗ ∂i in the canonical splitting (B.58) leads to the dual splitting (B.56)ofT ∗Y by means of a connection Γ . Definition B.10 A connection Γ on a fibre bundle Y → X equivalently is defined as a global section

1 λ i i λ i i Γ : Y → J Y,(x , y , yλ) ◦ Γ = (x , y ,Γλ ), (B.59) of the affine jet bundle J 1Y → Y . The following are corollaries of this definition. • Since J 1Y → Y is affine, a connection on a fibre bundle Y → X exists in accordance with Theorem B.4. • Connections on a fibre bundle Y → X make up an affine space modelled over a vector space of soldering forms on Y , i.e., sections of the vector bundle (B.46). 250 Appendix B: Differential Calculus on Fibre Bundles

• Connection components possess transition functions ∂ μ i x j i Γλ = (∂μ + Γμ ∂ j )y . ∂x λ

• Every connection Γ (B.59) on a ﬁbre bundle Y → X yields a ﬁrst order differential operator

: 1 → ∗ ⊗ , = λ −Γ ◦π 1 = ( i −Γ i ) λ ⊗∂ , DΓ J Y T X VY DΓ (1) 0 yλ λ dx i (B.60) Y Y

on Y called the covariant differential relative to a connection Γ .Ifs : X → Y is a section, from (B.60) we obtain its covariant differential

Γ 1 ∗ Γ i i λ ∇ s = DΓ ◦ J s : X → T X ⊗VY, ∇ s = (∂λs −Γλ ◦s)dx ⊗∂i , (B.61)

Γ Γ and the covariant derivative ∇τ = τ∇ along a vector ﬁeld τ on X. A section s is said to be an integral section for a connection Γ if it belongs to the kernel of the covariant differential DΓ (B.60), i.e.,

∇Γ s = 0orJ 1s = Γ ◦ s.

Theorem B.14 For any global section s : X → Y , there always exists a connection Γ such that s is an integral section for Γ .

Proof This connection Γ is an extension of the local section s(x) → J 1s(x) of the afﬁne jet bundle J 1Y → Y over a closed imbedded submanifold s(X) ⊂ Y in accordance with Theorem B.4.

Curvature and Torsion

Let Γ be a connection on a ﬁbre bundle Y → X.Itscurvature is deﬁned as the Nijenhuis differential

1 1 2 ∗ R = dΓ Γ = [Γ,Γ] : Y → ∧ T X ⊗ VY, (B.62) 2 2 FN 1 i λ μ i i i j i j i R = Rλμdx ∧ dx ⊗ ∂ , Rλμ = ∂λΓμ − ∂μΓλ + Γ ∂ Γμ − Γμ ∂ Γλ . 2 i λ j j This is a VY-valued horizontal two-form on Y . Given a connection Γ and a soldering form σ,thetorsion form of Γ with respect to σ is deﬁned as Appendix B: Differential Calculus on Fibre Bundles 251

2 ∗ T = dΓ σ = dσ Γ : Y → ∧ T X ⊗ VY, i j i i j λ μ T = (∂λσμ + Γλ ∂ j σμ − ∂ j Γλ σμ)dx ∧ dx ⊗ ∂i . (B.63)

Flat Connections

A connection g on a ﬁbre bundle Y → X is called the ﬂat connection if its curvature R (B.62) vanishes. The following fact is important for our consideration [100].

Theorem B.15 Let Γ be a flat connection on a fibre bundle Y → X. Then there exists a bundle coordinate atlas (xλ, yi ) on Y where transition functions of fibre coordinates yi → y i (y j ) are independent of base coordinates xλ (it is termed the atlas of constant local trivializations) such that, relative to these coordinates, the connection form (B.54) reads λ Γ = dx ⊗ ∂λ. (B.64)

Conversely, if a ﬁbre bundle Y → X admits an atlas of constant local trivializations, there exists a ﬂat connection on Y → X which takes the form (B.64) with respect to this atlas.

In particular, if Y → X is a trivial bundle, one can associate the ﬂat connection (B.64) to each its trivialization.

Theorem B.16 A connection Γ on a fibre bundle Y → X is flat if and only if the horizontal lift (B.55) of vector fields on X is functorial, i.e., the relation (B.28) holds.

Proof Given vector ﬁelds τ, τ on X and their horizontal lifts Γτ and Γτ (B.55)on Y , we have the relation

λ μ i R(τ, τ ) =−Γ [τ,τ ]+[Γτ,Γτ ]=τ τ Rλμ∂i .

Theorem B.17 A connection Γ on a ﬁbre bundle Y → X is ﬂat if and only if there exists its local integral section through any point y ∈ Y.

Proof The proof follows from the expression (B.64).

Linear Connections

A connection Γ on a vector bundle Y → X is called the linear connection if

1 λ i j Γ : Y → J Y,Γ= dx ⊗ (∂λ + Γλ j (x)y ∂i ), (B.65) is a linear bundle morphism over X. Let us note that linear connections are principal connections (Sect.8.1), and they always exist (Theorem 8.1). 252 Appendix B: Differential Calculus on Fibre Bundles

The curvature R (B.62) of a linear connection Γ (B.65) reads

1 i j λ μ R = Rλμ (x)y dx ∧ dx ⊗ ∂ , 2 j i i i i h i h i Rλμ j = ∂λΓμ j − ∂μΓλ j + Γλ j Γμ h − Γμ j Γλ h. (B.66)

The following are constructions of new linear connections from the old ones.

• Let Y → X be a vector bundle, coordinated by (xλ, yi ), and Y ∗ → X its dual, λ coordinated by (x , yi ). Any linear connection Γ (B.65) on a vector bundle Y → X deﬁnes the dual linear connection

∗ λ j i Γ = dx ⊗ (∂λ − Γλ i (x)y j ∂ ) (B.67)

on Y ∗ → X. • Let Y coordinated by (xλ, yi ) and Y coordinated by (xλ, ya) be vector bundles over the same base X. Their tensor product Y ⊗ Y is endowed with the bundle coordinates (xλ, yia). Linear connections Γ and Γ on Y → X and Y → X deﬁne the linear tensor product connection ∂ λ i ja a ib Γ ⊗ Γ = dx ⊗ ∂λ + (Γλ y + Γ y ) (B.68) j λ b ∂yia

on Y ⊗X Y → X. An important example of linear connections is a linear connection

λ μ ν ˙ Γ = dx ⊗ (∂λ + Γλ ν x˙ ∂μ) (B.69) on the tangent bundle TXof a manifold X. We agree to call it the world connection on X.Thedual world connection (B.67) on the cotangent bundle T ∗ X is

∗ λ μ ˙ ν Γ = dx ⊗ (∂λ − Γλ ν x˙μ∂ ). (B.70)

Then, using the construction of the tensor product connection (B.68), one can introduce the corresponding linear connection on an arbitrary tensor bundle T (B.16). A curvature of a world connection is deﬁned as the curvature R (B.66)ofthe connection Γ (B.69) on the tangent bundle TX. It reads

1 α β λ μ ˙ R = Rλμ β x˙ dx ∧ dx ⊗ ∂α, (B.71) 2 α α α γ α γ α Rλμ β = ∂λΓμ β − ∂μΓλ β + Γλ β Γμ γ − Γμ β Γλ γ .

By a torsion of a world connection is meant the torsion (B.63) of the connection Γ (B.69)onTXrelative to the canonical soldering form θJ (B.36): Appendix B: Differential Calculus on Fibre Bundles 253

1 ν λ μ ˙ ν ν ν T = Tμ λdx ∧ dx ⊗ ∂ν , Tμ λ = Γμ λ − Γλ μ. (B.72) 2 A world connection is called symmetric if its torsion (B.72) vanishes. Remark B.12 Every manifold X can be provided with a nondegenerate ﬁbre metric

2 1 λ μ g ∈ ∨ O (X), g = gλμdx ⊗ dx , 2 λμ g ∈ ∨ T (X), g = g ∂λ ⊗ ∂μ, in the tangent and cotangent bundles. We call it the world metric on X. For any world metric g, there exists a unique symmetric world connection Γ (B.69):

ν ν 1 νρ Γλ μ ={λ μ}=− g (∂λgρμ + ∂μgρλ − ∂ρ gλμ), (B.73) 2 termed the Christoffel symbols, such that ∇Γ g = 0. It is called the Levi–Civita connection associated to g.

B.4 Higher Order Jet Manifolds

The notion of first and second order jets of sections of a fibre bundle (Sect.B.2)is naturally extended to the higher order ones [53, 83, 133, 138]. Remark B.13 Let Y → X be a fibre bundle. Given its bundle coordinates (xλ, yi ), a multi-index Λ of the length |Λ|=k throughout denotes a collection of indices (λ1...λk ) modulo permutations. By Λ+Σ is meant a multi-index (λ1 ...λk σ1 ...σr ). For instance λ + Λ = (λλ1...λr ).ByΛΣ is denoted the union of collections (λ1 ...λk ; σ1 ...σr ) where the indices λi and σ j are not permitted. Summation over a multi-index Λ means separate summation over each its index λi . We use the compact ∂ = ∂ ◦···◦∂ Λ = (λ ...λ ) notation Λ λk λ1 , 1 k . Definition B.11 An r-order jet manifold J r Y of sections of a fibre bundle Y → X r → is defined as the disjoint union of equivalence classes jx s of sections s of Y X r such that sections s and s belong to the same equivalence class jx s if and only if

i i i i s (x) = s (x), ∂Λs (x) = ∂Λs (x), 0 < |Λ|≤r.

In brief, one can say that sections of Y → X are identiﬁed by the r + 1terms of their Taylor series at points of X. The particular choice of coordinates does not r matter for this deﬁnition. The equivalence classes jx s are called the r-order jets of sections. Their set J r Y is endowed with an atlas of the adapted coordinates

λ i i i (x , yΛ), yΛ ◦ s = ∂Λs (x), 0 ≤|Λ|≤r, (B.74) 254 Appendix B: Differential Calculus on Fibre Bundles possessing transition functions ∂ μ i x i y = dμ y , (B.75) λ+Λ ∂ xλ Λ where the symbol dλ stands for the higher order total derivative ∂ μ i Λ x dλ = ∂λ + yΛ+λ∂ , dλ = dμ. i ∂x λ 0≤|Λ|≤r−1

These derivatives act on exterior forms on J r Y and obey the relations

[dλ, dμ]=0, dλ ◦ d = d ◦ dλ, dλ(φ ∧ σ) = dλ(φ) ∧ σ + φ ∧ dλ(σ ), μ i i dλ(dx ) = 0, dλ(dyΛ) = dyλ+Λ.

= ◦···◦ Λ = (λ ...λ ) We use the compact notation dΛ dλr dλ1 , r 1 . The coordinates (B.74) brings a set J r Y into a manifold. They are compatible πr : r → k > with the natural surjections k J Y J Y , r k, which form a composite bundle

r r−1 1 π − π − π π πr : J r Y −→r 1 J r−1Y −→···r 2 −→0 Y −→ X, π k ◦ πr = πr ,πs ◦ πr = πr . s k s s

A glance at the transition functions (B.75), when |Λ|=r, shows that the ﬁbration

πr : r → r−1 r−1 J Y J Y (B.76) is an afﬁne bundle modelled over a vector bundle

r ∨ T ∗ X ⊗ VY → J r−1Y. (B.77) J r−1Y

r i In particular, J Y → Y is an afﬁne bundle with ﬁbre coordinates (yΛ), |Λ|= 1,...,r, which we agree to call the jet coordinates. For the sake of convenience, let us also denote J 0Y = Y .

Remark B.14 Let us recall that a base of any affine bundle is a strong deformation retract of its total space. Consequently, Y is a strong deformation retract of J 1Y , which in turn is a strong deformation retract of J 2Y , and so on. It follows that a fibre bundle Y is a strong deformation retract of any finite order jet manifold J r Y . Therefore, by virtue of the Vietoris–Begle theorem [23], there is an isomorphism

H ∗(J r Y ; R) = H ∗(Y ; R) (B.78) of cohomology groups of J r Y and Y with coefﬁcients in the constant sheaf R. Appendix B: Differential Calculus on Fibre Bundles 255

In the calculus in higher order jets, we have the r-order jet prolongation functor such that, given ﬁbre bundles Y and Y over X, every bundle morphism Φ : Y → Y (B.8) over a diffeomorphism f of X admits ther-order jet prolongation to a morphism of r-order jet manifolds

r Φ : r r → r (Φ ◦ ◦ −1) ∈ r . J J Y jx s j f (x) s f J Y

The jet prolongation functor is exact. If Φ is an injection or a surjection, so is J r Φ. It also preserves an algebraic structure. In particular, if Y → X is a vector bundle, J r Y → X is well. If Y → X is an affine bundle modelled over a vector bundle Y → X, then J r Y → X is an affine bundle modelled over a vector bundle J r Y → X. Every section s of a fibre bundle Y → X admits the r-order jet prolongation to a ( r )( ) = r r → section J s x jx s of a jet bundle J Y X. O∗ = O∗( k ) k Let k J Y be a DGA of exterior forms on a jet manifold J Y .Every φ k π k+i ∗φ exterior form on a jet manifold J Y gives rise to the pull-back form k on a jet manifold J k+i Y . We have a direct sequence of C∞(X)-algebras

π ∗ π 1∗ π 2∗ πr ∗ O∗( ) −→ O∗( ) −→0 O∗ −→···1 −→r−1 O∗. X Y 1 r

Remark B.15 By virtue of de Rham Theorem C.10, the cohomology of the de Rham O∗ ∗( k ; R) k complex of k equals the cohomology H J Y of J Y with coefﬁcients in the constant sheaf R. The latter in turn coincides with the sheaf cohomology H ∗(Y ; R) ∗ ( ) of Y (Remark B.14) and, thus, it equals the de Rham cohomology HDR Y of Y . Given a k-order jet manifold J k Y of Y → X, there exists the canonical bundle k k morphism r(k) : J TY → TJ Y over a surjection

J k Y × J k TX → J k Y × TX X X whose coordinate expression is i i i μ y˙Λ ◦ r(k) = (y˙ )Λ − (y˙ )μ+Σ (x˙ )Ξ , 0 ≤|Λ|≤k, where the sum is taken over all partitions Σ + Ξ = Λ and 0 < |Ξ|. In particular, we have the canonical isomorphism

k k i i r(k) : J VY → VJ Y,(y˙ )Λ =˙yΛ ◦ r(k), (B.79) over J k Y . As a consequence, every projectable vector field u (B.25) on a fibre bundle Y → X admits the k-order jet prolongation to a vector field 256 Appendix B: Differential Calculus on Fibre Bundles

k k k k J u = r( ) ◦ J u : J Y → TJ Y, k k = λ∂ + i ∂ + ( ( i − i μ) + i μ)∂Λ, J u u λ u i dΛ u yμu yμ+Λu i (B.80) 0<|Λ|≤k on J k Y (cf. (B.51)fork = 1). In particular, the k-order jet prolongation (B.80)ofa i vertical vector ﬁeld u = u ∂i on Y → X is a vertical vector ﬁeld k = i ∂ + i ∂Λ J u u i dΛu i 0<|Λ|≤k on J k Y → X owing to the isomorphism (B.79). Similarly to the canonical monomorphisms (B.47) and (B.48), there are the canonical bundle monomorphisms over J k Y :

k+1 ∗ k λ λ(k) : J Y −→ T X ⊗ TJ Y,λ(k) = dx ⊗ dλ, J k Y θ : k+1 −→ ∗ k ⊗ k ,θ= ( i − i λ) ⊗ ∂Λ. (k) J Y T J Y VJ Y (k) dyΛ yλ+Λdx i J k Y |Λ|≤k

The one-forms

i i i λ θΛ = dyΛ − yλ+Λdx

O∗ are called the local contact forms. They generate an ideal of a DGA k of exterior forms on J k Y which is termed the ideal of contact forms.

B.5 Differential Operators and Equations

Jet manifolds provides the conventional language of theory of differential equations and differential operators if they need not be linear [25, 53, 85].

Definition B.12 Asystemofk-order partial differential equations on a fibre bundle Y → X is defined as a closed subbundle E of a jet bundle J k Y → X. For the sake of brevity, we agree to call E the differential equation.

k λ i Let J Y be provided with the adapted coordinates (x , yΛ). There exists a local coordinate system (z A), A = 1,...,codimE, on J k Y such that E is locally given (in the sense of item (i) of Theorem B.1) by equations

A λ i E (x , yΛ) = 0, A = 1,...,codimE. (B.81)

Deﬁnition B.13 A solution of the k-order differential equation E (B.81)onY → X is deﬁned as a section s of the k-order jet bundle J k Y → X which lives in E.By Appendix B: Differential Calculus on Fibre Bundles 257 a classical solution of a differential equation E on Y → X is meant a section s of Y → X such that its k-order jet prolongation J k s takes its values into E, i.e., s obeys the differential equations

A λ i E (x ,∂Λs (x)) = 0.

In applications, differential equations are mostly associated to differential operators. There are several equivalent deﬁnitions of (nonlinear) differential operators. We start with the following.

Definition B.14 Let Y → X and E → X be fibre bundles, which are assumed to have global sections. A k-order E-valued differential operator on a fibre bundle Y → X is defined as a section E of the pull-back bundle

: k = k × → k . pr1 EY J Y E J Y (B.82) X

Given bundle coordinates (xλ, yi ) on Y and (xλ,χa) on E, the pull-back (B.82) λ j a is provided with coordinates (x , yΣ ,χ ),0 ≤|Σ|≤k. With respect to these E E ⊂ k coordinates, a differential operator seen as a closed imbedded submanifold EY is given by the equalities a a λ j χ = E (x , yΣ ). (B.83)

There is obvious one-to-one correspondence between the sections E (B.83)ofthe ﬁbre bundle (B.82) and the bundle morphisms

Φ : k −→ ,Φ= ◦ E ⇐⇒ E = ( k ,Φ). J Y E pr2 Id J Y (B.84) X

Therefore, there is the following equivalent deﬁnition of differential operators on Y .

Deﬁnition B.15 Let Y → X and E → X be ﬁbre bundles. A bundle morphism J k Y → E over X is called the E-valued k-order differential operator on Y → X.

It is readily observed that the differential operator Φ (B.84) sends each section s of Y → X onto a section Φ ◦ J k s of E → X. The mapping

k a a λ j ΔΦ : s → Φ ◦ J s,χ(x) = E (x ,∂Σ s (x)), is termed the standard form of a differential operator. Let e be a global section of a ﬁbre bundle E → X,akernel of a E-valued differential operator Φ is deﬁned as a kernel

−1 Ker eΦ = Φ (e(X)) (B.85) of the bundle morphism Φ (B.84). If it is a closed subbundle of a jet bundle J k Y → X, one says that Ker eΦ (B.85)isadifferential equation associated to the differential 258 Appendix B: Differential Calculus on Fibre Bundles operator Φ. By virtue of Theorem B.10, this condition holds if Φ is a bundle morphism of constant rank. If E → X is a vector bundle, by a kernel of a E-valued differential operator usually is meant its kernel with respect to the canonical zero-valued section 0of E → X. If Y → X and E → X are vector bundles, a k-order E-valued differential operator is called the linear differential operator if it is a linear morphism on fibres of J k Y → X. Then Serre–Swan Theorem A.10 states one-to-one correspondence between linear differential operators on vector bundles in accordance with Definition B.15 and differential operators (Definition A.3) on projective C∞(X)-modules of finite rank (Sect.A.2). Appendix C Calculus on Sheaves

In the book, we address algebraic systems on topological spaces. They are described in terms of sheaves. In this Appendix, the relevant material on sheaves of modules over a commutative ring is summarized. We follow the terminology of [23, 75].

C.1 Sheaf Cohomology

A sheaf on a topological space X is a continuous fibre bundle π : S → X (Remark B.2) in modules over a commutative ring K , where the surjection π is a local homeomorphism and fibres Sx , x ∈ X, called the stalks, are provided with the discrete topology. Global sections of a sheaf S make up a K -module S(X),termed the structure module of S. Any sheaf is generated by a presheaf. A presheaf S{U} on a topological space X is defined if a module SU over a commutative ring K is assigned to every open subset U ⊂ X (S∅ = 0) and if, for any pair of open subsets V ⊂ U, there exists a U : → restriction morphism rV SU SV such that

U = , U = V U , ⊂ ⊂ . rU Id SU rW rW rV W V U

Every presheaf S{U} on a topological space X yields a sheaf on X whose stalk Sx at a point x ∈ X is the direct limit of the modules SU , x ∈ U (Deﬁnition A.1), with U respect to a restriction morphisms rV . It means that, for each open neighborhood U of a point x, every element s ∈ SU determines an element sx ∈ Sx , called the germ of s at x. Two elements s ∈ SU and s ∈ SV belong to the same germ at x if and only ⊂ ∩ U = V if there exists an open neighborhood W U V of x such that rW s rW s .

0 Remark C.1 Let C{U} be a presheaf of local continuous real functions on a topological space X. Two such functions s and s deﬁne the same germ sx if they coincide 0 on an open neighborhood of x. Hence, we obtain a sheaf CX of germs of local ∞ continuous real functions on X. Similarly, a sheaf CX of germs of local smooth © Atlantis Press and the author(s) 2016 259 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0 260 Appendix C: Calculus on Sheaves

real functions on a manifold X is deﬁned. Let us also mention a presheaf of local real functions which are constant on connected open subsets of X. It generates the constant sheaf on X denoted by R.

Two different presheaves may generate the same sheaf. Conversely, every sheaf S deﬁnes a presheaf S({U}) of modules S(U) of its local sections. It is termed the canonical presheaf of a sheaf S. If a sheaf S is constructed from a presheaf S{U}, there are natural module morphisms

SU s → s(U) ∈ S(U), s(x) = sx , x ∈ U,

which are neither monomorphisms nor epimorphisms in general. For instance, it may happen that a nonzero presheaf defines a zero sheaf. A sheaf generated by the canonical presheaf of a sheaf S coincides with S. A direct sum and a tensor product of presheaves (as families of modules) and sheaves (as fibre bundles in modules) are naturally defined. By virtue of Theorem A.3, a direct sum (resp. a tensor product) of presheaves generates a direct sum (resp. a tensor product) of the corresponding sheaves.

Remark C.2 In the terminology of [144], a sheaf is introduced as a presheaf which satisﬁes the following additional axioms.

(S1) Suppose that U ⊂ X is an open subset and {Uα} is its open cover. If s, s ∈ SU U U ( ) = ( ) α = obey the condition rUα s rUα s for all U , then s s . (S2) Let U and {Uα} be as in previous item. Suppose that we are given a family of { ∈ } presheaf elements sα SUα such that

Uα Uλ ( α) = ( λ) rUα ∩Uλ s rUα ∩Uλ s

for all Uα, Uλ. Then there exists a presheaf element s ∈ SU such that sα = U ( ) rUα s . Canonical presheaves are in one-to-one correspondence with presheaves obeying these axioms. For instance, the presheaves of continuous, smooth and locally constant functions in Remark C.1 satisfy the axioms (S1) and (S2).

Remark C.3 The notion of a sheaf can be extended to sets, but not to noncommutative groups. One can consider a presheaf of such groups, but it generates a sheaf of sets because a direct limit of noncommutative groups need not be a group.

There is a useful construction of a sheaf on a topological space X from local sheaves on open subsets which make up a cover of X.

Theorem C.1 Let {Uζ } be an open cover of a topological space X and Sζ a sheaf on Uζ for every Uζ . Let us suppose that, if Uζ ∩ Uξ =∅, there is a sheaf isomorphism

ρ : | → | ζξ Sξ Uζ ∩Uξ Sζ Uζ ∩Uξ Appendix C: Calculus on Sheaves 261 and, for every triple (Uζ , Uξ , Uι), these isomorphisms fulﬁl the cocycle condition

ρ ◦ ρ ( | ) = ρ ( | ). ξζ ζι Sι Uζ ∩Uξ ∩Uι ξι Sι Uζ ∩Uξ ∩Uι

φ : | → Then there exists a sheaf S together with sheaf isomorphisms ζ S Uζ Sζ so that

φ | = ρ ◦ φ | . ζ Uζ ∩Uξ ζξ ξ Uζ ∩Uξ

A morphism of a presheaf S{U} to a presheaf S{U} on the same topological space γ : → X is deﬁned as a set of module morphisms U SU SU which commute with restriction morphisms. A morphism of presheaves yields a morphism of sheaves generated by these presheaves. This is a bundle morphism over X such that γx : Sx → γ ∈ Sx is the direct limit of morphisms U , x U. Conversely, any morphism of sheaves S → S on a topological space X yields a morphism of canonical presheaves of local sections of these sheaves. Let Hom (S|U , S |U ) be the commutative group of sheaf morphisms S|U → S |U for any open subset U ⊂ X. These groups are assembled into a presheaf, and deﬁne a sheaf Hom (S, S ) on X. There is a monomorphism

Hom (S, S )(U) → Hom (S(U), S (U)), (C.1) which need not be an isomorphism. By virtue of Theorem A.5, if a presheaf morphism is a monomorphism or an epimorphism, so is the corresponding sheaf morphism. Furthermore, the following holds.

Theorem C.2 A short exact sequence

→ → → → 0 S{U} S{U} S{U} 0(C.2) of presheaves on the same topological space yields a short exact sequence of sheaves generated by these presheaves

0 → S → S → S → 0, (C.3) where the quotient sheaf S = S/S is isomorphic to that generated by the quotient = / presheaf S{U} S{U} S{U}. If the exact sequence of presheaves (C.2) is split, i.e.,

=∼ ⊕ , S{U} S{U} S{U} the corresponding splitting S =∼ S ⊕ S of the exact sequence of sheaves (C.3) holds.

The converse is more intricate. A sheaf morphism induces a morphism of the corresponding canonical presheaves. If S → S is a monomorphism, S({U}) → S ({U}) also is a monomorphism. However, if S → S is an epimorphism, S({U}) → 262 Appendix C: Calculus on Sheaves

S ({U}) need not be so. Therefore, the short exact sequence (C.3)ofsheavesyields an exact sequence of canonical presheaves

0 → S ({U}) → S({U}) → S ({U}), (C.4) where S({U}) → S ({U}) is not necessarily an epimorphism. At the same time, there is a short exact sequence of presheaves

→ ({ }) → ({ }) → → , 0 S U S U S{U} 0 (C.5) = ({ })/ ({ }) where the quotient presheaf S{U} S U S U generates a quotient sheaf S = S/S , but need not be its canonical presheaf.

Theorem C.3 Let the exact sequence of sheaves (C.3) be split. Then

S({U}) =∼ S ({U}) ⊕ S ({U}), and the canonical presheaves make up the short exact sequence

0 → S ({U}) → S({U}) → S ({U}) → 0.

Let us turn now to sheaf cohomology. We follow its deﬁnition in [75]. Let S{U} be a presheaf of modules on a topological space X, and let U ={Ui }i∈I be an open cover of X. One constructs a cochain complex where a p-cochain is p p deﬁned as a function s which associates an element s (i ,...,i ) ∈ S ∩···∩ to 0 p Ui0 Ui p each (p + 1)-tuple (i0,...,i p) of indices in I . These p-cochains are assembled into p a module C (U, S{U}). Let us introduce a coboundary operator

p p p+1 δ : C (U, S{U}) → C (U, S{U}), p+1 δ p p( ,..., ) = (− )k Wk p( ,..., ,..., ), s i0 i p+1 1 rW s i0 ik i p+1 (C.6) k=0 = ∩···∩ , = ∩···∩ ∩···∩ . W Ui0 Ui p+1 Wk Ui0 Uik Ui p+1

One can verify that δ p+1 ◦ δ p = 0. Thus, we obtain a cochain complex of modules

δ0 δ p 0 p p+1 0 → C (U, S{U}) −→···C (U, S{U}) −→ C (U, S{U}) −→··· . (C.7)

Its cohomology groups

p p p−1 H (U; S{U}) = Ker δ /Im δ are modules. Of course, they depend on an open cover U of the topological space X. Appendix C: Calculus on Sheaves 263

Remark C.4 Throughout the book, only proper covers are considered, i.e., Ui = U j if i = j. A cover U is said to be a reﬁnement of a cover U if, for each U ∈ U , there exists U ∈ U such that U ⊂ U.

Let U be a reﬁnement of a cover U. Then there is a morphism of cohomology groups

∗ ∗ H (U; S{U}) → H (U ; S{U}).

∗ Let us take the direct limit of cohomology groups H (U; S{U}) with respect to these ∗ morphisms, where U runs through all open covers of X. This limit H (X; S{U}) is called the cohomology of X with coefficients in a presheaf S{U}. Let S be a sheaf on a topological space X. Cohomology of X with coefficients in S or, shortly, sheaf cohomology of X is defined as cohomology

H ∗(X; S) = H ∗(X; S({U})) with coefﬁcients in the canonical presheaf S({U}) of a sheaf S. p p p p In this case, a p-cochain s ∈ C (U, S({U})) is a collection s ={s (i0,...,i p)} p( ,..., ) ∩···∩ ( + ) of local sections s i0 i p of a sheaf S over Ui0 Ui p for each p 1 -tuple ( ,..., ) Ui0 Ui p of elements of the cover U. The coboundary operator (C.6) reads

p+1 p p k p δ s (i ,...,i + ) = (−1) s (i ,...,i ,...,i + )| ∩···∩ . 0 p 1 0 k p 1 Ui0 Ui p+1 k=0

For instance,

δ0 0( , ) =[ 0( ) − 0( )]| , s i j s j s i Ui ∩U j (C.8) δ1 1( , , ) =[ 1( , ) − 1( , ) + 1( , )]| . s i j k s j k s i k s i j Ui ∩U j ∩Uk

A glance at the expression (C.8) shows that a zero-cocycle is a collection s ={s(i)}I of local sections of a sheaf S over Ui ∈ U such that s(i) = s( j) on Ui ∩ U j .It follows from the axiom (S2) in Remark C.2 that s is a global section of a sheaf S, ( ) | while each s i is its restriction s Ui to Ui . Consequently, the cohomology group H 0(U, S({U})) is isomorphic to the structure module S(X) of global sections of a sheaf S. A one-cocycle is a collection {s(i, j)} of local sections of a sheaf S over overlaps Ui ∩ U j which satisfy the cocycle condition

[ ( , ) − ( , ) + ( , )]| = . s j k s i k s i j Ui ∩U j ∩Uk 0 264 Appendix C: Calculus on Sheaves

C.2 Abstract de Rham Theorem

If X is a paracompact space, the study of its sheaf cohomology is essentially simpliﬁed owing to the following fact [75].

Theorem C.4 Cohomology of a paracompact space X with coefﬁcients in a sheaf S equals cohomology of X with coefﬁcients in any presheaf generating a sheaf S.

Remark C.5 We follow the definition of a paracompact topological space in [75]as a Hausdorff space such that any its open cover admits a locally finite open refinement, i.e., any point has an open neighborhood which intersects only a finite number of elements of this refinement. A topological space X is paracompact if and only if any cover {Uξ } of X admits the subordinate partition of unity { fξ }, i.e.: (i) fξ are real positive continuous functions on X; (ii) supp fξ ⊂ Uξ ; (iii) each point x ∈ X has an open neighborhood which intersects only a finite number of the sets supp fξ ; (iv) fξ (x) = 1 for all x ∈ X. ξ

A key point of the analysis of sheaf cohomology is that short exact sequences of presheaves and sheaves yield long exact sequences of sheaf cohomology groups.

Let S{U} and S{U} be presheaves on the same topological space X.Itisreadily → observed that, given an open cover U of X, any morphism S{U} S{U} yields a cochain morphism of complexes

∗( , ) → ∗( , ) C U S{U} C U S{U} and the corresponding morphism

∗( , ) → ∗( , ) H U S{U} H U S{U} of cohomology groups of these complexes. Passing to the direct limit through all reﬁnements of U, we come to a morphism of the cohomology groups

∗( , ) → ∗( , ) H X S{U} H X S{U}

of X with coefﬁcients in the presheaves S{U} and S{U}. In particular, any sheaf morphism S → S yields a morphism of canonical presheaves S({U}) → S ({U}) and the corresponding cohomology morphism H ∗(X, S) → H ∗(X, S ). By virtue of Theorems A.7 and A.8, every short exact sequence

→ −→ −→ → 0 S{U} S{U} S{U} 0 of presheaves on the same topological space X and the corresponding exact sequence of complexes (C.7) yield the long exact sequence Appendix C: Calculus on Sheaves 265

→ 0( ; ) −→ 0( ; ) −→ 0( ; ) −→ 0 H X S{U} H X S{U} H X S{U} 1( ; ) −→··· p( ; ) −→ p( ; ) −→ H X S{U} H X S{U} H X S{U} p( ; ) −→ p+1( ; ) −→··· H X S{U} H X S{U} of the cohomology groups of X with coefﬁcients in these presheaves. This result however is not extended to an exact sequence of sheaves, unless X is a paracompact space. Let

0 → S −→ S −→ S → 0 be a short exact sequence of sheaves on X. It yields the short exact sequence of presheaves (C.5) where a presheaf S{U} generates a sheaf S .IfX is paracompact,

∗( ; ) = ∗( ; ) H X S{U} H X S by virtue of Theorem C.4, and we have the exact sequence of sheaf cohomology

0 → H 0(X; S ) −→ H 0(X; S) −→ H 0(X; S ) −→ H 1(X; S ) −→···H p(X; S ) −→ H p(X; S) −→ H p(X; S ) −→ H p+1(X; S ) −→··· .

Let us turn now to the abstract de Rham theorem which provides a powerful tool of studying algebraic systems on paracompact spaces. Let us consider an exact sequence of sheaves

h h0 h1 h p 0 → S −→ S0 −→ S1 −→···Sp −→··· . (C.9)

It is said to be a resolution of a sheaf S if each sheaf Sp≥0 is acyclic, i.e., its coho- k>0 mology groups H (X; Sp) vanish. Any exact sequence of sheaves (C.9) yields the sequence of their structure modules

0 1 p h∗ h∗ h∗ h∗ 0 → S(X) −→ S0(X) −→ S1(X) −→···Sp(X) −→··· (C.10) which is always exact at terms S(X) and S0(X) (see the exact sequence (C.4)). The p+1 p sequence (C.10) is a cochain complex because h∗ ◦ h∗ = 0. If X is a paracompact space and the exact sequence (C.9) is a resolution of S, the forthcoming abstract de Rham theorem establishes an isomorphism of cohomology of the complex (C.10)to cohomology of X with coefﬁcients in a sheaf S [75]. Theorem C.5 Given the resolution (C.9) of a sheaf S on a paracompact topological space X and the induced complex (C.10), there are isomorphisms

0 0 q q q−1 H (X; S) = Ker h∗, H (X; S) = Ker h∗/Im h∗ , q > 0. (C.11) 266 Appendix C: Calculus on Sheaves

We also refer to the following minor modiﬁcation of Theorem C.5 [56, 143]. Theorem C.6 Let

h h0 h1 h p−1 h p 0 → S −→ S0 −→ S1 −→···−→ Sp −→ Sp+1, p > 1, be an exact sequence of sheaves on a paracompact topological space X, where the sheaves Sq , 0 ≤ q < p, are acyclic, and let

0 1 p−1 p h∗ h∗ h∗ h∗ h∗ 0 → S(X) −→ S0(X) −→ S1(X) −→···−→ Sp(X) −→ Sp+1(X) be the corresponding cochain complex of structure modules of these sheaves. Then the isomorphisms (C.11) hold for 0 ≤ q ≤ p. In our book, we appeal to a fine resolution of sheaves, i.e., a resolution by fine sheaves. A sheaf S on a paracompact space X is called fine if, for each locally finite open cover U ={Ui }i∈I of X, there exists a system {hi } of endomorphisms hi : S → S such that: ⊂ ( ) = ∈/ (i) there is a closed subset Vi Ui and hi Sx 0ifx Vi , (ii) hi is the identity map of S. i∈I Theorem C.7 A fine sheaf on a paracompact space is acyclic. There are the following important examples of fine sheaves. Theorem C.8 Let X be a paracompact topological space which admits the partition of unity performed by elements of the structure module A(X) of some sheaf A of germs of local real functions on X. Then any sheaf S of A(X)-modules on X, including A itself, is fine.

0 Remark C.6 In particular, a sheaf CX of germs of local continuous real functions on a paracompact space is ﬁne, and so is any sheaf of C0(X)-modules. A smooth manifold X admits the partition of unity performed by smooth real functions (Remark ∞ B.1). It follows that the sheaf CX of germs of local smooth real functions on X is ﬁne, and so is any sheaf of C∞(X)-modules, e.g., the sheaves of sections of smooth vector bundles over X.

C.3 Local-Ringed Spaces

Local-ringed spaces are sheaves of local rings. In our book, graded manifolds are deﬁned as local-ringed spaces (Sect.6.3). A sheaf A on a topological space X is said to be the ringed space if its stalk Ax at each point x ∈ X is a commutative ring [144]. A ringed space often is denoted by Appendix C: Calculus on Sheaves 267 a pair (X, A) of a topological space X and a sheaf A of rings on X which are called the body and the structure sheaf of a ringed space, respectively.

Deﬁnition C.1 A ringed space is said to be the local-ringed space (the geometric space in the terminology of [144]) if it is a sheaf of local rings.

0 Remark C.7 A sheaf CX of germs of local continuous real functions on a topological 0 ∈ space X is a local-ringed space. Its stalk Cx , x X, contains an unique maximal 0 ideal of germs of functions vanishing at x. A sheaf CX of germs of local smooth real functions on a manifold X also is so.

Morphisms of local-ringed spaces are deﬁned to be particular morphisms of sheaves on different topological spaces as follows.

Let ϕ : X → X be a continuous map. Given a sheaf S on X, its direct image ϕ∗ S on X is generated by the presheaf of assignments

X ⊃ U → S(ϕ−1(U )) for any open subset U ⊂ X . Conversely, given a sheaf S on X , its inverse image ϕ∗ S on X is deﬁned as the pull-back onto X of a continuous ﬁbre bundle S over X , ϕ∗ = i.e., Sx Sϕ(x). This sheaf is generated by the presheaf which associates to any open V ⊂ X the direct limit of modules S (U) over all open subsets U ⊂ X such that V ⊂ f −1(U).

Remark C.8 Let i : X → X be a closed subspace of X . Then i∗ S is a unique sheaf on X such that

i∗ S|X = S, i∗ S|X \X = 0.

Indeed, if x ∈ X ⊂ X , then i∗ S(U ) = S(U ∩ X) for any open neighborhood U of this point. If x ∈/ X, there exists its neighborhood U such that U ∩ X is empty, i.e., i∗ S(U ) = 0. A sheaf i∗ S is termed the trivial extension of a sheaf S.

By a morphism of ringed spaces (X, A) → (X , A ) is meant a pair (ϕ, Φ) of a continuous map ϕ : X → X and a sheaf morphism Φ : A → ϕ∗A or, equivalently, a sheaf morphism ϕ∗A → A [144]. Restricted to each stalk, a sheaf morphism Φ is assumed to be a ring homomorphism. A morphism of ringed spaces is said to be: • a monomorphism if ϕ is an injection and Φ is an epimorphism, • an epimorphism if ϕ is a surjection, while Φ is a monomorphism. Let (X, A) be a local-ringed space. By a sheaf dA of derivations of the sheaf A is meant a subsheaf of endomorphisms of A such that any section u of dA over an open subset U ⊂ X is a derivation of a ring A(U). It should be emphasized that, since (C.1) is not necessarily an isomorphism, a derivation of a ring A(U) need not be a section of a sheaf dA|U . Namely, it may happen that, given open sets U ⊂ U, there is no restriction morphism d(A(U)) → d(A(U )). 268 Appendix C: Calculus on Sheaves

Given a local-ringed space (X, A), a sheaf P on X is called the sheaf of A- modules if every stalk Px , x ∈ X,isanAx -module or, equivalently, if P(U) is an A(U)-module for any open subset U ⊂ X. A sheaf of A-modules P is said to be locally free if there exists an open neighborhood U of every point x ∈ X such that P(U) is a free A(U)-module. If all these free modules are of finite rank (resp. of the same finite rank), one says that P is of finite type (resp. of constant rank). The structure module of a locally free sheaf is called the locally free module. The following is a generalization of Theorem C.8 [75].

Theorem C.9 Let X be a paracompact space which admits the partition of unity by elements of the structure module S(X) of some sheaf S of real functions on X. Let P be a sheaf of S-modules, i.e., the stalks Px ,x∈ X, of P areSx -modules. Then P is ﬁne and, consequently, acyclic.

Ok ∈ N In particular, all sheaves X , k +, of germs of exterior forms on X are ﬁne. These sheaves constitute the de Rham complex

→ R −→ ∞ −→d O1 −→···d Ok −→···d . 0 CX X X (C.12)

The corresponding complex of structure modules of these sheaves is the de Rham complex (A.18) of exterior forms on a manifold X. Due to the Poincaré lemma, the complex (C.12) is exact and, thereby, is a ﬁne resolution of the constant sheaf R on a manifold. Then a corollary of Theorem C.5 is the classical de Rham theorem.

Theorem C.10 There is an isomorphism

k ( ) = k ( ; R) HDR X H X (C.13) ∗ ( ) ∗( ; R) of the de Rham cohomology HDR X of a manifold X to the cohomology H X of X with coefﬁcients in the constant sheaf R.

The sheaf cohomology H k (X; R) in turn is related to cohomology of other types as follows. Let us consider a short exact sequence of constant sheaves

0 → Z −→ R −→ U(1) → 0, where U(1) = R/Z is a circle group of complex numbers of unit module. This exact sequence yields a long exact sequence of sheaf cohomology groups

0 → Z −→ R −→ U(1) −→ H 1(X; Z) −→ H 1(X; R) −→··· H p(X; Z) −→ H p(X; R) −→ H p(X; U(1)) −→ H p+1(X; Z) −→··· , H 0(X; Z) = Z, H 0(X; R) = R, H 0(X; U(1)) = U(1). Appendix C: Calculus on Sheaves 269

This exact sequence deﬁnes a homomorphism

H ∗(X; Z) → H ∗(X; R) (C.14) of cohomology with coefﬁcients in the constant sheaf Z to that with coefﬁcients in R. Combining the isomorphism (C.13) and the homomorphism (C.14) leads to a ∗( ; Z) → ∗ ( ) cohomology homomorphism H X HDR X . Its kernel contains all cyclic elements of cohomology groups H k (X; Z). Since manifolds are assumed to be paracompact, there is the following isomorphism between their sheaf cohomology and singular cohomology.

Theorem C.11 The sheaf cohomology H ∗(X; Z) (resp. H ∗(X; Q),H∗(X; R))of a paracompact topological space X with coefﬁcients in the constant sheaf Z (resp. Q, R) is isomorphic to the singular cohomology of X with coefﬁcients in a ring Z (resp. Q, R)[23, 139].

Let us note that singular cohomology of paracompact topological spaces coincide with the Chechˇ and Alexandery ones. Since singular cohomology is a topological invariant (i.e., homotopic topological spaces have the same singular cohomology) [139], the sheaf cohomology H ∗(X; Z), H ∗(X; Q), H ∗(X; R) and, consequently, the de Rham cohomology of smooth manifolds are topological invariants.

Appendix D Noether Identities of Differential Operators

Noether and higher-stage Noether identities of a Lagrangian system in Sect. 7.1 are particular Noether identities of differential operators which are described in homology terms as follows [61, 123]. Let E → X be a vector bundle, and let E be a E-valued k-order differential operator on a ﬁbre bundle Y → X in accordance with Deﬁnition B.14. It is represented by a section E a (B.83) of the pull-back bundle (B.82) endowed with bundle λ j a coordinates (x , yΣ ,χ ),0≤|Σ|≤k.

Deﬁnition D.1 One says that a differential operator E obeys Noether identities (NI) if there exist an r-order differential operator Φ on the pull-back bundle

EY = Y × E → X (D.1) X so that its projection to E is a linear differential operator and its kernel contains E : Φ = ΦΛχ a , ΦΛE a = . a Λ a Λ 0 0≤|Λ| 0≤|Λ|

Any differential operator admits NI, e.g., Φ = ΛΣ E bχ a , ΛΣ =− ΣΛ. Tab dΣ Λ Tab Tba (D.2) 0≤|Λ|,|Σ|

Therefore, NI must be separated into the trivial and nontrivial ones. For this purpose, we describe them as boundaries and cycles of some chain complex [61, 123].

Lemma D.1 One can associate to a differential operator E the chain complex (D.3) whose boundaries vanish on Ker E . ( , ) Proof Let us consider the composite graded manifold Y AEY modelled over the 0 vector bundle EY → Y .LetS∞[EY , Y ] be the ring (6.39) of graded functions on

© Atlantis Press and the author(s) 2016 271 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0 272 Appendix D: Noether Identities of Differential Operators an inﬁnite order jet manifold J ∞Y possessing the local generating basis (yi ,εa) of Grassmann parity [yi ]=0, [εa]=1 and, in accordance with the terminology of Chap. 7, antiﬁeld number Ant[yi ]=0, Ant[εa]=1. It is provided with a nilpotent a right odd derivation δ = ∂aE (Remark 6.10). Then we have a chain complex

δ δ 0 0 0←Im δ ←− S∞[EY , Y ]1 ←− S∞[EY , Y ]2 (D.3) of graded functions of antiﬁeld number k ≤ 2. By very deﬁnition, its one-boundaries 0 δΦ, Φ ∈ S∞[EY , Y ]2 vanish on Ker E .

Every one-cycle Φ = ΦΛεa ∈ S 0 [ , ] a Λ ∞ EY Y 1 (D.4) 0≤|Λ| of the complex (D.3) deﬁnes a linear differential operator on the pull-back bundle EY (D.1) such that it is linear on E and its kernel contains E , i.e., δΦ = , ΦΛ E a = . 0 a dΛ 0 (D.5) 0≤|Λ|

In accordance with Definition D.1, the one-cycles (D.4) define the NI (D.5)ofa differential operator E . These NI are trivial if a cycle is a boundary, i.e., it takes the form (D.2). Accordingly, nontrivial NI modulo the trivial ones are associated to elements of the homology H1(δ) of the complex (D.4). A differential operator is called degenerate if it obeys nontrivial NI. 0 One can say something more if the O∞Y -module H1(δ) is finitely generated, i.e., it possesses the following particular structure. There are elements Δ ∈ H1(δ) making ∞ up a projective C (X)-module C(0) of finite rank which, by virtue of Serre–Swan Theorem A.10, is isomorphic to the structure module of sections of some vector r bundle E0 → X.Let{Δ }: Δr = ΔΛr εa ,ΔΛr ∈ O0 , a Λ a ∞Y 0≤|Λ|

∞ be local bases for this C (X)-module. Then every element Φ ∈ H1(δ) factorizes as Φ = Ξ Δr , Ξ ∈ O , Gr dΞ Gr ∞Y (D.6) 0≤|Ξ| through elements of C(0), i.e., any NI (D.5) is a corollary of the NI ΔΛr E a = , a dΛ 0 (D.7) 0≤|Λ| called complete NI. Appendix D: Noether Identities of Differential Operators 273

Given an integer N = 2k or N = 2k + 1, k = 0, 1,..., let us denote

0 P{N}=S∞[EN ⊕ ···⊕E1 ⊕ E0 ⊕ EY , Y ×(E0 ⊕ E2 ⊕ ···⊕Ek )], (D.8) Y Y Y Y X X X X

0 where S∞[, ] is the graded commutative ring (6.39).

Lemma D.2 If the homology H1(δ) of the complex (D.3) is ﬁnitely generated, this complex can be extended to the one-exact complex (D.9) with a boundary operator whose nilpotency conditions are equivalent to complete NI. Proof Let us consider the graded commutative ring P{0} (D.8). It possesses the local generating basis {yi ,εa,εr } of Grassmann parity [εr ]=0 and antiﬁeld number Ant[εr ]=2. This ring is provided with the nilpotent graded derivation

r δ0 = δ + ∂r Δ .

Its nilpotency conditions are equivalent to the complete NI (D.7). Then the module P{0}≤3 of graded functions of antiﬁeld number ≤ 3 is decomposed into a chain complex δ δ δ 0 0 0 0←Im δ ←− S∞[EY , Y ]1 ←− P{0}2 ←− P{0}3. (D.9)

Let H∗(δ0) denote its homology. We have H0(δ0) = H0(δ) = 0. Furthermore, any one-cycle Φ up to a boundary takes the form (D.6) and, therefore, it is a δ0-boundary ⎛ ⎞ Φ = Ξ Δr = δ ⎝ Ξ εr ⎠ . Gr dΞ 0 Gr Ξ 0≤|Σ| 0≤|Σ|

Hence, H1(δ0) = 0, i.e., the complex (D.9) is one-exact.

Let us consider the second homology H2(δ0) of the complex (D.9). Its two-chains read Φ = + = Λεr + ΛΣ εa εb . G H Gr Λ Hab Λ Σ (D.10) 0≤|Λ| 0≤|Λ|,|Σ|

Its two-cycles deﬁne the ﬁrst-stage NI δ Φ = , Λ Δr + δ = . 0 0 Gr dΛ H 0 (D.11) 0≤|Λ|

Conversely, let the equality (D.11) hold. Then it is a cycle condition of the two-chain (D.10). The ﬁrst-stage NI (D.11) are trivial either if the two-cycle Φ (D.10)isa boundary or its summand G vanishes on Ker E . Lemma D.3 First-stage NI can be identiﬁed with nontrivial elements of the homol- 0 ogy H2(δ0) if and only if any δ-cycle Φ ∈ S∞[EY , Y ]2 is a δ0-boundary. 274 Appendix D: Noether Identities of Differential Operators

Proof The proof is similar to that of Theorem 7.3 [61, 123].

If the condition of Lemma D.3 is satisfied, let us assume that nontrivial first-stage NI are finitely generated as follows. There exists a graded projective C∞(X)-module C(1) ⊂ H2(δ0) of finite rank possessing a local basis Δ(1) : Λ Δr1 = Δ r1 εr + r1 , r Λ h 0≤|Λ| such that any element Φ ∈ H2(δ0) factorizes as Ξ r Φ = Φ dΞ Δ 1 (D.12) r1 0≤|Ξ| through elements of C(1). Thus, all nontrivial first-stage NI (D.11) result from the equalities Λ Δr1 Δr + δ r1 = , r dΛ h 0 (D.13) 0≤|Λ| called the complete first-stage NI.

Lemma D.4 If nontrivial first-stage NI are finitely generated, the one-exact complex (D.9) is extended to the two-exact one (D.14) with a boundary operator whose nilpotency conditions are equivalent to complete Noether and first-stage NI.

Proof By virtue of Serre–Swan Theorem A.10, a module C(1) is isomorphic to a module of sections of some vector bundle E1 → X. Let us consider the graded commutative ring P{1} (D.8) of graded functions on J ∞Y possessing the local generating bases {yi ,εa,εr ,εr1 } of Grassmann parity [εr1 ]=1 and antiﬁeld number Ant[εr1 ]=3. It can be provided with the nilpotent graded derivation

δ = δ + ∂ Δr1 . 1 0 r1

Its nilpotency conditions are equivalent to the complete NI (D.7) and the complete ﬁrst-stage NI (D.13). Then a module P{1}≤4 of graded functions of antiﬁeld number ≤ 4 is decomposed into a chain complex

δ δ δ δ 0 1 1 0 0←Im δ ←− S∞[EY , Y ]1 ←− P{0}2 ←− P{1}3 ←− P {1}4. (D.14)

Let H∗(δ1) denote its homology. It is readily observed that

H0(δ1) = H0(δ) = 0, H1(δ1) = H1(δ0) = 0. Appendix D: Noether Identities of Differential Operators 275

By virtue of the equality (D.12), any two-cycle of the complex (D.14) is a boundary ⎛ ⎞ Ξ r ⎝ Ξ r1 ⎠ Φ = Φ dΞ Δ 1 = δ Φ ε . r1 1 r1 Ξ 0≤|Ξ| 0≤|Ξ|

It follows that H2(δ1) = 0, i.e., the complex (D.14) is two-exact.

If the third homology H3(δ1) of the complex (D.14) is not trivial, its elements correspond to second-stage NI, and so on. Iterating the arguments, we come to the following. A degenerate differential operator E is called N-stage reducible if it admits ﬁnitely generated nontrivial N-stage NI, but no nontrivial (N + 1)-stage ones. It is characterized as follows [61, 123].

• There are vector bundles E0,...,EN over X, and a graded commutative ring 0 S∞[EY , Y ] is enlarged to a graded commutative ring P{N} (D.8) with the local generating basis (yi ,εa,εr ,εr1 ,...,εrN ) of Grassmann parity [εrk ]=k mod2 and rk antiﬁeld number Ant[εΛ ]=k + 2. • A graded commutative ring P{N} is provided with the nilpotent right graded derivation δ = δ = δ + ∂ Δrk , KT N 0 rk (D.15) ≤ ≤ 1 k N Λ r − ΞΣ r − Δrk = Δ rk ε k 1 + (h rk εa ε k 2 + ...), rk−1 Λ ark−2 Ξ Σ 0≤|Λ| 0≤Σ,0≤Ξ

of antiﬁeld number −1. • With this graded derivation, a module P{N}≤N+3 of graded functions of antiﬁeld number ≤ (N + 3) is decomposed into the exact KT complex

δ δ δ 0 0 1 0←Im δ ←− S∞[EY , Y ]1 ←− P{0}2 ←− P{1}3 ··· (D.16) δ δ δ N−1 KT 0 KT ←− P{N − 1}N+1 ←− P {N}N+2 ←− P{N}N+3,

which satisﬁes the homology regularity condition that any δk

Φ ∈ P{k}k+3 ⊂ P{k + 1}k+3

is a δk+1-boundary (cf. Deﬁnition 7.2). • δ2 = The nilpotentness KT 0 of the KT operator (D.15) is equivalent to the complete nontrivial NI (D.7) and the complete nontrivial (k ≤ N)-stage NI ⎛ ⎞ Λrk ⎝ Σrk− rk−2 ⎠ ΞΣrk a rk−2 Δ dΛ Δ 1 ε + δ h ε ε = 0. (D.17) rk−1 rk−2 Σ ark−2 Ξ Σ 0≤|Λ| 0≤|Σ| 0≤Σ,Ξ 276 Appendix D: Noether Identities of Differential Operators

Example

Let us study the following example of reducible NI of a differential operator which is relevant to topological BF theory (Chap. 12). Let us consider the ﬁbre bundles

n−1 Y = X × R, E = ∧ TX, 2 < n, (D.18)

λ λ μ ...μ coordinated by (x , y) and (x ,χ 1 n−1 ), respectively. We study the E-valued differential operator μ ...μ − μμ ...μ − E 1 n 1 =−ε 1 n 1 yμ, (D.19) where ε is the Levi–Civita symbol. It defines a first order differential equation dH y = 0 on the fibre bundle Y (D.18). Putting

n−1 EY = R × ∧ TX, X

∗ ∞ let us consider the graded commutative ring S∞[EY , Y ] of graded functions on J Y . μ ...μ μ ...μ It possesses the local generating basis (y,ε 1 n−1 ) of Grassmann parity [ε 1 n−1 ]= μ ...μ 1 and antiﬁeld number Ant[ε 1 n−1 ]=1. With the nilpotent derivation

∂ μ ...μ δ = E 1 n−1 , μ ...μ ∂ε 1 n−1 we have the complex (D.3). Its one-chains read μ ...μ Φ = ΦΛ ε 1 n−1 , μ1...μn−1 Λ 0≤|Λ| and the cycle condition δΦ = 0 takes a form

μ ...μ ΦΛ E 1 n−1 = 0. (D.20) μ1...μn−1 Λ

This equality is satisﬁed if and only if

λ ...λ μμ ...μ μλ ...λ λ μ ...μ Φ 1 k ε 1 n−1 =−Φ 2 k ε 1 1 n−1 . μ1...μn−1 μ1...μn−1

It follows that Φ factorizes as Ξ ν ...ν − Φ = G dΞ Δ 2 n 1 ω ν2...νn−1 0≤|Ξ| Appendix D: Noether Identities of Differential Operators 277 through graded functions

ν ...ν λ,ν ...ν α ...α Δ 2 n−1 = Δ 2 n−1 ε 1 n−1 = (D.21) α1...αn−1 λ λ ν ν − α1...αn−1 ν ν ...ν − δ δ 2 ···δ n 1 ε = dν ε 1 2 n 1 , α1 α2 αn−1 λ 1 which provide the complete NI

ν ν ...ν E 1 2 n−1 = . dν1 0 (D.22)

They can be written in a form dH dH y = 0. (D.23)

The graded functions (D.21) form a basis for a projective C∞(X)-module of finite rank which is isomorphic to the structure module of sections of a vector bundle n−2 0 E0 =∧ TX. Therefore, let us extend the graded commutative ring S∞[EY , Y ] to μ ...μ μ ...μ that P{0} (D.8) possessing the local generating basis (y,ε 1 n−1 ,ε 2 n−1 }, where μ ...μ ε 2 n−1 are even antifields of antifield number 2. We have the nilpotent graded derivation

∂ μ ...μ δ = δ + Δ 2 n−1 0 μ ...μ ∂ε 2 n−1

0 of P∞{0}. Its nilpotency is equivalent to the complete NI (D.22). Then we obtain the one-exact complex (D.9). Iterating the arguments, let us consider the vector bundles

n−k−2 EN=n−2 = X × R, Ek = ∧ TX, k = 1,...,n − 3, and the graded commutative ring P{N}, possessing the local generating basis

μ ...μ μ ...μ μ (y,ε 1 n−1 ,ε 2 n−1 ,...,ε n−1 ,ε)

μ ...μ of Grassmann parity [ε]=n, [ε k+2 n−1 ]=k mod 2 and of antiﬁeld number μ ...μ Ant[ε]=n,Ant[ε k+2 n−1 ]=k + 2. It is provided with the nilpotent graded derivation ∂ ∂ μ δ = δ + + ε n−1 , KT 0 μ ...μ dμn−1 (D.24) ∂ε k+2 n−1 ∂ε 1≤k≤n−3 μ ...μ μ μ ...μ Δ k+2 n−1 = ε k+1 k+2 n−1 , dμk+1 of antiﬁeld number −1. Its nilpotency results from the complete NI (D.22) and the equalities μ ...μ Δ k+2 n−1 = , = ,..., − , dμk+2 0 k 0 n 3 (D.25) 278 Appendix D: Noether Identities of Differential Operators which are the (k + 1)-stage NI (D.17). Then the KT complex (D.16) reads

δ δ δ 0 0 1 0←Im δ ←− S∞[EY , Y ]1 ←− P{0}2 ←− P{1}3 ··· (D.26) δ n−3 δKT δKT ←− P{n − 3}n−1 ←− P{n − 2}n ←− P{n − 2}n+1.

It obeys the above mentioned homology regularity condition as follows.

0 Lemma D.5 Any δk -cycle Φ ∈ P∞{k}k+3 up to a δk -boundary takes a form Λ ···Λ Φ = 1 i G 1 1 i i (D.27) μ + ...μ − ;...;μ + ...μ − k1 2 n 1 ki 2 n 1 (k1+···+ki +3i=k+3) (0≤|Λ1|,...,|Λi |) 1 1 i i μ + ...μ − μ + ...μ − Δ k1 2 n 1 ··· Δ ki 2 n 1 , =− , , ,..., − , dΛ1 dΛi k j 1 0 1 n 3

μ ...μ − μ ...μ − μ ...μ − where k j =−1 stands for ε 1 n 1 and Δ 1 n 1 = E 1 n 1 . It follows that Φ is a δk+1-boundary.

μ ...μ Proof Let us choose some basis element ε k+2 n−1 and denote it, simply, by ε.Let Φ contain a summand φ1ε, linear in ε. Then the cycle condition reads

[ε] δk Φ = δk (Φ − φ1ε) + (−1) δk (φ1)ε + φΔ = 0,Δ= δk ε.

It follows that Φ contains a summand ψΔ such that

[ε]+1 (−1) δk (ψ)Δ + φΔ = 0.

This equality implies a relation

[ε]+1 φ1 = (−1) δk (ψ) (D.28) because the reduction conditions (D.25) involve total derivatives of Δ, but not Δ. Φ = Φ + δ (ψε) Φ ε ε Hence, k , where contains no term linear in . Furthermore, let r be even and Φ have a summand φr ε polynomial in ε. Then the cycle condition leads to the equalities φr Δ =−δk φr−1, r ≥ 2. Since φ1 (D.28)isδk -exact, then φ2 = 0 and, consequently, φr>2 = 0. Thus, a cycle Φ up to a δk -boundary contains no term polynomial in c. It reads Λ ···Λ Φ = 1 i G 1 1 i i μ + ...μ − ;...;μ + ...μ − k1 2 n 1 ki 2 n 1 (k1+···+ki +3i=k+3) (0<|Λ1|,...,|Λi |) 1 1 i i μ + ...μ − μ + ...μ − ε k1 2 n 1 ···ε ki 2 n 1 . (D.29) Λ1 Λi

However, the terms polynomial in ε may appear under general coordinate transformations Appendix D: Noether Identities of Differential Operators 279 α ν + ν − ν ...ν ∂x ∂x k 2 ∂x n 1 μ ...μ ε k+2 n−1 = ··· ε k+2 n−1 det β μ μ ∂x ∂x k+2 ∂x n−1 of a chain Φ (D.29). In particular, Φ contains the summand ν1 ...ν1 νi ...νi + − k + n−1 1 1 i i ε k1 2 n 1 ···ε i 2 , Fν + ...ν − ;...;ν + ...ν − k1 2 n 1 ki 2 n 1 k1+···+ki +3i=k+3 which must vanish if Φ is a cycle. This takes place only if Φ factorizes through the μ ...μ graded densities Δ k+2 n−1 (D.24) in accordance with the expression (D.27).

Following the proof of Lemma D.5, one also can show that any δk -cycle Φ ∈ 0 P∞{k}k+2 up to a boundary takes a form Λ μ + ...μ − Φ = G dΛΔ k 2 n 1 , μk+2...μn−1 0≤|Λ| i.e., the homology Hk+2(δk ) of the complex (D.26) is ﬁnitely generated by the cycles μ ...μ Δ k+2 n−1 .

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Symbols L E , 161 ∞ (J Y, AJ ∞ F ), 118 L H , 39 (X, Y, AF ), 116 L", 54 (Z, AE ), 110 LYM, 174 ∗ (O∞Y, L), 15 Le, 146 ∗ (S∞[F; Y ], L), 129 NL , 29 0 ∗ CX , 225 P , 215 C∞(X), 225 PG , 171 ∞ CW, 194 P , 218 ⊗ DΓ , 250 P k , 217 EL , 70 PΣ , 191 EΓ , 69 P∞, 216 GL4, 190 P2k (FA), 202 ∗ H (X; S), 263 RGP, 164 ∗ H (X; R), 268 SU , 259 ∗ H (X; Z), 269 S{U}, 259 ∗ ( ) HDR X , 226 TTX, 242 H p(B∗), 219 TY, 236 HL , 29 TZ, 227 ∗ HN , 45 T Z, 227 HΓ , 38 TG P, 165 HΦ , 39 Tf, 228 Hp(B∗), 218 VY, 236 J ∗υ, 19 V Φ, 236 J 1 J 1Y , 246 V ∗, 234 J 1Y , 244 V ∗Y , 236 1 J Φ, 245 VG P, 165 J 1s, 245 Y (X), 234 1 J u, 246 ZY , 29 J ∞Y , 2 [A], 120 J ∞υ, 18 Γτ, 248 J k u, 255 Γ , 247 r J Y , 253 ΩT , 86 r J Φ, 255 ΩY , 38 r J s, 255 ΨT , 235 LX, 190 #Y , 37 ∗ ϒ L1, 151 Y , 37 © Atlantis Press and the author(s) 2016 287 G. Sardanashvily, Noether’s Theorems, Atlantis Studies in Variational Geometry 3, DOI 10.2991/978-94-6239-171-0 288 Index

i ΞL , 14 qΓ , 62 ΞT , 86 δ, 137 Ek , 8 ω, 241 Lu, 241 ωλ, 241 1, 213 ωμλ, 241 b, 155 ⊗P, 217 u, 146 π 1, 245 A π 1 E , 110 0 , 245 λ EH , 39 π , 28 E i L , 12 πi , 63 EΓ , 76 πr , 254 E k i , 12 πZ", 30 F , 170 π ji, 63 J υ , 23 ψξ , 230 ∗ O [dA ], 223 , 232 ∗ O A , 224 θi , 245 O∗ i ∞Y , 3 θΛ, 256 O∗ ∞, 3 θJ , 242 Ok X , 268 θZ , 242 Or ( ) Z , 240 θLX, 191 k,m O∞ , 5 υ , 18 ∗ H O , 255 υ , 18 k∗ V Q∞, 4 ξζ, 230 Q∗ [ ; ] ∞ F Y , 120 ϑH , 18 S ∗[ ; ] E Z , 113 ϑV , 18 S ∗ [ ; ] ∞ F Y , 118 ϑ f , 75 S ∗[ ; ] r F Y , 118 ∧P, 217 T (Z), 238 ∧Y , 234 VE , 112 0, 232 G (X), 171 H, 40 G L , 20 HL , 29 Diff s (P, Q), 221 Γ , 39 δ N , 143 Ju , 48 δKT, 143 τ, 237 ˙λ z , 228 {, }T , 86 z˙λ, 228 {, } , 74 ˙ V ∂λ, 239 {∂λ}, 228 ∂ Λ, 253 {∂i }, 236 A(Z), 110 {dyi }, 237 ◦ A(Z) , 183 {dzλ}, 228 AE , 110 ck (FA ), 202 AJ r F , 117 d, 240 EL , 13 dH , 5 ∗ Q∞, 4 dV , 5 ∗ Q∞[F; Y ], 120 dΛ, 254 γH , 39 dλ, 254 d(A , Q), 222 dt , 65 dA , 222 f ∗Y , 232 ∗ dAE , 113 f φ, 240 0 ∗ dO∞, 6 f s, 232 Hom A (P, Q), 215 h0, 5 SG 1 L , 133 jx s, 244 ∇Γ r , 250 jx s, 253 Index 289

i qΓ , 62 BRST extension of a Lagrangian, 161 uξ , 20 BRST operator, 155 Bundle affine, 237 A associated, 164 Absolute velocity, 63 atlas, 230 Action of a graded Lie group, 184 associated, 164 free, 185 holonomic, 235 regular, 185 Lorentz, 193 Action of a structure group of constant local trivializations, 251 on J 1 P, 167 automorphism, 232 on P, 164 vertical, 232 on TP, 165 continuous, 230 Action-angle coordinates, 88 coordinates, 231 global, 89 cotangent, 227 Affine bundle, 237 epimorphism, 232 morphism, 238 exterior, 234 Algebra, 213 G-bundle, 163 N-graded, 222 graded, 115 commutative, 223 isomorphism, 232 Z2-graded, 103 locally trivial, 230 commutative, 104 monomorphism, 232 ∗ O∞Y , 3 morphism, 231 ∗ O∞, 3 affine, 238 unital, 213 linear, 234 Algebraic Poincaré lemma, 9 natural, 239 Antibracket, 158 of principal connections, 168 Antifield, 137 of world connections, 194 first-stage, 142 P-associated, 164 k-stage, 143 principal, 164 Noether, 139 product, 232 smooth, 230 tangent, 227 B vertical, 236 Background field, 34 with a structure group, 163 Base of a fibred manifold, 229 Basic form, 241 Basis C for a graded bundle, 117 Canonical presheaf, 260 for a graded manifold, 111 Cartan equation, 31 for a module, 215 Cartesian coordinate, 60 Batchelor theorem, 110 Chain, 218 Bigraded exterior algebra, 105 Characteristic form, 202 Bimodule, 214 Chern form, 202 commutative, 214 total, 202 graded, 105 Chern–Simons form, 202 Body local, 202 of a graded manifold, 110 Chern–Simons Lagrangian, 203 of a ringed space, 267 local, 203 Body map, 104 Chevalley–Eilenberg Boundary, 218 coboundary operator, 224 Boundary operator, 218 graded, 108 BRST complex, 155 differential calculus, 224 290 Index

Grassmann-graded, 109 Noether, 24 minimal, 224 of integral of motion, 66 Christoffel symbols, 253 weak, 23 CIS, 84 Constant sheaf, 260 global, 89 Constrained Hamilton equation, 45 Closed map, 229 Constrained Lagrangian, 46 Coboundary, 219 Contact derivation, 17 Coboundary operator, 219 graded, 131 Cochain, 219 nilpotent, 132 Cochain morphism, 220 projectable, 18 Cocycle, 219 vertical, 18 Cocycle condition, 230, 263 graded, 131 Coframe, 228 Contact form Cohomology, 219 graded, 121 with coefficients in a presheaf, 263 local, 245 with coefficients in a sheaf, 263 of higher jet order, 256 Completely integrable system, 84 Contorsion, 195 autonomous, 86 Contraction, 241 commutative, 84 Coordinates globally, 89 adapted to a reference frame, 63 maximally, 87 bundle, 231 noncommutative, 84 fibre, 231 Complex, 219 fibred, 229 k-exact, 219 holonomic, 227 acyclic, 219 on homogeneous Legendre bundle, chain, 218 30 cochain, 219 on Legendre bundle, 28 exact, 219 on vertical cotangent bundle, 237 variational, 8 on vertical tangent bundle, 236 Component of a connection, 247 Cotangent bundle, 227 Composite bundle, 233 vertical, 236 Configuration space, 27 Covariant Connection, 247 derivative, 250 complete, 62 differential, 250 dual, 252 Hamilton equation, 40 equivariant, 167 Curvature, 250 flat, 251 of a world connection, 252 linear, 251 Cycle, 218 principal, 167 canonical, 169 Connection form, 247 D principal, 168 DBGA, 109 Conservation law De Donder form, 31 covariant, 177 De Rham cohomology, 223 differential, 23 of a graded manifold, 114 in mechanics, 65 of a manifold, 226 energy-momentum, 33 of a ring, 224 gauge, 24 De Rham complex, 223 Hamiltonian, 48 bigraded, 109 in mechanics, 81 of a ring, 224 integral, 23 of exterior forms, 226 Lagrangian, 23 of sheaves, 268 in mechanics, 67 De Rham theorem, 268 Index 291

abstract, 265 Energy-momentum Density, 241 conservation law, 33 Derivation current, 33 contact, 17 Hamiltonian, 49 projectable, 18 of gravity, 198 vertical, 18 tensor, 33 graded, 107 canonical, 34 right, 132 metric, 176 inner, 222 Equivalent bundle atlases, 230 with values in a module, 222 Equivariant Derivation module, 222 automorphism, 171 graded, 107 connection, 167 DGA, 223 function, 171 Differential vector ﬁeld, 165 covariant, 250 Euler–Lagrange equation, 13 exterior, 240 graded, 128 total, 5 second order, 28 graded, 122 Euler–Lagrange operator, 12 vertical, 5 second order, 27 graded, 122 Euler–Lagrange–Cartan operator, 30 Differential bigraded algebra, 109 Euler–Lagrange-type operator, 13 Differential calculus, 223 Evaluation element, 183 minimal, 223 Even element, 103 Differential equation, 256 Even morphism, 105 associated to a differential operator, 258 Exact sequence Differential form, 4 of modules, 215 Differential graded algebra, 223 of vector bundles, 235 Differential operator short, 235 as a morphism, 257 split, 235 as a section, 257 short, 216 degenerate, 272 Exterior graded, 106 algebra, 217 linear, 258 bigraded, 105 of standard form, 257 bundle, 234 on a module, 221 differential, 240 Direct image of a sheaf, 267 graded, 114 Direct limit, 216 form, 240 Direct sequence, 216 basic, 241 Direct sum graded, 113 of complexes, 219 horizontal, 241 of modules, 214 product, 217 Direct system of modules, 216 graded, 105 Directed set, 216 of differemtial forms, 240 Domain, 231 of vector bundles, 234 Dual vector bundle, 234

F E Fibration, 229 Ehresmann connection, 248 Fibre, 229 Energy function bundle, 230 canonical, 70 coordinates, 231 Hamiltonian, 84 afﬁne, 237 relative to a reference frame, 69 linear, 233 292 Index

Fibred coordinates, 229 Lie group, 184 Fibred manifold, 229 manifold, 110 Fibrewise morphism, 231 simple, 110 Field, 213 trivial, 111 Field-antifield theory, 158 module, 103 Finite dual, 183 dual, 106 Finite exactness, 11 morphism, 106 First variational formula, 19 even, 105 Flow, 238 odd, 105 Frölicher–Nijenhuis bracket, 242 ring, 104 Frame, 233 submanifold, 185 holonomic, 228 vector field, 112 Frame bundle, 190 vector space, 103 Frame field, 190 (n, m)-dimensional, 103 Yang–Mills Lagrangian, 187 Graded-homogeneous element, 104 G Grading automorphism, 103 Gauge Grassmann algebra, 104 operator, 151 Gravitational field, 193 higher-stage, 156 Group bundle, 171 parameters, 20 Group-like element, 184 potential, 168 supersymmetry, 149 algebraically closed, 157 H complete, 150 Hamilton equation, 76 first-stage, 151 constrained, 45 generalized, 157 covariant, 40 k-stage, 151 Hamilton operator, 39 symmetry, 20 Hamilton–De Donder equation, 31 Abelian, 157 Hamiltonian, 75 generalized, 22 covariant, 38 principal, 173 homogeneous, 78 transformation Hamiltonian connection, 37 infinitesimal, 171 Hamiltonian form, 38 principal, 171 associated to a Lagrangian, 41 General covariant transformation, 190 weakly, 42 infinitesimal, 190 constrained, 45 Generalized vector field, 18 Hamiltonian function, 76 Generating functions of a CIS, 86 Hamiltonian map, 39 Ghost, 146 Hamiltonian system, 76 Graded homogeneous, 78 bimodule, 105 Hamiltonian vector field, 85 bundle, 115 of a function, 86 over a smooth manifold, 115 Havas Lagrangian, 71 principal, 186 Helmholtz condition, 13 derivation, 107 Helmholtz–Sonin map, 13 derivation module, 107 Higgs field, 192 differential operator, 106 Hilbert–Einstein Lagrangian, 197 exterior differential, 114 Holonomic exterior form, 113 atlas, 235 exterior product, 105, 108 of a frame bundle, 191 function, 110 coframe, 228 interior product, 109, 113 coordinates, 227 Index 293

frame, 228 second order, 246 Homology, 218 Jet bundle, 245 Homology regularity condition, 143 affine, 245 Homotopy operator, 219 Jet coordinate, 254 Horizontal Jet manifold, 245 form, 241 graded, 117 graded, 121 higher order, 253 lift infinite order, 2 of a path, 248 graded, 118 of a vector field, 248 of a graded bundle, 117 projection, 5 infinite order, 118 splitting repeated, 246 canonical, 249 second order, 246 subbundle, 247 sesquiholonomic, 246 vector field, 61, 248 Jet prolongation, 255 of a morphism, 245 higher order, 255 I of a section, 245 Ideal, 213 higher order, 255 maximal, 213 of a vector field, 245 of contact forms, 256 higher order, 255 of nilpotents, 104 Juxtaposition rule, 223 prime, 214 proper, 213 Imbedded submanifold, 228 K Imbedding, 228 Kepler potential, 94 Immersion, 228 Kernel Induced coordinates, 235 of a bundle morphism, 232 Infinitesimal generator, 238 of a differential operator, 257 Infinitesimal transformation of a vector bundle morphism, 234 of a Lagrangian system, 17 Komar superpotential Initial data coordinates, 76 generalized, 199 Integral curve, 238 Koszul–Tate complex, 143 Integral of motion, 65 of a differential operator, 275 Hamiltonian, 80 Koszul–Tate operator, 143 Lagrangian, 68 KT-BRST complex, 160 Integral section for a connection, 250 KT-BRST operator, 160 Interior product, 241 graded, 109, 113 of vector bundles, 234 L Invariant of Poincaré–Cartan, 75 Lagrange equation, 64 Invariant submanifold, 87 Lagrange operator, 63 Inverse image of a sheaf, 267 Lagrangian, 12 Inverse limit, 218 almost regular, 29 topology, 2 characteristic, 40 Inverse sequence, 218 constrained, 46 degenerate, 138 first order, 27 J gauge invariant, 173 Jet graded, 128 first order, 244 hyperregular, 29 higher order, 253 regular, 29 infinite order, 2 semiregular, 29 294 Index

variationally trivial, 13 graded, 110 Lagrangian constraint space, 29 smooth, 227 Lagrangian system, 15 Mass tensor, 70 Abelian, 157 Master equation, 160 finitely degenerate, 138 Metric, 193 finitely reducible, 141 bundle, 193 graded, 129 connection, 196 irreducible, 141 Metricity condition, 196 N-stage reducible, 142 Minkowski metric, 193 reducible, 141 Mishchenko–Fomenko theorem, 88 Legendre bundle, 28 Module, 214 composite, 28 dual, 215 homogeneous, 30 finitely generated, 215 Legendre map, 28 free, 215 homogeneous, 30 graded, 103 Leibniz rule, 222 of finite rank, 215 graded, 223 over a Lie superalgebra, 105 Grassmann-graded, 107 projective, 215 Lepage equivalent, 14 Morphism of a graded Lagrangian, 129 of fibre bundles, 231 Lepage form, 14 of graded manifolds, 115 Levi–Civita connection, 253 of presheaves, 261 Levi–Civita symbol, 241 of ringed spaces, 267 Lie algebra bundle, 165 of sheaves, 261 Lie bracket, 238 Motion, 65 graded, 104 Multi-index, 253 Lie derivative Multisymplectic form, 38 graded, 109, 131 Multisymplectic Liouville form, 31 of a tangent-valued form, 243 of an exterior form, 241 N Lie superalgebra, 104 Nijenhuis differential, 243 of a graded Lie group, 184 Noether conservation law, 24 Lift of vector fields, 239 Noether current, 24 functorial, 239 Noether identities, 138 horizontal, 248 complete, 138 Liouville form first-stage, 140 canonical, 86 complete, 141 multisymplectic, 31 nontrivial, 140 polysymplectic, 37 trivial, 140 Local diffeomorphism, 228 higher-stage, 143 Local-ringed space, 267 irreducible, 141 Locally finite covering, 264 nontrivial, 138 Locally free module, 268 of a differential operator, 271 Lorentz trivial, 138 atlas, 193 Noether’s structure, 192 direct second theorem, 22 subbundle, 192 in a general setting, 151 first theorem, 23 in a general setting, 133 M inverse first theorem, 82 Manifold inverse second theorem, 145 fibred, 229 third theorem, 25 Fréchet, 2 Nonmetricity tensor, 195 Index 295

O complex, 219 Odd element, 103 module, 215 Odd morphism, 105 sheaf, 261 Open map, 228 Orbital momentum, 93 R Rank of a morphism, 228 P Reduced Paracompact space, 264 Lorentz structure, 192 Partition of unity, 264 principal subbundle, 192 Path, 248 structure, 192 Phase space, 36 Reference frame, 62 homogeneous, 38 complete, 63 of mechanics, 64 Refinement, 263 of mechanics, 64 Relative velocity, 63 Poincaré–Cartan form, 29 Resolution, 219 Poisson bracket fine, 266 canonical, 86 of a sheaf, 265 on vertical cotangent bundle, 74 Restriction morphism, 259 Polysymplectic Restriction of a bundle, 232 form, 37 Ring, 213 Liouville form, 37 graded, 104 manifold, 37 local, 213 structure, 36 of smooth functions, 225 Presheaf, 259 Ringed space, 266 Presymplectic form, 85 Rung–Lenz vector, 94 Primitive element, 183 Principal automorphism, 171 S of a connection bundle, 172 Saturated neighborhood, 87 of an associated bundle, 171 Section, 229 bundle, 164 global, 229 graded, 186 integral, 250 connection, 167 local, 229 canonical, 169 of a jet bundle, 245 vector field, 171 integrable, 245 associated, 171 zero-valued, 234 vertical, 171 Serre–Swan theorem, 225 Proper for graded manifolds, 111 cover, 263 Sheaf, 259 extension of a Lagrangian, 160 acyclic, 265 map, 229 fine, 266 solution of a master equation, 160 locally free, 268 Pull-back of constant rank, 268 bundle, 232 of finite type, 268 form, 240 of continuous functions, 259 section, 232 of derivations, 267 of graded derivations, 112 of modules, 268 Q of smooth functions, 260 Quotient Sheaf cohomology, 263 algebra, 214 Smooth manifold, 227 bundle, 235 Soldering form, 243 296 Index

basic, 243 vertical, 236 Solution of a differential equation, 256 Tangent morphism, 228 classical, 257 vertical, 236 Splitting of an exact sequence, 216 Tangent-valued form, 242 Spontaneous symmetry breaking, 191 canonical, 242 Standard one-form, 60 horizontal, 243 Standard vector field, 60 Tensor Strength, 168 algebra, 217 canonical, 170 bundle, 235 form, 170 field, 235 Structure module (m, k), 235 of a sheaf, 259 Tensor product of a vector bundle, 234 of complexes, 220 Structure ring of a graded manifold, 110 of graded modules, 105 Structure sheaf of modules, 214 of a graded manifold, 110 of vector bundles, 234 of a ringed space, 267 Tensor product connection, 252 of a vector bundle, 225 Tetrad Subbundle, 232 coframe, 193 Submanifold, 228 form, 193 Submersion, 228 frame, 193 Superbracket, 104 function, 193 Superintegrable system, 84 Topological invariant, 269 autonomous, 86 Torsion form, 250 Superpotential, 24 of a world connection, 252 Superspace, 106 tangent-valued, 195 Supersymmetry, 132 Total derivative, 245 gauge, 149 graded, 122 irreducible, 151 higher order, 254 Supersymmetry current, 133 infinite order, 6 Supervector space, 106 Total space, 229 SUSY, 132 Transgression formula, 202 Symmetry, 19 on a base, 202 classical, 20 Transition functions, 230 exact, 24 G-valued, 163 gauge, 20 Trivial extension of a sheaf, 267 generalized, 20 Trivialization chart, 230 Hamiltonian, 81 Trivialization morphism, 230 basic, 82 Typical fibre, 230 internal, 33 Lagrangian, 19 of a differential equation, 66 U of a differential operator, 66 Utiyama theorem, 174 Symmetry current, 23 Hamiltonian, 48 in mechanics, 81, 82 V Symplectic form, 85 Variational canonical, 86 bicomplex, 8 Symplectic manifold, 85 graded, 122 complex, 8 graded, 123 T derivative, 12 Tangent bundle, 227 operator, 7 Index 297

graded, 122 Vertical automorphism, 232 Variational formula, 14 Vertical splitting, 236 first, 19 of a vector bundle, 236 graded, 132 of an affine bundle, 238 graded, 129 Vertical-valued form, 243 Vector bundle, 233 characteristic, 110 density-dual, 136 W dual, 234 Weak graded, 136 conservation law, 23 density-dual, 136 equality, 66 Vector field, 238 transformation law complete, 238 under a background field, 36 equivariant, 165 under an external force, 68 fundamental, 165 Weil homomorphism, 202 generalized, 18 Whitney sum, 234 graded, 131 World graded, 112 connection, 252 Hamiltonian, 75, 85 on the cotangent bundle, 252 horizontal, 61, 248 symmetric, 253 principal, 171 manifold, 189 projectable, 238 metric, 253 standard, 60 pseudo-Riemannian, 193 vertical, 239 Vector space, 214 graded, 103 Y Vector-valued form, 244 Yang–Mills Velocity equation, 174 absolute, 63 gauge theory, 175 relative, 63 Lagrangian, 174 Velocity space, 60 graded, 187