DOCSLIB.ORG

Moment Maps, Cobordisms, and Hamiltonian Group Actions Mathematical Surveys and Monographs Volume 98

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Moment Maps, Cobordisms, and Hamiltonian Group Actions Mathematical Surveys and Monographs Volume 98 http://dx.doi.org/10.1090/surv/098 Moment Maps, Cobordisms, and Hamiltonian Group Actions Mathematical Surveys and Monographs Volume 98 Moment Maps, Cobordisms, and Hamiltonian Group Actions Victor Guillemin Viktor Ginzburg Yael Karshon American Mathematical Society EDITORIAL COMMITTEE Peter S. Landweber Tudor Stefan Ratiu Michael P. Loss, Chair J. T. Stafford 2000 Mathematics Subject Classification. Primary 53Dxx, 57Rxx, 55N91, 57S15. ABSTRACT. The book contains a systematic treatment of numerous symplectic geometry questions in the context of cobordisms of Hamiltonian group actions. The topics analyzed in the book in­ clude abstract moment maps, symplectic reduction, the Duistermaat-Heckman formula, geometric quantization, and the quantization commutes with reduction theorem. Ten appendices cover a broad range of related subjects: proper actions of compact Lie groups, equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem, the Kawasaki Riemann-Roch formula, and a variety of other results. The book can be used by researchers and graduate students working in symplectic geometry and its applications. Library of Congress Cataloging-in-Publication Data Guillemin, V., 1937- Moment maps, cobordisms, and Hamiltonian group actions / Victor Guillemin, Viktor Ginz- burg, Yael Karshon. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 98) Includes bibliographical references and index. ISBN 0-8218-0502-9 (alk. paper) 1. Symplectic geometry. 2. Cobordism theory. 3. Geometric quantization. I. Karshon, Yael, 1964- II. Ginzburg, Viktor L., 1962- III. Title. IV. Mathematical surveys and monographs ; no. 98. QA665.G85 2002 516.3/6-dc21 2002074590 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 Contents Chapter 1. Introduction 1. Topological aspects of Hamiltonian group actions 2. Hamiltonian cobordism 3. The linearization theorem and non-compact cobordisms 4. Abstract moment maps and non-degeneracy 5. The quantum linearization theorem and its applications 6. Acknowledgements Part 1. Cobordism Chapter 2. Hamiltonian cobordism 1. Hamiltonian group actions 2. Hamiltonian geometry 3. Compact Hamiltonian cobordisms 4. Proper Hamiltonian cobordisms 5. Hamiltonian complex cobordisms Chapter 3. Abstract moment maps 1. Abstract moment maps: definitions and examples 2. Proper abstract moment maps 3. Cobordism 4. First examples of proper cobordisms 5. Cobordisms of surfaces 6. Cobordisms of linear actions Chapter 4. The linearization theorem 1. The simplest case of the linearization theorem 2. The Hamiltonian linearization theorem 3. The linearization theorem for abstract moment maps 4. Linear torus actions 5. The right-hand side of the linearization theorems 6. The Duistermaat-Heckman and Guillemin-Lerman-Sternber Chapter 5. Reduction and applications 1. (Pre-)symplectic reduction 2. Reduction for abstract moment maps 3. The Duistermaat-Heckman theorem 4. Kahler reduction 5. The complex Delzant construction 6. Cobordism of reduced spaces vi CONTENTS 7. Jeffrey-Kirwan localization 82 8. Cutting 84 Part 2. Quantization 87 Chapter 6. Geometric quantization 89 1. Quantization and group actions 89 2. Pre-quantization 90 3. Pre-quantization of reduced spaces 96 4. Kirillov-Kostant pre-quantization 99 5. Polarizations, complex structures, and geometric quantization 102 6. Dolbeault Quantization and the Riemann-Roch formula 110 7. Stable complex quantization and Spinc quantization 113 8. Geometric quantization as a push-forward 117 Chapter 7. The quantum version of the linearization theorem 119 1. The quantization of Cd 119 2. Partition functions 125 3. The character of Q(Cd) 130 4. A quantum version of the linearization theorem 134 Chapter 8. Quantization commutes with reduction 139 1. Quantization and reduction commute 139 2. Quantization of stable complex toric varieties 141 3. Linearization of [Q,R]=0 145 4. Straightening the symplectic and complex structures 149 5. Passing to holomorphic sheaf cohomology 150 6. Computing global sections; the lit set 152 7. The Cech complex 155 8. The higher cohomology 157 9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 159 10. Overview of the literature 162 Part 3. Appendices 165 Appendix A. Signs and normalization conventions 167 1. The representation of G on C°°(M) 167 2. The integral weight lattice 168 3. Connection and curvature for principal torus bundles 169 4. Curvature and Chern classes 171 5. Equivariant curvature; integral equivariant cohomology 172 Appendix B. Proper actions of Lie groups 173 1. Basic definitions 173 2. The slice theorem 178 3. Corollaries of the slice theorem 182 4. The Mostow-Palais embedding theorem 189 5. Rigidity of compact group actions 191 Appendix C. Equivariant cohomology 197 1. The definition and basic properties of equivariant cohomology 197 CONTENTS vii 2. Reduction and cohomology 201 3. Additivity and localization 203 4. Formality 205 5. The relation between HQ and H^ 208 6. Equivariant vector bundles and characteristic classes 211 7. The Atiyah-Bott-Berline-Vergne localization formula 217 8. Applications of the Atiyah-Bott-Berline-Vergne localization formula 222 9. Equivariant homology 226 Appendix D. Stable complex and Spinc-structures 229 1. Stable complex structures 229 2. Spinc-structures 238 3. Spinc-structures and stable complex structures 248 Appendix E. Assignments and abstract moment maps 257 1. Existence of abstract moment maps 257 2. Exact moment maps 263 3. Hamiltonian moment maps 265 4. Abstract moment maps on linear spaces are exact 269 5. Formal cobordism of Hamiltonian spaces 273 Appendix F. Assignment cohomology 279 1. Construction of assignment cohomology 279 2. Assignments with other coefficients 281 3. Assignment cohomology for pairs 283 4. Examples of calculations of assignment cohomology 285 5. Generalizations of assignment cohomology 287 Appendix G. Non-degenerate abstract moment maps 289 1. Definitions and basic examples 289 2. Global properties of non-degenerate abstract moment maps 290 3. Existence of non-degenerate two-forms 294 Appendix H. Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization 301 1. The Hamiltonian cobordism ring and characteristic classes 301 2. Characteristic numbers 304 3. Characteristic numbers as a full system of invariants 305 4. Non-degenerate cobordisms 308 5. Geometric quantization 310 Appendix I. The Kawasaki Riemann-Roch formula 315 1. Todd classes 315 2. The Equivariant Riemann-Roch Theorem 316 3. The Kawasaki Riemann-Roch formula I: finite abelian quotients 320 4. The Kawasaki Riemann-Roch formula II: torus quotients 323 Appendix J. Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman 327 1. The Spin -Dirac operator and the Spinc-quantization 327 2. The summary of the results 329 viii CONTENTS 3. Transversally elliptic operators and their indexes 331 4. Index of the operator Ba 333 5. The model operator 335 6. Proof of Theorem 1 336 Bibliography 339 Index 349 Bibliography [Ad] J. F. Adams, Lectures on Lie groups, Benjamin Inc., New York, 1969. [Al] C. Albert, Le theoreme de reduction de Marsden-Weinstein en geometrie cosymplectique et de contact, (The Marsden-Weinstein reduction theorem in cosymplectic and contact geometry), J. Geom. Phys. 6 (1989), no. 4, 627-649. [ACG] J. M. Arms, R. H. Cushman, and M. J. Gotay, A universal reduction procedure for Hamiltonian group actions. The geometry of Hamiltonian systems (Berkeley, CA, 1989), 33-51, Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991. [Ar] V. I. Arnold, Mathematical methods of classical mechanics, Second Edition, Graduate Texts in Mathematics 60, Springer-Verlag, New York-Berlin, 1989. [Atl] M. F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics 400, Springer-Verlag, Berlin, Heidelberg, 1974. [At2] M. F. Atiyah, Elliptic operators, discrete groups and von Neuman algebras, Asterisque 32-33 (1976), 43-72. [At3] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. [AB1] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. Math. 86 (1967), 374-407. [AB2] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. [ABS] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), 3-38. [AM] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison- Wesley,
Load more

Recommended publications
  • Arxiv:1311.6429V2 [Math.AG]
    QUASI-HAMILTONIAN REDUCTION VIA CLASSICAL CHERN–SIMONS THEORY PAVEL SAFRONOV Abstract. This paper puts the theory of quasi-Hamiltonian reduction in the framework of shifted symplectic structures developed by Pantev, To¨en, Vaqui´eand Vezzosi. We compute the symplectic structures on mapping stacks and show how the AKSZ topological field theory defined by Calaque allows one to neatly package the constructions used in quasi- Hamiltonian reduction. Finally, we explain how a prequantization of character stacks can be obtained purely locally. 0. Introduction 0.1. This paper is an attempt to interpret computations of Alekseev, Malkin and Mein- renken [AMM97] in the framework of shifted symplectic structures [PTVV11]. Symplectic structures appeared as natural structures one encounters on phase spaces of classical mechanical systems. Classical mechanics is a one-dimensional classical field theory and when one goes up in the dimension shifted, or derived, symplectic structures appear. That is, given an n-dimensional classical field theory, the phase space attached to a d- dimensional closed manifold carries an (n−d−1)-shifted symplectic structure. For instance, if d = n − 1 one gets ordinary symplectic structures and for d = n, i.e. in the top dimension, one encounters (−1)-shifted symplectic spaces. These spaces can be more explicitly described as critical loci of action functionals. ∼ An n-shifted symplectic structure on a stack X is an isomorphism TX → LX [n] between the tangent complex and the shifted cotangent complex together with certain closedness conditions. Symplectic structures on stacks put severe restrictions on the geometry: for instance, a 0-shifted symplectic derived scheme is automatically smooth.
    [Show full text]
  • The Moment Map of a Lie Group Representation
    transactions of the american mathematical society Volume 330, Number 1, March 1992 THE MOMENT MAP OF A LIE GROUP REPRESENTATION N. J. WILDBERGER Abstract. The classical moment map of symplectic geometry is used to canoni- cally associate to a unitary representation of a Lie group G a G-invariant subset of the dual of the Lie algebra. This correspondence is in some sense dual to geometric quantization. The nature and convexity of this subset is investigated for G compact semisimple. Introduction Since Kirillov's fundamental paper [7], orbit theory has played an important role in the representation theory of Lie groups. The general philosophy is that there should be some connection between irreducible unitary representations of a Lie group G and orbits of the group in the dual 9* of its Lie algebra &. This connection should be functorial; the usual operations of restriction, induction, taking direct sums and tensor products of representations should correspond to geometric relationships between the associated coadjoint orbits. Properties of a representation such as square-integrability, temperedness etc. should be related to geometric aspects of the associated orbit. Kirillov's original theory showed that if G was a connected simply con- nected nilpotent Lie group one could establish such a bijection between the set of coadjoint orbits and the set of equivalence classes of irreducible unitary representations G by associating to each orbit a certain induced representa- tion. Kostant [9] noticed the close similarity between this association and what physicists refer to as quantization of a classical mechanical system, and he and Souriau [11] independently generalized this procedure to what is called geomet- ric quantization.
    [Show full text]
  • LECTURE 5 Let (M,Ω) Be a Connected Compact Symplectic Manifold, T A
    SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M; !) be a connected compact symplectic manifold, T a torus, T × M ! M a Hamiltonian action of T on M, and Φ: M ! t∗ the assoaciated moment map. Theorem 0.1 (The convexity theorem, Atiyah, Guillemin and Stern- berg, 1982). The image of Φ is a convex polytope. In this talk we will prove the convexity theorem. The proof given here is of Guillemin-Sternberg and is taken from [3]. We will then 1 discuss the case dim T = 2 dim M. Remark 0.1. We saw in Lecture 4 that for an Abelian group-action the equivariance condition amounts to the moment map being invariant; in case the group is a torus, the latter condition follows from Hamil- ξ ton's equation: dΦ = −ιξM ! for all ξ in the Lie algebra, see e.g., [4, Proposition 2.9]. 1. The local convexity theorem In the convexity theorem we consider torus actions on compact sym- plectic manifolds. However to understand moment maps locally, we will look at Hamiltonian torus actions on symplectic vector spaces. Example. T n = (S1)n acts on Cn by (a1; : : : ; an) · (z1; : : : ; zn) = (a1z1; : : : ; anzn): The symplectic form is n n X X ! = dxj ^ dyj = rjdrj ^ dθj j=1 j=1 iθj where zj = xj +iyj = rje . With a standard basis of the Lie algebra of T n, the action is generated by the vector fields @ for j = 1; : : : ; n and @θj n n the moment map is a map Φ = (Φ1;:::; Φn): C ! R . Hamilton's @ equation becomes dΦj = −ι( )!.
    [Show full text]
  • Gradient Flow of the Norm Squared of a Moment Map
    Gradient flow of the norm squared of a moment map Autor(en): Lerman, Eugene Objekttyp: Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 51 (2005) Heft 1-2: L'enseignement mathématique PDF erstellt am: 04.10.2021 Persistenter Link: http://doi.org/10.5169/seals-3590 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, www.library.ethz.ch http://www.e-periodica.ch L'Enseignement Mathématique, t. 51 (2005), p. 117-127 GRADIENT FLOW OF THE NORM SQUARED OF A MOMENT MAP by Eugene Lerman*) Abstract. We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map.
    [Show full text]
  • Symplectic Topology, Geometry and Gauge Theory Lisa Jeffrey
    Symplectic Topology, Geometry and Gauge Theory Lisa Jeffrey 1 Symplectic geometry has its roots in classical mechanics. A prototype for a symplectic manifold is the phase space which parametrizes the position q and momentum p of a classical particle. If the Hamiltonian (kinetic + potential en- ergy) is p2 H = + V (q) 2 then the motion of the particle is described by Hamilton’s equations dq ∂H = = p dt ∂p dp ∂H ∂V = − = − dt ∂q ∂q 2 In mathematical terms, a symplectic manifold is a manifold M endowed with a 2-form ω which is: • closed: dω = 0 (integral of ω over a 2-dimensional subman- ifold which is the boundary of a 3-manifold is 0) • nondegenerate (at any x ∈ M ω gives a map ∗ from the tangent space TxM to its dual Tx M; nondegeneracy means this map is invertible) If M = R2 is the phase space equipped with the Hamiltonian H then the symplectic form ω = dq ∧ dp transforms the 1-form ∂H ∂H dH = dp + dq ∂p ∂q to the vector field ∂H ∂H X = (− , ) H ∂q ∂p 3 The flow associated to XH is the flow satisfying Hamilton’s equations. Darboux’s theorem says that near any point of M there are local coordinates x1, . , xn, y1, . , yn such that Xn ω = dxi ∧ dyi. i=1 (Note that symplectic structures only exist on even-dimensional manifolds) Thus there are no local invariants that distin- guish between symplectic structures: at a local level all symplectic forms are identical. In contrast, Riemannian metrics g have local invariants, the curvature tensors R(g), such that if R(g1) =6 R(g2) then there is no smooth map that pulls back g1 to g2.
    [Show full text]
  • Hamiltonian Manifolds and Moment Map
    Hamiltonian manifolds and moment map. Nicole BERLINE and Mich`eleVERGNE July 4, 2011 Contents 1 Setup of Hamiltonian manifolds 5 1.1 Tangent and normal vector bundle . 5 1.2 Calculus on differential forms . 6 1.2.1 de Rham differential . 6 1.2.2 Contraction by vector fields . 8 1.2.3 Lie derivative with respect to a vector field. Cartan's Homotopy Formula . 8 1.3 Action of a Lie group on a manifold . 9 1.4 Symplectic manifold. Hamiltonian action . 10 1.4.1 Symplectic vector space . 10 1.4.2 Symplectic form. Darboux coordinates . 11 1.4.3 Hamiltonian vector field . 12 1.4.4 Moment map. Hamiltonian manifold . 13 2 Examples of Hamiltonian manifolds 15 2.1 Cotangent bundle . 15 2.2 Symplectic and Hermitian vector spaces . 16 2.3 Complex projective space . 17 2.4 Coadjoint orbits . 20 3 Reduced spaces 21 3.1 Fiber bundles . 21 3.1.1 Fibration . 21 3.1.2 Actions of compact Lie groups, linearization. 22 3.1.3 Free action of a Lie group . 23 1 3.1.4 Principal bundles. Basic differential forms . 24 3.2 Pre-Hamiltonian manifold . 26 3.2.1 Examples of pre-Hamiltonian manifolds . 26 3.2.2 Consequences of Hamilton equation. Homogeneous man- ifolds and coadjoint orbits . 27 3.3 Hamiltonian reduction . 29 4 Duistermaat-Heckman measure 31 4.1 Poincar´eLemma . 31 4.2 Pre-Hamiltonian structures on P × g∗. 33 4.3 Push-forward of the Liouville measure. 35 4.4 Push-forward of the Liouville measure and volume of the re- duced space .
    [Show full text]
  • The Moment Map and Equivariant Cohqmology
    MJ4&Y383/84 $3.00 + .w I( 1984 Pergamon Press Ltd THE MOMENT MAP AND EQUIVARIANT COHQMOLOGY M. F. ATIYAH and R. BOTT (Received20 December1982) $1. INTRODUCTION THE PURPOSEof this note is to present a de Rham version of the localization theorems of equivariant cohomology, and to point out their relation to a recent result of Duistermaat and Heckman and also to a quite independent result of Witten. To a large extent all the material that we use has been around for some time, although equivariant cohomology is not perhaps familiar to analysts. Our contribution is therefore mainly an expository one linking together various points of view. The paper of Duistermaat and Heckman [ 1 l] which was our initial stimulus concerns the moment mapf:M+R’ for the action of a torus T’ on a compact symplectic manifold M. Their theorem asserts that the push-forward byfof the symplectic (or Liouville) measure on A4 is a piece-wise polynomial measure on R’. An equivalent version is that the Fourier transform of this measure (which is the integral over A4 of e-i(c,fi) is exactly given by stationary phase approximation. For example when I = 1 (so that T’is the circle S) and the fixed points of the action are isolated points P, we have the exact formula , where (u is the symplectic 2-form on M and the e(P) are certain integers attached to the infinitesimal action of S at P. This principle, that stationary-phase is exact when the “Hamiltonian”fcomes from a circle action, is such an attractive result that it seemed to us to deserve further study.
    [Show full text]
  • Hamiltonian Actions of Unipotent Groups on Compact K\" Ahler
    Épijournal de Géométrie Algébrique epiga.episciences.org Volume 2 (2018), Article Nr. 10 Hamiltonian actions of unipotent groups on compact Kähler manifolds Daniel Greb and Christian Miebach Abstract. We study meromorphic actions of unipotent complex Lie groups on compact Kähler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable Kähler structures obtained by symplectic reduction. The relation of our complex-analytic theory to the work of Doran–Kirwan regarding the Geometric Invariant Theory of unipotent group actions on projective varieties is discussed in detail. Keywords. Unipotent algebraic groups; automorphisms of compact Kähler manifolds; Kähler metrics on fibre bundles and homogeneous spaces; moment maps; symplectic reduction; Geomet- ric Invariant Theory 2010 Mathematics Subject Classification. 32M05; 32M10; 32Q15; 14L24; 14L30; 37J15; 53D20 [Français] Titre. Actions hamiltoniennes des groupes unipotents sur les variétés kählériennes com- pactes Résumé. Nous étudions les actions méromorphes de groupes de Lie complexes unipotents sur les variétés kählériennes compactes en utilisant des techniques de type application moment. Nous introduisons des conditions de stabilité naturelles et nous montrons que l’ensemble des points semi-stables forme un ouvert de Zariski et admet des quotients géométriques munis de structures kählériennes compactifiables obtenues par réduction symplectique. Le parallèle entre notre théorie analytique complexe et les travaux de Doran–Kirwan concernant la Théorie Géométrique des Invariants des actions de groupes unipotents sur les variétés projectives est discuté en détails. arXiv:1805.02551v2 [math.CV] 8 Nov 2018 Received by the Editors on May 8, 2018, and in final form on September 6, 2018.
    [Show full text]
  • LECTURE 8: the MOMENT MAP Contents 1. Properties of The
    LECTURE 8: THE MOMENT MAP Contents 1. Properties of the moment map 1 2. Existence and Uniqueness of the moment map 4 3. Examples/Exercises of moment maps 7 4. Moment map in gauge theory 9 1. Properties of the moment map { Hamiltonian actions. Suppose a Lie group G acts smoothly on M. For simplicity we shall always assume G is connected. Associated to each vector X 2 g = Lie(G) one has a smooth vector field on M defined by d XM (m) = exp(tX) · m: dt t=0 Now suppose M is symplectic with symplectic form !. The G-action is called • symplectic if each element g 2 G acts on M by symplectomorphisms. { Equivalently, for each X 2 g, XM is symplectic, i.e. LXM ! = 0. • weakly Hamiltonian if each XM is Hamiltonian, with Hamiltonian functions depending nicely on X. More precisely, there exists functions µX on M, depending linearly on X 2 g, so that X dµ = ιXM !: There are two equivalent ways to group these functions µX 's together: { The comoment map is the linear map µ∗ : g ! C1(M); µ∗(X) = µX : { The moment map is the smooth map µ : M ! g∗; µX (·) = hµ(·);Xi: Remark. One can regard the moment map µ as the dual map of the linear map µ∗ restricted to M, where M is viewed as a subset of C1(M)∗ via hm; fi := f(m). 1 2 LECTURE 8: THE MOMENT MAP • Hamiltonian if it is weakly Hamiltonian, and the moment map is equivariant with respect to the G-action on M and the coadjoint G-action on g∗.
    [Show full text]
  • A Cohomological Framework for Homotopy Moment Maps 10
    A COHOMOLOGICAL FRAMEWORK FOR HOMOTOPY MOMENT MAPS YAËL FRÉGIER, CAMILLE LAURENT-GENGOUX, AND MARCO ZAMBON Abstract. Given a Lie group acting on a manifold M preserving a closed n + 1-form ω, the notion of homotopy moment map for this action was introduced in [6], in terms of L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of [6]. Contents Introduction 1 1. Closed forms 3 2. A double complex encoding moment maps 4 3. Closed forms and moment maps as cocycles 6 4. Equivariant cohomology 6 5. Obstruction theory 8 6. Actions on mapping spaces 8 7. Equivalences 10 Appendix A. Equivalences of moment maps and L∞-algebra morphisms 15 References 17 Introduction Recall that a symplectic form is a closed, non-degenerate 2-form. It is natural to consider arXiv:1409.3142v2 [math.DG] 31 Jul 2015 symmetries of a given symplectic manifold, that is, a Lie group acting on a manifold, preserving the symplectic form. Among such actions, a nice subclass is given by actions that admit a moment map; in that case the infinitesimal generators of the action are hamiltonian vector fields. Actions admitting a moment map enjoy remarkable geometric, algebraic and topological properties, that have been studied extensively in the literature (e.g.
    [Show full text]
  • Covariant Hamiltonian Field Theories on Manifolds with Boundary: Yang-Mills Theories
    JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2017002 c American Institute of Mathematical Sciences Volume 9, Number 1, March 2017 pp. 47{82 COVARIANT HAMILTONIAN FIELD THEORIES ON MANIFOLDS WITH BOUNDARY: YANG-MILLS THEORIES Alberto Ibort∗ ICMAT and Depto. de Matem´aticas,Univ. Carlos III de Madrid Avda. de la Universidad 30, 28911 Legan´es,Madrid, Spain Amelia Spivak Dept. of Mathematics, Univ. of California at Berkeley 903 Evans Hall, 94720 Berkeley CA, USA (Communicated by Janusz Grabowski) Abstract. The multisymplectic formalism of field theories developed over the last fifty years is extended to deal with manifolds that have boundaries. In par- ticular, a multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundaries is developed. This work is a geometric fulfillment of Fock's formulation of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [11]. This framework leads to a geomet- ric understanding of conventional choices for boundary conditions and relates them to the moment map of the gauge group of the theory. It is also shown that the natural interpretation of the Euler-Lagrange equa- tions as an evolution system near the boundary leads to a presymplectic Hamil- tonian system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions for evolution at the boundary are analyzed and the reduced phase space of the system is shown to be a sym- plectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations.
    [Show full text]
  • MATHEMATICAL USES of GAUGE THEORY S. K. Donaldson Imperial
    MATHEMATICAL USES OF GAUGE THEORY S. K. Donaldson Imperial College, London 1. Introduction 1.1. This article surveys some developments in pure mathematics which have, to vary- ing degrees, grown out of the ideas of gauge theory in Mathematical Physics. The realisation that the gauge fields of particle physics and the connections of differen- tial geometry are one and the same has had wide-ranging consequences, at different levels. Most directly, it has lead mathematicians to work on new kinds of questions, often shedding light later on well-established problems. Less directly, various funda- mental ideas and techniques, notably the need to work with the infinite-dimensional gauge symmetry group, have found a place in the general world-view of many math- ematicians, influencing developments in other fields. Still less direct, the workin this area—between geometry and mathematical physics—has been a prime exam- ple of the interaction between these fields which has been so fruitful over thepast thirty years. The body of this paper is divided into three sections: roughly corresponding to Analysis, Geometry and Topology. However the different topics come together in many different ways: indeed the existence of these links between the topics isone the most attractive features of the area. 1.2 Gauge transformations. We do not have space in this article to review the usual foundational mate- rial on connections, curvature and related differential geometric constructions: for these we refer to standard texts. We will however briefly recall the notions of gauge transformations and gauge fixing. The simplest case is that of abelian gauge theory—connections on a U(1)-bundle, say over R3.
    [Show full text]