Noether Theorems and Applications

21th International Summer School on Global Analysis and its Applications 15-19 August 2016, Star´aLesn´a,Slovakia

G. Sardanashvily Noether Theorems and Applications

1. Graded Lagrangian formalism. 2. First Noether theorem. 3. Gauge symmetries. 4. Noether identities. 5. Second Noether theorems. 6. Lagrangian BRST theory. 7. Applications. Topological BF theory.

Basic references1

G.Sardanashvily, Noether’s Theorems. Applications in Mechanics and Field The- ory (Springer, 2016). G.Sardanashvily, Higher-stage Noether identities and second Noether theorems, Adv. Math. Phys. 2015 (2015) 127481. G.Sardanashvily, Graded Lagrangian formalism, Int. J. Geom. Methods. Mod. Phys. 10 (2013) 1350016. G.Giachetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian ﬁeld theory, J. Math. Phys. 50 (2009) 012903. D.Bashkirov, G.Giachetta, L.Mangiarotti, G.Sardanashvily, The KT-BRST complex of degenerate Lagrangian systems, Lett. Math. Phys. 83 (2008) 237. G.Giachetta, L.Mangiarotti, G.Sardanashvily, Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology, Commun. Math. Phys. 259 (2005) 103.

1http://www.g-sardanashvily.ru

1 History

Classical Noether’s theorems are well known to treat symmetries of Lagran- gian systems.

First Noether’s theorem associates to a Lagrangian symmetry the • conserved symmetry current whose total di↵erential vanishes on-shell. Second Noether’s theorems provide the correspondence between the • gauge symmetries of a Lagrangian and the Noether identities which its Euler– Lagrange operator satisﬁes.

Let us refer for a rich history of Noether’s theorems to the brilliant book: Y.Kosmann-Schwarzbach, The Noether Theorems. Invariance and the Conservation Laws in the Twentieth Century (Springer, 2011).

However, one should go beyond classical Noether’s theorems because they do not provide a complete analysis of the degeneracy of a generic Lagrangian system, namely, reducible degenerate Lagrangian systems.

A problem has come from Quantum Field Theory (QFT) where an analysis of the degeneracy of a ﬁeld system is a preliminary step towards its quantization.

J.Fisch, M.Henneaux, Homological perturbation theory and algebraic structure of the antiﬁeld-antibracket formalism for gauge theories, Commun. Math. Phys. 128 (1990) 627-640. G.Barnich, F.Brandt, M.Henneaux, Local BRST cohomology in gauge theories. Phys. Rep. 338 (2000) 439-569.

2 Basic Problem

A problem is that, for any Lagrangian, its Euler–Lagrange operator sat- isﬁes Noether identities, which therefore must be separated into the trivial and non-trivial ones.

Moreover, these Noether identities obey ﬁrst-stage Noether identities, which in turn are subject to the second-stage ones, and so on.

Thus, there is a hierarchy of higher-stage Noether identities and, accordingly, gauge symmetries.

The corresponding higher-stage extension of Noether’s theorems therefore should be formulated.

3 Main Theses

We aim to formulate the generalized Noether theorems in a very general setting of reducible degenerate Lagrangian systems of graded (even and odd) variables on graded bundles. This formulation is based on the following.

Treating Noether identities in Lagrangian formalism, we follow the general notion of Noether identities of di↵erential operators on ﬁbre bundles.

G.Sardanashvily, Noether identities of a di↵erential operator. The Koszul– Tate complex Int. J. Geom. Methods Mod. Phys. 2 (2005) 873-886; arXiv: math/0506103.

A key point is that any di↵erential operator on a ﬁbre bundle sat- • isﬁes certain di↵erential identities, called the Noether identities, which thus must be separated into the trivial and non-trivial ones.

Furthermore, non-trivial Noether identities of a di↵erential operator obey • ﬁrst stage Noether identities, which in turn are subject to the second- stage ones, and so on. Thus, there is a hierarchy of higher-stage Noether identities of a di↵erential operator.

4 This hierarchy is described in terms of chain complexes whose bound- • aries are associated to trivial higher-stage Noether identities, but non-zero elements of their homology characterize non-trivial Noether identities modulo the trivial ones.

As a result, if a certain homology condition holds, one constructs an • exact chain complex, called the Koszul–Tate (KT) complex, with a KT boundary operator whose nilpotentness is equivalent to all complete non- trivial Noether and higher-stage Noether identities of a di↵erential operator.

A di↵erential operator is said to be degenerate if it admits non- • trivial Noether identities, and reducible if there exist non-trivial higher-stage Noether identities.

It should be noted that, though a di↵erential operator is defined on a fibre • bundle, the KT complex consists of graded (even and odd) elements, called antifields in accordance with the QFT terminology, and it is considered on graded manifolds and bundles.

5 II.

Given a Lagrangian system, we apply a general analysis of Noether identities of di↵erential operators to the corresponding Euler–Lagrange operator. A result is the associated KT chain complex with the boundary operator whose nilpotency condition reproduces all non-trivial Noether and higher- stage Noether identities of an Euler–Lagrange operator. In the case of a variational Euler–Lagrange operator, we obtain something more.

The extended inverse and direct second Noether theorems state • the relations between higher-stage Noether identities and gauge symmetries of a Lagrangian system. Namely, these theorems associate to the above- mentioned KT complex a certain cochain sequence whose ascent operator consists of gauge and higher-order gauge symmetries of a Lagrangian system. Therefore, it is called the gauge operator.

This cochain sequence, as like as the KT complex, consist of graded (even • and odd) elements, called the ghosts in accordance with the QFT terminology. A problem is that the gauge operator, unlike the KT boundary operator, is not nilpotent. Consequently, there is no self-consistent deﬁnition of non-trivial gauge symmetries, and therefore one has start just with Noether identities.

Nevertheless, if gauge symmetries are algebraically closed, the gauge op- • erator is extended to the nilpotent BRST operator which brings a cochain sequence into the BRST complex and provides a BRST extension of an original Lagrangian system by means of graded antiﬁelds and ghosts.

6 III.

Since the hierarchy of higher-stage Noether identities and gauge symmetries is described in the framework of graded homology and cohomology complexes, Lagrangian theory of graded even and odd variables is considered from the beginning. In QFT, this is the case of fermion ﬁelds, ghosts in gauge theory and SUSY extensions of Standard Model.

Lagrangian theory of even variables on a smooth manifold X convention- • ally is formulated in terms of ﬁbre bundles and jet manifolds. A key point is the classical Serre–Swan theorem which states the categorial equivalence between the projective modules of ﬁnite rank over a ring C1(X) of smooth real functions on X and the modules of global sections of vector bundles over X.

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (World Scientiﬁc, 2009). G.Sardanashvily, Advanced Di↵erential Geometry for Theoreticians. Fibre bundles, jet manifolds and Lagrangian theory (Lambert Aca- demic Publishing, 2013); arXiv: 0908.1886.

7 However, di↵erent geometric models of odd variables either on graded • manifolds or supermanifolds are discussed. It should be emphasized the dif- ference between graded manifolds and supermanifolds. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing sheaves on su- pervector spaces.

C.Bartocci, U.Bruzzo, D.Hern´andezRuip´erez, The Geometry of Super- manifolds (Kluwer, 1991).

We follow the graded extension of the Serre–Swan theorem. It • states that, if a graded commutative C1(X)-ring is generated by a projective

C1(X)-module of ﬁnite rank, it is isomorphic to the structure ring of graded functions on a graded manifold whose body is X. Therefore, we develop graded Lagrangian theory of even and odd variables in terms of graded manifolds and graded bundles.

A problem is that no conventional variational principle may be formu- • lated for odd variables because there is no measure on graded manifolds. Nevertheless, a Lagrangian theory on a ﬁbre bundle Y can be developed in algebraic terms of a variational bicomplex of di↵erential forms on an inﬁnite order jet manifold J 1Y of Y , without appealing to a variational principle. This technique has been extended to Lagrangian theory on graded bundles in terms of a graded variational bicomplex of graded di↵erential forms on graded jet manifolds.

8 IV.

Given a graded Lagrangian L, the cohomology of a variational bicomplex provides the global variational formula

dL = d ⌅ , EL H L where is the graded Euler–Lagrange operator and ⌅ is a Lepage equivalent EL L of a graded Lagrangian L.

The extended ﬁrst Noether theorem is a straightforward corollary of • the above global variational formula in a general case of graded Lagrangians and their supersymmetries.

It associates to a supersymmetry of a graded Lagrangian L the • current whose total di↵erential d vanishes on the shell L = 0. J H J If is a gauge supersymmetry of a graded Lagrangian L, the correspond- • ing current is a total di↵erential on-shell. This statement sometimes is J called the third Noether theorem.

9 1 Graded Lagrangian formalism

1.1. Di↵erential calculus over commutative rings 1.2. Lagrangian formalism on smooth ﬁbre bundles 1.3. Di↵erential calculus over graded commutative rings 1.4. Di↵erential calculus on graded manifolds 1.5. Lagrangian theory of even and odd variables on graded bundles

The di↵erential calculus, including formalism of linear di↵eren- • tial operators and the Chevalley–Eilenberg complex of di↵erential forms, can be formulated over any ring. A problem is that Euler–Lagrange operators need not be linear.

Theory of non-linear di↵erential operators and, in particular, La- • grangian formalism conventionally are formulated on smooth ﬁbre bundles over a smooth manifold X in terms of their jet manifolds.

In the framework of the di↵erential calculus over graded commutative • rings, Lagrangian formalism has been extended to graded manifolds and graded bundles over a smooth manifold X.

In applications, this is the case both of classical ﬁeld theory on • bundles over X, dim X>1, and non-relativistic mechanics on ﬁbre bundles over X = R. Relativistic mechanics and classical string theory can be formulated as Lagrangian theory of submanifolds.

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (World Scientiﬁc, 2009).

10 1.1 Di↵erential calculus over commutative rings

The di↵erential calculus conventionally is deﬁned over commutative rings. It straightforwardly is generalized to the case of graded commutative rings. However, this is not a particular case of the di↵erential calculus over non-commutative rings. A construction of the Chevalley–Eilenberg complex is generalized to an arbitrary ring, but an extension of the notion of di↵erential operators to non-commutative rings meets diculties. A key point is that multiplication in a non-commutative ring is not a zero-order di↵erential operator.

I.Krasil’shchik, V.Lychagin, A.Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Di↵erential Equations (Gordon and Breach, 1985). G.Sardanashvily, Lectures on Di↵erential Geometry of Modules and Rings (Lambert Academic Publishing, 2012); arXiv: 0910.1515.

In a case of graded commutative rings, one overcomes this diculty by means of reformulating the notion of di↵erential operators. In particular, derivations both of commutative and non-commutative rings obey A the Leibniz rule

@(ab)=@(a)b + a@(b), a, b , 2A whereas the graded Leibniz rule for a graded commutative ring reads

@(ab)=@(a)b +( 1)[a][@]a@(b), a, b , 2A where [a] = 1, [@] = 1 for odd elements a and derivations @. 2A

11 Therefore, supergeometry is not particular non-commutative geometry.

Remark 1.1: All algebras throughout are associative, unless they are Lie algebras and Lie superalgebras. By a ring is meant a unital algebra with a unit element 1 = 0. Given a commutative ring , an additive group P 6 A is called the -module if it is provided with a distributive multiplication A P P by elements of such that ap = pa for all a , p P .A A⇥ ! A 2A 2 module over a field is called the vector space. If a ring is module over a A commutative ring , it is said to be the -ring. A module P is called free K K if it admits a basis. A module is said to be of finite rank if it is the quotient of a free module with a finite basis. One says that a module P is projective if there exists a module Q such that P Q is a free module. ⇤ Let be a commutative ring, a commutative -ring, and let P and K A K Q be -modules. A -module Hom (P, Q)of -module homomorphisms A K K K :P Q can be endowed with two di↵erent -module structures ! A (a)(p)=a(p), ( a)(p)=(ap),a,pP. • 2A 2 Let us put =a a, a . a • 2A DEFINITION 1.1: An element Hom (P, Q) is called the linear s- 2 K order Q-valued di↵erential operator on P if ( )= 0 for a0 ··· as any tuple of s + 1 elements a ,...,a of . 0 s A ⇤

12 In particular, linear zero-order di↵erential operators obey conditions

(p)=a(p) (ap)=0,a,pP, a 2A 2 and, consequently, they coincide with -module morphisms P Q. A ! A linear ﬁrst-order di↵erential operator satisﬁes a relation

( )(p)=ba(p) b(ap) a(bp)+(abp)=0, a, b . b a 2A For instance, a ﬁrst-order di↵erential operator on P = obeys a condition A (ab)=b(a)+a(b) ba(1), a, b . 2A

DEFINITION 1.2: It is called a Q-valued derivation of if (1)=0, A i.e., it satisﬁes the Leibniz rule

(ab)=(a)b + a(b), a, b . 2A

⇤

If @ is a derivation of , then a@ is well for any a . Hence, derivations A 2A of constitute an -module d( ,Q), called the derivation module of . A A A A

If Q = , the derivation module d = d( , )of also is a Lie algebra A A A A A over a ring with respect to a Lie bracket K

[u, u0]=u u0 u0 u, u, u0 d . 2 A

13 A fact is that a linear s-order di↵erential operator on an -module P is A represented by a zero-order di↵erential operator on a module of s-order jets of P (Theorem 1.1 below).

DEFINITION 1.3: Given an -module P , let P be a tensor product A A⌦K of -modules and P . We put K A b(a p)=(ba) p a (bp),pP, a, b . ⌦ ⌦ ⌦ 2 2A Let us denote by µk+1 a submodule of P generated by elements A⌦K b0 bk (a p)=ab0 bk (1 p). ··· ⌦ ··· ⌦ A k-order jet module k(P ) of a module P is deﬁned as the quotient of a J k+1 -module P by µ . Its elements c k p are called the jets. ⇤ K A⌦K ⌦ There exists a module morphism

J k : P p 1 p k(P ) (1.1) 3 ! ⌦k 2J such that k(P ), seen as an -module, is generated by jets J kp, p P . J A 2 THEOREM 1.1: Any linear k-order Q-valued di↵erential operator on an -module P uniquely factorizes as A k f :P J k(P ) Q ! J ! through the morphism J k (1.1) and some -module homomorphism f : A k(P ) Q. J ! ⇤ In view of this fact, one says that jet modules k(P ) of a module P play J a role of the representative objects of linear di↵erential operators on P .

14 Since the derivation module d of a commutative -ring is a Lie - A K A K algebra, one can associate to the Chevalley–Eilenberg complex ⇤[d ] A O A of di↵erential forms over . A D.Fuks, Cohomology of Inﬁnite-Dimensional Lie Algebras (Consul- tants Bureau, 1986)

It consists of -modules k[d ] of -multilinear skew-symmetric maps A O A A k k k[d ] = Hom ( d , ) : d , (1.2) O A A ⇥ A A 3 ⇥ A!A provided both with the Chevalley–Eilenberg coboundary operator

k d(u ,...,u )= ( 1)iu ((u ,...,u ,...,u )) + (1.3) 0 k i 0 i k i=0 X i+j ( 1) ([ui,uj],u0,...,ui,...,buj,...,uk), i

0(u , ..., u ) = (1.4) ^ 1 r+s i1 irj1 js sgn1 ···r+s ··· (ui1 ,...,uir )0(uj1 ,...,ujs ), ··· i <

... where sgn... denotes the sign of a permutation. They obey relations

0 0 =( 1)| || |0 , ^ ^ d( 0)=d() 0 +( 1)| | d(0),,0 ⇤[d ], (1.5) ^ ^ ^ 2O A and bring ⇤[d ] into a di↵erential graded ring (DGR). O A

15 We also have the interior product u = (u), u d , 1[d ]. It c 2 A 2O A is extended as

(u )(u1,...,uk 1)=k(u, u1,...,uk 1),ud , ⇤[d ], (1.6) c 2 A 2O A to a DGR ⇤[d ], and obeys a relation O A u ( )=u +( 1)| | u . c ^ c ^ ^ c With the interior product (1.6), one deﬁnes a derivation (a Lie derivative)

L ()=d(u )+u d, ⇤[d ], u c c 2O A L ( )=L () + L , u ^ u ^ ^ u of a graded ring ⇤[d ] for any u d . Then one can think of elements of O A 2 A ⇤[d ] as being di↵erential forms over . O A A

The minimal Chevalley–Eilenberg complex ⇤ over a ring consists of O A A the monomials a da da , a . It is called the de Rham complex 0 1 ^···^ k i 2A of a -ring . K A

16 1.2 Lagrangian formalism on smooth ﬁbre bundles

In order to formulate Lagrangian formalism on smooth ﬁbre bundles, let us start with the di↵erential calculus on smooth manifolds.

Let X be a smooth manifold and C1(X) an R-ring of real smooth functions on X. The di↵erential calculus on a smooth manifold X is deﬁned as that over a real commutative ring C1(X).

Remark 1.2: A smooth manifold throughout is a ﬁnite-dimensional real manifold. It customarily is assumed to be a Hausdor↵and second-countable topological space. Consequently, it is a locally compact countable at inﬁn- ity space and paracompact space, which admits the partition of unity by smooth real functions. Let us emphasize that the paracompactness is very essential for a number of theorems in the sequel. ⇤

In a general setting, we follow the conventional deﬁnition of manifolds as local-ringed spaces, i.e., sheaves in local rings. This is the case both of smooth manifolds and graded manifolds in the sequel.

Let CX1 be a sheaf of germs of smooth real functions on a smooth manifold X, i.e., smooth functions are identiﬁed if they coincide on an open neighborhood of a point x X. Its stalk C1 of germs at x X has a unique 2 x 2 maximal ideal of germs of functions vanishing at x. Therefore, (X, CX1) is a local-ringed space. Though a sheaf CX1 exists on a topological space X, it ﬁxes a unique smooth manifold structure on X as follows.

17 THEOREM 1.2: Let X be a paracompact topological space and (X, A) a local-ringed space. Let X admit an open cover U such that a sheaf A { i} n restricted to each Ui is isomorphic to a local-ringed space (R ,CR1n ). Then X is an n-dimensional smooth manifold together with a natural isomorphism of local-ringed spaces (X, A) and (X, CX1). ⇤

One can think of this result as being an equivalent 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 C1(X) of smooth real functions on X.

Furthermore, the classical Serre–Swan theorem states the categorial equivalence between vector bundles over a smooth manifold X and projective modules of ﬁnite rank over the ring C1(X) of smooth real functions on X.

THEOREM 1.3: Let X be a smooth manifold. A C1(X)-module P is a projective module of ﬁnite rank i↵it is isomorphic to the structure module Y (X) of global sections of some vector bundle Y X over X. ! ⇤ The following are COROLLARIES of this theorem.

The derivation module of a real ring C1(X) coincides with a C1(X)- • module (X) of vector ﬁelds on X. T1 1 1 Its C1(X)-dual (X)= (X)⇤ is the structure module (X) of the • O T1 O cotangent bundle T ⇤X of X, i.e., a module of one-forms on X and, conversely, 1 (X)= (X)⇤. T1 O

It follows that the Chevalley–Eilenberg complex of a real ring C1(X) is • exactly the de Rham complex ( ⇤(X),d) of exterior forms on X. O 18 Let Y X be a vector bundle and Y (X) its structure module. An • ! r r-order jet module (Y (X)) of a C1(X)-module Y (X) is the structure J module J rY (X) of sections of an r-order jet bundle J rY X. ! Then by virtue of Theorem 1.1, a liner k-order di↵erential operator on a • projective C1(X)-module P of ﬁnite rank with values in a projective C1(X)- module Q of ﬁnite rank is represented by a linear bundle morphism J kY E ! of a jet bundle J kY X to a vector bundle E X where Y X and ! ! ! E X are smooth vector bundles with structure modules Y (X)=P and ! E(X)=Q in accordance with Serre–Swan Theorem 1.3.

Thus, the di↵erential calculus on a smooth manifold X leads to a Lie algebra of vector ﬁelds on X, a DGR of exterior forms on X, jet manifolds of a vector bundle over X as representative objects of linear di↵erential operators on this vector bundle.

A key point is that the construction of jet manifolds as representative objects of di↵erential operators is generalized to a case of non-linear di↵erential operators on ﬁbre bundles.

19 Let Y X be a smooth fibre bundle provided with bundle coordinates ! (x,yi). An r-order jet manifold J rY of sections of a fibre bundle Y X is ! defined as the disjoint union of equivalence classes jrs of sections s of Y X x ! which are identified by r + 1 terms of their Taylor series at points of X.

D.Saunders, The Geometry of Jet Bundles (Cambridge Univ. Press, 1989). G.Sardanashvily, Advanced Di↵erential Geometry for Theoreticians. Fibre bundles, jet manifolds and Lagrangian theory (Lambert Aca- demic Publishing, 2013); arXiv: 0908.1886.

A set J rY is endowed with an atlas of adapted coordinates µ i i i i @x i (x ,y⇤),y⇤ s = @⇤s (x),y0+⇤ = dµy⇤0 , 0 ⇤ r, (1.7) @0x | | where the symbol d stands for the higher order total derivative µ i ⇤ @x d = @ + y @ ,d0 = dµ. ⇤+ i @x 0 ⇤ r 1 0 |X| We use the compact notation d = d d ,⇤=( ... ). The coordi- ⇤ r ··· 1 r 1 nates (1.7) brings a set J rY into a smooth manifold.

Given ﬁbre bundles Y and Y 0 over X, every bundle morphism : Y Y 0 ! over a di↵eomorphism f of X admits the r-order jet prolongation to a morphism of r-order jet manifolds

r r r r 1 r J :J Y j s j ( s f ) J Y 0. 3 x ! f(x) 2 Every section s of a ﬁbre bundle Y X has the r-order jet prolongation ! to a section (J rs)(x)=jrs of a jet bundle J rY X. x ! 20 There are natural surjections of jet manifolds

r r r 1 ⇡r 1 : J Y J Y, ! which form the inverse sequence of ﬁnite order jet manifolds

r ⇡ 1 r 1 ⇡r 1 r Y J Y J Y J Y . (1.8) ··· ···

Its inverse limit J 1Y is a minimal set so that there exist surjections

k ⇡1 : J 1Y X,⇡1 : J 1Y Y,⇡1 : J 1Y J Y, ! 0 ! k ! k obeying the relations ⇡1 = ⇡ ⇡1, r

A set J 1Y is provided with the inverse limit topology. This is the coarsest topology such that the surjections ⇡r1 are continuous. Its base consists of r inverse images of open subsets of J Y , r =0,..., under the maps ⇡r1. With the inverse limit topology, J 1Y is a paracompact Fr´echet manifold. A bundle i coordinate atlas U , (x ,y ) of Y X provides J 1Y with a manifold { Y } ! coordinate atlas µ 1 i i @x i (⇡01) (UY ), (x ,y⇤) 0 ⇤ ,y0+⇤ = dµy⇤0 . { } | | @x0

One calls J 1Y the inﬁnite order jet manifold.

F.Takens, A global version of the inverse problem of the calculus of variations, J. Di↵. Geom. 14 (1979) 543.

21 DEFINITION 1.4: Let Y X and E X be smooth fibre bundles. ! ! A bundle morphism : J kY E over X is called the E-valued k-order ! di↵erential operator on Y . This di↵erential operator sends each section s of Y X to the section J ks of E X. ! ! ⇤ Jet manifolds J kY of a fibre bundle Y X constitute the inverse sequence ! (1.8) whose inverse limit is an infinite order jet manifold J 1Y . Then any k- order E-valued di↵erential operator on a fibre bundle Y is defined by a continuous bundle map

⇡r1 : J 1Y E. !X

In particular, di↵erential operators in Lagrangian theory on ﬁbre bundles, e.g., Euler–Lagrange and Helmholtz–Sonin operators are represented by certain exterior forms on ﬁnite order jet manifolds and, consequently, on J 1Y .

The inverse sequence (1.8) of jet manifolds yields the direct sequence of r DGRs ⇤ = ⇤(J Y ) of exterior forms on ﬁnite order jet manifolds Or O 1 r ⇡⇤ ⇡0⇤ ⇡r 1⇤ ⇤(X) ⇤(Y ) 1⇤ r⇤ 1 r⇤ , (1.9) O ! O ! O ! ···O ! O ! ···

r where ⇡r 1⇤ are the pull-back monomorphisms. Its direct limit ⇤ Y con- O1 sists of all exterior forms on ﬁnite order jet manifolds modulo the pull-back identiﬁcation. It is a DGR which inherits operations of the exterior di↵erential d and the exterior product of DGRs ⇤. ^ Or

22 , ADGR ⇤ Y is split into a variational bicomplex ⇤ ⇤Y . Lagrangians L, O1 O1 Euler–Lagrange operators = L and the variational operator are deﬁned EL 0,n 1,n as elements L , L Y , n = dim X, and the coboundary operator 2O1 E 2O1 of this bicomplex. Its cohomology provides the global variational formula

dL = d ⌅ , EL H L where is an Euler–Lagrange operator and ⌅ is a Lepage equivalent of L. EL L

G.Sardanashvily, Cohomology of the variational complex in the class of exterior forms of ﬁnite jet order, Int. J. Math. & Math. Sci. 30 (2002) 39.

We reproduce these results below in the framework of a graded Lagrangian formalism

It should be emphasized that we deal with a variational bicomplex of the

DGR ⇤ Y of di↵erential forms of bounded jet order on an infinite order O1 jet manifold J 1Y . They are exterior forms on finite order jet manifolds J rY modulo the pull-back identification. One also considers a variational bicomplex of a DGR ⇤ Y of di↵erential forms of locally finite jet order Q1 on J 1Y , which are di↵erential forms on finite order jet manifolds only locally on an open neighborhood of each point of J 1Y .

I.Anderson, Introduction to the variational bicomplex. Contemp. Math. 132, 51-73 (1992)

23 A fact is that J 1Y is a paracompact topological space which admits the partition of unity by elements of a ring 0 Y , but not 0 Y . Therefore, one Q1 O1 can apply the abstract de Rham theorem in order to ﬁnd the cohomology of

⇤ Y . Then we proved that the cohomology of ⇤ Y equals that of ⇤ Y . Q1 O1 Q1 In a di↵erent way, variational sequences of ﬁnite jet order are considered.

D.Krupka, Introduction to Global Variational Geometry (Springer, 2015). M.Palese, O.Rossi, E.Winterroth, J.Musilov´a, Variational Sequences, Repre- sentation Sequences and Applications in Physics, SIGMA 12 (2016) 045.

24 1.3 Di↵erential calculus over graded commutative rings

If there is no danger of confusion, by the term graded throughout is meant

Z2-graded.

Let be a commutative ring. A -module Q is called graded (i.e., Z2- K K graded) if it is decomposed into a direct sum Q = Q = Q0 Q1 of modules Q0 ⇤ and Q1, called the even and odd parts of Q , respectively. A Z2-graded - ⇤ K module is said to be free if it has a basis composed by homogeneous elements.

A morphism : P Q of graded -modules is said to be an even ⇤ ! ⇤ K (resp. odd) morphism if preserves (resp. changes) the Z2-parity of all homogeneous elements. A morphism : P Q of graded -modules is ⇤ ! ⇤ K called graded if it is represented by a sum of even and odd morphisms. A set Hom (P, Q) of these graded morphisms is a graded -module. K K

DEFINITION 1.5:A -ring is called graded (i.e., Z2-graded) if it is K A a graded -module , and a product of its homogeneous elements ↵↵0 is a K A⇤ homogeneous element of degree ([a]+[a0])mod 2. In particular, [1] = 0. Its even part is a -ring, and the odd one is an -module. A0 K A1 A0 ⇤

DEFINITION 1.6: A graded ring is called graded commutative if A⇤

[a][a0] aa0 =( 1) a0a, a, a0 . 2A⇤ ⇤

A graded commutative ring can admit di↵erent graded commutative structures in general. A⇤ 25 By automorphisms of a graded commutative ring are meant au- A⇤ tomorphisms of a -ring which are graded -module morphisms of . K A K A⇤ Obviously, they are even, and they preserve a graded structure of . How- A ever, there exist automorphisms of a -ring which do not possess this K A property in general. Then and (A ) are isomorphic, but di↵erent graded A⇤ ⇤ commutative structures of a ring . Moreover, it may happen that a -ring A K admits non-isomorphic graded commutative structures. A Example 1.3: Given a graded commutative ring and its odd element A⇤ , an automorphism

: a a, a a(1 + ), A0 3 ! A1 3 ! of a -ring does not preserve its original graded structure . ⇤ K A A⇤ Given a graded commutative ring ,agraded -module Q is deﬁned A⇤ A⇤ ⇤ as an ( )-bimodule which is a graded -module such that AA K [aq] = ([a]+[q])mod 2,qa=( 1)[a][q]aq, a ,qQ . 2A⇤ 2 ⇤ The following are constructions of new graded modules from the old ones.

A direct sum of graded modules and a graded factor module are • deﬁned just as those of modules over a commutative ring.

A tensor product P Q of graded -modules P and Q is their • ⇤ ⌦ ⇤ A⇤ ⇤ ⇤ tensor product as -modules such that A [p q] = ([p]+[q]) mod 2,pP ,qQ , ⌦ 2 ⇤ 2 ⇤ ap q =( 1)[p][a]pa q =( 1)[p][a]p aq, a . ⌦ ⌦ ⌦ 2A⇤ 26 In particular, the tensor algebra • k P = P ( P ) ⌦ A ··· ⌦ ··· A of an -module P is deﬁned just as that of a module over a commutative A⇤ ⇤ ring. Its quotient P with respect to the ideal generated by elements ^ ⇤

[p][p0] p p0 +( 1) p0 p, p, p0 P , ⌦ ⌦ 2 ⇤ is the exterior algebra of a graded module P with respect to the graded ⇤ exterior product

[p][p0] p p0 = ( 1) p0 p. (1.10) ^ ^ A graded morphism :P Q of graded -modules is their • ⇤ ! ⇤ A⇤ graded morphism as graded -modules which obeys the relations K (ap)=( 1)[][a]a(p),pP ,a . (1.11) 2 ⇤ 2A⇤ These morphisms form a graded -module Hom (P ,Q ). A graded - A⇤ A ⇤ ⇤ A⇤ module P ⇤ = Hom (P , ) is called the dual of a graded -module P . A ⇤ A⇤ A⇤ ⇤

In the sequel, we are concerned with graded manifolds. They are sheaves in Grassmann algebras, whose derivations form a Lie superalgebra. They are deﬁned as follows.

A graded commutative -ring ⇤ is said to be the Grassmann algebra K ⇤ if the following hold. It is finitely generated in degree 1, i.e., it is a free -module of finite • K rank so that ⇤ = ⇤2 and, consequently, 0 K 1 ⇤= R, R =⇤ (⇤ )2, K 1 1 27 where R is the ideal of nilpotents of a ring ⇤. A surjection :⇤ is !K called the body map. It is isomorphic to the exterior algebra (R/R2)ofa -module R/R2 • ^ K where R is the ideal of nilpotents of ⇤ . ⇤ An exterior algebra Q of a free -module Q of finite rank is a Grassmann ^ K algebra. Conversely, a Grassmann algebra admits a structure of an exterior algebra Q by a choice of its minimal generating -module Q ⇤ , and all ^ K ⇢ 1 these structures are mutually isomorphic if is a field. K Let be a graded commutative ring. A graded -algebra g is called A⇤ A⇤ ⇤ the Lie -superalgebra if its product [., .], called the Lie superbracket, A⇤ obeys the rules

["]["0] [","0]= ( 1) ["0,"], ["]["00] ["0]["] ["00]["0] ( 1) [", ["0,"00]] + ( 1) ["0, ["00,"]] + ( 1) ["00, [","0]] = 0.

Clearly, an even part g0 of a Lie superalgebra g is a Lie 0-algebra. Given ⇤ A an -superalgebra, a graded -module P is called a g -module if it is A⇤ A⇤ ⇤ ⇤ provided with an -bilinear map A⇤ g P (", p) "p P , ["p] = (["]+[p])mod 2, ⇤ ⇥ ⇤ 3 ! 2 ⇤ ["]["0] [","0]p =(" "0 ( 1) "0 ")p.

28 The di↵erential calculus over graded commutative rings is deﬁned similarly to that over commutative rings, but it di↵ers from the di↵erential calculus over non-commutative rings.

G.Sardanashvily, Advanced Di↵erential Geometry for Theoreticians. Fibre bundles, jet manifolds and Lagrangian theory (Lambert Aca- demic Publishing, 2013); arXiv: 0908.1886.

Let be a commutative ring and a graded commutative -ring. Let P K A K and Q be graded -modules. A graded -module Hom (P, Q) of graded - A K K K module homomorphisms : P Q admits two graded -module structures ! A (a)(p)=a(p), ( a)(p)=(ap),a,pP. • 2A 2 Let us put

=a ( 1)[a][] a, a . a • 2A

DEFINITION 1.7: An element Hom (P, Q) is said to be the Q-valued 2 K graded di↵erential operator of order s on P if

=0 a0 ··· as for any tuple of s + 1 elements a ,...,a of . A set Di↵ (P, Q) of these 0 s A s operators is a graded -module. A ⇤ In particular, zero-order graded di↵erential operators coincide with graded -module morphisms P Q. A !

29 A ﬁrst-order graded di↵erential operator satisﬁes a relation

(p)=ab(p) ( 1)([b]+[])[a]b(ap) ( 1)[b][]a(bp)+ a b ( 1)[b][]+([]+[b])[a](bap)=0, a, b ,pP. 2A 2 For instance, a ﬁrst-order graded di↵erential operator on P = fulﬁls a A condition

(ab)=(a)b +( 1)[a][]a(b) ( 1)([b]+[a])[]ab(1), a, b . 2A

DEFINITION 1.8: It is called the Q-valued graded derivation of if A (1) = 0, i.e., if it obeys the graded Leibniz rule

(ab)=(a)b +( 1)[a][]a(b), a, b . 2A

⇤

If @ is a graded derivation of , then a@ is so for any a . Hence, graded A 2A derivations of constitute a graded -module d( ,Q), called the graded A A A derivation module.

If Q = , the graded derivation module d = d( , ) also is a Lie - A A A A K superalgebra with respect to the superbracket

[u][u0] [u, u0]=u u0 ( 1) u0 u, u, u0 . 2A

30 Since the graded derivation module d of a graded commutative ring A A is a Lie -superalgebra, one can consider the Chevalley–Eilenberg complex K ⇤[d ] whose cochains are graded -modules O A A k Ck[d ; ] = Hom ( d , ) A A A ^ A A k of -linear graded morphisms of graded exterior products d of a graded A ^ A -module d to , seen as a d -module. A A A A k Let us bring homogeneous elements of d into a form ^ A " " ✏ ✏ ,"d ,✏ d . 1 ^··· r ^ r+1 ^···^ k i 2 A0 j 2 A1

Then a Chevalley–Eilenberg coboundary operator d of a complex ⇤[d ], O A called the graded exterior di↵erential reads

dc("1 "r ✏1 ✏s)= r ^···^ ^ ^···^ i 1 ( 1) " c(" " " ✏ ✏ )+ i 1 ^··· i ···^ r ^ 1 ^··· s i=1 Xs ( 1)r✏ c(" b" ✏ ✏ ✏ )+ j 1 ^···^ r ^ 1 ^··· j ···^ s j=1 X ( 1)i+jc([" ," ] " " b " " ✏ ✏ )+ i j ^ 1 ^··· i ··· j ···^ r ^ 1 ^···^ s 1 i

31 A graded module ⇤[d ] is provided with the graded exterior product O A

0(u , ..., u )= ^ 1 r+s i1 irj1 js Sgn1 ···r+s ··· (ui1 ,...,uir )0(uj1 ,...,ujs ), ··· i <

i1 irj1 js u1 ur+s = Sgn1 ···r+s ··· ui1 uir uj1 ujs . ^···^ ··· ^···^ ^ ^···^

A graded di↵erential d and a graded exterior product bring ⇤[d ] into ^ O A a di↵erential bigraded ring (DBGR) whose elements obey relations

0 +[][0] 0 =( 1)| || | 0 , d( 0)=d 0 +( 1)| | d0. ^ ^ ^ ^ ^ It is called the graded Chevalley–Eilenberg di↵erential calculus over a graded commutative -ring . K A 1 In particular, [d ] = Hom (d , )=d ⇤ is the dual of the derivation O A A A A A module d ⇤. One can extend this duality relation to the graded interior A product of u d with any element ⇤[d ] by the rules 2 A 2O A u (bda)=( 1)[u][b]bu(a), a, b , c 2A +[][u] u ( 0)=(u ) 0 +( 1)| | (u 0). c ^ c ^ ^ c As a consequence, a graded derivation u d of yields a graded derivation 2 A A

L = u d + d(u ),⇤[d ],ud , u c c 2O A 2 A [u][] L ( 0)=L () 0 +( 1) L (0), u ^ u ^ ^ u termed the graded Lie derivative of a DBGR ⇤[d ]. O A 32 The minimal graded Chevalley–Eilenberg di↵erential calculus

⇤ ⇤[d ] over a graded commutative ring consists of monomials O A⇢O A A a da da , a . The corresponding complex 0 1 ^···^ k i 2A 0 d 1 d k d !K!A ! O A ! ···O A ! ··· is called the de Rham complex of a graded commutative -ring . K A Let us note that, if = 0 is a commutative ring, graded di↵erential op- A A erators and the graded Chevalley–Eilenberg di↵erential calculus are reduced to the familiar commutative di↵erential calculus.

33 1.4 Di↵erential calculus on graded manifolds

As was mentioned above, we follow the Serre – Swan theorem extended to graded manifolds. It states that, if a graded commutative C1(X)-ring is generated by a projective C1(X)-module of ﬁnite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is X. Therefore, we aim to develop Lagrangian formalism of graded (even and odd) variables in terms just of graded manifolds.

Accordingly to a general concept of a manifold as a local-ringed space, a graded manifold has been deﬁned as a sheaf on a smooth body manifold in local graded commutative algebras which are real Grassmann algebras.

C. Bartocci, U.Bruzzo, D.Hern´andezRuip´erez, The Geometry of Super- manifolds (Kluwer Academic Publ., 1991).

We start with the conventional notion of a Z2-graded manifold. It is a local-ringed space (Z, A) where Z is an n-dimensional smooth manifold, and A = A A is a sheaf in real Grassmann algebras ⇤such that: 0 1 there is the exact sequence of sheaves • 2 0 A C1 0, = A +(A ) , !R! ! Z ! R 1 1 where CZ1 is the sheaf of smooth real functions on Z; 2 / is a locally free sheaf of C1-modules of ﬁnite rank (with respect •R R Z to pointwise operations), and the sheaf A is locally isomorphic to the exterior 2 product C ( / ). ^ Z1 R R

34 A sheaf A is called the structure sheaf of a graded manifold (Z, A), its stalk (a ﬁbre) at a point z Z is the tensor product Cz1 ⇤. A manifold 2 ⌦R Z is said to be the body of (Z, A). Sections of the sheaf A are termed the graded functions on a graded manifold (Z, A). They make up a graded commutative C1(Z)-ring A(Z) called the structure ring of (Z, A).

By virtue of the well-known Batchelor theorem, graded manifolds possess the following structure.

THEOREM 1.4: 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) as a sheaf in real rings is isomorphic to the structure sheaf

A = E⇤ of germs of sections of the exterior bundle E ^ Z 2 E⇤ =(Z R) E⇤ E⇤ E⇤ , ^ ⇥ Z Z ^ Z ^ ··· whose typical ﬁbre is the Grassmann algebra ⇤= V ⇤. ^ ⇤ Note that Batchelor’s isomorphism in Theorem 1.4 is not canonical.

Combining Batchelor Theorem 1.4 and classical Serre–Swan Theorem, we come to the above mentioned Serre–Swan theorem for graded manifolds.

THEOREM 1.5: Let Z be a smooth manifold. A graded commutative

C1(Z)-ring is isomorphic to the structure ring of a graded manifold with A a body Z if and only if it is the exterior algebra of some projective C1(Z)- module of ﬁnite rank. ⇤

35 In fact, the structure sheaf AE of a Z2-graded manifold (Z, A) in Theorem

1.4 is a sheaf in N-graded commutative rings ⇤⇤ = V whose N-graded ^ structure is ﬁxed.

Remark 1.4: Let be a commutative ring. A direct sum of -modules K K i P = P ⇤ = P (1.12) iN 2 is called the N-graded -module.A -ring is called N-graded if it is K K A an N-graded -module ⇤ (1.12) so that a product of homogeneous elements K A ↵↵0 is a homogeneous element of degree [↵]+[↵0]. Any N-graded -module K P (1.12) admits the associated Z2-graded structure

2i 2i+1 P = P0 P1,P0 = P ,P1 = P . iN iN 2 2 Accordingly, an N-graded ring ⇤ also is the Z2-graded one . The con- A A⇤ verse is not true. An N-graded -ring ⇤ is said to be graded commu- K A tative if

[↵][] ↵ =( 1) ↵,↵, ⇤. 2A

An N-graded commutative ring ⇤ possesses an associated Z2-graded com- A mutative structure

2k 2k+1 0 = , 1 = ,kN, A k A A k A 2 ↵ =( 1)[↵][]↵,↵, . 2A⇤ The converse need not be true. However, a Grassmann algebra ⇤possess 1 an associated N-graded structure of an exterior algebra ⇤=⇤⇤ = ⇤ , which ^ is not unique. ⇤

36 Hereafter, we consider a Z2-graded manifold (Z, A) when its Batchelor’s isomorphism (Z, A = AE) holds. Why?

Though Batchelor’s isomorphism is not canonical, in applications it is • ﬁxed from the beginning as a rule.

From the mathematical viewpoint, we restrict our considerations to mor- • phisms which preserve a ﬁxed N-graded structure.

Example 1.5: Let ⇤be a real Grassmann algebra. Its associated N-graded structure is deﬁned by a choice of a minimal generating vector space ⇤1 ⇤ . ⇢ 1 Given a basis ci for ⇤1, elements of a Grassmann algebra ⇤take a form { }

i1 ik i1 a = ai1 ik c c ,ai1 ik c R. ··· ··· ··· 2 k=0X,1,... X We call ci the generating basis for a Grassmann algebra ⇤. Then one can { } show that any ring automorphism of ⇤is a compositions of automorphisms

i i i j i c c0 = ⇢ c + b , (1.13) ! j where ⇢ is an automorphism of a vector space ⇤1 and bi are odd elements of ⇤>2, and of morphisms

i i i c c0 = c (1 + ),⇤ . (1.14) ! 2 1 i Automorphisms (1.13), where b = 0, preserve the N-graded structure ⇤⇤.If i b = 0, they keep a Z2-graded structure of ⇤, but not the N-graded one ⇤⇤. 6 Automorphisms (1.14) preserve an even sector ⇤0 of ⇤, but not the odd one

⇤1. However, one can show that di↵erent N- and Z2-graded structures of a real Grassmann algebra are mutually isomorphic. ⇤

37 Thus, we further deal with graded manifolds (Z, AE) which are N- graded.

A graded manifold (Z, AE) is said to be modelled over a vector bundle E Z, and E is called its characteristic vector bundle. Its structure ! ring is the structure module = E⇤(Z) of sections of the exterior AE AE ^ bundle E⇤. ^

A key point is that automorphisms of an N-graded manifold (Z, AE) are restricted to those induced by automorphisms of its characteristic vector bundle E Z. !

Remark 1.6: One can treat a local-ringed space (Z, A0 = CZ1) as a trivial graded manifold whose characteristic vector bundle is E = Z 0 . Its ⇥{ } structure module is a ring C1(Z) of smooth real functions on Z. ⇤

A a Given a graded manifold (Z, AE), every trivialization chart (U; z ,y )of its characteristic vector bundle E Z yields a splitting domain (U; zA,ca) ! of (Z, AE). Graded functions on such a chart are ⇤-valued functions

m 1 f = f (z)ca1 cak , (1.15) k! a1...ak ··· Xk=0 a where fa a (z) are smooth functions on U and c is the ﬁbre basis for E⇤. 1··· k { } One calls zA,ca the local basis for a graded manifold (Z, A ). Transition { } E a a A b functions y0 = ⇢ (z )y of bundle coordinates on E Z induce the cor- b ! a a A b responding transformation c0 = ⇢b (z )c of the associated local basis for a graded manifold (Z, AE) and the according coordinate transformation law of graded functions (1.15).

38 The following is an essential peculiarity of an N-graded manifold (Z, AE) in comparison with the Z2-graded ones.

THEOREM 1.6: Derivations of the structure module of a graded ma- AE nifold (Z, AE) are represented by sections of a vector bundle

E = E⇤ TE Z. (1.16) V ^ ⌦E !

⇤

Due to the canonical splitting VE= E E, the vertical tangent bundle VE ⇥ of E Z can be provided with ﬁbre bases @ , which are the duals of bases ! { a} ca . Then graded derivations of on a splitting domain (U; zA,ca)of { } AE (Z, AE) read

A a u = u @A + u @a, [@A]=0, [@a]=1, (1.17) @ @ = @ @ ,@@ = @ @ ,@@ = @ @ , A B B A A a a A a b b a where u,ua are local graded functions on U possessing a transformation law

A A a a j A a j u0 = u ,u0 = ⇢j u + u @A(⇢j )c .

The graded derivations (1.17) are called graded vector ﬁelds on a graded manifold (Z, A ). They act on graded functions f A (U) by a rule E 2 E u(f ca cb)=uA@ (f )ca cb + ukf @ (ca cb). a...b ··· A a...b ··· a...b kc ··· In accordance with Theorem 1.6, the graded derivation module d AE is isomorphic to the structure module (Z) of global sections of the vector VE bundle Z (1.16). It is a real Lie superalgebra. VE ! 39 Given the structure ring of graded functions on a graded manifold AE (Z, A ) and the real Lie superalgebra d of its graded derivations, we con- E AE sider the graded Chevalley–Eilenberg di↵erential calculus ⇤[E; Z]= ⇤[d ]: S O AE d 1 d k d 0 R E [E; Z] [E; Z] , (1.18) ! !A ! S ! ···S ! ··· over 0[E; Z]= . Since a graded derivation module d is the structure S AE AE module of sections of a vector bundle Z, elements of ⇤[E; Z] are VE ! S represented by sections of the exterior bundle of the -dual ^VE AE

E = E⇤ T ⇤E Z (1.19) V ^ ⌦E !

A b of . With respect to the dual ﬁbre bases dz for T ⇤Z and dc for E⇤, VE { } { } sections of (1.19) take a local form VE A a 1b 1b a j = Adz + adc ,a0 = ⇢ ab,A0 = A + ⇢ a@A(⇢j )bc .

1 The duality relation [E; Z]=d ⇤ is given by a graded interior product S AE

u = uA +( 1)[a]ua . c A a The graded exterior di↵erential reads

d = dzA @ + dca @ . ^ A ^ a

Elements of a DBGR ⇤[E; Z] are called the graded di↵erential forms S on a graded manifold (Z, A ). Seen as an -ring, ⇤[E; Z] on a splitting E AE S domain (zA,ca) is locally generated by even and odd one-forms dzA and dci.

40 Cohomology of the DBGR ⇤[E; Z] (1.18) is called the de Rham coho- S mology of a graded manifold (Z, AE). It equals the de Rham cohomology of its body Z. In particular, there exist both a cochain monomorphism

⇤(Z) ⇤[E; Z] and a cochain body epimorphism ⇤[E; Z] ⇤(Z). O !S S !O G.Sardanashvily, Graded inﬁnite order jet manifolds, Int. J. Geom. Meth- ods Mod. Phys. 4 (2007) 1335-1362.

A morphism of graded manifolds (Z, A) (Z0, A0) is deﬁned as that ! of local-ringed spaces : Z Z0, :A0 A, where is a manifold ! ! ⇤ morphism and is a sheaf morphism of A0 to the direct image A of A onto b ⇤ Z0. This morphism of graded manifolds is said to be: b a monomorphism if is an injection and is an epimorphism, and • an epimorphism if is a surjection and is a monomorphism. • b

An epimorphism of graded manifolds (Z, A) (Z0, A0) where Z Z0 is b! ! a ﬁbre bundle is called the graded bundle.

D.Hernández Ruipérez,J.Muñoz Masqué,Global variational calculus on graded manifolds. J. Math. Pures Appl. 63 (1984) 283-309. T.Stavracou, Theory of connections on graded principal bundles, Rev. Math. Phys. 10 (1998) 47-79.

41 In particular, let (Y,A) be a graded manifold whose body is a ﬁbre bundle

Y X. Let us consider a trivial graded manifold (X, A = C1). Then we ! 0 X have a graded bundle

(Y,A) (X, C1). (1.20) ! X 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 take a form (x,yi,ca) where (x,yi) are bundle coordinates of Y X. Therefore, we agree to call the graded ! bundle (1.20) over a trivial graded manifold (X, CX1) the graded bundle over a smooth manifold.

Note that a graded manifold (X, A) itself can be treated as the graded bundle (X, X, A) (1.20) associated to the identity smooth bundle X X. ! Let E Z and E0 Z0 be vector bundles and : E E0 their bundle ! ! ! morphism over a morphism : Z Z0. A bundle morphism (,) induces ! a morphism of graded manifolds

(Z, A ) (Z0, A ). (1.21) E ! E0 It is a monomorphism (resp. epimorphism) if is a bundle injection (resp. surjection). In particular, the graded manifold morphism (1.21) is a graded bundle if is a ﬁbre bundle. Let be the corresponding pull- AE0 !AE back monomorphism of the structure rings. It yields a monomorphism of the DBGRs

⇤[E0; Z0] ⇤[E; Z]. S !S

42 In order to formulate Lagrangian theory both of even and odd variables, let us consider graded manifolds whose body is a ﬁbre bundle Y X. ! Let (Y,A ) be a graded manifold modelled over a vector bundle F Y . F ! This is a graded bundle (X, Y, AF ):

(Y,A ) (X, C1), (1.22) F ! X modelled over a composite bundle

F Y X. (1.23) ! !

The structure ring of graded functions on a graded manifold (Y,AF ) is the graded commutative C1(X)-ring = F ⇤(Y ). Let the composite bundle AF ^ (1.23) be provided with adapted bundle coordinates (x,yi,qa) possessing transition functions

µ i µ j a a µ j b x0 (x ),y0 (x ,y ),q0 = ⇢b (x ,y )q .

i a Then the corresponding basis for a graded manifold (Y,AF ) is (x ,y ,c ) a a µ j b together with transition functions c0 = ⇢b (x ,j )c . We call it the local basis for a graded bundle (X, Y, AF ) (1.22).

As was shown above, the di↵erential calculus on a ﬁbre bundle Y X is ! formulated in terms of jet manifolds J ⇤Y of Y . Being ﬁbre bundles over X, k they can be regarded as trivial graded bundles (X, J Y,CJ1kY ). We describe their partners in a case of graded bundles as follows.

43 Let us note that, given a graded manifold (X, AE) and its structure ring 1 , one can deﬁne the jet module J of a C1(X)-ring . It is a module AE AE AE 1 1 of global sections of the jet bundle J ( E⇤). A problem is that J fails ^ AE to be a structure ring of some graded manifold. By this reason, we have suggested a di↵erent construction of jets of graded manifolds (Deﬁnition 1.9), though it is applied only to N-graded manifolds. Let (X, ) be a graded manifold modelled over a vector bundle E X. AE ! Let us consider a k-order jet manifold J kE of E. It is a vector bundle over k X. Then let (X, k ) be a graded manifold modelled over J E X. AJ E !

DEFINITION 1.9: We call (X, k ) the graded jet manifold of a AJ E graded manifold (X, ). AE ⇤ Given a splitting domain (U; x,ca) of a graded manifold (Z, ), the AE adapted splitting domain of a graded jet manifold (X, k ) reads AJ E a a a a a a a a a (U; x ,c ,c,c12 ,...c1...k ),c01...r = ⇢b (x)c1...r + @⇢b (x)c1...r .

As was mentioned above, a graded manifold is a particular graded bundle over its body. Then Deﬁnition 1.9 of graded jet manifolds is generalized to graded bundles over smooth manifolds as follows.

Let (X, Y, AF ) be the graded bundle (1.22) modelled over the composite bundle F Y X (1.23). It is readily observed that the jet manifold ! ! J rF of F X is a vector bundle J rF J rY coordinated by (x,yi ,qa ), ! ! ⇤ ⇤ r 0 ⇤ r. Let (J Y,A r ) be a graded manifold modelled over this vector | | J F bundle. Its local generating basis is (x,yi ,ca ), 0 ⇤ r. ⇤ ⇤ | |

44 r DEFINITION 1.10: We call (J Y,AJ rF ) the graded 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, C1) modelled over a composite bundle Y 0 Y X. Y ⇥{ }! ! r Then its graded jet manifold is a trivial graded bundle (X, J Y,CJ1rY ), i.e., the jet manifold J rY of Y . Thus, Definition 1.10 of graded jet manifolds of graded bundles is compatible with the definition of jets of fibre bundles. The ane bundles J r+1Y J rY and the corresponding fibre bundles ! J r+1F J rF also yield the graded bundles ! r+1 r (J Y,A r+1 ) (J Y,A r ). J F ! J F As a consequence, we have the inverse sequence of graded manifolds

1 r 1 r (Y,A ) (J Y,A 1 ) (J Y,A r 1 ) (J Y,A r ) . (1.24) F J F ··· J F J F ···

One can think of its inverse limit (J 1Y,AJ 1F ) as being the graded inﬁ- nite order jet manifold whose body is an inﬁnite order jet manifold J 1Y and whose structure sheaf AJ 1F is a sheaf of germs of graded functions on graded manifolds (J ⇤Y,AJ ⇤F ). However (J 1Y,AJ 1F ) is not a graded manifold in a strict sense because J 1Y is not a smooth manifold.

The inverse system of graded jet manifolds (1.24) yields a direct system

r ⇡⇤ ⇡r⇤ 1 ⇤[F ; Y ] 1⇤[F ; Y ] r⇤ 1[F ; Y ] r⇤[F ; Y ] ,(1.25) S ! S ! ···S ! S ! ··· r+1 k k ⇡ ⇤ : ⇤[F ; Y ] ⇤ [F ; Y ], ⇤[F ; Y ]= ⇤[J F ; J Y ], r Sr !Sr+1 Sk S

r+1 where ⇡r ⇤ are the pull-back monomorphisms of DBGRs.

45 The DBGR ⇤ [F ; Y ] associated to a graded bundle (X, Y, AF ) is deﬁned S1 as the direct limit of the direct system (1.25). It include all graded dif- r ferential forms ⇤[F ; Y ] on graded manifolds (J Y,A r ) modulo the 2Sr J F r+1 monomorphisms ⇡r ⇤.

THEOREM 1.7: There is an isomorphism H⇤( ⇤ [F ; Y ]) = HDR⇤ (Y ) of the S1 cohomology H⇤( ⇤ [F ; Y ]) of the de Rham complex S1 0 d 1 d k 0 R [F ; Y ] [F ; Y ] [F ; Y ] ! ! S 1 ! S 1 ··· ! S 1 ! ··· of a DBGR ⇤ [F ; Y ] to the de Rham cohomology HDR⇤ (Y ) of Y . ⇤ S1

r In particular, monomorphisms ⇤(J Y ) ⇤[F ; Y ] yield a cochain monomor- O !Sr phism of complexes ⇤ Y ⇤ [F ; Y ], and body epimorphisms r⇤[F ; Y ] O1 !S1 S ! r⇤Y deﬁne a cochain epimorphism ⇤ [F ; Y ] ⇤ Y . O S1 !O1

One can think of elements of ⇤ [F ; Y ] as being graded di↵erential S1 forms on an inﬁnite order jet manifold J 1Y in the sense that ⇤ [F ; Y ] S1 is a submodule of the structure module of sections of some sheaf on J 1Y . In particular, one can restrict ⇤ [F ; Y ] to a coordinate chart of J 1Y so that S1 0 ⇤ [F ; Y ] as an Y -algebra is locally generated by the elements S1 O1 (ca ,dx,✓a = dca ca dx,✓i = dyi yi dx), 0 ⇤ , ⇤ ⇤ ⇤ +⇤ ⇤ ⇤ +⇤ | |

a a i i a where c⇤, ✓⇤ are odd and dx , ✓⇤ are even. We agree to call (y ,c ) the A local generating basis for ⇤ [F ; Y ]. Let the collective symbol s stand for S1 its elements. We further denote [A]=[sA].

46 1.5 Lagrangian theory of even and odd variables on graded bundles

r Similarly to DGR ⇤ Y of di↵erential forms on jet manifolds J Y in La- O1 grangian formalism on a ﬁbre bundle Y X, a DBGR ⇤ [F ; Y ] of graded ! S1 di↵erential forms is split into a graded variational bicomplex which provides Lagrangian theory of graded (even and odd) variables.

Let (X, Y, AF ) be a graded bundle modelled over a composite bundle

F Y X over an n-dimensional smooth manifold X, and let ⇤ [F ; Y ] ! ! S1 be the associated DBGA of graded exterior forms on graded jet manifolds 0 of (X, Y, AF ). A DBGA ⇤ [F ; Y ] is decomposed into [F ; Y ]-modules S1 S1 k,r[F ; Y ] of k-contact and r-horizontal graded forms together with the cor- S1 responding projections

k, m ,m hk : ⇤ [F ; Y ] ⇤[F ; Y ],h: ⇤ [F ; Y ] ⇤ [F ; Y ]. S1 !S1 S1 !S1

Accordingly, the graded exterior di↵erential d on ⇤ [F ; Y ] falls into a sum S1 d = dV + dH of the vertical and total graded di↵erentials

m m m A ⇤ dV h = h d h ,dV ()=✓⇤ @A, ⇤ [F ; Y ], ^ 2S1 d h = h d h ,dh = h d, d ()=dx d (), H k k k H 0 0 H ^ A ⇤ d = @ + s+⇤@A, X where d are graded total derivatives. These di↵erentials obey the nilpotent relations

d d =0,dd =0,dd + d d =0. H H V V H V V H

47 A DBGA ⇤ [F ; Y ] also is provided with the graded projection morphism S1 1 n >0,n >0,n % = % hk h : ⇤ [F ; Y ] ⇤ [F ; Y ], k S1 !S1 Xk>0 ⇤ A ⇤ >0,n %()= ( 1)| |✓ [d⇤(@A )], [F ; Y ], ^ c 2S1 X such that % d = 0, and with the nilpotent graded variational operator H ,n +1,n = % d : ⇤ [F ; Y ] ⇤ [F ; Y ]. S1 !S1

, With these operators a DBGA ⇤ [F ; Y ] is decomposed into the graded S1 variational bicomplex ......

d 6 d 6 d 6 6 V V V 1,0 dH 1,1 dH 1,n % 0 E1 0 !S1 !S1 !···S1 ! ! d 6 d 6 d 6 6 V V V dH dH 0 R 0 0,1 0,n 0,n ! !S1 !S1 !···S1 ⌘S1 6 6 6 d d d 0 R 0(X) 1(X) n(X) 0 ! !O !O !···O ! 6 6 6 00 0

k,n where ⇤ = ⇤ [F ; Y ] and Ek = %( [F ; Y ]). S1 S1 S1

48 We restrict our consideration to a short variational subcomplex

0 dH 0,1 dH 0,n 0 R [F ; Y ] [F ; Y ] [F ; Y ] E1 (1.26) ! !S1 ! S 1 ··· ! S 1 ! of this bicomplex and its subcomplex of one-contact graded forms

1,0 dH 1,1 dH 1,n % 0 [F ; Y ] [F ; Y ] [F ; Y ] E1 0. (1.27) !S1 ! S 1 ··· ! S 1 ! ! They possess the following cohomology.

THEOREM 1.8: Cohomology of the complex (1.26) equals the de Rham cohomology of Y . The complex (1.27) is exact. ⇤

Decomposed into a variational bicomplex, the DBGA ⇤ [F ; Y ] describes S1 graded Lagrangian theory on a graded bundle (X, Y, AF ). Its graded Lagrangian is deﬁned as an element

L = ! 0,n[F ; Y ],!= dx1 dxn, (1.28) L 2S1 ^···^ of the graded variational complex (1.26). Accordingly, a graded exterior form

A ⇤ A ⇤ 1,n L = ✓ A! = ( 1)| |✓ d⇤(@AL)! %( [F ; Y ]) (1.29) ^E ^ 2 S1 X is said to be its graded Euler–Lagrange operator. Its kernel yields an Euler–Lagrange equation

⇤ A ⇤ L =0, = ( 1)| |✓ d (@ L)=0. (1.30) EA ^ ⇤ A X 0,n We call a pair ( [F ; Y ],L) the graded Lagrangian system and ⇤ [F ; Y ] S1 S1 its structure algebra.

The following are corollaries of Theorem 1.8.

49 COROLLARY 1.9: Any -closed (i.e., variationally trivial) graded La- grangian L 0,n[F ; Y ] is a sum 2S1 0,n 1 L = h0 + dH ⇠,⇠ [F ; Y ], 2S1 where is a closed form on Y . ⇤

COROLLARY 1.10: Given a graded Lagrangian L, there is the global variational formula

n 1 dL = L dH ⌅L, ⌅ [F ; Y ], (1.31) 2S1 A ⌫s...⌫1 ⌅L = L + ✓ F !, (1.32) ⌫s...⌫1 ^ A s=0 X F ⌫k...⌫1 = @⌫k...⌫1 d F ⌫k...⌫1 + ⌫k...⌫1 ,k=1, 2,..., A A L A A

⌫ (⌫k⌫k 1)...⌫1 where local graded functions obey relations A = 0, A = 0. ⇤

The form ⌅L (1.32) provides a global Lepage equivalent of a graded

Lagrangian L. In particular, one can locally choose ⌅L (1.32) where all graded functions vanish.

50 2 First Noether theorem

Given a graded Lagrangian system ( ⇤ [F ; Y ],L), by its inﬁnitesimal trans- S1 formations are meant contact graded derivations of a real graded commutative ring 0 [F ; Y ]. They constitute a 0 [F ; Y ]-module d 0 [F ; Y ] which S1 S1 S1 is a real Lie superalgebra relative to a Lie superbracket. The following holds.

THEOREM 2.1: The derivation module d 0 [F ; Y ] is isomorphic to the S1 0 1 1 [F ; Y ]-dual ( [F ; Y ])⇤ of the module of graded one-forms [F ; Y ]. ⇤ S1 S1 S1

COROLLARY 2.2: A DBGA ⇤ [F ; Y ] is the Chevalley–Eilenberg minimal S1 di↵erential calculus over a real graded commutative ring 0 [F ; Y ]. ⇤ S1 Let # , # d 0 [F ; Y ], 1 [F ; Y ], denote the corresponding interior c 2 S1 2S1 product in accordance with Theorem 2.1. Extended to a DBGR ⇤ [F ; Y ], S1 it obeys the rule

+[][#] # ( )=(# ) +( 1)| | (# ),, ⇤ [F ; Y ]. c ^ c ^ ^ c 2S1

Restricted to a coordinate chart of J 1Y , a DBGR ⇤ [F ; Y ] is a free S1 0 A A [F ; Y ]-module generated by graded one-forms dx and ✓⇤ ,[✓⇤ ]=[A]. S1 Due to the isomorphism stated in Theorem 2.1, any graded derivation # 2 d 0 [F ; Y ] reads S1 A A ⇤ # = # @ + # @A + #⇤@A, (2.1) 0< ⇤ X| | ⇤ ⇤ where the graded derivations @A,[@A]=[A], obey the relations

@⇤(sB)=@⇤ dsB = B⇤. A ⌃ Ac ⌃ A ⌃ 51 Every graded derivation # (2.1) of a graded commutative ring 0 [F ; Y ] S1 yields a graded derivation (called the graded Lie derivative) L# of a DBGA

⇤ [F ; Y ] given by the relations S1

L# = # d + d(# ),⇤ [F ; Y ], c c 2S1 L ( )=L () +( 1)[#][] L (). # ^ # ^ ^ # The graded derivation # (2.1) is called contact if the graded Lie derivative L# preserves the ideal of contact graded forms of the DBGA ⇤ [F ; Y ] S1 generated by contact graded one-forms ✓A.

THEOREM 2.3: With respect to the local generating basis (sA) for a

DBGA ⇤ [F ; Y ], any its contact graded derivation takes a form S1 # = # + # = d +[A@ + d (A sAµ)@⇤], (2.2) H V A ⇤ µ A ⇤ >0 |X| where #H and #V denotes the horizontal and vertical parts of #. ⇤

A glance at the expression (2.2) shows that a contact graded derivation # is the inﬁnite order jet prolongation # = J 1 of its restriction

= @ + A@ = + = d +(uA@ sA@ ) (2.3) A H V A A to a graded commutative ring 0[F ; Y ]. We call (2.3) the generalized S graded vector field on a 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 time, any graded vector field u on (Y,AF ) is the generalized graded vector field (2.3) generating the contact graded derivation J 1u.

52 In particular, the vertical contact graded derivation (2.3) reads

A A ⇤ # = @A + d⇤ @A. (2.4) ⇤ >0 |X|

THEOREM 2.4: Any vertical contact graded derivation (2.4) obeys the relations

# dH = dH (# ), L#(dH )=dH (L#),⇤ [F ; Y ]. c c 2S1

⇤

The vertical contact graded derivation # (2.4) is said to be nilpotent if

B A L (L )= (B@⌃(A)@⇤ +( 1)[s ][ ]BA@⌃@⇤) =0 # # ⌃ B ⇤ A ⌃ ⇤ B A 0 ⌃ ,0 ⇤ | X| | | 0, for any horizontal graded form ⇤[F, Y ]. 2S1 THEOREM 2.5: The vertical contact graded derivation (2.4) is nilpotent only if it is odd. ⇤

Remark 2.1: If there is no danger of confusion, the common symbol further stands for a generalized graded vector ﬁeld (2.3), the contact graded derivation # = J 1 determined by , and the Lie derivative L#. We call all these operators, in brief, a graded derivation of the structure algebra of a graded Lagrangian system. ⇤

53 Remark 2.2: For the sake of convenience, right graded derivations A = @A also are considered. They act on graded di↵erential forms on the right by the rules

()=d +d( ),✓@⌃B = A⌃, b b ⇤Ab B ⇤ [0] ( 0)=( 1) () 0 + (0). ^ ^ ^

⇤

DEFINITION 2.1: Let ( ⇤ [F ; Y ],L) be a graded Lagrangian system. A S1 generalized graded vector ﬁeld is called the supersymmetry (or, simply, symmetry) of a graded Lagrangian L if a graded Lie derivative L#L of L along the contact graded derivation # = J 1 is dH -exact, i.e., L#L = dH . ⇤

A corollary of the graded variational formula (1.31) is the graded ﬁrst variational formula for a graded Lagrangian.

THEOREM 2.6: The graded Lie derivative of a graded Lagrangian along any contact graded derivation # fulﬁls the graded ﬁrst variational formula

L L = L + d (h (# ⌅ )) + d ( !) , (2.5) # V c H 0 c L V H c L where ⌅L is a Lepage equivalent of a graded Lagrangian L. ⇤

A glance at the expression (2.5) shows the following.

54 THEOREM 2.7: (i) A generalized graded vector field is a symmetry only if it is projected onto X. (ii) Any projectable generalized graded vector field is a symmetry of a variationally trivial graded Lagrangian. (iii) A generalized graded vector field is a symmetry if and only if its vertical part V is well. (iv) It is a symmetry if and only if the graded density L is d -exact. V c H ⇤

Symmetries of a graded Lagrangian L constitute a real vector subspace 0 L of the graded derivation module d [F ; Y ]. By virtue of item (ii) SG S1 of Theorem 2.7, the Lie superbracket

L[#,#0] =[L#, L#0 ] of symmetries is a symmetry, and their vector space is a real Lie superalgebra. An immediate corollary of the graded ﬁrst variational formula (2.5) is the ﬁrst Noether theorem.

THEOREM 2.8: If the generalized graded vector ﬁeld is a symmetry of a graded Lagrangian L, the ﬁrst variational formula (2.5) leads to a weak conservation law 0 d ( h (# ⌅ )+) (2.6) ⇡ H 0 c L of the symmetry current

= µ! = h (# ⌅ )+ (2.7) J J# µ 0 c L on-shell. ⇤

55 3 Gauge symmetries

Treating gauge symmetries of Lagrangian theory, one traditionally is based on gauge theory of principal connections on principal bundles.

This notion of gauge symmetries has been generalized to Lagrangian theory on an arbitrary ﬁbre bundle Y X. Gauge symmetry is deﬁned as a ! di↵erential operator on sections of some vector bundle E X as gauge ! parameters with values in a space of symmetries of a Lagrangian L.

Let us generalize this construction of gauge symmetries to Lagrangian theory on graded bundles.

Let ( ⇤ [F ; Y ],L) be a graded Lagrangian system on a graded bundle • S1 (X, Y, A ) modelled over a composite bundle F Y X, and let (sA) F ! ! be its local generating basis. To deﬁne gauge symmetries of this Lagrangian system, one extends a graded manifold (X, Y, AF ) as follows. Let us consider a composite bundle

E = E1 E0 E0 X X ! !

0 r and a graded bundle (X, E , AE) modelled over it, and let (c ) be its local 0 generating basis. Then we deﬁne the product (X, E Y,AE F ) of graded ⇥X ⇥X 0 bundles (X, Y, AF ) and (X, E , AE) which is modelled over the product

F E Y E0 X ⇥X ! ⇥X ! of composite bundles E and F , and which possesses a generating basis (sA,cr).

56 Let us consider the corresponding DBGR ⇤ [E F ; E0 Y ] together • S1 ⇥X ⇥X with the monomorphisms of DBGRs

0 0 0 ⇤ [F ; Y ] ⇤ [E F ; E Y ], ⇤ [E; E ] ⇤ [E F ; E Y ]. S1 !S1 ⇥X ⇥X S1 !S1 ⇥X ⇥X Given a graded Lagrangian L 0,n[F ; Y ], let us deﬁne its pull-back 2S1 0,n 0 L [F ; Y ] ⇤ [E F ; E Y ], (3.1) 2S1 ⇢S1 ⇥X ⇥X and consider an extended Lagrangian system

0 ( ⇤ [E F ; E Y ],L) (3.2) S1 ⇥X ⇥X provided with the local generating basis (sA,cr).

DEFINITION 3.1:Agauge transformation of the Lagrangian L (3.1) is deﬁned to be a contact graded derivation # of the ring 0 [E F ; E0 Y ] S1 ⇥X ⇥X such that a derivation # equals zero on a subring

0 [E; E0] 0 [E F ; E0 Y ]. S1 ⇢S1 ⇥X ⇥X A gauge transformation # is called the gauge symmetry if it is a symmetry of the Lagrangian L (3.1). ⇤

In view of the condition in Deﬁnition 3.1, the variables cr of the ex- • tended Lagrangian system (3.2) are treated as graded gauge parameters of a gauge symmetry #. Furthermore, we additionally assume that a gauge r r symmetry is linear in gauge parameters c and their jets c⇤. Then its generalized graded vector ﬁeld reads

= ⇤(xµ)cr @ + A⇤(xµ,sB)cr @ . (3.3) 0 r ⇤1 0 r ⌃ ⇤1 A 0 ⇤ m 0 ⇤ m X| | X| | @ A @ A 57 Remark 3.1: In a general setting, one can deﬁne gauge symmetries which are non-linear in gauge parameters. However, the direct second Noether theorem is not relevant for them because Euler–Lagrange operator in this case satisﬁes the identities depending on gauge parameters. ⇤

If a gauge transformation (3.3) is a symmetry, it deﬁnes a weak con- • servation law in accordance with the ﬁrst Noether theorem. The peculiarity of this conservation law is that the symmetry current (3.4) is the total J di↵erential on-shell, i.e., it is reduced to a superpotential.

THEOREM 3.1: If u (3.3) is a gauge symmetry of a graded Lagrangian L, the corresponding symmetry current (up to a d -closed term) takes a Ju H form = W + d U =(W µ + d U ⌫µ)! , (3.4) Ju H ⌫ µ where a term W vanishes on-shell and U ⌫µ = U µ⌫ is a superpotential which reads

µ i µµk...µN r Aµ r =( u c + u c ) A Ju V r µk...µN V r E 1

G.Sardanashvily, Gauge conservation laws in a general setting. Superpoten- tial, Int. J. Geom. Methods Mod. Phys. 6(2009) 1047-1056

One sometimes treats Theorem 3.1 as the third Noether theorem.

58 4 Noether identities

We follow the general analysis of Noether identities (NI) and higher-stage NI of di↵erential operators on ﬁbre bundles when trivial and non-trivial NI are represented by boundaries and cycles of a chain complex.

It should be noted that the notion of higher-stage Noether identities came 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.

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 di↵erential operators, unlike constraints, are di↵erential equations. 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 a certain homology regularity condition.

59 Let ( ⇤ [F ; Y ],L) be a graded Lagrangian system on a graded bundle S1 (X, Y, A ), modelled over a composite bundle F Y X. F ! !

One can associate to a graded Lagrangian system ( ⇤ [F ; Y ],L) the • S1 chain complex (4.1) whose one-boundaries vanish on the shell L = 0. Let us consider the density-dual

n VF = V ⇤F T ⇤X F ⌦F ^ ! of the vertical tangent bundle VF F , and let us enlarge an original DBGR ! A ⇤ [F ; Y ] with the generating basis (s ) to ⇤ [VF; Y ] with the generat- S1 P1 A ing basis (s , sA), [sA]=[A]+1. Following the terminology of BRST theory, we call its elements sA the antiﬁelds of antiﬁeld number Ant[sA] = 1.

A DBGR ⇤ [VF; Y ] is endowed with the nilpotent right graded derivation P1 A =@ A, where A = AL are the variational derivatives. Then we have a E E chain complex

0,n 0,n 0 Im [VF; Y ]1 [VF; Y ]2 (4.1) P 1 P 1 of graded densities of antiﬁeld number 2. Its one-boundaries ,  2 0,n [VF; Y ]2, by very deﬁnition, vanish on-shell. Any one-cycle of the P1 complex (4.1) is a di↵erential operator on a bundle VF such that its kernel contains the graded Euler–Lagrange operator L, i.e.,

=0, A,⇤d ! =0. (4.2) ⇤EA 0 ⇤ X| | Refereing to a notion of NI of a di↵erential operator, we say that one-cycles deﬁne the Noether identities (4.2) of an Euler–Lagrange operator L.

60 One-chains are necessarily NI if they are boundaries. Therefore, these • NI are called trivial. They are of the form

= T (A⇤)(B⌃)d s !,T (A⇤)(B⌃) = ( 1)[A][B]T (B⌃)(A⇤). ⌃EB ⇤A 0 ⇤ , ⌃ X| | | | Accordingly, non-trivial NI modulo trivial ones are associated to elements of the ﬁrst homology H1() of the complex (4.1).

A Lagrangian L is called degenerate if there exist non-trivial NI.

Non-trivial NI can obey first-stage NI. To describe them, let us assume • that a module H1() is finitely generated. Namely, there exists a graded projective C1(X)-module H () of finite rank possessing a local basis C(0) ⇢ 1 ! such that any element H () factorizes as { r } 2 1 r,⌅ r,⌅ 0 = d⌅r!, [F ; Y ], (4.3) 2S1 0 ⌅ X| | through elements ! of . Thus, all non-trivial NI (4.2) result from the r C(0) NI = A,⇤d =0, (4.4) r r ⇤EA 0 ⇤ X| | called the complete NI. Then the complex (4.1) can be extended to the chain complex (4.5) with a boundary operator whose nilpotency is equivalent to the complete NI (4.4) as follows.

61 By virtue of the Serre–Swan theorem, a graded module is isomorphic to C(0) that of sections of the density-dual E of some graded vector bundle E X. 0 0 ! Let us enlarge ⇤ [VF; Y ] to a DBGR P1

⇤ 0 = ⇤ [VF E0; Y ] P1{ } P1 ⇥X A with the generating basis (s , sA, cr) where cr are antiﬁelds such that [cr]=

[r] + 1 and Ant[cr] = 2. This DBGR admits a derivation

r 0 = + @ r which is nilpotent if and only if the complete NI (4.4) hold. Then 0 is a boundary operator of a chain complex

0,n 0 0,n 0 0,n 0 Im [VF; Y ]1 0 2 0 3 (4.5) P1 P1 { } P1 { } of graded densities of antiﬁeld number 3. One can show that its homology  H1(0) vanishes, i.e., the complex (4.5) is one-exact.

Let us consider the second homology H ( ) of the complex (4.5). Its • 2 0 two-cycles deﬁne the ﬁrst-stage NI

=0, Gr,⇤d ! = H. (4.6) 0 ⇤ r 0 ⇤ X| | Conversely, let the equality (4.6) hold. Then it is a cycle condition. The first- stage NI (4.6) are trivial either if the two-cycle is a 0-boundary or its summand G vanishes on-shell. Therefore, non-trivial first-stage NI fails to exhaust the second homology H2(0) of the complex (4.5) in general. One can show that non-trivial first-stage NI modulo trivial ones are identified 0,n with elements of H2(0) i↵any -cycle 0 2 is a 0-boundary. 2 P1 { } 62 A degenerate Lagrangian is called reducible if it admits non-trivial first- stage NI.

Non-trivial ﬁrst-stage NI can obey second-stage NI, and so on. It- • erating the arguments, we say that a degenerate graded Lagrangian system

( ⇤ [F ; Y ],L) is N-stage reducible if it admits non-trivial N-stage NI, but S1 no non-trivial (N + 1)-stage ones. It is characterized as follows.

(i) There are graded vector bundles E0,...,EN over X, and ⇤ [VF; Y ] is P1 enlarged to a DBGR

⇤ N = ⇤ [VF E0 EN ; Y ] (4.7) P1{ } P1 ⇥X ⇥X ···⇥X

A with a local generating basis (s , sA, cr, cr1 ,...,crN ) where crk are k-stage antiﬁelds of antiﬁeld number Ant[crk ]=k +2.

(ii) The DBGR (4.7) is provided with a nilpotent right graded derivation

r A,⇤ rk KT = N = + @ r s⇤A + @ rk , (4.8) 0 ⇤ 1 k N X| | X rk 1,⇤ rk ! = rk c⇤rk 1 ! + (4.9) 0 ⇤ X| | 0,n (rk 2,⌃)(A,⌅) (hrk c⌃rk 2 s⌅A + ...)! k 1 k+1, 2 P1 { } 0 ⌃ , ⌅ X| | | | of antiﬁeld number 1. The index k = 1 here stands for s . The nilpotent A derivation KT (4.8) is called the Koszul–Tate (KT) operator.

63 0,n (iii) With this KT operator, a module N N+3 of densities of anti- P1 { } ﬁeld number (N + 3) is split into the exact Koszul–Tate (KT) chain  complex

0,n 0 0,n 1 0,n 0 Im [VF; Y ]1 0 2 1 3 (4.10) P 1 P1 { } P1 { } ··· N 1 0,n KT 0,n KT 0,n N 1 N+1 N N+2 N N+3 P1 { } P1 { } P1 { } which satisﬁes the following homology regularity condition.

0,n 0,n CONDITION 4.1: Any k

(iv) The nilpotentness of the KT operator (4.8) is equivalent to complete non-trivial NI (4.4) and complete non-trivial (k N)-stage NI 

rk 1,⇤ rk 2,⌃ (rk 2,⌃)(A,⌅) rk d⇤( rk 1 c⌃rk 2 )= ( hrk c⌃rk 2 s⌅A). (4.11) 0 ⇤ 0 ⌃ 0 ⌃ , ⌅ X| | X| | X| | | |

It may happen that a graded Lagrangian system possesses non-trivial NI of any stage. However, we restrict our consideration to N-reducible Lagrangians for a ﬁnite integer N. In this case, the KT operator (4.8) and the gauge operator (5.4) below contain a ﬁnite number of terms.

It also should be emphasized that, in order to describe a hierarchy of NI, we suppose that NI and higher-stage NI are finitely generated (i.e., they form projective modules of finite rank) and that homology Condition 3.1 is satisfied. These are not true for any Lagrangian.

64 5 Second Noether theorems

Di↵erent variants of the second Noether theorem have been suggested in order to relate reducible NI and gauge symmetries.

G.Barnich, F.Brandt, M.Henneaux, Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000) 439. R.Fulp, T.Lada, J. Stashe↵, Noether variational Theorem II and the BV formalism, Rend. Circ. Mat. Palermo (2) Suppl. No. 71 (2003) 115.

The extended inverse second Noether theorem, that we formulate in homology terms, associates to the KT complex (4.10) of non-trivial NI the cochain sequence (5.3) with the ascent operator u (5.4) whose components are gauge and higher-stage gauge symmetries of a Lagrangian system.

Given a DBGR ⇤ N (4.7), let us consider the DBGRs P1{ }

P ⇤ N = P ⇤ [F E0 EN ; Y ], (5.1) 1{ } 1 ⇥X ⇥X ···⇥X

A r r1 rN rk possessing the generating bases (s ,c ,c ,...,c ), [c ]=[crk ] + 1, and the DBGRs

⇤ N = ⇤ [VF E0 EN E0 EN ; Y ] (5.2) P1{ } P1 ⇥X ⇥X ···⇥X ⇥X ⇥X ···⇥X

A r r1 rN with the generating bases (s , sA,c ,c ,...,c , cr, cr1 ,...,crN ). Their elements crk are called the k-stage ghosts of ghost number gh[crk ]=k+1 and antiﬁeld number Ant[crk ]= (k + 1). The KT operator (4.8) naturally KT is extended to a graded derivation of a DBGR ⇤ N . P1{ } 65 THEOREM 5.1: Given the KT complex (4.10), a module of graded densities P 0,n N is decomposed into a cochain sequence 1 { } 0 0,n[F ; Y ] u P 0,n N 1 u P 0,n N 2 u (5.3) !S1 ! 1 { } ! 1 { } ! ··· graded in ghost number. Its ascent operator

(1) (N) A @ r @ rN 1 @ u = u + u + + u = u + u + + u , (5.4) A r rN 1 ··· @s @c ··· @c is an odd graded derivation of ghost number 1 where @ u = uA ,uA = cr ⌘(A)⇤, (5.5) @sA ⇤ r 0 ⇤ X| | r (c r) A A! = u A! = dH 0, (5.6) sA E E is a symmetry of a graded Lagrangian L and the derivations

(k) rk 1 @ rk rk 1 ⇤ @ u = u = c⇤ ⌘(r ) ,k=1,...,N, (5.7) @crk 1 k @crk 1 0 ⇤ X| | obey the relations

rk (rk 2,⌃)(A,⌅) rk 1 rk 2,⌅ c hrk c⌃rk 2 d⌅ A! + u rk 1 c⌅rk 2 ! = dH k0 . (5.8) E X X ⇤

A glance at the symmetry u (5.5) shows that it is a derivation of a ring P 0 [0] which satisﬁes Deﬁnition 3.1 of gauge transformations. Consequently, 1 u (5.5) is a gauge symmetry of a graded Lagrangian L associated to the complete non-trivial NI (4.4). Therefore, it is a non-trivial gauge symmetry.

66 Turn now to the relation (5.8). The variational derivative of both its sides with respect to crk 2 leads to the equality

rk 1 @ rk 2 rk 2 d⌃u rk 1 u = (↵ ), (5.9) @c⌃ Xrk 2 (rk 2)(A,⌅) ⌃ rk ↵ = ⌘(h ) d⌃(c s⌅A), rk X For k = 1, it takes a form

r ⌃ A A d⌃u @r u = (↵ ) X of a ﬁrst-stage gauge symmetry condition on-shell which the non-trivial gauge symmetry u (5.5) satisﬁes. Therefore, one can treat the odd graded derivation

(1) r r r1 r ⇤ u = u @r,u= c⇤ ⌘(r1 ) , X as a ﬁrst-stage gauge symmetry associated to a complete ﬁrst-stage NI

r,⇤ A,⌃ (B,⌃)(A,⌅) d⇤( s⌃A)= ( h s⌃Bs⌅A). r1 r r1 X X X

Iterating the arguments, one comes to the relation (5.9) providing a k- stage gauge symmetry condition which is associated to the complete non-trivial k-stage NI (4.11). The odd graded derivation u(k) (5.7) is called the k-stage gauge symmetry.

Thus, components of the ascent operator u (5.4) in Theorem 5.1 are non- trivial gauge and higher-stage gauge symmetries. Therefore, we agree to call it the gauge operator.

67 The correspondence of gauge and higher-stage gauge symmetries to NI and higher-stage NI in Theorem 5.1 is unique due to the following direct second Noether theorem.

THEOREM 5.2: If u (5.5) is a gauge symmetry, the variational derivative of the d -exact • H density uA ! (5.6) with respect to ghosts cr leads to the equality EA A ⇤ A⇤ (u !)= ( 1)| |d [u ] = (5.10) r EA ⇤ r EA ⇤ A ⇤ ⇤ A ⇤ ( 1)| X|d (⌘( ) )= ( 1)| |⌘(⌘( )) d =0, ⇤ r EA r ⇤EA X X which reproduces the complete NI (4.4). Given the k-stage gauge symmetry condition (5.9), the variational deriva- • tive of the equality (5.8) with respect to ghosts crk leads to the equality, reproducing the k-stage NI (4.11). ⇤

68 6 Lagrangian BRST theory

In contrast with the KT operator (4.8), the gauge operator u (5.3) need not be nilpotent.

Let us study its extension to a nilpotent graded derivation

(k) rk 1 @ b = u + = u + = u + (6.1) @crk 1 1 k N+1 1 k N+1  X  X @ @ @ @ = uA + r + urk + rk+1 @sA @cr @crk @crk+1 ✓ ◆ 0 k N 1 ✓ ◆  X of ghost number 1 by means of antiﬁeld-free terms (k) of higher polynomial degree in ghosts cri and their jets cri ,0 i

If a BRST operator exists, the cochain sequence (5.3) is brought into a BRST complex

0 0,n[F ; Y ] b P 0,n N 1 b P 0,n N 2 b . (6.2) !S1 ! 1 { } ! 1 { } ! ··· One can show the following.

The gauge operator (5.3) admits the BRST extension (6.1) only if the • gauge symmetry conditions (5.8) and the higher-stage NI (4.11) are satisﬁed o↵-shell.

Gauge symmetries need not form an algebra. Therefore, we replace the • notion of the algebra of gauge symmetries with some conditions on the gauge

69 operator. Gauge symmetries are said to be algebraically closed if the gauge operator admits the nilpotent BRST extension (6.1).

J.Gomis, J.Par´ıs,S.Samuel, Antibracket, antiﬁelds and gauge theory quantization, Phys. Rep. 295 (1995) 1.

A nilpotent BRST operator provides a BRST extension of an original • Lagrangian system by means of graded antiﬁelds and ghosts as follows.

The DBGR P ⇤ N (5.2) is a particular ﬁeld-antiﬁeld theory of the fol- 1{ } lowing type. Let us consider a pull-back composite bundle

W = Z Z0 Z X ⇥X ! ! where Z0 X is a vector bundle. Let us regard it as an odd graded vector ! bundle over Z. The density-dual VW of the vertical tangent bundle VW of W X is a graded vector bundle ! n VW = ((Z0 V ⇤Z) T ⇤X) Z0 Z ⌦Z ^ Y over Z. Let us consider the DBGR ⇤ [VW; Z] with the local generating P1 a a a basis (z , za), [za]=[z ] + 1. Its elements z and za are called ﬁelds and antiﬁelds, respectively. Graded densities of this DBGR are endowed with the antibracket

L L L L 0 [L0]([L0]+1) 0 L!, L0! = a +( 1) a !. (6.3) { } 2 za z za z 3 4 5

70 Then one associates to any even Lagrangian L! the odd vertical graded derivations

a L @ a @ L L = @a = a , L =@ a = a , (6.4) E za @z E @za z l [a]+1 L @ L @ #L = L + =( 1) + , (6.5) L za @z za @z ✓ a a ◆ such that #L(L0!)= L!, L0! . { } THEOREM 6.1: The following conditions are equivalent.

(i) The antibracket of a Lagrangian L! is dH -exact, i.e.,

L L L!, L! =2 a ! = dH . (6.6) { } za z

(ii) The graded derivation #L (6.5) is nilpotent. ⇤

The equality (6.6) is called the classical master equation. A solution of the master equation (6.6) is called non-trivial if both the derivations (6.4) do not vanish.

Being an element of the DBGA ⇤ N (5.2), an original Lagrangian L P1{ } obeys the master equation (6.6) and yields the graded derivations L =0,

L = (6.4), i.e., it is a trivial solution of the master equation. Therefore, let us consider its extension

L = L + L + L + E 1 2 ··· by means of even densities L , i 2, of zero antiﬁeld number and polynomial i degree i in ghosts. Then the following is a corollary of Theorem 6.1.

71 COROLLARY 6.2: A Lagrangian L is extended to a non-trivial solution

LE of the master equation only if the gauge operator u (5.3) admits the nilpotent extension #E (6.5). ⇤

However, one can say something more.

THEOREM 6.3: If the gauge operator u (5.3) can be extended to the BRST operator b (6.1), then the master equation has a non-trivial proper solution

rk 1 LE = L + b c crk 1 ! + dH , (6.7) 0 k N ! X such that b = E is the graded derivation deﬁned by the Lagrangian LE (6.7). ⇤

The Lagrangian LE (6.7) is said to be the BRST extension of an original Lagrangian L.

This extension is a preliminary step towards quantization of reducible degenerate Lagrangian theories.

72 7 Applications. Topological BF theory

The most of basic Lagrangian models in ﬁeld theory and mechanics are irreducible degenerate.

G.Sardanashvily, Noether’s Theorems. Applications in Mechanics and Field Theory (Springer, 2016). G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (World Scientific, 2009). G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010). G.Sardanashvily, Classical field theory. Advanced mathematical formulation, Int. J. Geom. Methods Mod. Phys. 5 (2008) 1163-1189; arXiv: 0811.0331

Gauge theory of principal connections on principal bundles (irreducible • degenerate Lagrangian system)

Gauge gravitation theory on natural bundles (irreducible degenerate • Lagrangian system whose gauge symmetries are general covariant transformations)

G.Sardanashvily, Gauge gravitation theory. Gravity as a Higgs ﬁeld, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1650086

73 Topological Chern–Simons gauge theory (irreducible degenerate Lagran- • gian system, whose gauge symmetries are variational, but not exact)

Topological BF theory (reducible degenerate Lagrangian system) • G.Sardanashvily, Higher-stage Noether identities and second Noether theorems, Adv. Math. Phys. 2015 (2015) 127481

SUSY gauge theory on principal graded bundles (irreducible degenerate • graded Lagrangian system)

Covariant (polysymplectic) Hamiltonian ﬁeld theory, formulated as par- • ticular Lagrangian theory on a phase space

G.Sardanashvily, Polysymplectic Hamiltonian ﬁeld theory, arXiv: 1505.01444

Lagrangian and Hamiltonian non-autonomous mechanics on ﬁbre bun- • dles over R

G.Sardanashvily, Noether’s ﬁrst theorem in Hamiltonian mechanics, arXiv: 1510.03760

Relativistic mechanics as a Lagrangian theory of one-dimensional sub- • manifolds

G.Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics, J. Geom. Mech. 4 (2012) 99-110

74 We address topological BF theory of two exterior forms A and B of form degree A + B = dim X 1 on a smooth manifold X because it | | | | exempliﬁes reducible degenerate Lagrangian theory which satisﬁes homology regularity Condition 4.1.

D.Birmingham, M.Blau, Topological ﬁeld theory, Phys. Rep. 209 (1991) 129.

Its dynamic variables A and B are sections of a ﬁbre bundle

p q Y = T ⇤X T ⇤X, p + q = n 1 > 1, ^ ^ coordinated by (x ,Aµ1...µp ,B⌫1...⌫q ). Without a loss of generality, let q be even and q p. The corresponding DGR is ⇤ Y . O1 There are the canonical p- and q-forms

A = A dxµ1 dxµp ,B= B dx⌫1 dx⌫q µ1...µp ^···^ ⌫1...⌫q ^···^ on Y . A Lagrangian of topological BF theory reads

L = A d B = ✏µ1...µn A d B !, (7.1) BF ^ H µ1...µp µp+1 µp+2...µn where ✏ is the Levi–Civita symbol. It is a reduced ﬁrst order Lagrangian. Its ﬁrst order Euler–Lagrange operator

L = µ1...µp dA ! + ⌫p+2...⌫n dB !, EA µ1...µp ^ EB ⌫p+2...⌫n ^ µ ...µ µ ...µ 1 p = ✏µ1...µn d B , p+2 n = ✏µ1...µn d A , EA µp+1 µp+2...µn EB µp+1 µ1...µp satisﬁes the Noether identities

d µ1...µp =0,d⌫1...⌫q =0. (7.2) µ1 EA ⌫1 EB 75 Given a family of vector bundles

p k 1 q k 1 Ek = T ⇤X T ⇤X, 0 k

⇤ q 1 = ⇤ [VY E0 Eq 1 E0 Eq 1; Y ]. (7.3) P1{ } P1 Y ···Y Y Y ···Y It possesses a local generating basis

A ,B ," ,...," ,",⇠ ,...,⇠ ,⇠, { µ1...µp ⌫1...⌫q µ2...µp µp ⌫2...⌫q ⌫q µ ...µ ⌫ ...⌫ ⌫ ...⌫ ⌫ A 1 p , B 1 q , "µ2...µp ,...,"µp , ", ⇠ 2 q ,...,⇠ q , ⇠ } of Grassmann parity

["µk...µp ]=[⇠⌫k...⌫q ]=(k + 1)mod 2, ["]=p mod 2, [⇠]=0, ⌫ ...⌫ ["µk...µp ]=[⇠ k q ]=k mod 2, ["]=(p + 1)mod 2, [⇠]=1, of ghost number

gh["µk...µp ] = gh[⇠⌫k...⌫q ]=k, gh["]=p +1, gh[⇠]=q +1, and of antiﬁeld number

µ ...µ ⌫ ...⌫ Ant[A 1 p ] = Ant[B p+1 q ]=1, ⌫ ...⌫ Ant["µk...µp ] = Ant[⇠ k q ]=k +1, Ant["]=p, Ant["]=q.

76 One can show that homology regularity Condition 4.1 holds, and the

DBGR ⇤ q 1 (7.3) is endowed with the KT operator P1{ }

@ µ1...µp @ ⌫1...⌫q @ µk...µp KT = µ ...µ + ⌫ ...⌫ + + 1 p EA 1 q EB @"µk...µp A @A @B 2 k p X

@ @ ⌫ ...⌫ @ ⌫q d "µp + k q + d ⇠ , @" µp ⌫k...⌫q B ⌫q 2 k q @⇠ @⇠   µ ...µ µ1...µXp µ ...µ 2 p = d A , k+1 p = d "µkµk+1...µp , 2 k < p, A µ1 A µk  ⌫ ...⌫ ⌫ ⌫ ...⌫ ⌫2...⌫q = d B 1 q , ⌫k+1...⌫q = d ⇠ k k+1 q , 2 k

µk...µp dµk A =0,k=2,...,p,

⌫k...⌫q d⌫k B =0,k=2,...,q.

It follows that topological BF theory is (q 1)-reducible. Applying inverse second Noether Theorem 5.1, one obtains the gauge operator (5.4) which reads @ @ u = dµ1 "µ2...µp + d⌫1 ⇠⌫2...⌫q + (7.4) @Aµ1µ2...µp @B⌫1⌫2...⌫q @ @ d " + + d " + µ2 µ3...µp @" ··· µp @"  µ2µ3...µp µp @ @ d ⇠ + + d ⇠ . ⌫2 ⌫3...⌫q @⇠ ··· ⌫q @⇠  ⌫2⌫3...⌫q ⌫q

In particular, a gauge symmetry of the Lagrangian LBF (7.1) is @ @ u = dµ1 "µ2...µp + d⌫1 ⇠⌫2...⌫q . @Aµ1µ2...µp @B⌫1⌫2...⌫q

77 It also is readily observed that the gauge operator u (7.4) is nilpotent.

Thus, it is the BRST operator b = u. As a result, the Lagrangian LBF is extended to the non-trivial solution of the master equation LE (6.7) which reads

µ ...µ 1 p µk...µp µp LE = LBF + "µ2...µp dµ1 A + "µk+1...µp dµk " + "dµp " + 1