The Breadth of Symplectic and Poisson Geometry: Feschrift in Honor of Alan Weinstein

Progress in Mathematics Volume 232 Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein The Breadth of Symplect...

Author: Jerrold E. Marsden | Tudor S. Ratiu

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Progress in Mathematics Volume 232

Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein

The Breadth of Symplectic and Poisson Geometry Festschrift in Honor of Alan Weinstein

Jerrold E. Marsden Tudor S. Ratiu Editors

Birkh¨auser Boston • Basel • Berlin

Jerrold E. Marsden California Institute of Technology Department of Engineering and Applied Science Control and Dynamical Systems Pasadena, CA 91125 U.S.A.

Tudor S. Ratiu Ecole Polytechnique F´ed´erale de Lausanne D´epartement de Math´ematiques CH-1015 Lausanne Switzerland

AMS Subject Classifications: 53Dxx, 17Bxx, 22Exx, 53Dxx, 81Sxx Library of Congress Cataloging-in-Publication Data The breadth of symplectic and Poisson geometry : festschrift in honor of Alan Weinstein / Jerrold E. Marsden, Tudor S. Ratiu, editors. p. cm. – (Progress in mathematics ; v. 232) Includes bibliographical references and index. ISBN 0-8176-3565-3 (acid-free paper) 1. Symplectic geometry. 2. Geometric quantization. 3. Poisson manifolds. I. Weinstein, Alan, 1943- II. Marsden, Jerrold E. III. Ratiu, Tudor S. IV. Progress in mathematics (Boston, Mass.); v. 232. QA665.B74 2004 516.3’.6-dc22

ISBN 0-8176-3565-3

2004046202

Printed on acid-free paper.

c 2005 Birkh¨auser Boston

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 www.birkhauser.com

SPIN 10958261

(JLS/HP)

Margo, Alan, and Asha in Paris at the lovely Fontaine des Quatre Parties du Monde.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Academic genealogy of Alan Weinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii About Alan Weinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Students of Alan Weinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Alan Weinstein’s publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Dirac structures, momentum maps, and quasi-Poisson manifolds Henrique Bursztyn, Marius Crainic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Construction of Ricci-type connections by reduction and induction Michel Cahen, Simone Gutt, Lorenz Schwachhöfer . . . . . . . . . . . . . . . . . . . . . . . 41 A mathematical model for geomagnetic reversals J. J. Duistermaat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization Kurt Ehlers, Jair Koiller, Richard Montgomery, Pedro M. Rios . . . . . . . . . . . . . 75 Thompson’s conjecture for real semisimple Lie groups Sam Evens, Jiang-Hua Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 The Weinstein conjecture and theorems of nearby and almost existence Viktor L. Ginzburg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Simple singularities and integrable hierarchies Alexander B. Givental, Todor E. Milanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Momentum maps and measure-valued solutions (peakons, ﬁlaments, and sheets) for the EPDiff equation Darryl D. Holm, Jerrold E. Marsden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart, Gerstenhaber, and Batalin–Vilkovisky algebras Johannes Huebschmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Localization theorems by symplectic cuts Lisa Jeffrey, Mikhail Kogan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Reﬁnements of the Morse stratiﬁcation of the normsquare of the moment map Frances Kirwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Quasi, twisted, and all that… in Poisson geometry and Lie algebroid theory Yvette Kosmann-Schwarzbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Minimal coadjoint orbits and symplectic induction Bertram Kostant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces Camille Laurent-Gengoux, Ping Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Duality and triple structures Kirill C. H. Mackenzie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Star exponential functions as two-valued elements Y. Maeda, N. Miyazaki, H. Omori, A. Yoshioka . . . . . . . . . . . . . . . . . . . . . . . . . . 483 From momentum maps and dual pairs to symplectic and Poisson groupoids Charles-Michel Marle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds Yong-Geun Oh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 The universal covering and covered spaces of a symplectic Lie algebra action Juan-Pablo Ortega, Tudor S. Ratiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraic topics related to Alan’s interests Jim Stasheff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Dirac submanifolds of Jacobi manifolds Izu Vaisman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Quantum maps and automorphisms Steve Zelditch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

Preface

Alan Weinstein is one of the top mathematicians in the world working in the area of symplectic and differential geometry. His research on symplectic reduction, Lagrangian submanifolds, groupoids, applications to mechanics, and related areas has had a profound inﬂuence on the ﬁeld. This area of research remains active and vibrant today and this volume is intended to be a reﬂection of that vigor. In addition to reﬂecting the vitality of the ﬁeld, this is a celebratory volume to honor Alan’s 60th birthday. His birthday was also celebrated in August, 2003 with a wonderful week-long conference held at the ESI: the Erwin Schrödinger International Institute for Mathematical Physics in Vienna. Alan was born in New York in 1943. He was an undergraduate at MIT and a graduate student at UC Berkeley, where he was awarded his Ph.D. in 1967 under the direction of S. S. Chern. After spending postdoctoral years at IHES near Paris, MIT, and the University of Bonn, he joined the faculty at UC Berkeley in 1969, becoming a full Professor in 1976. Alan has received many honors, including an Alfred P. Sloan Foundation Fellowship, a Miller Professorship (twice), a Guggenheim Fellowship, election to the American Academy of Arts and Sciences in 1992, and an honorary degree at the University of Utrecht in 2003. At the ESI conference, S. S. Chern, Alan’s advisor, sent the following words to celebrate the occasion: “I am glad about this celebration and I think Alan richly deserves it. Alan came to me in the early sixties as a graduate student at the University of California at Berkeley. At that time, a prevailing problem in our geometry group, and the geometry community at large, was whether on a Riemannian manifold the cut locus and the conjugate locus of a point can be disjoint. Alan immediately showed that this was possible. The result became part of his Ph.D. thesis, which was published in the Annals of Mathematics. He received his Ph.D. degree in a short period of two years. I introduced him to IHES and the French mathematical community. He stays close with them and with the mathematical ideas of Charles Ehresmann. He is original and

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often came up with ingenious ideas. An example is his contribution to the solution of the Blaschke conjecture. I am very proud to count him as one of my students and I hope he will remain interested in mathematics up to my age, which is now 91.’’ Alan’s technical contributions are wide ranging and deep. As many of his early papers in his publication list illustrate, he started off in his thesis and the years immediately following in pure differential geometry, a topic he has come back to from time to time throughout his career. Already starting with his postdoc years and his early career at Berkeley, he became interested in symplectic geometry and mechanics. In this area he rapidly established himself as one of the world’s authorities, producing important and deep results ranging from reduction theory to Lagrangian and Poisson manifolds to studies of periodic orbits in Hamiltonian systems. He also did important work in ﬂuid mechanics and plasma physics and through this work, he established warm relations with the Berkeley physicists Allan Kaufman and Robert Littlejohn. Alan’s important work on periodic orbits in Hamiltonian systems led him eventually to formulate the “Weinstein conjecture,’’ namely that for a given Hamiltonian ﬂow on a symplectic manifold, there must be at least one closed orbit on a regular compact contact type level set of the Hamiltonian. Along with Arnold’s conjecture, the Weinstein conjecture has been one of the driving forces in symplectic topology over the last two decades. Alan kept up his interest in symplectic reduction theory throughout his later work. For instance, he laid some important foundation stones in the theory of semidirect product reduction as well as in singular reduction through his work on Satake’s V -manifolds, along with ﬁnding important links with singular structures in moduli spaces. Intertwined with his work on symplectic geometry and mechanics, he did extensive work on geometric PDE, eigenvalues, the Schrödinger operator and geometric quantization. Alan took the point of view of microlocal analysis and phase space structures in his work in this area, emphasizing the links with quantum mechanics.

Preface

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His work on the limit distribution of eigenvalue clusters in terms of the geodesic Radon transform of the potential inspired a large number of related articles. He showed that the geodesic ﬂow of a Zoll surface was symplectically equivalent to that of a round sphere, and hence that its Laplacian could be conjugated globally to the round Laplacian plus a pseudodifferential potential. This work inspired many other results on conjugacies. One of Alan’s fundamental contributions to Poisson geometry was the introduction of symplectic groupoids in 1987, which marks the ofﬁcial beginning of his “oids’’ period. In these works, he makes sweeping generalizations about a wide variety of constructions in symplectic geometry, including (with Courant) the important notion of Dirac structures. During this period of generalizations he constantly returned to speciﬁc topics in symplectic and Poisson geometry, such as geometric phases and Poisson Lie groups, in addition to making other key links. For instance, symplectic groupoids are used to link Poisson geometry to noncommutative geometry, and groupoids are also intimately related to many other areas, including symmetries and reduction, dual pairs, quantization and the theory of sigma models. One of the central ideas is that the usual theory of Hamiltonian actions, momentum maps, and symplectic reduction makes sense in the more general context of actions of symplectic groupoids; in this setting, momentum maps are Poisson maps taking values in general Poisson manifolds, rather than just Lie–Poisson manifolds (that is, duals of Lie algebras). Alan has raised the question of whether this framework can be further extended to include new notions of momentum maps such as quasi-Poisson manifolds with group-valued momentum maps as well as optimal momentum maps. Alan is well known not only for his brilliant papers and conjectures, but also for his general philosophy, such as the symplectic creed: Everything is a Lagrangian submanifold . Those of us who know him well also appreciate his very special insight. For example, in the middle of a discussion (for instance, as we both had in our joint works on semidirect product reduction as well as stability theory) he will say something like what you are really doing is. . . and then give us some usually very special insight that invariably substantially improves the whole project. Alan also has a very interesting and charming sense of humor that even makes its way into his papers from time to time. For instance, Alan had great fun in his papers with the “East Coast–West Coast’’ discussions of whether one should use the term momentum map or moment map. He also gave us a good laugh with the term symplectic bones as it relates to the French translation of Poisson as Fish. Alan is a great educator. His lectures, even on Calculus, are always a treat and are very inspiring for their special insight, their wit and lively presentation. His enthusiasm for mathematics is infectious. One story that comes to mind on the education front is this: during the days when he was exceptionally keen about groupoids, he was preparing a lecture for undergraduates on the subject. Some of us convinced him to present it as a colloquium lecture for faculty, keeping in mind the old advice “no colloquium talk can be too simple.’’ It was, in fact, not only a beautiful colloquium talk, but was perfectly pitched for the faculty, and it became a popular article in the Notices of the American Mathematical Society. Part of being a good educator is being

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cognizant of history. Alan excels in this area. For instance, his research into the history of Lie is what led directly to the introduction of the term “Lie–Poisson’’ bracket. The papers in this volume were selected by invitation and all of them underwent a rigorous refereeing process. While this process took some time, it resulted in high quality papers. We thank all of the referees for their diligent and helpful work. The authors of this volume represent some of the best workers in the subject and their contributions span a wide range of the topics covered by symplectic and Poisson geometry and mechanics, broadly interpreted. The intended audience for the book includes active researchers in the general area of symplectic geometry and mechanics, as well as aspiring graduate students who wish to learn where the subject is headed and what some of the current research topics are. Alan and Margo have a special relationship to Paris. They have spent many happy times there, and we wish them all the best and many more happy visits in the years to come. We wish to thank Ann Kostant for her expert editorial guidance throughout the production of this volume. Of course, we also thank all the authors for their contributions as well as their helpful guidance and advice. The referees are also thanked for their valuable comments and suggestions.

Jerry Marsden and Tudor Ratiu September, 2004

Academic genealogy of Alan Weinstein

Otto Mencken Universität Leipzig, 1668 Thomae Hobbesii Epicureismum historice delineatum sistit Johann C. Wichmannshausen Universität Leipzig, 1685 Disputationem Moralem De Divortiis Secundum Jus Naturae Christian A. Hausen Martin-Luther-Universität Halle-Wittenberg, 1713 De corpore scissuris ﬁgurisque non cruetando ductu Abraham G. Kaestner Universität Leipzig, 1739 Theoria radicum in aequationibus Johann F. Pfaff Georg-August-Universität Gottingen, 1786 Commentatio de ortibus et occasibus siderum apud auctores classicos commemoratis August F. Möbius Universität Leipzig, 1815 De computandis occultationibus ﬁxarum per planetas Otto W. Fiedler Universität Leipzig, 1859

Johannes Frischauf Universität Wien, 1861

Karl Friesach Universität Wien, 1846

Gustav Ritter von Escherich Technische Universität Graz, 1873 Die Geometrie auf Flachen constanter negativer Krummung

Emil Weyr University of Prague, 1870

Wilhelm Wirtinger Universität Wien, 1887 Uber eine spezielle Tripelinvolution in der Ebene

Wilhelm Blaschke Universität Wien, 1908 Shiing-Shen Chern Universität Hamburg, 1936 Eine Invariantentheorie der Dreigewebe aus r-dimensionalen Mannigfaltigkeiten im R2r Alan D. Weinstein University of California at Berkeley, 1967 The Cut Locus and Conjugate Locus of a Riemannian Manifold

About Alan Weinstein

Alan David Weinstein Ph.D.: University of California at Berkeley, 1967 Dissertation: The Cut Locus and Conjugate Locus of a Riemannian Manifold Advisor: Shiing-Shen Chern

Students of Alan Weinstein 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Jair Koiller, Studies on the Spring-Pendulum, 1975 Otto Ruiz, Existence of Brake-Orbits in Finsler Mechanical Systems, 1975 Yilmaz Akyildiz, Dynamical Symmetries of the Kepler Problem, 1976 Gerald Chachere, Numerical Experiments Concerning the Eigenvalues of the Laplacian on a Zoll Surface, 1977 John Jacob, Geodesic Symmetries of Homogeneous Kahler Manifolds, 1977 Steven Zelditch, Reconstruction of Singularities for Solutions of Schrödinger’s Equations, 1981 Enrique Planchart, Analogies in Symplectic Geometry of Some Results of Cartan in Representation Theory, 1982 Barry Fortune, A Symplectic Fixed Point Theorem for Complex Projective Spaces, 1984 Stephen Omohundro (Department of Physics), Geometric Perturbation Theory in Physics, 1985 Theodore Courant, Dirac Manifolds, 1987 Yong-Geun Oh, Nonlinear Schrödinger Equations with Potentials: Evolution, Existence, and Stability of Semi-Classical Bound States, 1988 Viktor Ginzburg, On Closed Characteristics of 2-Forms, 1990 Milton Lopes Filho, Microlocal Regularity and Symbols for Distributions, 1990 Jiang-Hua Lu, Multiplicative and Afﬁne Poisson Structures on Lie Groups, 1990 Ping Xu, Morita Equivalence of Poisson Manifolds, 1990

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16. Sean Bates, Symplectic End Invariants and C 0 Symplectic Topology, 1994 17. Agust Egilsson, On Embedding a Stratiﬁed Symplectic Space in a Smooth Poisson Manifold , 1995 18. Dong Yan, Yang–Mills Theory on Symplectic Manifolds, 1995 19. Zhao-Hui Qian, Groupoids, Midpoints and Quantizations, 1997 20. Vinay Kathotia, Universal Formulae for Deformation Quantization and The Campbell–Baker–Hausdorff Formula, 1998 21. Dmitry Roytenberg, Courant Algebroids, Derived Brackets And Even Symplectic Supermanifolds, 1999 22. Mélanie Bertelson-Volckaert (Stanford University), Foliations Associated to Regular Poisson Structures, 2000 23. Benjamin Davis, On Poisson Spaces Associated to Finitely Generated Poisson R-Algebras, 2001 24. Henrique Bursztyn, Morita Equivalence in Deformation Quantization, 2001 25. Olga Radko, Some Invariants of Poisson Manifolds, 2002 26. Xiang Tang, Quantization of Noncommutative Poisson Manifolds, 2003 27. Marco Zambon, Submanifold Averaging in Riemannian, Symplectic, and Contact Geometry, 2003 28. Chenchang Zhu, Integrating Lie Algebroids via Stacks and Applications to Jacobi Manifolds, 2003

Alan Weinstein’s publications [1] Weinstein, A., On the homotopy type of positively pinched manifolds, Arch. Math., 18 (1967), 523–524. [2] Weinstein, A., A ﬁxed point theorem for positively curved manifolds, J. Math. Mech., 18 (1968), 149–153. [3] Weinstein, A., The cut locus and conjugate locus of a riemannian manifold, Ann. Math. 87 (1968), 29–41. [4] Weinstein, A., Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc., 75 (1969), 1040–1041. [5] Weinstein, A., and Simon, U., Anwendungen der de rhamschen Zerlegung auf Probleme der localen Flächentheorie, Manuscripta Math., 1 (1969), 139–146. [6] Marsden, J., and Weinstein, A., A comparison theorem for hamiltonian vector ﬁelds, Proc. Amer. Math. Soc., 26 (1970), 629–631. [7] Weinstein,A., The generic conjugate locus, in Global Analysis, Proceedings of Symposia on Pure Mathematics, Vol. 15, American Mathematical Society, Providence, RI, 1970, 299–301. [8] Weinstein, A., Positively curved n-manifolds in R n+2 , J. Differential Geom., 4 (1970), 1–4. [9] Weinstein, A., Positively curved deformations of invariant Riemannian metrics, Proc. Amer. Math. Soc., 26 (1970), 151–152. [10] Weinstein, A., Sur la non-densité des géodésiques fermées, C. R. Acad. Sci. Paris, 271 (1970), 504. [11] Roels, J., and Weinstein, A., Functions whose Poisson brackets are constants, J. Math. Phys., 12 (1971), 1482–1486.

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[12] Weinstein, A., Singularities of families of functions, in Proceedings of the Conference “Differentialgeometrie im Grossen,’’ Mathematische Forschungsinstitut, Oberwolfach, Germany, 1971, 323–330. [13] Weinstein, A., Perturbation of periodic manifolds of Hamiltonian systems, Bull. Amer. Math. Soc., 77 (1971), 814–818. [14] Weinstein, A., Remarks on curvature and the Euler integrand, J. Differential Geom., 6 (1971), 259–262. [15] Weinstein, A., Symplectic manifolds and their lagrangian submanifolds, Adv. Math., 6 (1971), 329–346. [16] Weinstein, A., On the invariance of Poincaré’s generating function for canonical transformations, Invent. Math., 16 (1972), 202–213. [17] Weinstein, A., Distance spheres in complex projective spaces, Proc. Amer. Math. Soc., 39 (1973), 649–650. [18] Weinstein, A., Lagrangian submanifolds and hamiltonian systems, Ann. Math., 98 (1973), 377–410. [19] Weinstein, A., Normal modes for nonlinear hamiltonian systems, Invent. Math., 20 (1973), 47–57. [20] Marsden, J., and Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121–130. [21] Weinstein, A., Application des opérateurs intégraux de Fourier aux spectres des variétés riemanniennes, C. R. Acad. Sci. Paris, 279 (1974), 229–230. [22] Weinstein, A., On the volume of manifolds, all of whose geodesics are closed, J. Differential Geom., 9 (1974), 513–517. [23] Weinstein, A., Quasi-classical mechanics on spheres, Sympos. Math., 14 (1974), 25–32. [24] Weinstein, A., On Maslov’s quantization condition, in Proceedings of the International Symposium on Fourier Integral Operators (Nice, May 1974), Lecture Notes in Mathematics, Vol. 469, Springer-Verlag, New York, 1975, 341–372. [25] Guillemin, V., and Weinstein, A., Eigenvalues associated with a closed geodesic, Bull. Amer. Math. Soc., 82 (1976), 92–94. [26] Weinstein, A., Fourier integral operators, quantization, and the spectrum of a Riemannian manifold, in Géométrie Symplectique et Physique Mathématique, Colloque Internationale de Centre National de la Recherche Scientiﬁque No. 237, CNRS, Paris, 1976, 289–298. [27] Weinstein, A., The principal symbol of a distribution, Bull. Amer. Math. Soc., 82 (1976), 548–550. [28] Weinstein, A., Symplectic V -manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265–271. [29] Weinstein, A., Lectures on Symplectic Manifolds, Regional Conference Series in Mathematics, Vol. 29, American Mathematical Society, Providence, RI, 1977. [30] Weinstein, A., Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., 44 (1977), 883–892. [31] Weinstein, A., The order and symbol of a distribution, Trans. Amer. Math. Soc., 241 (1978), 1–54. [32] Weinstein, A., Simple periodic orbits, Amer. Inst. Phys. Conf. Proc., 46 (1978), 260–263. [33] Weinstein, A., Eigenvalues of the laplacian plus a potential, in Proceedings of the International Congress of Mathematicians, Helsinki, 1978, 803–805. [34] Weinstein, A., A universal phase space for particles in Yang–Mills ﬁelds, Lett. Math. Phys., 2 (1978), 417–420. [35] Weinstein, A., Periodic orbits for convex hamiltonian systems, Ann. Math., 108 (1978), 507–518.

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[36] Weinstein, A., Bifurcations and Hamilton’s principle, Math. Z., 159 (1978), 235–248. [37] Weinstein, A., On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979), 353–358. [38] Marsden, J. E., and Weinstein, A., Review of Geometric Asymptotics and Symplectic Geometry and Fourier Analysis, Bull. Amer. Math. Soc., 1 (1979), 545–553. [39] Marsden, J., and Weinstein, A., Calculus, Benjamin/Cummings, San Francisco, 1980. [40] Weinstein, A., Fat bundles and symplectic manifolds, Adv. Math., 37 (1980), 239–250. [41] Weinstein, A., Nonlinear stabilization of quasimodes, in Proceedings of the AMS Symposium on Geometry of the Laplacian, Hawaii, 1979, Proceedings of Symposia on Pure Mathematics, Vol. 36, American Mathematical Society, Providence, RI, 1980, 301–318. [42] Marsden, J., and Weinstein, A., Calculus Unlimited , Benjamin/Cummings, San Francisco, 1981. [43] Croke, C., and Weinstein, A., Closed curves on convex hypersurfaces and periods of nonlinear oscillations, Invent. Math., 64 (1981), 199–202. [44] Marsden, J., Morrison, P., and Weinstein, A., Comments on: The Maxwell–Vlasov equations as a continuous Hamiltonian system, Phys. Lett., 96A (1981), 235–236. [45] Stanton, R. J., and Weinstein A., On the L4 norm of spherical harmonics, Math. Proc. Cambridge Philos. Soc., 89 (1981), 343–358. [46] Weinstein, A., Symplectic geometry, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 1–13. [47] Weinstein, A., Neighborhood classiﬁcation of isotropic embeddings, J. Differential Geom., 16 (1981), 125–128. [48] Marsden. J., and Weinstein, A., The hamiltonian structure of the Maxwell–Vlasov equations, Physica, 4D (1982), 394–406. [49] Weinstein, A., Gauge groups and Poisson brackets for interacting particles and ﬁelds, Amer. Inst. Phys. Conf. Proc., 88 (1982), 1–11. [50] Weinstein, A., What is microlocal analysis?, Math. Intel., 4 (1982), 90–92. [51] Weinstein, A., The symplectic “category,’’ in Doebner, H.-D., Andersson, S. I., and Petry, H. R., eds., Differential Geometric Methods in Mathematical Physics (Clausthal, Germany 1980), Lecture Notes in Mathematics, Vol. 905, Springer-Verlag, Berlin, 1982, 45–50. [52] Weinstein, A., and Zelditch, S., Singularities of solutions of some Schrödinger equations on R n , Bull. Amer. Math. Soc., (1982). [53] Gotay, M. J., Lashof, R., Sniatycki, J., and Weinstein, A., Closed forms on symplectic ﬁbre bundles, Comm. Math. Helv., 58 (1983), 617–621. [54] Marsden, J., Ratiu, T., Schmid, R., Spencer, R., and Weinstein, A., Hamiltonian systems with symmetry, coadjoint orbits, and plasma physics, Atti Acad. Sci. Torino, 117Supplemento (1983), 289–340. [55] Marsden, J., and Weinstein, A., Coadjoint orbits, vortices and Clebsch variables for incompressible ﬂuids, Physica, 7D (1983), 305–323. [56] Sniatycki, J., and Weinstein, A., Reduction and quantization for singular momentum mappings, Lett. Math. Phys., 7 (1983), 159–161. [57] Weinstein, A., A symplectic rigidity theorem, Duke Math. J., 50 (1983), 1121–1125. [58] Weinstein, A., Hamiltonian structure for drift waves and geostrophic ﬂow, Phys. Fluids, 26, (1983), 388–390. [59] Weinstein, A., Sophus Lie and symplectic geometry, Expos. Math., 1 (1983), 95–96. [60] Weinstein, A., Removing intersections of lagrangian immersions, Illinois J. Math., 27 (1983), 484–500. [61] Weinstein, A., The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523–557.

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[62] Marsden, J., Morrison, P., and Weinstein, A., The Hamiltonian structure of the BBGKY hierarchy equations, Contemp. Math., 28 (1984), 115–124. [63] Marsden, J., Ratiu, T., and Weinstein, A., Reduction and Hamiltonian structures on duals of semidirect product Lie algebras, Contemp. Math., 28 (1984), 55–100. [64] Marsden. J., Ratiu, T., and Weinstein, A., Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147–177. [65] Weinstein, A., Equations of plasma physics (notes by Stephen Omohundro), in Chern, S. S., ed., Seminar on Nonlinear Partial Differential Equations, MSRI Publications, Springer-Verlag, New York, 1984, 359–373. [66] Weinstein, A., Stability of Poisson-Hamiltonian equilibria, Contemp. Math., 28 (1984), 3–13. [67] Weinstein, A., C 0 perturbation theorems for symplectic ﬁxed points and lagrangian intersections, Travaux en Cours, 3 (1984), 140–144. [68] Fortune, B., and Weinstein, A., A symplectic ﬁxed point theorem for complex projective spaces, Bull. Amer. Math. Soc., 12 (1985), 128–130. [69] Holm. D. D., Marsden, J. E., Ratiu, T., and Weinstein, A., Nonlinear stability of ﬂuid and plasma equilibria, Phys. Rep., 123-1–2 (1985), 1–116. [70] Marsden, J., and Weinstein, A., Calculus I, II, III, 2nd ed., Springer-Verlag, New York, 1985. [71] Weinstein, A., A symbol calculus for some Schrödinger equations on R n , Amer. J. Math., (1985), 1–21. [72] Weinstein, A., A global invertibility theorem for manifolds with boundary, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1985), 283–284. [73] Weinstein, A., Periodic nonlinear waves on a half-line, Comm. Math. Phys., 99 (1985), 385–388. [74] Weinstein, A., Poisson structures and Lie algebras, Astérisque, hors série (1985), 421– 434. [75] Weinstein, A., Symplectic reduction and ﬁxed points, in Séminaire Sud-Rhodanien de Géométrie, Rencontre de Balaruc I, Travaux en Cours, Hermann, Paris, 1985, 140–148. [76] Weinstein, A., Three dimensional contact manifolds with vanishing torsion tensor (appendix to a paper by S. S. Chern and R. Hamilton), in Hirzebruch, F., Schwermer J., and Suter S., eds., Proceedings of the Meeting held by the Max-Planck-Institut für Mathematik, Bonn. June 15–22, 1984, Lecture Notes in Mathematics, Vol. 1111, SpringerVerlag, Berlin, 1985, 306–308. [77] Floer, A., and Weinstein A., Nonspreading wave packets for the nonlinear Schrödinger equation with a bounded potential, J. Functional Anal., 69 (1986), 397–408. [78] Weinstein, A., On extending the Conley-Zehnder theorem to other manifolds, Proc. Sympos. Pure Math., 45 (1986), 541–544. [79] Weinstein, A., Critical point theory, symplectic geometry, and hamiltonian systems, in Proceedings of the 1983 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 1986, 261–289. [80] Coste, A., Dazord, P., and Weinstein, A., Groupoïdes symplectiques, Publ. Dép. Math. Univ. Claude Bernard-Lyon I, 2A (1987), 1–62. [81] Weinstein, A., ed., Some Problems in Symplectic Geometry, Séminaire Sud-Rhodanien de Géométrie VI, Travaux en Cours, Hermann, Paris, 1987. [82] Weinstein, A., Standing and travelling waves for nonlinear wave equations, Transport Theory Stat. Phys., 16 (1987), 267–277. [83] Weinstein, A., The Geometry of Poisson Brackets (Notes by K. Ono and K. Sugiyama), Surveys in Geometry, Tokyo, 1987.

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[84] Weinstein, A., Poisson geometry of the principal series and nonlinearizable structures, J. Differential Geom., 25 (1987), 55–73. [85] Weinstein, A., Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16, (1987), 101–104. [86] Courant, T. J., and Weinstein, A., Beyond Poisson structures, in Séminaire SudRhodanien de Géométrie VIII, Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39–49. [87] Mikami, K., and Weinstein, A., Moments and reduction for symplectic groupoid actions, Publ. RIMS Kyoto Univ., 24 (1988), 121–140. [88] Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705–727. [89] Weinstein, A., Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo. Sect. 1A Math., 36 (1988), 163–167. [90] Lu, J.-H., and Weinstein, A., Groupoïdes symplectiques doubles des groupes de LiePoisson, C. R. Acad. Sci. Paris, 309 (1989), 951–954. [91] Weinstein, A., Cohomology of symplectomorphism groups and critical values of hamiltonians, Math. Z., 201 (1989), 75–82. [92] Weinstein, A., Blowing up realizations of Heisenberg–Poisson manifolds, Bull. Sci. Math., 113 (1989), 381–406. [93] Lu, J.-H., and Weinstein, A., Poisson Lie groups, dressing transformations, and the Bruhat decomposition, J. Differential Geom., 31 (1990), 501–526. [94] Weinstein, A., Connections of Berry and Hannay type for moving lagrangian submanifolds, Adv. Math., 82 (1990), 133–159. [95] Weinstein, A., Afﬁne Poisson structures, Internat. J. Math., 1 (1990), 343–360. [96] Dazord, P., Lu, J.-H., Sondaz, D., and Weinstein, A., Afﬁnoïdes de Poisson, C. R. Acad. Sci. Paris, 312 (1991), 523–527. [97] Hofer, H., Weinstein, A., and Zehnder, E., Andreas Floer, 1956–1991 (obituary), Notices Amer. Math. Soc., 38 (1991), 910–911. [98] Lu, J.-H., and Weinstein, A., Classiﬁcation of SU (2)-covariant Poisson structures on S 2 (appendix to a paper of A. J.-L. Sheu), Comm. Math. Phys., 135 (1991), 229–232. [99] Weinstein, A., Symplectic groupoids, geometric quantization, and irrational rotation algebras, in Symplectic Geometry, Groupoids, and Integrable Systems: Séminaire SudRhodanien de Géométrie à Berkeley (1989), Dazord, P., and Weinstein, A., eds., Springer–MSRI Series, Springer-Verlag, New York, 1991, 281–290. [100] Weinstein, A., Contact surgery and symplectic handlebodies, Hokkaido Math. J., 20 (1991), 241–251. [101] Weinstein, A., Noncommutative geometry and geometric quantization, in Donato, P., Duval, C., Elhadad, J., and Tuynman, G. M., eds., Symplectic Geometry and Mathematical Physics: Actes du Colloque en l’Honneur de Jean-Marie Souriau, Progress in Mathematics, Birkhäuser, Basel, 1991, 446–461. [102] Weinstein, A., and Xu, P., Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., 417 (1991), 159–189. [103] Ginzburg, V. L., and Weinstein, A., Lie–Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc., 5 (1992), 445–453. [104] Weinstein, A., and Xu, P., Classical solutions of the quantum Yang–Baxter equation, Comm. Math. Phys., 148 (1992), 309–343. [105] Mardsen, J. E., Tromba,A. J., and Weinstein,A., Basic Multivariable Calculus, SpringerVerlag and W. H. Freeman, New York, 1993. [106] Weinstein, A., Traces and triangles in symmetric symplectic spaces, Contemp. Math., 179 (1994), 261–270.

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[107] Maeda, Y., Omori, H., and Weinstein, A., eds., Symplectic Geometry and Quantization: Two Symposia on Symplectic Geometry and Quantization Problems, July 1993, Japan, Contemporary Mathematics, Vol. 179, American Mathematical Society, Providence, RI, 1994. [108] Weinstein, A., Classical theta functions and quantum tori, Publ. RIMS Kyoto Univ., 30 (1994), 327–333. [109] Birnir, B., McKean, H., and Weinstein, A., The rigidity of sine-Gordon breathers, Comm. Pure Appl. Math., 47 (1994), 1043–1051. [110] Scovel, C., and WeinsteinA., Finite dimensional Lie–Poisson approximations to Vlasov– Poisson equations, Comm. Pure Appl. Math., 47 (1994), 683–709. [111] Emmrich, C., and Weinstein, A., The differential geometry of Fedosov’s quantization, in Brylinski, J. L., Brylinski, R., Guillemin, V., and Kac, V., eds., Lie Theory and Geometry: In Honor of B. Kostant, Progress in Mathematics, Birkhäuser, Boston, 1994, 217–239. [112] Weinstein, A., Deformation quantization, Astérisque, 227 (1995) (Séminaire Bourbaki, 46ème année, 1993–94, no. 789), 389–409. [113] Bates, S., and Weinstein, A., Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, American Mathematical Society, Providence, RI, 1997. [114] Weinstein, A., The symplectic structure on moduli space, in Hofer, H., Taubes, C., Weinstein, A., and Zehnder, E., eds., The Floer Memorial Volume, Birkhäuser, Basel, 1995, 627–635. [115] Weinstein, A., Lagrangian mechanics and groupoids, in Shadwick, W. F., Krishnaprasad, P. S., and Ratiu, T. S., eds., Mechanics Day, Fields Institute Communications, Vol. 7., American Mathematical Society, Providence, RI, 1995, 207–231. [116] Emmrich, C., and Weinstein, A., Geometry of the transport equation in multicomponent WKB approximations, Comm. Math. Phys., 176 (1996), 701–711. [117] Weinstein, A., Groupoids: Unifying internal and external symmetry, Notices Amer. Math. Soc., 43 (1996), 744–752; reprinted in Contemp. Math., 282 (2001), 1–19. [118] Reshetikhin, N., Voronov, A. A., and Weinstein, A., Semiquantum geometry, Algebraic geometry 5, J. Math. Sci., 82 (1996), 3255–3267. [119] Guruprasad, K., Huebschmann, J., Jeffrey, L., and Weinstein, A., Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377– 412. [120] Weinstein, A., Tangential deformation quantization and polarized symplectic groupoids, in Gutt, S., Rawnsley, J., and Sternheimer, D., eds., Deformation Theory and Symplectic Geometry, Mathematical Physics Studies, Vol. 20, Kluwer, Dordrecht, the Netherlands, 1997, 301–314. [121] Liu, Z.-J., Weinstein, A., and Xu, P., Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547–574. [122] Weinstein, A., The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379–394. [123] Weinstein, A., Some questions about the index of quantized contact transformations, RIMS Kôkyûroku, 1014 (1997), 1–14. [124] Weinstein, A., and Xu, P., Hochschild cohomology and characteristic classes for starproducts, in Khovanskii, A., Varchenko, A., and Vassiliev, V., eds., Geometry of Differential Equations, American Mathematical Society, Providence, RI, 1997, 177–194. [125] Liu, Z.-J., Weinstein, A., and Xu, P., Dirac structures and Poisson homogeneous spaces, Comm. Math. Phys., 192 (1998), 121–144. [126] Weinstein, A., Poisson geometry, Differential Geom. Appl., 9 (1998), 213–238. [127] Roytenberg, D., and Weinstein, A., Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys., 46 (1998), 81–93.

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[128] Weinstein, A., From Riemann Geometry to Poisson Geometry and Back Again, Lecture at Chern Symposium, Mathematical Sciences Research Institute, Berkeley, CA, 1998; available from http://msri.org/publications/video/contents.html and on CD-ROM. [129] Cannas da Silva, A., and Weinstein, A., Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, American Mathematical Society, Providence, RI, 1999. [130] Nistor, V., Weinstein, A., and Xu., P., Pseudodifferential operators on differential groupoids, Paciﬁc J. Math. 189 (1999), 117–152. [131] Evens, S., Lu, J.-H., and Weinstein, A., Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quart. J. Math., 50 (1999), 417–436. [132] Fuchs, D., Eliashberg, Y., Ratiu, T., and Weinstein, A., eds., Northern California Symplectic Geometry Seminar, American Mathematical Society, Providence, RI, 1999. [133] Weinstein, A., Almost invariant submanifolds for compact group actions, J. European Math. Soc., 2 (2000), 53–86. [134] Mikami, K., and Weinstein, A., Self-similarity of Poisson structures on tori, in Poisson Geometry, Banach Center Publications, Vol. 51, Polish Scientiﬁc Publishers PWN, Warsaw, 2000, 211–217. [135] Weinstein, A., Linearization problems for Lie algebroids and Lie groupoids, Lett. Math. Phys., 52 (2000), 93–102. [136] Weinstein, A., Omni-Lie algebras, RIMS Kôkyûroku, 1176 (2000), 95–102. [137] Weinstein, A., Review of Riemannian Geometry During the Second Half of the Twentieth Century by Marcel Berger, Bull. London Math. Soc., 33 (2001), 11. [138] Kinyon, M. K., and Weinstein, A., Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math., 123 (2001), 525–550. [139] Weinstein, A., Poisson geometry of discrete series orbits, and momentum convexity for noncompact group actions, Lett. Math. Phys., 56 (2001), 17–30. [140] Marsden, J., and Weinstein, A., Some comments on the history, theory, and applications of symplectic reduction, in Landsman, N. P., Pﬂaum, M., and Schlichenmaier, M., eds., Quantization of Singular Symplectic Quotients, Birkhäuser, Basel, 2001, 1–19. [141] Hirsch, M. W., and Weinstein, A., Fixed points of analytic actions of supersoluble Lie groups on compact surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 1783–1787. [142] Ševera, P., and Weinstein, A., Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl. Ser., 144 (2002), 145–154. [143] Weinstein, A., Linearization of regular proper groupoids, J. Inst. Math. Jussieu, 1 (2002), 493–511. [144] Newton, P. K., Holmes, P., and Weinstein,A., eds., Geometry, Mechanics, and Dynamics: Special Volume in Honor of the 60th Birthday of J. E. Marsden, Springer-Verlag, New York, 2002. [145] Bursztyn, H., and Weinstein, A., Picard groups in Poisson geometry, Moscow Math. J., 4 (2004), 39–66. [146] Weinstein, A., The geometry of momentum, in Proceedings of the Conference on “Geometry in the 20th Century: 1930–2000,’’ (Paris, September 2001), to appear; math.SG/0208108. [147] Bursztyn, H., Crainic, M., Weinstein, A., and Zhu, C., Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549–607. [148] Tang, X., and Weinstein, A., Quantization and Morita equivalence for constant Dirac structures on tori, Ann. Inst. Fourier, 54 (2004), to appear; math.QA/0305413. [149] Weinstein, A., The Maslov gerbe, Lett. Math. Phys., to appear; math.SG/0312274.

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[150] Bursztyn, H., and Weinstein, A., Poisson geometry and Morita equivalence, in Poisson Geometry, Deformation Quantization, and Group Representations, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, to appear; preprint math.SG/0402347. [151] Weinstein, A., Integrating the nonintegrable, in Proceedings of the Workshop “Feuilletages: Quantiﬁcation Géométrique,’’ Maison des Science de l’Homme, Paris, 2004.

Dirac structures, momentum maps, and quasi-Poisson manifolds Henrique Bursztyn1 and Marius Crainic2 1 Department of Mathematics

University of Toronto Toronto, ON M5S 3G3 Canada [email protected] 2 Department of Mathematics Utrecht University P. O. Box 80.010, 3508 TA Utrecht The Netherlands [email protected] Dedicated to Alan Weinstein for his 60th birthday. Abstract. We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and Hamiltonian actions, to the realm of Dirac geometry.As an example, we show how Hamiltonian quasi-Poisson manifolds ﬁt into this framework by constructing an “inversion’’ procedure relating quasi-Poisson bivectors to twisted Dirac structures.

1 Introduction This paper builds on three ideas pursued by Alan Weinstein in some of his many fundamental contributions to Poisson geometry: First, Lie algebroids play a prominent role in the study of Poisson manifolds [8, 30]; second, Poisson maps can be regarded as generalized momentum maps for actions of symplectic groupoids [25, 31]; third, Poisson structures on manifolds are particular examples of more general objects called Dirac structures [12, 13, 28]. The main objective of this paper is to combine these three ideas in order to extend the notion of “momentum map’’ to the realm of Dirac geometry. As an application, we obtain an alternative approach to Hamiltonian quasi-Poisson manifolds [2] which answers many of the questions posed in [28, 31], shedding light on the relationship between various notions of generalized Poisson structures, Hamiltonian actions and reduced spaces. Let g be a Lie algebra, and consider its dual g∗ , equipped with its Lie–Poisson structure. The central ingredients in the formulation of classical Hamiltonian g-actions

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H. Bursztyn and M. Crainic

are a Poisson manifold (Q, πQ ) and a Poisson map J : Q → g∗ , which we use to deﬁne an action of g on Q by Hamiltonian vector ﬁelds: g −→ X (Q),

v → XJv := idJv (πQ ),

(1.1)

where Jv ∈ C ∞ (Q) is given by Jv (x) = J (x), v. For the global picture, we assume that J is a complete Poisson map [8, Sec. 6.2], in which case the inﬁnitesimal action (1.1) can be integrated to an action of the connected, simply connected Lie group G with Lie algebra g, in such a way that J becomes G-equivariant with respect to the coadjoint action of G on g∗ . The map J is called a momentum map for the G-action on Q, and we refer to the G-action as Hamiltonian. A key observation, described in [25, 31], is that this construction of a Hamiltonian action out of a Poisson map holds in much more generality: one may replace g∗ by any Poisson manifold, as long as Lie groups are replaced by symplectic groupoids [29]. In this sense, any Poisson map can be seen as a “Poisson-manifold valued moment map.’’ In this paper, we show that the correspondence between Poisson maps and Hamiltonian actions by symplectic groupoids can be further extended to the context of Dirac geometry: in this setting, Poisson maps must be replaced by special types of Dirac maps called Dirac realizations (see Deﬁnition 3.11); for the associated global actions, twisted presymplectic groupoids [6] (alternatively called quasi-symplectic groupoids [33]) play the role of symplectic groupoids. Our main results show that various important notions of generalized Hamiltonian actions, such as the “quasi’’ objects of [2, 3], ﬁt nicely into the Dirac geometry framework. We organize our results as follows. In Section 2, we discuss important connections between Lie algebroids and bivector ﬁelds. Our main result is that, just as ordinary Poisson structures give rise to Lie algebroid structures on their cotangent bundles, a quasi-Poisson manifold [2, Def. 2.1] (M, π ) deﬁnes a Lie algebroid structure on T ∗ M ⊕ g, where g is the Lie algebra of the Lie group acting on M. The leaves of this Lie algebroid coincide with the leaves of the “quasi-Hamiltonian foliation’’ of [2, Sec. 9] in the Hamiltonian case, though, in our framework, we make no assumption about the existence of group-valued moment maps. In Section 3, we study Hamiltonian actions in the context of Dirac geometry at the inﬁnitesimal level. We observe that Dirac realizations, like Poisson maps, are always associated with Lie algebroid actions. (This is, in fact, the guiding principle in our definition of Dirac realizations.) After discussing how classical notions of inﬁnitesimal Hamiltonian actions ﬁt into this framework, we prove the main result of the section: Dirac realizations of Cartan–Dirac structures on Lie groups [6, 28] are equivalent to quasi-Poisson g-manifolds carrying group-valued moment maps. This equivalence involves an “inversion’’procedure relating twisted Dirac structures and quasi-Poisson bivectors, revealing that these two objects are in a certain sense “mirror’’ to one another. The main ingredients in this discussion are the Lie algebroids of Section 2 and the bundle maps which appear in [6] as inﬁnitesimal versions of multiplicative 2forms. This result explains, in particular, the relationship between Cartan–Dirac and quasi-Poisson structures on Lie groups; on the other hand, it recovers the correspondence proved in [2, Thm. 10.3] between “nondegenerate’’ Hamiltonian quasi-Poisson

Dirac structures, momentum maps, and quasi-Poisson manifolds

3

manifolds (i.e., those for which the Lie algebroids of Section 2 are transitive) and quasi-Hamiltonian spaces [3]. In Section 4, we study moment maps in Dirac geometry from a global point of view. We show that complete Dirac realizations “integrate’’to presymplectic groupoid actions, which are natural extensions of those studied in [33]. As our main example, we show that the “integration’’ of Dirac realizations of Cartan–Dirac structures on Lie groups results in Hamiltonian quasi-Poisson G-manifolds. Finally, we show that the natural reduction procedure in the setting of Dirac geometry encompasses various classical reduction theorems [21, 24, 25] as well as their “quasi’’ counterparts [2, 3, 33]. We remark, following an observation of E. Meinrenken, that the results concerning quasi-Poisson manifolds in this paper only require the Lie algebras to be quadratic, in contrast with some of the constructions in [2], in which the positivity of the bilinear forms plays a key role. (In particular, our results hold for quasi-Poisson G-manifolds when G is a noncompact semisimple Lie group.) Most of our constructions can be carried out in the more general setting of [1], but this will be discussed in a separate paper. A work which gave initial motivation and is closely related to the present paper is that of Xu [33], in which a Morita theory of quasi-symplectic groupoids is developed in order to compare “moment map theories.’’ Our results show that twisted Dirac structures complement Xu’s picture in two ways: on the one hand, by providing the inﬁnitesimal framework for Morita equivalence; on the other hand, by leading to more general “modules’’ (i.e., Hamiltonian spaces). It is a pleasure to dedicate this paper to Alan Weinstein, whose work and insightful ideas have been an unlimited source of inspiration to us. Notation. We use the following conventions for bundle maps: if π is a bivector ﬁeld on M, then π : T ∗ M → T M, α → π(α, ·); if ω is a 2-form, then ω : T M → T ∗ M, X → ω(X, ·). The space of k-multivector ﬁelds on M is denoted by X k (M). On a Lie group G, with Lie algebra g, (·, ·)g will denote a bi-invariant nondegenerate quadratic form; we write φ G for the associated Cartan 3-form, and χG ∈ 3 g for the dual trivector. The Lie algebra g is identiﬁed with right-invariant vector ﬁelds on G.

2 Lie algebroids, bivector ﬁelds, and Poisson geometry 2.1 Lie algebroids A Lie algebroid over a manifold M is a vector bundle A → M together with a Lie algebra bracket [·, ·] on the space of sections (A), and a bundle map ρ : A → T M, called the anchor, satisfying the Leibniz identity [ξ, f ξ ] = f [ξ, ξ ] + Lρ(ξ ) (f )ξ

for ξ, ξ ∈ (A)

and

f ∈ C ∞ (M). (2.1)

Whenever there is no risk of confusion, we will write Lρ(ξ ) simply as Lξ .

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If M is a point, then a Lie algebroid over M is a Lie algebra in the usual sense. An important feature of Lie algebroids A → M is that the image of the anchor, ρ(A) ⊆ T M, deﬁnes a generalized integrable distribution, determining a singular foliation of M. The leaves of this foliation are the orbits of the Lie algebroid. The following example plays a key role in the study of Hamiltonian actions and moment maps. Example 2.1 (transformation Lie algebroids). Consider an inﬁnitesimal action of a Lie algebra g on a manifold M, given by a Lie algebra homomorphism ρ¯ : g → X (M). The transformation Lie algebroid associated with this action is the trivial vector bundle M × g, with anchor (x, v) → ρ(x, v) := ρ(v)(x) ¯ and Lie bracket on (M × g) = C ∞ (M, g) deﬁned by [u, v](x) := [u(x), v(x)]g + (ρ(u(x)) ¯ · v)(x) − (ρ(v(x)) ¯ · u)(x).

(2.2)

We often denote a transformation Lie algebroid by g M. Note that [·, ·] is uniquely determined by the condition that it coincides with [·, ·]g on constant functions and the Leibniz identity. The orbits of g M are the g-orbits on M. The remainder of this section is devoted to examples of Lie algebroids closely related to Poisson manifolds. 2.2 Bivector ﬁelds and Poisson structures If (M, π ) is a Poisson manifold, then T ∗ M carries a Lie algebroid structure with anchor π : T ∗ M → T M, β(π (α)) = π(α, β), (2.3) and bracket [α, β] = Lπ (α) (β) − Lπ (β) (α) − dπ(α, β),

(2.4)

uniquely characterized by [df, dg] = d{f, g} and the Leibniz identity (2.1). Here, as usual, {f, g} = π(df, dg) is the Poisson bracket on C ∞ (M). In this case, the orbits of T ∗ M are the symplectic leaves of M, i.e., the integral manifolds of the distribution deﬁned by the Hamiltonian vector ﬁelds Xf = π (df ). Example 2.2 (Lie–Poisson structures). Let (g, [·, ·]) be a Lie algebra, and consider g∗ equipped with the associated Lie–Poisson structure {f, g}(µ) := µ, [df (µ), dg(µ)], µ ∈ g∗ .

(2.5)

Under the identiﬁcation T ∗ g∗ ∼ = g∗ ×g, one can see that the Lie algebroid structure on ∗ ∗ T g induced by (2.5) is that of a transformation Lie algebroid g g∗ (see Example 2.1) and a direct computation reveals that the action in question is the coadjoint action.

Dirac structures, momentum maps, and quasi-Poisson manifolds

5

If π ∈ X 2 (M) is an arbitrary bivector ﬁeld, let us consider π , [·, ·], {·, ·} and Xf as deﬁned by the previous formulas, and let χπ ∈ X 3 (M) be the trivector ﬁeld deﬁned by χπ := [π, π], (2.6) i.e., χπ satisﬁes 1 χπ (df, dg, dh) = {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = {f, {g, h}} + c.p., 2 where we use c.p. to denote cyclic permutations. Lemma 2.3. For any bivector ﬁeld π on M, one has 1 π ([α, β]) = [π (α), π (β)] − iα∧β (χπ ), 2 1 [α, [β, γ ]] + c.p. = (Liα∧β (χπ ) (γ ) + c.p.) − d(χπ (α, β, γ )), 2

(2.7) (2.8)

for α, β, γ ∈ 1 (M). As a result, the following are equivalent: (i) π is a Poisson tensor; (ii) π : 1 (M) → X (M) preserves the brackets; (iii) the bracket [·, ·] on 1 (M) satisﬁes the Jacobi identity; (iv) (T ∗ M, π , [·, ·]) is a Lie algebroid. Proof. The key remark is that the difference between the left- and right-hand sides of each of (2.7) and (2.8) is C ∞ (M)-multilinear in α, β and γ . So it is enough to prove the identities on exact forms, which is immediate.

Example 2.4 (twisted Poisson manifolds). Consider a closed 3-form φ ∈ 3 (M). A φ-twisted Poisson structure on M [19, 27] consists of a bivector ﬁeld π ∈ X 2 (M) satisfying 1 [π, π] = π (φ). 2 Here, we abuse notation and write π to denote the map induced by (2.3) on exterior algebras. We know from Lemma 2.3 that the bracket (2.4) induced by π is not preserved by π and does not satisfy the Jacobi identity. However, π ([α, β] + iπ (α)∧π (β) (φ)) = [π (α), π (β)]. Hence, if we deﬁne a “twisted’’ version of the bracket (2.4), [α, β]φ := [α, β] + iπ (α)∧π (β) (φ), then π will preserve this new bracket, and [·, ·]φ satisﬁes the Jacobi identity. As a result, (T ∗ M, π , [·, ·]φ ) is a Lie algebroid. We leave it to the reader to prove a “twisted’’ version of Lemma 2.3.

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2.3 The Lie algebroid of a quasi-Poisson manifold Let G be a Lie group with Lie algebra g, equipped with a bi-invariant nondegenerate quadratic form (·, ·)g . Let φ G be the bi-invariant Cartan 3-form on G, and let χG ∈ 3 g be its dual trivector. On Lie algebra elements u, v, w ∈ g, we have φ G (u, v, w) = χG (u∨ , v ∨ , w∨ ) =

1 (u, [v, w])g , 2

where u∨ , v ∨ , w∨ are dual to u, v, w via (·, ·)g ; when (·, ·)g is a metric and ea is an orthonormal basis of g, we can write1 χG =

1 (ea , [eb , ec ])g ea ∧ eb ∧ ec . 12

A quasi-Poisson G-manifold [2] consists of a G-manifold M together with a Ginvariant bivector ﬁeld π satisfying χπ = ρM (χG ),

(2.9)

where ρM : g −→ X (M) is the associated inﬁnitesimal action, and we keep the same notation for the induced maps of exterior algebras. When M is just a g-manifold, we call the corresponding object a quasi-Poisson g-manifold. The two notions are related by the standard procedure of integration of inﬁnitesimal actions; in particular, they coincide if M is compact and G is simply connected. In analogy with ordinary or twisted Poisson manifolds, are quasi-Poisson structures also associated with Lie algebroids? As we now discuss, the answer is yes. Let us consider a more general setup: let M be a g-manifold and let π ∈ X 2 (M) be an arbitrary bivector ﬁeld. Motivated by [2, Sec. 9], we consider on T ∗ M ⊕ g the “anchor’’ map r : T ∗ M ⊕ g −→ T M, r(α, v) = π (α) + ρM (v),

(2.10)

combining the bivector ﬁeld and the action. On sections of T ∗ M ⊕ g, we consider the bracket deﬁned by 1 [(α, 0), (β, 0)] = [α, β], iρM∗ (α∧β) (χG ) , (2.11) 2 [(0, v), (0, v )] = (0, [v, v ]), (2.12) [(0, v), (α, 0)] = (LρM (v) (α), 0), (2.13) for all 1-forms α, β ∈ 1 (M) and all v, v ∈ g (thought of as constant sections in C ∞ (M, g)). As in Example 2.1, the deﬁnition of the bracket on general elements in (T ∗ M ⊕ g) = 1 (M) ⊕ C ∞ (M, g) is obtained from the Leibniz formula (2.1). With these deﬁnitions, we obtain a quasi-Poisson analogue of Lemma 2.3. 1 More generally, with no positivity assumptions on (·, ·) , we can write χ g G = 1 (e , [e , e ]) f ∧ f ∧ f , where f is a basis of g satisfying (f , e ) = δ . a c g a c a a g b b b ab 12

A similar observation holds for (2.25).

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7

Theorem 2.5. Let M be a g-manifold equipped with a bivector ﬁeld π . The following are equivalent: (i) (M, π ) is a quasi-Poisson g-manifold; (ii) r : 1 (M) ⊕ C ∞ (M, g) → X (M) preserves brackets; (iii) the bracket [·, ·] on 1 (M) ⊕ C ∞ (M, g) satisﬁes the Jacobi identity; (iv) (T ∗ M ⊕ g, r, [·, ·]) is a Lie algebroid. Proof. Note that r preserves the bracket (2.12), since ρM is an action. From the identity (2.7) in Lemma 2.3, it follows that r preserves the bracket of type (2.11) if and only if χπ = ρM (χG ). On the other hand, r preserves the bracket of type (2.13) if and only if π LρM (v) (ξ ) = LρM (v) π (ξ ), which is equivalent to the g-invariance of π. This shows that (i) and (ii) are equivalent. Let us prove that (i) implies (iii); from the proof, the converse will be clear. Assuming (i), we must show that [·, ·] on 1 (M) ⊕ C ∞ (M, g) satisﬁes the Jacobi identity. On elements of type (0, v), this reduces to the Jacobi identity for g (or, alternatively, for g M). On elements (0, v), (0, w) and (α, 0), the Jacobi identity of [·, ·] reduces to the fact that ρM is an action. Computing the “jacobiator’’ for elements of type (0, v), (α, 0), (β, 0), we see that the ﬁrst component is [LρM (v) (α), β] + [α, LρM (v) (β)] − LρM (v) ([α, β]).

(2.14)

Using the Leibniz identity, we see that the C ∞ (M)-linearity of (2.14) with respect to β is equivalent to π Lρ(v) (β) = Lρ(v) π (β), i.e., to the g-invariance of π . Hence, if π is invariant, (2.14) is C ∞ (M)-linear on α and β, and then one can check that it is zero by looking at the particular case when α and β are exact. The second component of the jacobiator of (0, v), (α, 0), (β, 0) can be computed similarly. To complete the proof that (i) implies (iii), we must deal with the Jacobi identity for elements of type (α, 0), (β, 0), (γ , 0). To this end, we ﬁrst need to ﬁnd the expression for the bracket between elements of type (0, v) ˜ and (α, 0), with v˜ ∈ C ∞ (M, g) not 1 necessarily constant: pairing d v˜ ∈ (M; g) with an element µ ∈ C ∞ (M, g∗ ) gives us a 1-form on M, denoted by Av˜ (µ), satisfying the following two properties: Af v˜ (µ) = f Av˜ (µ) + µ(v)df, ˜ and

Av˜ (f µ) = f Av˜ (µ),

for f ∈ C ∞ (M). We claim that ∗ [(0, v), ˜ (α, 0)] = (LρM (v) ˜ ˜ (α) − Av˜ (ρM (α)), −Lπ (α) (v)).

(2.15)

To see this, note that (2.15) holds when v˜ is constant, and the difference between the left- and right-hand sides is C ∞ (M)-linear in v. ˜ We remark that Aiµ∧µ (χG ) (µ ) + c.p. = 2d(χG (µ, µ , µ )).

(2.16)

Again, it is easy to check this identity when µ, µ and µ are constant, so (2.16) follows from C ∞ (M)-linearity. Also, denoting χM := ρM (χG ), a direct computation shows that

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ρM (iρM∗ (α∧β) (χG )) = iα∧β (χM ). We now turn to the computation of the jacobiator of the elements (α, 0), (β, 0) and (γ , 0), that we denote by Jac(α, β, γ ). For the ﬁrst component of Jac(α, β, γ ), we obtain 1 1 ∗ (γ ))+c.p.). (2.17) ([α, [β, γ ]]+c.p.)− (Liα∧β (χM ) (γ )+c.p.)+ (Aiα∧β (χM ) (ρM 2 2 Combining the second identity of Lemma 2.3 with (2.16), we get that (2.17) equals d((ρM (χG ) − χπ )(α, β, γ )), which vanishes by the condition χπ = ρM (χG ). So we are left with proving that the second component of Jac(α, β, γ ) vanishes, which amounts to showing that iρM∗ ([α,β]∧γ ) (χG ) + c.p. = Lπ (γ ) iρM∗ (α∧β) (χG ) + c.p..

(2.18)

In order to do that, consider the operators iρM∗ ([α,β]) and Lπ (α) iρM∗ (β) − Lπ (β) iρM∗ (α) acting on C ∞ (M, g), for α, β ∈ 1 (M). Claim 2.6. On g, seen as constant functions in C ∞ (M, g), we have iρM∗ ([α,β]) = Lπ (α) iρM∗ (β) − Lπ (β) iρM∗ (α) .

(2.19)

Proof. Both operators are derivations of degree −1 on g, hence it sufﬁces to show (2.19) for elements v ∈ g. As we now check, this follows from the deﬁnition of the bracket induced by π and the invariance of π: on the one hand, iρM∗ ([α,β]) (v) = [α, β](ρM (v)) = iρM (v) Lπ (α) (β) − iρM (v) Lπ (β) (α) − iρM (v) dπ(α, β).

(2.20)

Using that i[X,Y ] = LX iY − iY LX for vector ﬁelds X, Y , we have iρM (v) Lπ (α) (β) = Lπ (α) (β(ρM (v))) − β([π (α), ρM (v)]) = Lπ (α) iρM∗ (β) (v) − π(LρM (v) (α), β),

(2.21)

where the last equality follows from the g-invariance of π . Using the identity (2.21) (and its analogue for α and β interchanged) in (2.20), (2.19) follows.

Using the claim, we see that iρM∗ ([α,β]∧γ ) + c.p. = iρM∗ (γ ) iρM∗ ([α,β]) + c.p. = (iρM∗ (γ ) Lπ (α) iρM∗ (β) − iρM∗ (γ ) Lπ (β) iρM∗ (α) ) + c.p., (2.22) when restricted to constant elements in C ∞ (M, g) . On the other hand, it follows from (2.19) that, on g, we can write iρM∗ ([α,β]) = [Lπ (α) , iρM∗ (β) ] − [Lπ (β) , iρM∗ (α) ]

(2.23)

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9

since the Lie derivatives are zero on constant functions. But both sides of (2.23) are C ∞ (M)-linear, so this equality is valid for all C ∞ (M, g). So we can write iρM∗ ([α,β]∧γ ) + c.p. = −iρM∗ ([α,β]) iρM∗ (γ ) + c.p. = −([Lπ (α) , iρM∗ (β) ] − [Lπ (β) , iρM∗ (α) ])iρM∗ (γ ) + c.p., from which we deduce that iρM∗ ([α,β]∧γ ) + c.p. = 2(Lπ (α) iρM∗ (β∧γ ) + c.p.) − (iρM∗ (γ ) Lπ (α) iρM∗ (β) − iρM∗ (β) Lπ (α) iρM∗ (γ ) + c.p.). On constant functions, we can use (2.22) to conclude that (iρM∗ ([α,β]∧γ ) + c.p.) = 2(Lπ (α) iρM∗ (β∧γ ) + c.p.) − (iρM∗ ([α,β]∧γ ) + c.p.), i.e., iρM∗ ([α,β]∧γ ) + c.p. = Lπ (α) iρM∗ (β∧γ ) . Evaluating this identity at χG proves (2.18), showing that (i) implies (iii). Looking back at the proof, one can check that the same formulas show the converse, so that (i) and (iii) are equivalent. Since (iii) and the Leibniz identity for [·, ·] are together equivalent to (iv), it follows that (i)–(iv) are equivalent to each other.

Corollary 2.7. If (M, π ) is a quasi-Poisson g-manifold, then the generalized distribution π (α) + ρM (v) ⊆ T M, for α ∈ T ∗ M, v ∈ g, is integrable. This result shows that the singular distribution discussed in [2, Thm. 9.2] in the context of Hamiltonian quasi-Poisson manifolds is integrable even without the presence of a moment map (and without the positivity of (·, ·)g ). As in the case of ordinary Poisson manifolds, we call a quasi-Poisson manifold nondegenerate if its associated Lie algebroid is transitive (i.e., its anchor map is onto). Example 2.8 (quasi-Poisson structures on Lie groups). Let G be a Lie group with Lie algebra g, which we assume to be equipped with an invariant nondegenerate quadratic form (·, ·)g . We consider G acting on itself by conjugation. As shown in [2, Sec. 3], the bivector ﬁeld πG , deﬁned on left invariant 1-forms by πG (dlg∗−1 (µ), dlg∗−1 (ν)) :=

1 ((Ad g −1 − Ad g )(µ∨ ), ν ∨ )g , 2

(2.24)

where lg denotes left multiplication by g ∈ G, µ, ν ∈ g∗ , and µ∨ is the element in g dual to µ via (·, ·)g , makes G into a quasi-Poisson G-manifold. If (·, ·)g is a metric, then we can write 1 l πG = (2.25) ea ∧ ear , 2 where ea is an orthonormal basis of g and ear (respectively, eal ) are the corresponding right (respectively, left) translations.

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In this example, the image of πG is tangent to the G-orbits, so the leaves of the corresponding foliation are the conjugacy classes. The formula for the Lie algebroid bracket on T ∗ G ⊕ g bears close resemblance with the one for the bracket in the “double’’ of the Lie quasi-bialgebra of G, as in [4]. We will discuss this connection in a separate work.

3 Moment maps in Dirac geometry: The inﬁnitesimal picture 3.1 Dirac manifolds Let φ be a closed 3-form on a manifold M. A φ-twisted Dirac structure on M [28] is a subbundle L ⊂ E = T M ⊕ T ∗ M satisfying the following two conditions: 1. L is maximal isotropic with respect to the symmetric pairing ·, ·+ : (E) × (E) → C ∞ (M), (X, α), (Y, β)+ := β(X) + α(Y );

(3.1)

2. The space of sections (L) is closed under the bracket [[·, ·]]φ : (E) × (E) → (E), [[(X, α), (Y, β)]]φ := ([X, Y ], LX β − iY dα + iX∧Y φ). (3.2) Since the pairing (3.1) has zero signature, condition 1 is equivalent to requiring that L has rank equal to dim(M) and that ·, ·+ |L = 0. The bracket (3.2) is the φ-twisted Courant bracket considered in [28]. When φ = 0, this bracket is a nonskew-symmetric version of Courant’s original bracket introduced in [12]. Twisted Dirac structures are always associated with Lie algebroids. Indeed, the restriction of the Courant bracket [[·, ·]]φ to a Dirac subbundle L ⊂ T M ⊕ T ∗ M deﬁnes a Lie algebra bracket on the space of sections (L), making L → M into a Lie algebroid with anchor ρ = pr 1 |L : L → T M, where pr 1 is the ﬁrst projection. The orbits of this algebroid are also called the leaves of L. Example 3.1 (twisted Poisson structures). If π is a bivector ﬁeld on M, then Lπ := graph(π ) ⊂ T M ⊕ T ∗ M satisﬁes condition 1, and Lπ is a φ-twisted Dirac structure if and only if π is a φ-twisted Poisson structure in the sense of Example 2.4. In this case, the second projection pr 2 |L : L → T ∗ M establishes an isomorphism of Lie algebroids, where T ∗ M is equipped with the Lie algebroid structure described in Example 2.4. Setting φ = 0, we obtain a one-toone correspondence between ordinary Poisson structures on M and Dirac structures satisfying the extra condition L ∩ T M = {0}.

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Example 3.2 (twisted presymplectic forms). Similarly, the graph associated with a 2-form ω ∈ 2 (M), Lω = graph(ω ), is a φ-twisted Dirac structure if and only if dω + φ = 0, and we refer to ω as a φ-twisted presymplectic form. In this case, setting φ = 0, we have an identiﬁcation of closed 2-forms on M with Dirac structures satisfying L ∩ T ∗ M = {0}. In general, the leaves of a φ-twisted Dirac structure L carry twisted presymplectic forms deﬁned as follows: at each x ∈ M, we deﬁne a skew symmetric bilinear form θx on ρ(L)x = pr 1 (L)x by θx (X1 , X2 ) = α(X2 ),

(3.3)

where α is any element in Tx∗ M satisfying (X1 , α) ∈ Lx . The fact that L is maximal isotropic with respect to (3.1) guarantees that (3.3) is independent of the choice of α, and these forms ﬁt together into a smooth leafwise 2-form θ. Using that (L) is closed with respect to (3.2), one can show that on each leaf ι : O → M, the 2-form θ satisﬁes dθ + ι∗ φ = 0. At each point x ∈ M, the kernel of θ coincides with Lx ∩ Tx M, which shows that the leafwise presymplectic forms are nondegenerate if and only if L comes from a φ-twisted Poisson structure. We will denote the distribution L ∩ T M on M by ker(L). Since Dirac structures are always associated with Lie algebroids, it is natural to consider how to obtain Dirac structures from them. The following is a useful construction, see [6]: for a Lie algebroid A over M with anchor ρ : A −→ T M, we deﬁne a φ-IM form of A to be any bundle map2 σ : A −→ T ∗ M satisfying the following properties: (3.4) σ (ξ ), ρ(ξ ) = −σ (ξ ), ρ(ξ ); σ ([ξ, ξ ]) = Lξ (σ (ξ )) − Lξ (σ (ξ )) + dσ (ξ ), ρ(ξ ) + iρ(ξ )∧ρ(ξ ) (φ), (3.5) for ξ, ξ ∈ (A) (here ·, · denotes the usual pairing between a vector space and its dual). Let Lσ ⊂ T M ⊕ T ∗ M be the image of the map (ρ, σ ) : A −→ T M ⊕ T ∗ M. Then the following is immediate. Lemma 3.3. If σ is a φ-IM form of A and rank(Lσ ) = dim(M), then Lσ is a φ-twisted Dirac structure on M. Of course, any Dirac structure can be realized as the image of an IM form by taking A = L, viewed as an algebroid with ρ = pr 1 |L, and σ = pr 2 |L. The following is a key example. 2 These bundle maps are inﬁnitesimal versions of multiplicative 2-forms on groupoids; see

[6]; the terminology “IM’’ stands for “inﬁnitesimal multiplicative.’’

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Example 3.4 (Cartan–Dirac structures on Lie groups). Cartan–Dirac structures on Lie groups play a role in Dirac geometry analogous to the one played by Lie–Poisson structures (Example 2.2) in Poisson geometry. Just as Lie–Poisson structures on the dual of Lie algebras are completely determined by the Kostant–Kirillov–Souriau (KKS) symplectic forms along coadjoint orbits, Cartan–Dirac structures on Lie groups “assemble’’ certain 2-forms deﬁned on conjugacy classes deﬁned as follows. Let G be a Lie group with Lie algebra g, and let (·, ·)g be a bi-invariant nondegenerate quadratic form, which we use to identify T G and T ∗ G. For v ∈ g, let vG = vr −vl be the inﬁnitesimal generator of the action of G on itself by conjugation. On each conjugacy class, we deﬁne ι : C → G, a 2-form θ by 1 (Ad g − Ad g −1 )u, v , g ∈ C. θg (uG , vG ) := (3.6) 2 g Direct computations show that dθ − ι∗ φ G = 0, where φ G is the bi-invariant Cartan 3-form on G, and that θg is nondegenerate at a point g if and only if (Ad g + 1) is invertible. The 2-forms (3.6) appear in [17] in the study of symplectic structures of moduli spaces. Since these 2-forms are not symplectic, but twisted presymplectic, they should correspond to a −φ G -twisted Dirac structure LG on G rather than a Poisson structure. A simple computation shows that 1 vr − vl , (vr + vl ) , v ∈ g ⊂ T G ⊕ T G. (3.7) LG = 2 (Recall that we are identifying T G with T ∗ G via (·, ·)g .) We call LG the Cartan– Dirac structure on G associated with (·, ·)g [28, 6]. Note that ρ(v) = vr − vl is the anchor of the action Lie algebroid (Example 2.1) g G with respect to the action by conjugation, and the map σ : G × g −→ T G, σ (v) =

1 (vr + vl ) 2

(3.8)

satisﬁes the conditions of Lemma 3.3. So σ is a −φ G -IM form of g G, and the Cartan–Dirac structure LG arises as the image of (ρ, σ ). In this case, (ρ, σ ) actually establishes an isomorphism between g G and LG . (Note the analogy with Example 2.2, which shows that Lie algebroids of Lie–Poisson structures are isomorphic to the action Lie of algebroids for the coadjoint action!) Let us ﬁnally recall an important operation involving Dirac structures: if L is a φ-twisted Dirac structure on M and B ∈ 2 (M), then τB (L) := {(X, α + B (X)) | (X, α) ∈ L}

(3.9)

deﬁnes a (φ − dB)-twisted Dirac structure on M [28]. The operation τB is called a gauge transformation associated with B, and it has the effect of modifying L by adding the pullback of B to the presymplectic form on each leaf.

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3.2 Dirac maps Since Dirac structures generalize both Poisson and presymplectic structures, we have two possible deﬁnitions of Dirac maps; see [7]. Let (M, LM ) and (N, LN ) be twisted Dirac manifolds.Asmooth map f : N → M is a forward Dirac map, or f-Dirac in short, if LN and LM are related as follows: LM = {(df (Y ), α) | Y ∈ T N, α ∈ T ∗ M and (Y, df ∗ (α)) ∈ LN }.

(3.10)

If LM and LN are associated with twisted Poisson structures, then an f-Dirac map is equivalent to a Poisson map. The terminology “forward’’ is due to the fact that, at each point, (3.10) extends the usual notion of “push-forward’’ of a linear bivector. For this reason, we may write LM = f∗ LN instead of (3.10), in analogy with the notation for “f -related’’ bivector ﬁelds on a manifold. Similarly, f : N → M is a backward Dirac map, or simply b-Dirac, if LN = {(Y, df ∗ α) | Y ∈ T N, α ∈ T ∗ M and (df (Y ), α) ∈ LM }.

(3.11)

If LM and LN are associated with twisted presymplectic structures ωM and ωN , then a b-Dirac map is just a map satisfying f ∗ ωM = ωN . As before, we will write LN = f ∗ LM to denote that f is a b-Dirac map. Note that f ∗ LM is always a well-deﬁned, though not necessarily smooth, subbundle of T N, in contrast with f∗ LN , which may not be well deﬁned at all. In fact, f ∗ LM deﬁnes a Dirac structure on N provided it is smooth, which is the case, e.g., when f is a submersion. However, as illustrated in the next example, f ∗ LM may deﬁne a Dirac structure even when f is not a submersion. Example 3.5 (inclusion of presymplectic leaves). Let L be a twisted Dirac structure on M, and consider a presymplectic leaf O, equipped with Dirac structure Lθ associated with the twisted presymplectic form θ. Denoting by ι : O → M the inclusion map, it follows from the deﬁnition of θ that Lθ = {(X, iX θ ) | X ∈ T O} = {(X, (dι)∗ α) | (dι(X), α) ∈ L} = ι∗ L.

(3.12)

So ι : (O, Lθ ) → (M, L) is a b-Dirac map. On the other hand, at each point of M, we have ι∗ Lθ = {(dι(X), α) | (X, (dι)∗ α) ∈ Lθ }. By the second equality in (3.12), it follows that ι∗ Lθ ⊆ L, but since they have the same dimension, we get ι∗ Lθ = L, (3.13) so ι is f-Dirac as well.

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Note that the fact that the inclusion of presymplectic leaves into a Dirac manifold is an f-Dirac map is a direct generalization of the fact that the inclusion of symplectic leaves into a Poisson manifold is a Poisson map. As a simple consequence, we have the following. Corollary 3.6. Let (N, LN ) and (M, LM ) be twisted Dirac manifolds. A map J : N → M is f-Dirac if and only if its restriction to each presymplectic leaf of N is f-Dirac. We remark that Example 3.5 is very special in that the inclusion map of presymplectic leaves is both forward and backward Dirac (see also Remark 4.12). In general, f-Dirac maps need not be b-Dirac, or the other way around. 3.3 Poisson maps as inﬁnitesimal Hamiltonian actions The usual notion of Lie algebra action can be extended to the realm of Lie algebroids, the main difference being that algebroids, rather than acting on manifolds, act on maps from manifolds into their base [18]: An action of a Lie algebroid A → M on a map J : N → M consists of a Lie algebra homomorphism ρN : (A) → X (N ) satisfying dJ ◦ ρN (ξ ) = ρ(ξ ), for all ξ ∈ (A), (3.14) and such that, for f ∈ C ∞ (M) and ξ ∈ (A), ρN (f ξ ) = J ∗ fρN (ξ ) (i.e., the induced map (J ∗ A) → X (N ) comes from a vector bundle morphism J ∗ A → T N, where J ∗ A = A ×M N is the pullback of the vector bundle A by J ). Example 3.7 (actions of transformations Lie algebroids). Consider an inﬁnitesimal action ρ of g on a manifold M. Then an action ρN of the transformation Lie algebroid A = g M on a map J : N → M is equivalent to an inﬁnitesimal action ρN of g on N for which J is g-equivariant. Indeed, ρN and ρN are related by the formula ρN (v)y = ρN (v(J (y)))y , where v ∈ C ∞ (M, g), y ∈ N,

(3.15)

and the g-equivariance of J corresponds to (3.14). In Poisson geometry, Poisson maps are always associated with Lie algebroid actions: If (Q, πQ ) and (P , πP ) are Poisson manifolds, then any Poisson map J : Q → P induces a Lie algebroid action of T ∗ P on Q by

1 (P ) −→ X (Q),

α → πQ (J ∗ α).

(3.16)

When the target P is the dual of a Lie algebra, we recover a familiar example. Example 3.8 (inﬁnitesimal Hamiltonian actions). Consider g∗ equipped with its Lie– Poisson structure. As remarked in Example 2.2, the Lie algebroid structure on T ∗ g∗ induced by (2.5) is that of a transformation Lie algebroid g g∗ with respect to the coadjoint action. If J : Q → g∗ is a Poisson map, then it induces an action of T ∗ g∗ via (3.16), which, by Example 3.7, is equivalent to an ordinary g-action on Q for which J is equivariant. A simple computation shows that the g-action arising in this way is just a Hamiltonian action in the usual sense, making Q into a Hamiltonian Poisson g-manifold having J as a momentum map.

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Recall that a Poisson map J : Q → P is called a symplectic realization if Q is symplectic. The following is an immediate consequence. Proposition 3.9. There is a one-to-one correspondence between Poisson maps into g∗ and Hamiltonian Poisson g-manifolds, and this correspondence restricts to a one-to-one correspondence between symplectic realizations of g∗ and Hamiltonian symplectic g-manifolds. Remark 3.10. An analogue of Proposition 3.9 holds more generally in the context of Poisson–Lie groups [20, 22]. Let (G, π) be a simply connected Poisson–Lie group, and let G∗ be its dual. The Lie algebroid structure on T ∗ G∗ ∼ = G∗ × g induced from the dual Poisson structure is a transformation Lie algebroid, now associated with the inﬁnitesimal dressing action of g on G∗ . For a Poisson map J : Q → G∗ , the general Lie algebroid action described by (3.16) reduces to a Poisson g-action for which J is an equivariant momentum map in the sense of Lu [21]. In order to extend this discussion to Dirac geometry, let us consider LπP = graph(πP ), the associated Dirac structure on (P , πP ). Using the Lie algebroid isomorphism T ∗ P ∼ = LπP , we can rewrite the inﬁnitesimal action (3.16) as (LπP ) → X (Q), (X, α) → Y,

(3.17)

where Y ∈ X (Q) is uniquely determined by the condition (Y, J ∗ α) ∈ LπQ . Also note that Y is related to X by dJ (Y ) = X, since J is a Poisson map. The question of whether this procedure can be carried out for f-Dirac maps leads us to the notion of Dirac realization. 3.4 Dirac realizations If (N, LN ) and (M, LM ) are twisted Dirac manifolds, then, by deﬁnition, a smooth J : N → M is an f-Dirac map if and only if, given (X, α) ∈ (LM )J (y) , there exists a Y ∈ Ty N with the property that (Y, dJ ∗ α) ∈ (LN )y and X = (dJ )y (Y ).

(3.18)

It is natural to try to deﬁne an action of LM on N as in the case of Poisson maps, see (3.17), except that (3.18) does not determine Y uniquely in general. In fact, this is the case if and only if the following extra “nondegeneracy’’ condition holds: ker(dJ ) ∩ ker(LN ) = {0}.

(3.19)

A similar argument as in [6, Section 7.1] shows that (3.19) is equivalent to J : ker(LN ) → ker(LM ) being an isomorphism. Deﬁnition 3.11. A Dirac realization of a φ-twisted Dirac manifold (M, LM ) is an f-Dirac map J : (N, LN ) → (M, LM ), where LN is a J ∗ φ-twisted Dirac structure on N, satisfying (3.19).

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As a consequence of Deﬁnition 3.11, we have the following. Corollary 3.12. Let J : N → M be a Dirac realization. Then the map (LM ) → X (N), (X, α) → Y , where Y is determined by the conditions in (3.18), is a Lie algebroid action. Dirac realizations J : N → M for which N is presymplectic were studied in [6, Sec. 7.1] under the name of presymplectic realizations. As a result of Corollary 3.6, we have the following. Corollary 3.13. A map J : N → M is a Dirac realization if and only if its restriction to each presymplectic leaf of N is a presymplectic realization. Similar to Poisson geometry, the connection between Dirac realizations and “Hamiltonian actions’’ is established by a suitable choice of “target’’ M. Following the analogy between Lie–Poisson structures and Cartan–Dirac structures, it is natural to study the “Hamiltonian spaces’’ associated with Dirac realizations of Cartan– Dirac structures. The particular case of presymplectic realizations is discussed in [6, Sec. 7.2]. Example 3.14 (presymplectic realizations of Cartan–Dirac structures). Let G be a Lie group with Lie algebra g, equipped with a bi-invariant nondegenerate quadratic form (·, ·)g . Let LG be the associated Cartan–Dirac structure on G. If (M, ωM ) is a twisted presymplectic manifold, then the conditions for J : M → G being a presymplectic realization can be expressed as follows: 1. ωM is g-invariant and satisﬁes dωM = J ∗ φ G ; 2. at each x ∈ M, Ker(ωM )x = {(ρM )x (v) : v ∈ Ker(Ad J (p) + 1)}; 3. the map J satisﬁes the moment map condition ω ρM = J ∗ σ.

(3.20)

The invariance of ωM in 1 is with respect to the g-action ρM induced by J (recall that LG ∼ = g G, see Example 3.4, so an LG -action deﬁnes an ordinary g-action), for which J is equivariant; in 3, σ is the IM-form of the Cartan–Dirac structure (3.8), σ : g −→ T ∗ G,

σ (v) =

1 (vr + vl )∨ , 2

(3.21)

where v −→ v ∨ denotes the isomorphism T G −→ T ∗ G induced by the quadratic form. The “relative closedness’’ of ωM in 1 expresses that the associated Dirac structure is −J ∗ φ G -twisted, while condition 2 is the “nondegeneracy’’condition (3.19) applied to this particular case; ﬁnally, condition 3 follows from J being an f-Dirac map. Conditions 1, 2, and 3 are exactly the deﬁning axioms of a quasi-Hamiltonian gspace, in the sense of [3], for which J is the group-valued moment map. Conversely, any group-valued moment map of a quasi-Hamiltonian g-space is a presymplectic realization of (G, LG ).

Dirac structures, momentum maps, and quasi-Poisson manifolds

17

We summarize Example 3.14 in the next result, analogous to Proposition 3.9; see [6, Thm. 7.6]. Theorem 3.15. There is a one-to-one correspondence between presymplectic realizations of G endowed with the Cartan–Dirac structure, and quasi-Hamiltonian gmanifolds. Combining Corollary 3.13 with Theorem 3.15, we conclude that general Dirac realizations of Cartan–Dirac structures must be “foliated’’ by quasi-Hamiltonian gmanifolds. Since Hamiltonian quasi-Poisson manifolds, in the sense of [2], also have this property [2, Sec. 10], we are led to investigate the relationship between these objects. 3.5 Dirac realizations and Hamiltonian quasi-Poisson g-manifolds 3.5.1 The equivalence theorem For a quasi-Poisson g-manifold (M, π), a momentum map is a g-equivariant map J : M −→ G (with respect to the inﬁnitesimal action by conjugation on G) satisfying the condition [2, Lem. 2.3] π J ∗ = ρM σ ∨ , (3.22) where

1 (dr −1 (ξg∨ ) + dlg −1 (ξg∨ )) (3.23) 2 g is the adjoint of σ (3.21) with respect to the form (·, ·)g , and ρM : g → T M is the gaction. Here lg and rg denote the left and right translations by g, respectively, and, as in Example 3.14, the symbol ∨ on elements of T ∗ G is used to denote the corresponding element in T G via the identiﬁcation induced by (·, ·)g (and vice versa). The following is our main result in this section. σ ∨ : T ∗ G −→ g,

σ ∨ (ξg ) =

Theorem 3.16. There is a one-to-one correspondence between Dirac realizations of G, endowed with the Cartan–Dirac structure, and Hamiltonian quasi-Poisson gmanifolds. Before proving Theorem 3.16, let us collect some useful formulas relating the maps σ , σ ∨ , ρ : g → T G, ρ(v) = vr − vl , and, for symmetry, the dual of ρ with respect to (·, ·)g , ρ ∨ : T G −→ g,

ρ ∨ (Vg ) = drg −1 (Vg ) − dlg −1 (Vg ).

(3.24)

The following lemma follows from a straightforward computation. Lemma 3.17. The following formulas hold true: 4σ ∨ σ + ρ ∨ ρ = 4Id g , 4σ σ ∨ + (ρρ ∨ )∗ = 4Id T ∗ G ,

(3.25) (3.26)

18

H. Bursztyn and M. Crainic

σ ∗ ρ = −ρ ∗ σ, σρ ∨ = −(ρ ∨ )∗ σ ∗ , ∨

∨ ∗

∨

∨ ∗

ρ (σ ) = −σ (ρ ) , ρσ ∨ = −(ρσ ∨ ).∗

(3.27) (3.28) (3.29) (3.30)

Motivated by the equivalence between quasi-Hamiltonian manifolds and nondegenerate Hamiltonian quasi-Poisson manifolds [2, Thm. 10.3], which will also follow from Theorem 3.16, it is natural to combine the two moment map conditions (3.20) and (3.22). By applying π to (3.20) and using (3.22), we obtain, in particular, the relation ρM σ ∨ σ = π ω ρM . This suggests the importance of writing ρM σ ∨ σ as the composition of some operator C : T M → T M with ρM in general. Using (3.25) and the equivariance of J , written as ρ = dJ ◦ ρM , it is easy to ﬁnd an expression for C (which already appears in [2, Lem. 10.2]). Lemma 3.18. For any manifold M equipped with an inﬁnitesimal action ρM : g −→ T M, and any g-equivariant map J : M −→ G, the operator 1 C = 1 − ρM ρ ∨ (dJ ) : T M −→ T M 4 and its dual C ∗ : T ∗ M −→ T ∗ M satisfy the formulas ρM σ ∨ σ = CρM , and J ∗ σ σ ∨ = C ∗ J ∗ .

(3.31)

Theorem 3.16 follows from the next two propositions, each one describing explicitly one direction of the asserted one-to-one correspondence. Proposition 3.19. Let M be a quasi-Poisson g-manifold, and let A = T ∗ M ⊕ g be its associated Lie algebroid, with anchor r. Then any moment map J : M −→ G induces a −J ∗ φ G -IM form of A by s : A −→ T ∗ M,

s(α, v) = C ∗ (α) + J ∗ σ (v),

(3.32)

so that the image L of the map (r, s) : A −→ T M ⊕ T ∗ M is a −J ∗ φ G -twisted Dirac structure on M, and J : (M, L) −→ (G, LG ) is a Dirac realization of the Cartan–Dirac structure on G. This proposition also suggests the converse construction. Proposition 3.20. Let J : (M, L) −→ (G, LG ) be a Dirac realization of the Cartan– Dirac structure on G. Then (i) for any v ∈ g there is an unique vector V ∈ T M satisfying dJ (V ) = ρ(v) and (V , J ∗ σ (v)) ∈ L;

(3.33)

Dirac structures, momentum maps, and quasi-Poisson manifolds

19

(ii) for any α ∈ T ∗ M, there is an unique vector X ∈ T M satisfying dJ (X) = −(ρM σ ∨ )∗ α, ∗

(X, C (α)) ∈ L.

(3.34) (3.35)

Moreover, v → ρM (v) := V deﬁnes a g-action on M, and α → π (α) := X deﬁnes a quasi-Poisson tensor π on M so that (M, π) is a Hamiltonian quasi-Poisson gmanifold with moment map J . Note that the g-action deﬁned by (3.33) is the one induced by the inﬁnitesimal LG = g G-action with “moment’’ J : M → G. It is simple to check that the constructions in Propositions 3.19 and 3.20 are inverses to one another. For example, if L is obtained from π as in Proposition 3.19, then L = {(π (α) + ρM (v), C ∗ (α) + J ∗ σ (v)) | α ∈ T ∗ M, v ∈ g}, and it is clear that, given α ∈ T ∗ M, X = π (α) satisﬁes conditions (3.34) (which is the dual of the moment map condition (3.22)) and (3.35), so Proposition 3.20 constructs π back. Since the proofs of Propositions 3.19 and 3.20 involve long computations, we will postpone them to the next subsection; we will discuss some examples and implications of the results ﬁrst. Example 3.21 (nondegenerate quasi-Poisson and quasi-Hamiltonian g-manifolds). It is clear from the correspondence constructed in Proposition 3.19 that the singular foliation associated with π, tangent to Im(π ) + Im(ρM ) ⊆ T M, coincides with the singular foliation of the Dirac structure L, tangent to pr 1 (L) ⊆ T M. In other words, the Lie algebroids associated with π and L have the same leaves, so one is transitive if and only if the other one is. Note that, for Dirac structures, pr 1 (L) = T M means exactly that L is deﬁned by a 2-form. As a result, it follows that the correspondence established by Theorem 3.16 restricts to a one-to-one correspondence between nondegenerate Hamiltonian quasi-Poisson manifolds and presymplectic realizations of Cartan–Dirac structures on Lie groups. Combining Example 3.21 with Theorem 3.15, we obtain Corollary 3.22. There is a one-to-one correspondence between (i) nondegenerate quasi-Poisson Hamiltonian g-manifolds, (ii) quasi-Hamiltonian g-manifolds, (iii) presymplectic realizations of G endowed with the Cartan–Dirac structure. Of course, in general, the leaves of a Hamiltonian quasi-Poisson g-manifold are quasi-Hamiltonian g-manifolds, which can now be seen as a particular case of Dirac structures having presymplectic foliations (see also Corollary 3.13). The equivalence of (i) and (ii) can be found in [2]. The next example answers a question posed in [28, Ex. 5.2].

20

H. Bursztyn and M. Crainic

Example 3.23 (Cartan–Dirac and quasi-Poisson structures on Lie groups). If G is a Lie group with Lie algebra g equipped with a bi-invariant nondegenerate quadratic form (·, ·)g , then one can regard it as a −φ G -twisted Dirac manifold with respect to the Cartan–Dirac structure LG , or as a Hamiltonian quasi-Poisson g-manifold. In the latter case, we consider G acting on itself by conjugation, the quasi-Poisson tensor is πG , deﬁned in Example 2.8, and the moment map J : G → G is the identity map; see [2, Sec. 3]. We claim that these two structures are “dual’’ to each other in the sense of Theorem 3.16. Indeed, starting with πG and using Proposition 3.19, we see that the corresponding Dirac structure is L = {(πG (α) + ρ(v), C ∗ (α) + σ (v)), | (α, v) ∈ T ∗ M ⊕ g}.

Since LG is the image of (ρ, σ ), it is clear that LG ⊆ L, which implies that LG = L since they have the same dimension. To complete the “duality’’ picture between Dirac realizations of Cartan–Dirac structures and Hamiltonian quasi-Poisson g-manifolds, we note that the correspondence established in Theorem 3.16 preserves maps. If (Mi , πi ) is a Hamiltonian quasi-Poisson g-manifold with moment map Ji : Mi → G, i = 1, 2, then a map f : M1 → M2 is a Hamiltonian quasi-Poisson map if f∗ π1 = π2 , f is g-equivariant, and J2 ◦ f = J1 . Suppose that Li is the Dirac structure on Mi corresponding to πi , i = 1, 2, via Theorem 3.16. Proposition 3.24. A map f : (M1 , π1 ) → (M2 , π2 ) is a Hamiltonian quasi-Poisson map if and only if f : (M1 , L1 ) → (M2 , L2 ) is f-Dirac and commutes with the realization maps, J2 ◦ f = J1 . Proof. Suppose f : M1 → M2 is a Hamiltonian quasi-Poisson map. In order to check that f is f-Dirac, we have to compare L2 , at each point of M2 , with f∗ L1 = {(df (X), β) | (X, df ∗ (β)) ∈ L1 }.

(3.36)

To simplify the notation, we will denote the inﬁnitesimal actions of g on Mi by ρi , and Ci = 1 − (1/4)ρi ρ ∨ dJ , i = 1, 2. Since L2 corresponds to π2 , we have L2 = {(π2 (β) + ρ2 (v), C2∗ (β) + J2∗ σ (v)) | β ∈ T ∗ M2 , v ∈ g}.

Using that df π1 df ∗ = π2 (which is another way of writing f∗ π1 = π2 ) and dfρ1 = ρ2 (which is f g-equivariance), we deduce that

π2 (β) + ρ2 (v) = df (π1 (df ∗ (β)) + ρ1 (v)).

(3.37)

On the other hand, using the g-equivariance of f and J2 ◦ f = J1 , it follows that df C1 = C2 df , and we obtain df ∗ (C2∗ (β) + J2∗ σ (v)) = C1∗ (df ∗ (β)) + J1∗ σ (v).

(3.38)

Dirac structures, momentum maps, and quasi-Poisson manifolds

Since

21

(π1 (df ∗ (β)) + ρ1 (v), C1∗ (df ∗ (β)) + J1∗ σ (v)) ∈ L1 ,

for L1 corresponds to π1 , it follows that, at each point, L2 ⊆ f∗ L1 . But since they have equal dimension, we conclude that L2 = f∗ L1 , so f is forward Dirac. For the converse, suppose that f∗ L1 = L2 and J2 ◦ f = J1 . It is easy to check that, in this case, f is automatically g-equivariant with respect to (3.33). From this, it follows that df C1 = C2 df . In order to prove that f is a Hamiltonian quasi-Poisson map, we must still check that f∗ π1 = π2 , or, equivalently, that df π1 df ∗ = π2 . By Proposition 3.20, it sufﬁces to prove that, for β ∈ T ∗ M2 , Y = df π1 df ∗ (β) ∈ T M2 satisﬁes (3.39) (Y, C2∗ (β)) ∈ L2 and dJ2 (Y ) = −(ρ2 σ ∨ )∗ β. Since f is an f-Dirac map, the ﬁrst condition in (3.39) holds since (π1 df ∗ (β), df ∗ (C2∗ (β))) = (π1 df ∗ (β), C1∗ (df ∗ (β))) ∈ L1 ,

for L1 corresponds to π1 . The second condition holds since dJ2 (df π1 df ∗ (β)) = dJ1 (π1 df ∗ (β)) = −(ρ1 σ ∨ )∗ df ∗ (β) = −(ρ2 σ ∨ )∗ (β).

We now proceed to the proofs of Propositions 3.19 and 3.20. 3.5.2 Proofs Proof of Proposition 3.19. To simplify our formulas, we set T = ρM ρ ∨ dJ, and we denote by ·, · the pairing between vector spaces and their duals. First, we have to show that s(ξ ), r(ξ ) is antisymmetric in ξ, ξ ∈ A = T ∗ M ⊕g. We check this on elements of type (α, 0), (β, 0), and we leave the other cases to the reader. Since π is antisymmetric, we only have to show that T ∗ β, π (α) is antisymmetric in α and β. Using that dJ π = −(σ ∨ )∗ (ρM )∗ , which is the adjoint of the moment map condition (3.22), we see that ∗ ∗ (β), ρ ∨ (σ ∨ )∗ ρM (α), π(α, T ∗ β) = T ∗ β, π (α) = −ρM

(3.40)

which is antisymmetric by (3.29). We now turn to proving that s satisﬁes (3.5). For sections of A of type ξ = (0, u), ξ = (0, v), with u, v ∈ g, (3.5) follows from the next lemma. Lemma 3.25. Given a g-manifold M and an equivariant map J : M −→ G, then, for any u, v ∈ g, J ∗ σ ([u, v]) = LρM (u) (J ∗ σ (v)) − LρM (v) (J ∗ σ (u)) + dJ ∗ σ (v), ρM (u) − iρM (u)∧ρM (v) (J ∗ φ G ).

22

H. Bursztyn and M. Crainic

Proof. Using the equivariance of J , dJρM = ρ, we immediately see that this equation is the pullback by J of (3.5) for σ (for instance, the last term in the equation equals J ∗ iu∧v (φ G )).

Next we consider the case ξ = (0, u) and ξ = (α, 0), which is handled by the next result. Lemma 3.26. Given a g-manifold M, an equivariant map J : M −→ G, and a bivector π on M satisfying the moment map condition (3.22), then, for all v ∈ g and α ∈ T ∗ M, C ∗ LρM (v) (α) = LρM (v) (C ∗ (α)) − Lπ (α) (J ∗ σ (v)) + dJ ∗ σ (v), π (α) − iρM (v)∧π (α) (J ∗ φ G ). Although Lemma 3.26 is more difﬁcult than Lemma 3.25, it is still simpler than the next case, treated in Lemma 3.27 below. Since a formula that holds under the same assumptions and is proved by the same method is proved in detail in Claim 3.37 below, we will omit its proof. Let us now consider ξ = (α, 0) and ξ = (β, 0). Comparing with Lemmas 3.25 and 3.26, the greater technical difﬁculty of this case comes from the fact that now the formulas involve both χG and φ G , the bracket [·, ·] induced by π on 1-forms, and require more than just the moment map condition. Lemma 3.27. Given a g-manifold M, an equivariant map J : M −→ G, and an invariant bivector π on M satisfying the moment map condition (3.22), then, for all α, β ∈ T ∗ M, 1 C ∗ ([α, β]) + J ∗ σ (i(ρM )∗ (α∧β) χG ) 2 = Lπ (α) C ∗ (β) − Lπ (β) C ∗ (α) − dC ∗ (β), π (α) − iπ (α)∧π (β) (J ∗ φ G ). Proof. Since C = 1 − 14 T , using the deﬁnition of s and of the bracket in (A), we see that we can rewrite the equation in the lemma as T ∗ ([α, β]) − Lπ (α) (T ∗ (β)) + Lπ (β) (T ∗ (α)) + dπ(α, T ∗ (β)) = 2J ∗ σ iρM∗ (α)ρM∗ (β) χG + 4iπ (α)∧π (β) J ∗ (φ G ).

(3.41)

Let us evaluate all the terms of (3.41) on an arbitrary vector ﬁeld X ∈ X (M). To simplify the formulas, we set ∗ a = ρM (α),

∗ b = ρM (β),

V = J (X)

and we consider the Hom(g∗ , g)-valued function on G given by D = ρ ∨ (σ ∨ )∗ .

(3.42)

Dirac structures, momentum maps, and quasi-Poisson manifolds

23

Claim 3.28. The following formula holds: dπ(α, T ∗ (β)), X = a, LV (Db) − b, LV (Da) − a, LV (D)b.

(3.43)

Proof. The left-hand side of (3.43) is LX π(α, T ∗ β). Hence, using (3.40), it equals ∗ ∗ ∗ ∗ (β), ρ ∨ (σ ∨ )∗ ρM (α) − ρM (β), LdJ (X) (ρ ∨ (σ ∨ )∗ )ρM (α). −LJ (X) ρM

(3.44)

With the notation of (3.42), and using D ∗ = −D (i.e. (3.29)) to rewrite the ﬁrst term, we see that (3.44) equals DLV (b), a − b, LV (Da). To obtain (3.43), we write DLV (b) = LV (Db) − LV (D)(b).

From the deﬁnition of [α, β] (2.4), we have T ∗ ([α, β]) = T ∗ Lπ (α) (β) − T ∗ Lπ (β) (α) − T ∗ dπ(α, β).

(3.45)

Claim 3.29. The following formula holds: T ∗ dπ(α, β), X = π(LT (X) (α), β) + π(α, LT (X) (β)).

(3.46)

Proof. This follows from the invariance of π and the fact that the image of T sits inside that of ρM .

Using (3.45) and (3.46), we can split the left-hand side of (3.41) as a difference of two terms which are symmetric to each other. The next claim deals with such a term. Claim 3.30. The following formula holds: T ∗ Lπ (α) (β) − Lπ (α) (T ∗ (β)), X = π(LT (X) (α), β) + b, LV (Da) + b, dρ ∨ ((σ ∨ )∗ (a), V ).

(3.47)

Proof. The left-hand side of (3.47) equals −β, [π (α), T (X)] + T ([π (α), X]).

(3.48)

To rewrite [π (α), T (X)], we note that [π (α), ρM (v)] ˜ = −π LρM (v) ˜ ˜ (α) + ρM Lπ (α) (v),

(3.49)

for all v˜ ∈ C ∞ (M, g): Indeed, due to C ∞ (M)-linearity with respect to v, ˜ it sufﬁces to check (3.49) for v˜ constant; in this case, the equation is just the invariance of π . We now use (3.49) for v˜ = ρ ∨ J (X) to get [π (α), T (X)] = −π LT (X) (α) + ρM Lπ (α) (ρ ∨ dJ (X)). We deduce that (3.48) equals

24

H. Bursztyn and M. Crainic ∗ ∗ β, π LT (X) (α) − ρM (β), Lπ (α) (ρ ∨ dJ (X)) + ρM (β), ρ ∨ J [π (α), X] ∗ (β), Lπ (α) (ρ ∨ J )(X). = π(LT (X) (α), β) − ρM

(3.50)

On the other hand, for all vector ﬁelds Y on M and g-valued 1-forms ν on G, we have iX LY (J ∗ ν) = LdJ (X) (ν(dJ (Y ))) + (dν)(dJ (Y ), dJ (X)). Using this identity for Y = π α and ν = ρ ∨ in (3.50), together with the dual of the moment map condition, dJ π = −((σ )∨ )∗ (ρM )∗ , we obtain (3.47) and prove the claim.

Combining the formulas of Claims 3.28, 3.29, and 3.30, we conclude that the left-hand side of (3.41) evaluated at a vector ﬁeld X is b, dρ ∨ ((σ ∨ )∗ a, V ) − a, dρ ∨ ((σ ∨ )∗ b, V ) − a, LV (D)b.

(3.51)

On the other hand, the right-hand side of (3.41) applied to X equals ∗ ∗ 2(χG (ρM (α), ρM (β), σ ∗ dJ (X)) + 4φ G (dJ π (α), dJ π (β), dJ (X)))

= 2(χG (a, b, σ ∗ V ) + 4φ G ((σ ∨ )∗ a, (σ ∨ )∗ b, V )),

(3.52)

where we have used again that dJ π = −((σ )∨ )∗ (ρM )∗ . To conclude the proof of the lemma, it sufﬁces to show that χG and φ G are related as follows. Claim 3.31. For all a, b ∈ g∗ and all vector ﬁelds V on G, one has 1 χG (a, b, σ ∗ V ) + φ G ((σ ∨ )∗ a, (σ ∨ )∗ b, V ) 2 1 = −a, dρ ∨ ((σ ∨ )∗ b, V ) + b, dρ ∨ ((σ ∨ )∗ a, V ) − a, LV (D)b . (3.53) 4 Proof. It sufﬁces to prove (3.53) on elements of type a = u∨ , b = v ∨ , V = wr , where u, v, w ∈ g, and we recall that u∨ ∈ g∗ denotes the dual of u with respect to the quadratic form, and wr is the vector ﬁeld on G obtained from w by right translations. We will also denote by Ad(u) ∈ C ∞ (G, g) the function g → Ad g (u), and we deﬁne Ad −1 (u) similarly. We will need the explicit formulas for σ ∗ and (σ ∨ )∗ : σ ∗ (wr ) =

1 (w + Ad −1 (w))∨ , 2

(σ ∨ )∗ (u∨ ) =

1 (ur + ul ). 2

Using these formulas, combined with the invariance of χG and φ G , the formula ul = Ad(ur ), and the explicit formulas for χG and φ G on elements of g, one can check that the left-hand side of (3.53) is

Dirac structures, momentum maps, and quasi-Poisson manifolds

1 ([u, v], w + Ad −1 (w))g + ([u + Ad(u), v + Ad(v)], w)g . 8

25

(3.54)

Since ρ ∨ is the difference between the right and left Maurer–Cartan forms on G, we have (dρ ∨ )(ur , vr ) = −[u, v] − Ad −1 [u, v]. By the invariance of (·, ·)g with respect to Ad, we get that a, dρ ∨ ((σ ∨ )∗ b, V ) = −

1 (u, [v, w] + Ad −1 ([v, w]))g 2 + (u, [Ad(v), w] + [v, Ad −1 (w)])g

1 = − ([u + Ad(u), v + Ad(v)], w)g . 2

(3.55)

Since D = 12 (Ad − Ad −1 ), and Lwr (D)(v) = [w, Ad(v)] + Ad −1 ([w, v]), it follows that 1 a, LV (D)b = ([u, Ad(v)], w)g + ([Ad(u), v], w)g . 2 Hence the right-hand side of (3.53) equals 1 2([u + Ad(u), v + Ad(v)], w)g − ([u, Ad(v)], w)g − ([Ad(u), v], w)g , 8 which is easily seen to coincide with (3.54). This concludes the proof of the claim. Using Claim 3.31, we conclude that (3.51) and (3.52) coincide, and this proves Lemma 3.27.

From Lemmas 3.25, 3.26 and 3.27, it follows that s is a −J ∗ φ G -IM form for A. To conclude that L = Im(r, s) is a Dirac structure, we must still prove that L has rank n = dim(M). This follows from the next lemma (which also serves as inspiration for the proof of Proposition 3.20). Lemma 3.32. The sequence j

(r,s)

0 −→ T ∗ G −→ A −→ L −→ 0 is exact, where j (a) = (−J ∗ a, σ ∨ (a)), a ∈ T ∗ G. Proof. The fact that (r, s) ◦ j = 0 is equivalent to (3.22) and the second formula in (3.31). We deﬁne the maps U : A −→ T ∗ G, i : L −→ A,

1 ∗ U (α, v) = − (ρ ∨ )∗ ρM (α) + σ (v), 4 1 ∨ i(X, α) = α, ρ dJ (X) . 4

(3.56) (3.57)

26

H. Bursztyn and M. Crainic

We claim that U ◦ j = Id,

(r, s) ◦ i = Id,

and j ◦ U + i ◦ (r, s) = Id,

(3.58)

and these identities imply that the sequence is exact. For the ﬁrst identity in (3.58), write 1 ∗ ∗ U (j (a)) = ρ ∨ ρM J a + σ σ ∨ a, 4 and then, using that dJρM = ρ and (3.26), we see that U ◦ j = Id. The second identity is immediate from the ﬁrst and the last ones. To prove the last identity, we evaluate j (U (α, v)). The ﬁrst component gives 1 ∨ ∗ ∗ 1 ∗ ∗ ∨ ∗ −J − (ρ ) ρM (α) + σ (v) = −J σ (v) + α − 1 − (ρM ρ J ) α 4 4 ∗ ∗ = −(J σ (v) + C (α)) + α. (3.59) The second component is

1 ∗ σ ∨ − (ρ ∨ )∗ ρM α + σ (v) . 4

(3.60)

∗ = −ρ ∨ (σ ∨ )∗ ρ ∗ = ρ ∨ dJ π , where we used (3.29) and the moment But σ ∨ (ρ ∨ )∗ ρM M map condition (3.22). Expressing σ ∨ σ using (3.25), we see that (3.60) is

1 v − ρ ∨ dJ (ρM (v) + π (α)). 4 Hence

1 ∨ j (U (α, v)) = (α, v) − s(α, v), ρ dJ r(α, v) , 4

i.e., j ◦ U + i ◦ (r, s) = Id. This concludes the proof of Proposition 3.19.

Proof of Proposition 3.20. Let J : (M, L) → (G, LG ) be a Dirac realization. Identifying LG with g G, we know that there is an induced action of g on M, denoted by ρM . Spelling out the deﬁnition, for v ∈ g, ρM (v) is the unique vector ﬁeld satisfying the equations in (3.33). From the second condition in (3.33) and the fact that L is isotropic, we immediately deduce the following. Lemma 3.33. For all (X, α) ∈ L, ∗ ρM (α) + σ ∗ dJ (X) = 0.

Inspired by Lemma 3.32, we prove the following. Lemma 3.34. There is an exact sequence i

U

0 −→ L −→ T ∗ M ⊕ g −→ T ∗ G −→ 0, where U and i are given by (3.56) and (3.57), respectively.

(3.61)

Dirac structures, momentum maps, and quasi-Poisson manifolds

27

Proof. First of all, U is surjective since, as in Lemma 3.32 (and keeping the same notation), U ◦ j = Id. Next, U ◦ i = 0 is an immediate consequence of (3.28) and (3.61). Finally, using the nondegeneracy condition (3.19) for a Dirac realization, it follows that i is injective. By a dimension argument, it follows that the sequence is exact.

We now concentrate on constructing the quasi-Poisson bivector ﬁeld π . Following (ii) of Proposition 3.20, we have the following. Claim 3.35. π is well deﬁned. Proof. We ﬁrst show that (3.34) and (3.35) have a solution X, for any given α: the point is that the element 1 ∨ ∨ ∗ ∗ ∗ −C (α), ρ (σ ) ρM (α) 4 is in the kernel of U ; this is a simple computation using (3.26) and (3.29). Hence it must be in the image of i. More explicitly, we ﬁnd that there exists an X such that ∗ (X, C ∗ (α)) ∈ L, ρ ∨ (dJ (X) + (σ ∨ )∗ ρM (α)) = 0.

(3.62)

On the other hand, applying Lemma 3.33 to (X, C ∗ (α)), and then using the ﬁrst equation in (3.31) to replace CρM , we ﬁnd that ∗ σ ∗ (dJ (X) + (σ ∨ )∗ ρM (α)) = 0.

(3.63)

Since Ker(σ ∗ ) ∩ Ker(ρ ∨ ) = 0, equations (3.62) and (3.63) imply (3.34). The uniqueness of X follows from the nondegeneracy condition (3.19).

Claim 3.36. π deﬁnes a bivector ﬁeld π which satisﬁes the moment map condition (3.22). Proof. We have to show that α(π (β)) + β(π (α)) = 0 for all 1-forms α and β. Let X = π (α) and Y = π (β). Using (3.35) for (α, X) and (β, Y ), the fact that L is isotropic, and the deﬁnition of C, we ﬁnd that 4(α(Y ) + β(X)) = α(ρM ρ ∨ dJ (Y )) + β(ρM ρ ∨ dJ (X)).

(3.64)

Let us show that the right-hand side of (3.64) is zero: using (3.34), (3.64) becomes α(ρM ρ ∨ (ρM σ ∨ )∗ (β)) + β(ρM ρ ∨ (ρM σ ∨ )∗ (α)), and this is zero due to (3.29). On the other hand, (3.34) shows that dJ π = −(ρM σ ∨ )∗ ; dualizing it (and using (π )∗ = −π , which holds by the ﬁrst part of the lemma), we obtain the moment map condition.

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Claim 3.37. The bivector ﬁeld π is g-invariant. Proof. We have to show that LρM (v) (π (α)) = π (LρM (v) (α)) for v ∈ g, and 1-forms α. For this, it sufﬁces to show that LρM (v) (π (α)) satisﬁes (3.34) and (3.35), i.e., dJ (LρM (v) (π (α))) = −(ρM σ ∨ )∗ LρM (v) (α), ∗

(LρM (v) (π (α)), C LρM (v) (α)) ∈ L.

(3.65) (3.66)

These conditions are related to Lemma 3.26. Let us ﬁrst prove (3.66). Using (3.35), (3.33), and the fact that L is isotropic, we conclude that ([ρM (v), π (α)], LρM (v) (C ∗ α) − Lπ (α) (J ∗ σ (v)) + dJ ∗ σ (v), π (α) − iρM (v)∧π (α) (J ∗ φ G )) ∈ L. Using Lemma 3.26, we see that this expression is precisely (LρM (v) (π (α)), C ∗ LρM (v) (α)). Formula (3.65) is closely related to the one in Lemma 3.26: the proofs are similar and hold under the same hypothesis (which might be a bit surprising since (3.65) says that, although the invariance condition on π is not assumed, it must be satisﬁed modulo the kernel of J ). Since we have omitted the proof of Lemma 3.26, we will give the details for (3.65). We evaluate both sides of (3.65) on an arbitrary 1-form µ ∈ 1 (G). The left-hand side gives J ∗ µ, [ρM (v), π (α)] = d(J ∗ µ)(ρM (v), π (α)) + LρM (v) J ∗ µ, π (α) − Lπ (α) J ∗ µ, ρM (v) ∗ ∗ (α)) − LρM (v) µ, (σ ∨ )∗ ρM (α) = −(dµ)(ρ(v), (σ ∨ )∗ ρM + L(σ ∨ )∗ ρM∗ (α) µ, ρ(v). (3.67)

Evaluating µ on the right-hand side, we get −LρM (v) (α), ρM σ ∨ µ = −LρM (v) α, ρM σ ∨ µ + α, [ρM (v), ρM σ ∨ µ]. Now, using [ρM (v), ρM (v)] ˜ = ρM ([v, v]) ˜ + ρM LρM (v) (v) ˜ for v˜ = σ ∨ µ ∈ ∞ C (M, g), we get ∗ ∗ ∗ −LρM (v) ρM (α), σ ∨ µ + ρM (α), [v, σ ∨ µ] + ρM (α), LρM (v) (σ ∨ µ).

(3.68)

We have to show that this coincides with the r.h.s. of (3.67). Comparing the two ∗ α replaced by any formulas, we see that the resulting equation makes sense for ρM element in C ∞ (M, g∗ ). On the other hand, since the equation is C ∞ (M)-linear with respect to this element, we may assume that the element is a constant a ∈ g∗ (and the remaining appearances of ρM become ρ). The identity to be proved, relating the r.h.s. of (3.67) and (3.68), becomes −(dµ)(ρ(v), (σ ∨ )∗ a) − Lρ(v) µ, (σ ∨ )∗ a + L(σ ∨ )∗ a µ, ρ(v) = a, [v, σ ∨ µ],

Dirac structures, momentum maps, and quasi-Poisson manifolds

or, equivalently,

−µ([ρ(v), (σ ∨ )∗ a]) = a, [v, σ ∨ µ].

29

(3.69)

We may assume that µ is the dual (with respect to the quadratic form) of the vector ﬁeld wr for some w ∈ g, and that a is the dual of an element u ∈ g. Equation (3.69) becomes (after multiplying by 2): −(wr , [vr − vl , ur + ul ]) = (u, [v, w + Ad −1 (w)]), and this can be proved to hold from the invariance of the quadratic form and the identities [vl , ul ] = −[v, u]l (see (4.1) for the convention), [vr , ul ] = [vl , ur ].

Claim 3.38. The bivector ﬁeld π is a quasi-Poisson tensor. Proof. We must show that π ([α, β]) = [π (α), π (β)] + 12 iα∧β (ρM (χG )). Using the deﬁnition of π ((ii) of Proposition 3.20) evaluated at [α, β], we have to show that 1 J [π (α), π (β)] + iα∧β (ρM (χG ) = −(ρM σ ∨ )∗ [α, β], (3.70) 2 1 ∗ (3.71) [π (α), π (β)] + iα∧β (ρM (χG )), C ([α, β]) ∈ L. 2 Similar to the discussion in the previous claim, these conditions are related to Lemma 3.27. The ﬁrst equation holds under the same assumptions, and it is proved by the same method, so it will be left to the reader. (Similar to the discussion in the previous proof, the equation tells us that, although the quasi-Poisson condition is not assumed, it must be satisﬁed modulo the kernel of J .) We now prove (3.71). First, we use that (π (α), C ∗ (α)) ∈ L, (π (β), C ∗ (β)) ∈ L, the fact that L is isotropic, and then apply the formula in Lemma 3.27, to conclude that 1 [π (α), π (β)], C ∗ ([α, β]) − J ∗ σ iρM∗ (α∧β) (χG ) ∈ L. (3.72) 2 On the other hand, applying the action in (3.33) to v = iρM∗ (α∧β) (χG ) and observing that ρM (v) = iα∧β (ρM (χG )), we ﬁnd that (iα∧β (ρM (χG )), J ∗ σ iρM∗ (α∧β) (χG )) ∈ L. Since L, at each point, is a vector space, (3.72) and (3.73) imply (3.71).

(3.73)

4 Moment maps in Dirac geometry: The global picture 4.1 Integrating Lie algebroids and inﬁnitesimal actions Lie groupoids are the global counterparts of Lie algebroids. In order to ﬁx our notation, we recall that a Lie groupoid over a manifold M consists of a manifold G together with

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surjective submersions t, s : G → M, called target and source, a partially deﬁned multiplication m : G (2) → G, where G (2) := {(g, h) ∈ G × G | s(g) = t(h)}, a unit section ε : M → G and an inversion G → G, all related by the appropriate axioms; see, e.g., [8]. To simplify our notation, we will often identify an element x ∈ M with its image ε(x) ∈ G. For a Lie groupoid G, the associated Lie algebroid A(G) consists of the vector bundle ker(ds)|M → M, (4.1) with anchor ρ = dt : ker(ds)|M → T M and bracket induced from the Lie bracket on X (G) via the identiﬁcation of sections (ker(ds)|M ) with right-invariant vector ﬁelds on G tangent to the s-ﬁbers. An integration of a Lie algebroid A is a Lie groupoid G together with an isomorphism A ∼ = A(G). Unlike Lie algebras, not every Lie algebroid admits an integration, see [14] for a description of the obstructions. On the other hand, if a Lie algebroid is integrable, then there exists a canonical source-simply-connected integration G(A); see [14]. If M is a point, then a Lie groupoid over M is a Lie group, and the associated Lie algebroid is its Lie algebra. Example 4.1 (transformation Lie groupoids). Let G be a Lie group acting from the left on a manifold M. The associated transformation Lie groupoid , denoted by GM, is a Lie groupoid over M with underlying manifold G × M, source map s(g, x) = x, target map t(g, x) = g · x, and multiplication (g, x) · (g , x ) = (gg , x ). In this case, A(G M) = g M, the transformation Lie algebroid associated with the inﬁnitesimal action of g on M corresponding to the given G-action. (However, even if a g-action does not come from a global action of a Lie group, one can always ﬁnd a Lie groupoid integrating the transformation Lie algebroid g M; see [16, 26].) Similar to inﬁnitesimal actions, Lie groupoids act on maps into their identity sections: if G is a Lie groupoid over M, then a (left) action of G on a map J : N → M is a map mN : G ×M N → N , (g, y) → g · y, satisfying 1. J (g · y) = t(g), 2. (gg )y = g(g y), 3. J (y) · y = y. Here G ×M N := {(g, y) ∈ G × N | s(g) = J (y)}. For reasons that will be clear in the next two subsections, the map J : N → M is often referred to as the moment map of the action mN [25]. Example 4.2 (actions of transformation Lie groupoids). Analogous to Example 3.7, an action mN of a transformation Lie groupoid G = G M on a map J : N → M is equivalent to an ordinary action mN of the Lie group G on N for which J is G-equivariant. Indeed, mN and mN are related by mN ((g, J (y)), y) = mN (g, y),

where g ∈ G

and

y ∈ N.

(4.2)

Dirac structures, momentum maps, and quasi-Poisson manifolds

31

The link between inﬁnitesimal and global actions is based on the following notion: An inﬁnitesimal action ρN of a Lie algebroid A is called complete if ρN (ξ ) ∈ X (N ) is a complete vector ﬁeld whenever ξ ∈ (A) has compact support. As in the case of Lie algebras, a complete action of a Lie algebroid A can be integrated to an action of its canonical source-simply-connected integration G(A); see, e.g., [26]. 4.2 Poisson maps as moment maps for symplectic groupoid actions A 2-form ω on a Lie groupoid G is called multiplicative if the graph of the groupoid multiplication m : G (2) → G is an isotropic submanifold of (G, ω)×(G, ω)×(G, −ω). Equivalently, the multiplicativity condition for ω can be written as m∗ ω = pr ∗1 ω + pr ∗2 ω,

(4.3)

where pr i : G (2) → G, i = 1, 2, are the canonical projections. A symplectic groupoid [29] is a Lie groupoid together with a multiplicative symplectic form. Symplectic groupoids are the global counterparts of Poisson manifolds in the following sense: If π is a Poisson structure on a manifold P inducing an integrable Lie algebroid structure on A = T ∗ P (as in Section 2.2), then the associated sourcesimply-connected groupoid G(P ) := G(A) carries a natural multiplicative symplectic structure [9, 15, 23]; on the other hand, on any symplectic groupoid (G, ω) over a manifold P , condition (4.3) automatically implies that P has an induced Poisson structure uniquely determined by the condition that the target map t : G → P (respectively, source map s : G → P ) is a Poisson map (respectively, anti-Poisson map) [11]. An integration of a Poisson manifold (P , π) is a symplectic groupoid (G, ω) over P for which the induced Poisson structure coincides with π . Note that the symplectic form ω deﬁnes a vector bundle map ker(ds)|P −→ T ∗ P ,

ξ → iξ ω|T P ,

(4.4)

inducing an isomorphism of Lie algebroids A(G) ∼ = T ∗ P [11]. This immediately implies that dim(G) = 2 dim(P ). Example 4.3 (integrating Lie–Poisson structures). Let us consider g∗ , equipped with its Lie–Poisson structure. If G is a Lie group with Lie algebra g, then the transformation groupoid G = G g∗ , with respect to the coadjoint action, integrates T ∗ g∗ = g g∗ . The identiﬁcation G × g∗ ∼ = T ∗ G by right translations induces a multiplicative symplectic form ω on G, in such a way that (G, ω) is a symplectic groupoid integrating g∗ . Remark 4.4. The construction of the symplectic groupoid in the previous example can be extended to the context of Poisson–Lie groups, see Remark 3.10: If (G, π ) is a simply connected Poisson–Lie group and G∗ is its dual, then, assuming that the dressing action is complete, the transformation groupoid G G∗ carries a symplectic structure making it into a symplectic groupoid integrating G∗ . (This symplectic

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structure is basically the one associated with the semidirect product Poisson structure on G × G∗ induced from the action of G on itself by right multiplication.) For a more general construction when the actions are not complete, see [22]. Let us assume that P is an integrable Poisson manifold. We have seen that any Poisson map J : Q → P induces a Lie algebroid action of T ∗ P on Q. Analogous to the case of Lie algebras, when this action is complete, it can be “integrated’’ to an action of G(P ), the canonical source-simply-connected symplectic groupoid of P . We remark that the completeness of the T ∗ P action in the Lie algebroid sense coincides with the notion of J : Q → P being complete as a Poisson map, i.e., if f ∈ C ∞ (P ) has compact support (or if Xf is complete), then XJ ∗ (f ) is complete. The global action mN : G(P ) ×P Q → Q arising in this way is compatible with the Poisson structure on Q in the sense that graph(mN ) is a Lagrangian submanifold of (G(P ), π ) × (Q, πQ ) × (Q, −πQ ),3 where π is the Poisson structure associated with the symplectic form ω on G(P ). Since inclusions of symplectic leaves of Poisson manifolds are Poisson maps, an equivalent way to express this compatibility is that the restricted action mN : G(P ) ×P S → S to each symplectic leaf (S, ωS ) → (Q, πQ ) satisﬁes m∗N ωS = pr ∗G ω + pr ∗S ωS , (4.5) where pr G : G(P ) ×P S → G(P ) and pr S : G(P ) ×P S → S are the natural projections, see [25, 32]. On the other hand, if (Q, πQ ) is a Poisson manifold and mN is an action of a symplectic groupoid G on J : Q → P compatible with πQ in the sense just described, then J is automatically a Poisson map (this is just a leafwise version of [25, Thm. 3.8]). The next example is the global version of Example 3.8. Example 4.5 (global Hamiltonian actions). Consider g∗ with its Lie–Poisson structure, and let G be the simply connected Lie group with Lie algebra g. As in Example 3.8, the starting point is a Poisson map J : Q → g∗ . Note that J is complete as a Poisson map if and only if the associated inﬁnitesimal g-action is by complete vector ﬁelds. In this case, the global action of the symplectic groupoid T ∗ G ∼ = G g∗ is equivalent, in the sense of Example 3.7, to the Hamiltonian G-action obtained by integrating the inﬁnitesimal Hamiltonian g-action on Q. So, in the previous example, the “moment’’ J : Q → g∗ for the symplectic groupoid action of T ∗ G∗ is just a momentum map for a Hamiltonian G-action in the ordinary sense. Remark 4.6. Analogous to the previous example and following Remarks 3.10 and 4.4, a Poisson map J : Q → G∗ , where G∗ is the dual group to a complete simply connected Poisson Lie group, can be “integrated’’ to an action of the symplectic groupoid G G∗ , which is equivalent to a G-action on Q for which J is equivariant 3 A submanifold C of a Poisson manifold (P , π) is Lagrangian if, at each x ∈ P , the intersection of Tx C with π (Tx∗ P ), the tangent space to the symplectic leaf at x, is a Lagrangian subspace of π (Tx∗ P ).

Dirac structures, momentum maps, and quasi-Poisson manifolds

33

(with respect to the dressing action on G∗ ). The “moment’’ J in this case coincides with Lu’s momentum map [21] for a Poisson action of a Poisson–Lie group on a Poisson manifold. 4.3 Dirac realizations as moment maps for presymplectic groupoid actions In order to describe the global actions “integrating’’ Dirac realizations, we should ﬁrst identify the global objects integrating Dirac manifolds, generalizing symplectic groupoids. This was done in [6]: if φ is a closed 3-form on M, then a φ-twisted presymplectic groupoid over M is a Lie groupoid G over M equipped with a multiplicative 2-form ω such that 1. dω = s∗ φ − t∗ φ, 2. dim(G) = 2 dim(M), 3. ker(ωx ) ∩ ker(dx s) ∩ ker(dx t) = {0}, for all x ∈ M. (Twisted presymplectic groupoids are called quasi-symplectic groupoids in [33].) The multiplicativity of ω and condition 1 in this deﬁnition guarantee that the map σω : A → T ∗ M, ξ → iξ ω|T M

(4.6)

is a φ-IM form for A, while 2 and 3 are the extra conditions needed in Lemma 3.3 to insure that the image L of (ρ, σω ) is a φ-twisted Dirac structure. When (G, ω) is a symplectic groupoid, such L is precisely the Dirac structure associated with the induced Poisson structure on M. As proved in [6], L is uniquely determined by the condition that t is an f-Dirac map (respectively, s is an anti-f-Dirac map). Conversely, the canonical groupoid G(L) integrating the Lie algebroid associated with a φ-twisted Dirac structure (assuming it is integrable) is naturally a φ-twisted presymplectic groupoid [6, Sec. 5]. This correspondence generalizes the one between Poisson manifolds and symplectic groupoids [9, 15, 23] (see also [10] for the integration of twisted Poisson structures). We now have all the ingredients to generalize the “integration’’ procedure of Poisson maps to symplectic groupoid actions, explained in Section 4.2, to the context of Dirac geometry. Let LM be a φ-twisted Dirac structure on M associated with an integrable Lie algebroid. We call a Dirac realization J : N → M complete if the induced Lie algebroid action of LM on N is complete, in which case it integrates to an action mN : G(LM ) ×M N → N , where (G(LM ), ω) is the canonical twisted presymplectic groupoid associated with LM . In this situation, we will simply say that the action mN integrates the realization J . Theorem 4.7. Let (M, LM ) be a φ-twisted Dirac manifold and assume that LM is integrable. A complete Dirac realization J : N → M integrates to an action mN : G(LM ) ×M N → N satisfying m∗N LN = τpr∗G ω (pr ∗N LN ),

(4.7)

where pr G and pr N are the projections from G(LN ) ×M N onto G(LN ) and N , respectively, and τpr∗G ω denotes a gauge transformation.

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Conversely, if mN is an action of G(LM ) on J : N → M satisfying (4.7), then J is f-Dirac; if J also satisﬁes (3.19), then it is a Dirac realization whose integration is mN . In order to prove the theorem, we need the following result. Lemma 4.8. Let (M, LM ) be a φ-twisted Dirac manifold and assume that LM is integrable. Let mN : G(LM ) ×M N → N be an action of G(LM ) on J : N → M, and assume that N is equipped with a J ∗ φ-twisted presymplectic form ωN . Then J is an f-Dirac map if and only if m∗N ωN = pr ∗N ωN + pr ∗G ω.

(4.8)

Proof. To simplify the notation, let G = G(LM ), and let us denote by A the corresponding Lie algebroid (which is just LM ). The source and target maps in G are denoted by s and t. Also, let ω1 = m∗N ωN − pr ∗N ωN and ω2 = pr ∗G ω. With these deﬁnitions, our goal is to show that J is f-Dirac if and only if ω1 = ω2 . The key observation is that if we regard G ×M N as a transformation Lie groupoid over N, with source pr N and target mN , a direct computation shows that both ω1 and ω2 are multiplicative. Hence, by [6, Thm. 2.5], ω1 = ω2 if and only if the corresponding bundle maps σωi : A ×M N → T ∗ N, ξy → σωi = (iξy ωi )|T N i = 1, 2, see (4.6), coincide. For ξy ∈ A ×M N (ξ ∈ Ax and x = J (y)) and Y ∈ T N (as usual, we identity T N with T ε(N ), where ε : N → G ×M N is the identity section), we have σω1 (ξy , Y ) = ωN (dmN (ξy ), Y ) and σω2 (ξy , Y ) = ω(ξ, dJ (Y )).

(4.9)

For the ﬁrst identity in (4.9), we used that iξy pr ∗N ωN = 0 for ξy ∈ A ×M N , since pr N is the source map in G ×M N , and A ×M N is its Lie algebroid, which is tangent to the source ﬁbres along the identity section. Since LM = {(dt(ξ ), iξ ω|T M ) | ξ ∈ A}, J : N → M being f-Dirac means that {(dt(ξ ), iξ ω|T M ) | ξ ∈ A} = {(dJ (Y ), α) | iY ωN = J ∗ α}.

(4.10)

But, for ξy ∈ A ×M N , we have dJ (dmN (ξy )) = dt(ξ ). It then follows from (4.10) that ω(ξ, dJ (Y )) = ωN (dmN (ξy ), Y ) for all Y ∈ T N, which implies that σω1 = σω2 , i.e., ω1 = ω2 . The converse follows from the same arguments, reversing the steps. We can now prove Theorem 4.7.

Dirac structures, momentum maps, and quasi-Poisson manifolds

35

Proof. We keep writing G for G(LM ). Suppose that mN integrates a Dirac realization J : N → M. The bundles m∗N LN and pr ∗N LN , seen as subbundles of T (G ×M N ) ⊕ T ∗ (G ×M N ), have the same projection onto the ﬁrst factor: at a point (g, y), they both coincide with Tg G × Ty O, where O is the leaf of LN through y. Note that, since pr N is a submersion, pr ∗N LN is a smooth subbundle, so it is an honest Dirac structure. By Corollary 3.13, since J is a Dirac realization of M, its restriction to any leaf of LN , (O, θ), is a presymplectic realization, and mN is tangent to the leaves. By Lemma 4.8, m∗N θ = pr ∗N θ + pr ∗G ω, which implies the compatibility (4.7). Conversely, (4.7) implies that J is tangent to the leaves of LN . Restricting mN to these leaves, (4.7) amounts to (4.8). So, by Lemma 4.8, J is an f-Dirac map when restricted to each leaf, which implies that J is f-Dirac by Corollary 3.6. The last statement follows from Corollary 3.13 and a direct check.

Remark 4.9. The presymplectic groupoid actions resulting from presymplectic realizations are exactly the “modules’’ considered in the Morita theory developed in [33] to compare various notions of moment maps. More general Dirac realizations give rise to more general “Hamiltonian spaces’’ which still ﬁt with the constructions in [33]. We will discuss examples of the “integration’’ in Theorem 4.7 related to “quasi’’Hamiltonian actions in Section 4.5. 4.4 Reduction in Dirac geometry As in Poisson geometry, one can also carry out reduction in the context of Dirac manifolds. The general construction described in this section recovers reduction procedures in various settings, including [2, 3, 25, 33]. The setup is as follows. Let J : N → M be a Dirac realization of a φ-twisted Dirac manifold (M, LM ). Let x ∈ M be a regular value of J , and consider the submanifold ι : C = J −1 (x) → N . Following [25, 33], let lx = ker(ρ)x be the isotropy Lie algebra of LM at x. Since the anchor ρ is the projection pr 1 |LM , it follows that lx = (LM ∩ T ∗ M)x .

(4.11)

The induced Lie algebroid action of LM on J : N → M deﬁnes a vector bundle morphism LM ×M N → T N, and a simple computation shows that this morphism gives rise to an action of the Lie algebra lx on C. Our object of interest is the orbit space C/lx . Lemma 4.10. If the stabilizer algebras of the lx -action on C have constant dimension (on each component), then ι∗ LN is a (untwisted) Dirac structure on C. Proof. As mentioned in Section 3.2, the conclusion in the lemma holds as long as we show that ι∗ LN is a smooth subbundle of T C ⊕ T ∗ C. As a vector bundle, ι∗ LN is naturally identiﬁed with

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LN ∩ (T C ⊕ T ∗ N ) , (LN ∩ T C ◦ )

(4.12)

see [12], and ι∗ LM will be smooth if we show that both bundles in (4.12) are smooth. For this, it sufﬁces to show that each one has constant dimension. But since their quotient ι∗ LN has constant dimension, it sufﬁces to show that either LN ∩(T C⊕T ∗ N ) or LN ∩ T C ◦ has constant dimension. We will prove that for LN ∩ T C ◦ . On the one hand, LN ∩ T C ◦ = {(0, β) ∈ LN | ι∗ β = 0} = {(0, dJ ∗ α) ∈ LN | α ∈ T ∗ M}. It follows from J being an f-Dirac map that if (0, dJ ∗ α) ∈ LN , then (0, α) ∈ LM . So, if ρN is the inﬁnitesimal action of LM on J , we can write ker(ρN ) ∩ LM ∩ T ∗ M LN ∩ T C ◦ = {(0, dJ ∗ α) ∈ LN | α ∈ T ∗ M} ∼ . = ker(dJ ∗ ) But ker(ρN ) ∩ LM ∩ T ∗ M is the stabilizer of the lx -action on C, which is assumed to have constant dimension. Since x is a regular value, dJ has maximal rank on C, so ker(dJ ∗ ) also has constant dimension. As a result, the dimension of (4.12) is constant, and ι∗ LN is a smooth bundle. Finally, note that ι∗ LN is a (ι∗ J ∗ φ)-Dirac structure on C, but ι∗ J ∗ φ = 0. So ι∗ LN is an ordinary Dirac structure.

We now show that the quotient C/lx carries a natural Poisson structure. Theorem 4.11. Suppose that the orbit space C/lx is a smooth manifold so that projection C −→ C/lx is a submersion. Then there is a unique Poisson structure πred on C/lx for which the projection (C, ι∗ LN ) → (C/lx , πred ) is an f-Dirac map. Remark 4.12. The projection (C, ι∗ LN ) → (C/lx , πred ) is also a b-Dirac map, and this property characterizes πred uniquely as well. Proof. It follows from our assumptions that the lx -orbits on C have constant dimension, so the same holds for the stabilizer algebras. By Lemma 4.10, (C, ι∗ LN ) is a Dirac manifold. The admissible functions on (C, ι∗ LN ), i.e., the set of functions on C whose differential vanish on ker(ι∗ LN ) = ι∗ LN ∩ T C form a Poisson algebra (see [12, Sec. 2.5]) under the bracket {f, g} := LXf g, where Xf is a local vector ﬁeld such that (Xf , df ) ∈ ι∗ LN . We will show that this Poisson algebra induces a Poisson structure on C/lx by showing that the kernel of ι∗ LN coincides with the lx -orbits, i.e., ker(ι∗ LN ) = ρN (lx ). On the one hand,

(4.13)

Dirac structures, momentum maps, and quasi-Poisson manifolds

37

ι∗ LN ∩ T Q = {Y ∈ T Q | ∃β ∈ T ∗ N with (Y, β) ∈ LN , ι∗ β = 0} = {Y ∈ T Q | ∃α ∈ T ∗ M with (Y, dJ ∗ α) ∈ LN }. But since J is f-Dirac and dJ (Y ) = 0, we can write ι∗ LN ∩ T Q = {Y ∈ T Q | ∃α ∈ LM ∩ T ∗ M with (Y, dJ ∗ α) ∈ LN }. On the other hand, ρN (lx ) = {Y ∈ T N | ∃α ∈ LM ∩ T ∗ M with (Y, dJ ∗ α) ∈ LN , dJ (Y ) = 0} = {Y ∈ T Q | ∃α ∈ LM ∩ T ∗ M with (Y, dJ ∗ α) ∈ LN }. So (4.13) follows. The fact that the projection (C, ι∗ LN ) → (C/lx , πred ) is an f-Dirac map and the claim in Remark 4.12 follow from a direct computation; see, e.g., [7].

Of course, if the Dirac realization J : N → M is complete, one can state Theorem 4.11 in terms of the action of the isotropy group of G(LM ) at x ∈ M on C = J −1 (x). Versions of Theorem 4.11 can also be derived when this action is locally free and the quotient is an orbifold, as well as for more general “intertwiner spaces’’ in the sense of [33]. Remark 4.13 (other reductions). The following are important particular cases of the reduction in Theorem 4.11: •

•

•

If M is Poisson and J : N → M is a symplectic realization, we recover [25, Thm. 3.12]; in particular, when M = g∗ , this reduces to the Marsden–Weinstein classical theorem [24], and when M = G∗ , the dual of a Poisson–Lie group, we get Lu’s reduction [21]. If J : N → M is a Poisson map, we get the “Poisson version’’ of these results. If M is φ-twisted Dirac and J : N → M is a presymplectic realization, then we obtain Xu’s reduction [33, Thm. 3.17]; in particular, when M is a Lie group equipped with Cartan–Dirac structure, one recovers the quasi-Hamiltonian reduction of [3]. If J : N → G is a general Dirac realization of a Lie group with Cartan–Dirac structure, then we recover the reduction of quasi-Poisson manifolds of [2] via the identiﬁcation established in Theorem 3.16; see Remark 4.16 below.

4.5 AMM-groupoids and Hamiltonian quasi-Poisson G-manifolds We now discuss global actions, in the sense of Theorem 4.7, associated with complete Dirac realizations of Cartan–Dirac structures. Let G be a Lie groups equipped with a Cartan–Dirac structure LG with respect to a bi-invariant nondegenerate quadratic form (·, ·)g . The ﬁrst step is to identify G(LG ), the canonical presymplectic groupoid integrating LG .

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H. Bursztyn and M. Crainic

As shown in [6, Sec. 7], G(LG ) is closely related to the AMM-groupoids of [5]: if G = G G is the transformation groupoid with respect to the conjugation action, then the 2-form [3] ω(g,x) =

1 (Ad x pg∗ λ, pg∗ λ)g + (pg∗ λ, px∗ (λ + λ))g , 2

where pg , px : G × G → G are the ﬁrst and second projections, and λ and λ are the left and right Maurer–Cartan forms, makes G into a φ G -twisted presymplectic groupoid. If G is simply connected, then (G, ω) is isomorphic to G(LG ), the canonical source-simply-connected integration of LG . In general, G(LG ) is obtained from the G, where G is the universal cover of G [6, AMM groupoid by pulling back ω to G Thm. 7.6]. As a result, just as Lie–Poisson structures “integrate’’ to cotangent bundles of Lie groups, see Example 4.3, Cartan–Dirac structures “integrate’’ to the “double’’ (G × G, ω) in the sense of [3]. For simplicity, let G be simply connected. A complete Dirac realization J : M → G induces a presymplectic groupoid action of (G, ω), as in Theorem 4.7, which is equivalent to a G-action on M for which J is G-equivariant, see Example 4.2; this Gaction is just an integration of the inﬁnitesimal g-action which makes M into a quasiPoisson g-manifold, as constructed in Proposition 3.20. So M becomes a Hamiltonian quasi-Poisson G-manifold for which J : M → G is the group-valued moment map [2]. This construction yields the following global version of Theorem 3.16. Theorem 4.14. There is a one-to-one correspondence between complete Dirac realizations of (G, LG ) and Hamiltonian quasi-Poisson G-manifolds. Corollary 4.15. There is a one-to-one correspondence between compact Dirac realizations of (G, LG ) and compact Hamiltonian quasi-Poisson G-manifolds. Of course, a global version of Proposition 3.24 also holds. Remark 4.16 (reduction). Given a Dirac realization of (G, LG ), J : M → G, the Dirac reduction of Theorem 4.11 produces Poisson spaces J −1 (g)/Gg , where Gg is the centralizer of g ∈ G. Using Remark 4.13 and [2, Prop. 10.6], one can check that these are the same Poisson spaces obtained by quasi-Poisson reduction [2, Thm.6.1] if we regard M as a Hamiltonian quasi-Poisson G-manifold instead. Acknowledgments We would like to thank many people for helpful discussions concerning this work, including A. Alekseev, Y. Kosmann-Schwarzbach, E. Meinrenken, D. Roytenberg, A. Weinstein, P. Xu, and the referees. Our collaboration was facilitated by invitations to the conferences “PQR2003,’’ in Brussels, and “AlanFest’’ and “Symplectic Geometry and Moment Maps,’’ held at the Erwin Schrödinger Institute, where most of the results in this paper were announced; we thank the organizers of these meetings, in particular A. Alekseev, S. Gutt, J. Koiller, J. Marsden, and T. Ratiu. For ﬁnancial support, H. B. thanks DAAD (German Academic Exchange Service) and M. C. thanks KNAW (Dutch Royal Academy of Arts and Sciences). H. B. thanks Freiburg University for its hospitality while part of this work was being done.

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39

References [1] A.Alekseev and Y. Kosmann-Schwarzbach, Manin pairs and moment maps, J. Differential Geom., 56 (2000), 133–165. [2] A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken, Quasi-Poisson manifolds, Canadian J. Math., 54 (2002), 3–29. [3] A. Alekseev, A. Malkin, and E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), 445–495. [4] M. Bangoura and Y. Kosmann-Schwarzbach, The double of a Jacobian quasi-bialgebra, Lett. Math. Phys., 28 (1993), 13–29. [5] K. Behrend, P. Xu, and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris, 336 (2003), 251–256. [6] H. Bursztyn, M. Crainic, A. Weinstein, and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123-3 (2004), 549–607. [7] H. Bursztyn and O. Radko, O, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble), 53 (2003), 309–337. [8] A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, American Mathematical Society, Providence, RI, 1999. [9] A. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, Progress in Mathematics, Vol. 198, Birkhäuser, Basel, 2001, 61–93. [10] A. Cattaneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49-2 (2004), 187–196. [11] A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, in Publications du Département de Mathématiques. Nouvelle Série A, Vol. 2, i–ii, Université Claude-Bernard, Lyon, 1987, 1–62. [12] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631–661. [13] T. Courant and A. Weinstein, Beyond Poisson Structures, Séminaire sud-rhodanien de géométrie VIII, Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39–49. [14] M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. Math., 157 (2003), 575–620. [15] M. Crainic and R. Fernandes, Integrability of Poisson Brackets, Math.DG/ 0210152. [16] P. Dazord, Groupoïde d’holonomie et géométrie globale, C. R. Acad. Sci. Paris, 324 (1997), 77–80. [17] K. Guruprasad, J. Huebschmann, L. Jeffrey, andA. Weinstein, Groups systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377–412. [18] P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194–230. [19] C. Klimˇcik and T. Ströbl, WZW-Poisson manifolds, J. Geom. Physics, 4 (2002), 341–344. [20] J.-H. Lu, Multiplicative and Afﬁne Poisson Structures on Lie Groups, Ph.D. thesis, University of California at Berkeley, Berkeley, 1990. [21] J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Springer-Verlag, New York, 1991, 291–311. [22] J.-H. Lu and A. Weinstein, Groupoïdes symplectiques double des groupes de Lie-Poisson, C. R. Acad. Sci. Paris, 309 (1989), 951–954. [23] K. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445–467. [24] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121–130.

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[25] K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoid actions, Publ. RIMS Kyoto Univ., 24 (1988), 121–140. [26] I. Moerdijk and J. Mrˇcun., On integrability of inﬁnitesimal actions, Amer. J. Math., 124 (2002), 567–593. [27] J.-S. Park, Topological open p-branes, in Symplectic geometry and Mirror symmetry (Seoul 2000), World Scientiﬁc Publishing, River Edge, NJ, 2001, 311–384. [28] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl., 144 (2001), 145–154. [29] A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101–104. [30] A. Weinstein, Poisson geometry. Symplectic geometry, Differential Geom. Appl., 9 (1998), 213–238. [31] A. Weinstein, The geometry of momentum, in Proceedings of the Conference on “Geometry in the 20th Century: 1930–2000,’’ to appear; Math.SG/0208108. [32] P. Xu,Morita equivalent symplectic groupoids, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Springer-Verlag, New York, 1991, 291–311. [33] P. Xu, Morita Equivalence and Momentum Maps, preprint; Math.SG/0307319.

Construction of Ricci-type connections by reduction and induction∗ Michel Cahen1 , Simone Gutt1,2 , and Lorenz Schwachhöfer3 1 Université Libre de Bruxelles

Campus Plaine, CP 218 BE-1050 Brussels Belgium [email protected] 2 Université de Metz Ile du Saulcy F-57045 Metz Cedex 01 France [email protected] 3 Mathematisches Institut Universität Dortmund Vogelppothsweg 87 D-44221 Dortmund Germany [email protected] It is a pleasure to dedicate this paper to Alan Weinstein on the occasion of his 60th birthday. Abstract. Given the Euclidean space R2n+2 endowed with a constant symplectic structure and the standard ﬂat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen [1] have deﬁned a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold (M, ω) of dimension 2n (n ≥ 2) endowed with a symplectic connection of Ricci type is locally given by a local version of such a reduction. We also consider the reverse of this reduction procedure, an induction procedure: we construct globally on a symplectic manifold endowed with a connection of Ricci type (M, ω, ∇) a circle or a real line bundle which embeds in a ﬂat symplectic manifold (P , µ, ∇ 1 ) as the zero set of a function whose third covariant derivative vanishes in such a way that (M, ω, ∇) is obtained by reduction from (P , µ, ∇ 1 ). We further develop the particular case of symmetric symplectic manifolds with Ricci-type connections.

∗ This research was partially supported by an Action de Recherche Concertée de la Commu-

nauté française de Belgique.

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M. Cahen, S. Gutt, and L. Schwachhöfer

1 Introduction A symplectic connection ∇ on a symplectic manifold (M, ω) of dimension 2n is a linear connection which is torsion free and for which ω is parallel. The space of symplectic connections on (M, ω) is inﬁnite dimensional. Selecting some particular class of connections by curvature conditions has, a priori, two interests. The “moduli space’’ of such particular connections may be ﬁnite dimensional and, on some compact symplectic manifolds which do admit a connection of the chosen class, this connection may be “rigid.’’ In this paper, we describe completely the local behavior of symplectic connections of Ricci type (see deﬁnition below) and give some global description of simply connected symplectic manifolds admitting a connection of Ricci type. We denote by R the curvature of ∇ and by R the symplectic curvature tensor R(X, Y, Z, T ) := ω(R(X, Y )Z, T ). For any point x ∈ M, we have the symmetry properties (i) R x (X, Y, Z, T ) = −R x (Y, X, Z, T ) (ii) R x (X, Y, Z, T ) = R x (X, Y, T , Z) (iii) + R x (X, Y, Z, T ) = 0. X,Y,Z

From (i) and (ii), R x ∈ 2 Tx∗ M ⊗ 2 Tx∗ M, where k V is the symmetrized ktensor product of the vector space V . We denote by r the Ricci tensor of the connection ∇ (i.e., rx (X, Y ) = tr [Z → Rx (X, Z)Y ], where X, Y, Z are in Tx M); this tensor r is symmetric. We denote by ρ the corresponding endomorphism of the tangent bundle ω(X, ρY ) := r(X, Y ) so that ρx belongs to the symplectic algebra sp(Tx M, ωx ) and tr ρ = 0. The space Rx of symplectic curvature tensors at x is Rx = ker a ⊂ 2 Tx∗ M ⊗ 2 Tx∗ M, where a is the skew-symmetrization map a : p Tx∗ M ⊗ q Tx∗ M → p+1 Tx∗ M ⊗ q−1 Tx∗ M a(u1 ∧ · · · ∧ up ⊗ v1 . . . vq ) :=

q

u1 ∧ · · · ∧ up ∧ vi ⊗ v1 . . . vˆi . . . vq .

i=1

The group Sp(Tx M, ωx ) acts on Rx . Under this action the space Rx , in dimension 2n ≥ 4, decomposes into two irreducible subspaces [6], Rx = Ex ⊕ Wx , and the decomposition of the curvature tensor R x into its Ex component (denoted Ex ) and its Wx component (denoted Wx ), R x = Ex + Wx , is given by

Construction of Ricci-type connections by reduction and induction

Ex (X, Y, Z, T ) = −

43

1 2ωx (X, Y )rx (Z, T ) + ωx (X, Z)rx (Y, T ) 2(n + 1)

+ ωx (X, T )rx (Y, Z) − ωx (Y, Z)rx (X, T ) − ωx (Y, T )rx (X, Z) . A connection ∇ is said to be of Ricci type if, at each point x, Wx = 0. (Let us mention that such connections were called reducible by Vaisman in [6].) In dimension 2 (n = 1), the space W vanishes identically. We shall assume in what follows that the manifold has dimension m = 2n > 2. Let us ﬁrst recall two interesting features of such connections. •

When a symplectic connection is of Ricci type, it satisﬁes the equations + (∇X r)(Y, Z) = 0.

X,Y,Z

•

Those are the Euler–Lagrange equations of any natural variational principle whose Lagrangian is a second degree invariant polynomial in the curvature (r 2 or R 2 ). Connections which are solutions of those equations are called preferred. They are completely described in dimension 2. The condition to be of Ricci type is the condition on a symplectic connection ∇ to have an integrable almost complex structure J ∇ on the twistor space over M which is the bundle of all compatible almost complex structures on M [2].

In this paper, we show that any symplectic manifold (M, ω) of dimension 2n (n ≥ 2) admitting a symplectic connection of Ricci type has a local model given by a reduction procedure (as introduced by Baguis and Cahen in [1]) from the Euclidean space R2n+2 endowed with a constant symplectic structure and the standard ﬂat connection. We also consider the reverse of this reduction procedure, an induction procedure: we construct globally on a simply connected symplectic manifold endowed with a connection of Ricci type (M, ω, ∇) a circle or a real line bundle N which embeds in a ﬂat symplectic manifold (P , µ, ∇ 1 ) as the zero set of a function whose third covariant derivative vanishes in such a way that (M, ω, ∇) is obtained by reduction from (P , µ, ∇ 1 ). We ﬁnally describe completely the symmetric symplectic manifolds whose canonical connection is of Ricci type. Those were already studied in [4] in collaboration with John Rawnsley.

2 Some properties of the curvature of a Ricci-type connection Let (M, ω) be a smooth symplectic manifold of dim 2n (n ≥ 2) and let ∇ be a smooth Ricci-type symplectic connection. The following results follow directly from the deﬁnition (and Bianchi’s second identity). Lemma 2.1 ([3]). The curvature endomorphism reads

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M. Cahen, S. Gutt, and L. Schwachhöfer

R(X, Y ) = −

1 [−2ω(X, Y )ρ −ρY ⊗X +ρX ⊗Y −X ⊗ρY +Y ⊗ρX], (1) 2(n + 1)

where X denotes the 1-form i(X)ω ( for X a vector ﬁeld on M) and where, as before, ρ is the endomorphism associated with the Ricci tensor [r(U, V ) = ω(U, ρV )]. Furthermore, (i) there exists a vector ﬁeld u such that ∇X ρ = −

1 [X ⊗ u + u ⊗ X]; 2n + 1

(2)

(ii) there exists a function f such that ∇X u = −

2n + 1 2 ρ X + f X; 2(n + 1)

(3)

(iii) there exists a real number K such that trρ 2 +

4(n + 1) f = K. 2n + 1

(4)

3 Construction by reduction of manifolds with Ricci-type connections Let A be a nonzero element in the symplectic Lie algebra sp(R2n+2 , ), where is the standard symplectic structure on R2n+2 . Let A be the closed hypersurface A ⊂ R2n+2 with equation (5)

(x, Ax) = 1; in order for A to be nonempty we replace, if necessary, A by −A. Let ∇˙ be the standard ﬂat symplectic afﬁne connection on R2n+2 . If X, Y are vector ﬁelds tangent to A , deﬁne A (∇X Y )(x) = (∇˙ X Y )(x) − (AX, Y )x;

(6)

this is a torsion free linear connection on A . The vector ﬁeld Ax is an afﬁne vector ﬁeld for this connection; it is clearly complete and we denote by φt the 1-parametric group of diffeomorphisms of A generated by this vector ﬁeld; clearly this ﬂow is given by the restriction to A of the action of exp tA on R2n+2 . Since the vector ﬁeld Ax is nowhere 0 on A , for any x0 ∈ A , there exists • • • •

a neighborhood Ux0 (⊂ A ), a ball D ⊂ R2n of radius r0 , centered at the origin, a real interval I = (−, ), and a diffeomorphism χ : D × I → Ux0 such that χ (0, 0) = x0 and χ (y, t) = φt (χ (y, 0)).

(7)

Construction of Ricci-type connections by reduction and induction

We shall denote π : Ux0 → D,

45

π = p1 ⊗ χ −1 .

If we view A as a constraint manifold in R2n+2 , D is a local version of the Marsden– Weinstein reduction of A around the point x0 . If x ∈ A , Tx A =Ax⊥ , where v1 , . . . , vp denotes the subspace spanned by v1 , . . . , vp and ⊥ denotes the orthogonal relative to ; let Hx (⊂ Tx A ) =x, Ax⊥ ; then Tx R2n+2 = (Hx ⊕ RAx) ⊕ Rx and π∗x deﬁnes an isomorphism between Hx and the tangent space Ty D for y = π(x). A vector belonging to Hx will be called horizontal. A symplectic form on D, ω, is deﬁned by ¯ Y¯ ), ωy (X, Y ) = x (X,

y = π(x),

(8)

where X¯ (respectively, Y¯ ) denotes the horizontal lift of X (respectively, Y ). A symplectic connection ∇ on D is deﬁned by A ¯ ¯ ¯ ∇X Y (x) = ∇X ¯ Y (x) + (X, Y )Ax.

(9)

Proposition 3.1 ([1]). The manifold (D, ω) is a symplectic manifold and ∇ is a symplectic connection of Ricci type. Furthermore, a direct computation shows that the corresponding ρ, u and f are given by ¯ ρX(x) = −2(n + 1)Ax X, u(x) ¯ = ∗

−2(n + 1)(2n + 1)A2x x, 2

(π f )(x) = 2(n + 1)(2n + 1) (A x, Ax),

(10) (11) (12)

where Akx is the map induced by Ak with values in Hx : Akx (X) = Ak X + (Ak X, x)Ax − (Ak X, Ax)x.

4 Local models for symplectic connections of Ricci type The properties of a symplectic connection of Ricci type, as stated in Lemma 2.1, imply in particular that • • • •

the curvature tensor is determined by ρ; its covariant derivative is determined by u; its second covariant derivative is determined by ρ and f , hence by ρ and K with K a constant; the third covariant derivative of the curvature is determined by u, ρ, K, and similarly for all orders. Hence we have the following.

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M. Cahen, S. Gutt, and L. Schwachhöfer

Corollary 4.1. Let (M, ω) be a smooth symplectic manifold of dimension 2n (n ≥ 2) and let ∇ be a smooth Ricci-type connection. Let p0 ∈ M; then the curvature Rp0 and its covariant derivatives (∇ k R)p0 (for all k) are determined by (ρx0 , ux0 , K). Corollary 4.2. Let (M, ω, ∇) (respectively, (M , ω , ∇ )) be two real analytic symplectic manifolds of the same dimension 2n (n ≥ 2), each endowed with a real analytic symplectic connection of Ricci type. Assume that there exists a linear map b : Tx0 M → Tx0 M such that (i) b∗ ωx = 0

ωx0 ; (ii) bux0 = u x ; (iii) b ◦ ρx0 ◦ b−1 = ρx . Assume further that K = K . 0 0 Then the manifolds are locally afﬁnely symplectically isomorphic, i.e., there exists a normal neighborhood of x0 (respectively, x0 ), Ux0 (respectively, Ux ), and a sym0

plectic afﬁne diffeomorphism ϕ : (Ux0 , ω, ∇) → (Ux , ω , ∇ ) such that ϕ(x0 ) = x0 0 and ϕ∗x0 = b.

This follows from classical results, see for instance Theorem 7.2 and Corollary 7.3 in Kobayashi–Nomizu Vol. 1 [5]. Consider now (M, ω, ∇) a real analytic symplectic manifold of dimension 2n (n ≥ 2) endowed with an analytic Ricci-type symplectic connection; denote as before by u, ρ, f and K the associated quantities (see Lemma 2.1). Let p0 be a point in M and choose ξ0 a symplectic frame of Tp0 M, i.e., a linear symplectic isomorphism ξ0 : (R2n , ) → (Tp0 , ωp0 ), where is the standard symplectic form on R2n . Denote by u(ξ ˜ 0 ) the element of R2n corresponding to u(p0 ), i.e., u(ξ ˜ 0 ) = (ξ0 )−1 u(p0 ) and by ρ(ξ ˜ 0 ) the element of sp(R2n , ) corresponding to ρ(p0 ), i.e., ρ(ξ ˜ ) = (ξ0 )−1 ρ(p0 ) ξ0 . Deﬁne an element A of sp((R2n+2 , ) as ⎛ ⎞ −u(ξ ˜ 0) f (p0 ) ⎜ 0 2(n + 1)(2n + 1) 2(n + 1)(2n + 1) ⎟ ⎟ ⎜ ⎟ ⎜ 0 0 A = ⎜1 ⎟, ⎟ ⎜ ⎠ ⎝ −ρ(ξ ˜ 0) −u(ξ ˜ 0) 0 2(n + 1)(2n + 1) 2(n + 1) where u(ξ ˜ 0 ) = i(u(ξ ˜ 0 )) and where we have chosen a basis {e0 , e0 , e1 , . . . , e2n } of the symplectic vector space R 2n+2 relative to which the symplectic form has matrix ⎛ ⎞ 0 1 0 0 In ⎝ ⎠

= .

= −1 0 0 , −In 0 0 0 Consider the local reduction procedure described in Section 3 from the element A deﬁned above around the point x0 = e0 ∈ A = {x ∈ R2n+2 | (x, Ax) = 1}.

Construction of Ricci-type connections by reduction and induction

47

From what we saw in Section 3 this yields a symplectic manifold with a Ricci-type connection (M , ω , ∇ ). Denote by π the map π : Ue0 → M , where Ue0 is the neighborhood of e0 in A ⊂ R2n+2 considered in Section 3 and consider y0 = π (e0 ). Then He0 = e0 , Ae0 = e0 ⊥ =e1 , . . . , e2n is isomorphic under π∗ to Ty0 M . Introduce the injection j : R2n → R2n+2 with j (x1 , . . . , x2n ) = (0, 0, x1 , . . . , x2n ) so that j (R2n ) = He0 and denote by b : Tp0 M → Ty0 M the map given by b = π∗ e ◦ j ◦ ξ0−1 . 0

This map b is a linear symplectic isomorphism since ωy 0 (bX, bY ) = (j ξ0−1 X, j ξ0−1 Y ) = (ξ0−1 X, ξ0−1 Y ) = ωp0 (X, Y ). Furthermore, u (y0 ) = π∗ e (u¯ (x0 )) 0

= π∗ e (−2(n + 1)(2n + 1)(A2 e0 − (A2 e0 , Ae0 )e0 )) 0

˜ 0 )) = π∗ e (ξ0 )−1 u(p0 ) = bu(p0 ) = π∗ e (j u(ξ 0

0

ρ (y0 )bX = π∗ e ρ (y0 )X(e0 ) = π∗ e (−2(n + 1)Ae0 (j ξ0−1 (X))) 0

=

0

˜ 0 )ξ0−1 (X)) π∗ e (j ρ(ξ 0

so that ρ (y0 )b = bρ(p0 ) (f )(y0 ) = 2(n + 1)(2n + 1) (A2 e0 , Ae0 ) = f (p0 ). Hence we have the following. Theorem 4.3. Any real analytic symplectic manifold with a Ricci-type connection is locally symplectically afﬁnely isomorphic to the symplectic manifold with a Riccitype connection obtained by a local reduction procedure around e0 = (1, 0, . . . , 0) from a constraint surface A deﬁned by a second order polynomial in the standard ˙ ﬂat symplectic manifold (R2n+2 , , ∇).

5 Construction of a contact manifold which is a global circle or line bundle over M Consider (M, ω, ∇) a smooth symplectic manifold of dimension 2n > 2 with a π smooth Ricci-type connection and let B(M) → M be the Sp(R2n , )-principal bundle of symplectic frames over M. (An element in the ﬁber over a point p ∈ M is a symplectic isomorphism ξ : (R2n , ) → (Tp M, ωp )). As before, we consider u˜ : B(M) → R2n the Sp(R2n , ) equivariant function given by u(ξ ˜ ) = ξ −1 u(x) where x = π(ξ )

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and ρ˜ : B(M) → sp(R2n , ) the Sp(R2n , ) equivariant function given by ρ(ξ ˜ ) = ξ −1 ρ(x)ξ and we deﬁne the Sp(R2n , ) equivariant map A˜ : B(M) → sp(R2n+2 , ) ⎛ ⎞ −u(ξ ˜ ) (π ∗ f )(ξ ) ⎜ 0 2(n + 1)(2n + 1) 2(n + 1)(2n + 1) ⎟ ⎟ ⎜ ⎟ ˜ )=⎜ 0 0 A(ξ ⎟, ⎜1 ⎟ ⎜ ⎠ ⎝ −ρ(ξ ˜ ) −u(ξ ˜ ) 0 2(n + 1)(2n + 1) 2(n + 1)

(13)

where V = i(V ) for V in R2n . We inject the symplectic group Sp(R2n , ) into Sp(R2n+2 , ) as the set of matrices I2 0 ˜ , A ∈ Sp(R2n , ). j (A) = 0 A Lemma 5.1. Deﬁne the 1-form α on B(M), with values in sp(R2n+2 , ) by ⎛ ⎞ ) −ρ(X)(ξ −ωx (u, X) ⎟ ⎜ 0 ⎜ 2(n + 1)(2n + 1) 2(n + 1) ⎟ ⎜ hor ˜ ) ⎟ ⎟, 0 0 −X(ξ αξ (X ) = ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ −ρ(X)(ξ ) ˜ 0 X(ξ ) 2(n + 1)

(14)

hor

where X ∈ Tx M with x = π(ξ ) and X is the horizontal lift of X in Tξ B(M), and by α(C ∗ ) = j˜∗ (C) (15) for all C ∈ sp(R2n , ), where C ∗ denotes the fundamental vertical vector ﬁeld on d B(M) associated with C (Cξ∗ = dt ξ. exp tC|0 ). This form has the following properties: (i) Rh∗ α = Ad(j˜(h−1 ))α ∀h ∈ Sp(R2n , ); ˜ (ii) d A˜ = −[α, A]; ˜ ∗ ω. (iii) dα + [α, α] = −2Aπ p

When one has a G-principal bundle P → M, an embedding of the group G in a larger group G , j : G → G , and a 1-form α with values in the Lie algebra of G , such that α(C ∗ ) = j∗ (C) for all C in the Lie algebra of G and Rh∗ α = Ad(j (h−1 )) α for p

all h in G, one can build the G -principal bundle P = P ×G G → M and the unique connection 1-form on P , α satisfying i ∗ α = α, where i : P → P ; ξ → [(ξ, 1)]. In our situation we build the Sp(R2n+2 , )-principal bundle B (M) = B(M) ×Sp(R2n , ) Sp(R2n+2 , )

Construction of Ricci-type connections by reduction and induction

49

whose elements are equivalence classes of pairs (ξ, g), ξ ∈ B(M), g ∈ Sp(R2n+2 , ), with (ξ, g) equivalent to (ξ h, j˜(h−1 )g) ∀h ∈ Sp(R2n , ). The projection π : B(M) → M maps [(ξ, g)] to π(ξ ). The connection 1-form α is characterized by the fact that α[ξ,1] ([X

hor

, 0]) = αξ (X

hor

)

and the equations above give the following. Lemma 5.2. The curvature 2-form of the connection 1-form α is equal to −2A˜ π ∗ ω, where A˜ is the unique Sp(R2n+2 , )-equivariant extension of A˜ to B (M). This curvature 2-form is invariant by parallel transport (d α curv(α ) = 0). Thus the holonomy algebra of α is of dimension 1. Corollary 5.3. Assume M is simply connected. The holonomy bundle of α is a circle π

or a real line bundle over M, N → M. This bundle has a natural contact structure ν given by the restriction to N ⊂ B(M) of the 1-form −α (viewed as real valued since it is valued in a one-dimensional algebra). One has dν = 2π ∗ ω. It is enlightening to point out the link between the holonomy bundle N over M and the constraint surface A when one sees M as obtained (locally) by reduction. The link is only local since A is in general not a principal bundle over M; in fact in most cases the quotient of A by the action of the group exp tA is at best an orbifold. Let A be a nonzero element of sp(R2n+2 , ) and let A = {y ∈ R2n+2 |

(y, Ay) = 1}; we assume as before that it is not empty. Assume that (M, ω, ∇) is obtained by reduction from A (as before, we restrict ourselves to some open set in A ). Let y0 be a point in A , let x0 = π(y0 ) ∈ M and choose a symplectic frame ξ0 at x0 . Let γ (t) be a curve in M such that γ (0) = x0 . Let ξ(t) be the symplectic frame at γ (t) obtained by parallel transport along γ from ξ0 and let y(t) be the horizontal curve in A lifting γ (t) from y0 (i.e., π(y(t) = γ (t) and (y(t), y(t)) ˙ = 0). Deﬁne the element C(t) of Sp(R2n+2 , ) as the matrix whose columns are C(t) = (y(t) Ay(t) ξ(t)), where ξ(t) consists of the 2n vectors which are the horizontal lifts at the point y(t) of the vectors of the frame ξ(t) (the image under the map ξ(t) of the usual basis of R2n ). Then d hor C(t)|s = C(s).αξ(s) (γ˙ (s) ), dt hor

where α is the 1-form on B(M) deﬁned in (14) and where X is the horizontal lift hor of X in B(M); hence γ˙ (s) = ξ˙ (s). Let B (M) be the Sp(R2n+2 , )-principal bundle over M considered above and let [(ξ0 , 0 )] (where 0 is an element in Sp(R2n+2 , )) be a point of B (M) above x0 . The horizontal lift of γ (t) to B (M) starting from [(ξ0 , 0 )] lives in the holonomy subbundle containing this point; it reads

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[(ξ(t), D(t))], where ξ(t) has been deﬁned above and where D(t) obeys the differential equation d D(t)|s = −αξ(s) (ξ˙ (s)).D(s) dt and has initial value 0 . Deﬁne the map above γ which sends y(t) to [(ξ(t), D(t))], where D(t) = C −1 (t)C(0)0 ; this map sends elements of A to elements in the holonomy bundle through [(ξ0 , 0 )]. The map from the holonomy bundle through [(ξ0 , 0 )] to R2n+2 given by [(ξ, D)] → C0 0 D −1 e0 , where C0 is a ﬁxed element in Sp(R2n+2 , ) has value in the hypersurface A , ˜ 0 )C −1 . where A = C0 A(ξ 0

6 Embedding of the contact manifold in a ﬂat symplectic manifold Let (M, ω) be a smooth symplectic manifold of dim 2n (n ≥ 2) and let ∇ be a smooth symplectic connection of Ricci type. Let (N, α) be a smooth (2n + 1)-dimensional contact manifold (i.e., α is a smooth 1-form such that α ∧(dα)n = 0 everywhere). Let X be the corresponding Reeb vector ﬁeld (i.e., i(X)dα = 0 and α(X) = 1). Assume there exists a smooth submersion π : N → M such that dα = 2π ∗ ω. Then at each point x ∈ N, Ker(π∗x ) = RX and LX α = 0. Remark that such a contact manifold always exists if M is simply connected as we saw in the previous section. If U is a vector ﬁeld on M we can deﬁne its “horizontal lift’’ U on N by (i) π∗ U = U , (ii) α(U ) = 0. Let us denote by ν the 2-form ν = dα = 2π ∗ ω on N . Deﬁne a connection ∇ N on N by ∇ N U V = ∇U V − ν(U , V )X 1 ∇N XU = ∇N U X = − ρU 2(n + 1) 1 ∇N XX = − u, 2(n + 1)(2n + 1) where ρ is the Ricci endomorphism of (M, ∇) and where u is the vector ﬁeld on M appearing in ∇ρ; see Lemma 2.1. Then ∇ N is a torsion free connection on N and the Reeb vector ﬁeld X is an afﬁne vector ﬁeld for this connection.

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51

The curvature of this connection has the following form: 1 [ν(ρV , W )U − ν(ρU , W )V ] 2(n + 1) 1 R N (U , V )X = [ν(u, V )U − ν(u, U )V ] 2(n + 1)(2n + 1) 1 1 R N (U , X)V = ν(u, V )U + ν(U , ρV )X 2(n + 1)(2n + 1) 2(n + 1) 1 R N (U , X)X = [−π ∗ f U + ν(U , u)X], 2(n + 1)(2n + 1)

R N (U , V )W =

where f is the function appearing in Lemma 2.1. Consider now the embedding of the contact manifold N into the symplectic manifold (P , µ) of dimension 2n + 2, where P = N × R, and, if we denote by s the variable along R and let θ = e2s p1∗ α (p1 : P → N ), we set µ = dθ = 2e2s ds ∧ α + e2s dα and let i : N → P , x → (x, 0). Obviously, i ∗ µ = ν. We now deﬁne a connection ∇ 1 on P as follows. If Z is a vector ﬁeld along N , we denote by the same letter the vector ﬁeld on P such that (i) Zi(x) = i∗x Z, (ii) [Z, ∂s ] = 0. The formulas for ∇ 1 are ∇Z1 Z = ∇ N Z Z + γ (Z, Z )∂s , where γ (Z, Z ) = γ (Z , Z) 1 π ∗f γ (X, X) = 2(n + 1)(2n + 1) 1 γ (X, U ) = − ν(u, U ) 2(n + 1)(2n + 1) 1 γ (U , V ) = ν(U , ρV ), 2(n + 1) and ∇Z1 ∂s = ∇∂1s Z = Z ∇∂1s ∂s = ∂s .

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Theorem 6.1. The connection ∇ 1 on (P , µ) is symplectic and has zero curvature. Proposition 6.2. Let ψ(s) be a smooth function on P . Then ψ has vanishing third covariant differential if and only if ∂s2 ψ − 2∂s ψ = 0.

(16)

In particular the function e2s has this property. The procedure described above is called the induction. Let (P , µ, ∇ 1 ) be as above and let = N be the constrained submanifold deﬁned by e2s = 1. Let Y be the vector ﬁeld transversal to such that i(Y )µ = α, thus Y = ∂s . Let H be the 1-parametric group generated by X. Then /H can be identiﬁed with M and (M, ω) is the classical Marsden–Weinstein reduction of (P , µ) for the constraint . The connection ∇ on M is obtained from the ﬂat connection ∇ 1 on (P , µ) by reduction. Hence we have the following. Corollary 6.3. Any smooth simply connected symplectic manifold with a Ricci-type connection (M, ω, ∇) can be obtained by reduction from an hypersurface in a ﬂat symplectic manifold (P , µ, ∇ 1 ) deﬁned by the 1-level set of a function ψ on P whose third covariant derivative vanishes. Corollary 6.4. Any smooth simply connected symplectic manifold with a Ricci-type connection (M, ω, ∇) is automatically analytic. Proof. Since (P , µ, ∇ 1 ) is locally symmetric, P , µ and ∇ 1 are real analytic and the explicit construction given preserves analyticity.

7 Symmetric symplectic spaces with Ricci-type connections Lemma 7.1. The reduction construction described in Section 3 yields a locally symmetric symplectic space (i.e., such that the curvature tensor is parallel) if and only if the element 0 = A ∈ sp(R2n+2 , ) satisﬁes A2 = λI for a constant λ ∈ R. Proof. Indeed the connection ∇ has parallel curvature tensor if and only if ∇ρ = 0 hence iff u = 0. From the formulas above, this is true iff A2x (x) = A2 x − (A2 x, Ax)x = 0 for any x ∈ A . When u = 0, f is a constant (cf. Lemma 2.1) and it follows from Lemma 3.1 that (A2 x, Ax) is a constant λ. Since A contains a basis of R2n+2 , this yields A2 = λI .

Construction of Ricci-type connections by reduction and induction

53

Proposition 7.2. If 0 = A ∈ sp(R2n+2 , ) satisﬁes A2 = λI for a constant λ ∈ R, the quotient of A by the action of exp tA is a manifold and the natural projection map A → M is a submersion which endows A with a structure of circle or real line bundle over M. Proof. Consider 0 = A ∈ sp(R2n+2 , ) so that A2 = λI . Case 1: λ > 0, say λ = k 2 with k > 0. Then there exists a basis of R2n+2 in which kIn+1 0 0 In A= ,

= 0 −kIn+1 −In 0 so that A = {(u, v) u, v ∈ Rn+1 | −2ku · v = 1}. The ﬂow of the vector ﬁeld Ax is given by φt = etA . The map π : A → T S n = {(u , v ) u , v ∈ Rn+1 |u · u = 1, u · v = 0} deﬁned by u u , u(v + π(u, v) = ) u 2ku2 induces a diffeomorphism between M = A /φt and T S n . M is a noncompact simply connected manifold and A is a R-bundle over T S n . Case 2: λ < 0, say λ = −k 2 with k > 0. One splits V C (V = R2n+2 ) into the eigenspaces relative to A, V C = Vik ⊕ V−ik and observe that those subspaces are Lagrangian. Choosing a basis {z1 , . . . , zn+1 } for Vik ωkl := (zk , zl ); then , consider i iω is a Hermitian matrix. A change of basis (zj = zi Uj ) yields ω =t U ωU so we can ﬁnd a basis for Vik so that 0 Ip ω = −2iIp,n+1−p = −2i . 0 −In+1−p In the basis of R2n+2 given by ej = 12 (zj + zj ), fj = 2i1 (zj − zj ), we have 0 −kI 0 Ip,n+1−p A= ,

= kI 0 −Ip,n+1−p 0 so that A = {(u, v) u, v ∈ Rn+1 | k i≤p ((ui )2 +(v i )2 )−k i>p ((ui )2 +(v i )2 ) = ∼

1}. We assume p ≥ 1 or replace A by −A so that A = S 2p−1 × R2n−2p+2 is nonempty. The ﬂow φt is given by the action of cos ktI − sin ktI exp tA = . sin ktI cos ktI Then M = A /φt = (S 2p−1 × R2n−2p+2 )/U (1), so this reduced manifold is • • •

M = R2n if p = 1; M is a complex line bundle of rank q := n + 1 − p over the complex projective space Pp−1 (C) = S 2p−1 /U (1) if 1 < p ≤ n; M = Pn (C) if p = n + 1.

In all those cases, M is simply connected and A is a circle bundle over M; the only compact case is M = Pn (C).

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Case 3: λ = 0, so A2 = 0 with A = 0. Let us denote by p the rank of A. One splits V = R2n+2 into V = V0 ⊕ V1 ⊕ V2 , where V1 = Im A (dim V1 = p), V0 ⊕V1 = Ker A (so dim V0 = 2n+2−2p and V0 is symplectic, since V0 ⊕V1 = V1⊥ ) and V2 is a Lagrangian subspace of V0⊥ supplementary to V1 . Choose a basis of V2 and a corresponding basis (dual for ) in V1 and a symplectic basis of V0 so that in those basis ⎛ ⎞ ⎛ ⎞ 00 0

1 0 0 A = ⎝ 0 0 A ⎠ ,

= ⎝ 0 0 Ip ⎠ 00 0 0 −Ip 0 the basis and A is symmetric. Changing the basis of V2 and correspondingly of V1 , one can bring A to the form A = Ir,p−r so that (x, Ax) = i≤r (w i )2 − r 1 and A consists of two copies of R2n+1 if r = 1. The action of φt on (u, v, w) is given by φt (u, v, w) = (u, v + tA w, w) so the reduced manifold is • •

two copies of R2n (if r = 1); or M = S r−1 × R2n+1−r if r > 1.

In all cases, M is a noncompact manifold and A is a line bundle over M.

Proposition 7.3. If 0 = A ∈ sp(R2n+2 , ) satisﬁes A2 = λI for a constant λ ∈ R, the quotient manifold is a symmetric space and the connection obtained by reduction is the canonical symmetric connection. Proof. Any linear symplectic transformation B of R2n+2 which commutes with A obviously induces a symplectic afﬁne transformation β(B) of the reduced space M = A /φt . If π denotes the canonical projection π : A → M, then β(B) ◦ π = π ◦ B. In particular the symmetry at the point x = π(y), y ∈ A is induced by By u = −u + 2 (u, Ay)y − 2 (u, y)Ay.

The list of all symmetric spaces whose canonical connection is of Ricci type and the fact that the only compact simply connected one is Pn (C) already appears in [4]. What we describe here is a construction of all those manifolds by global reduction from the ﬂat Euclidean space. We shall now describe the transvection group of M (i.e., the group G of afﬁne transformations of M generated by the composition of two symmetries). Let us denote by G the group G = {B ∈ Sp(R 2n+2 , ) | BA = AB}. The transvection group of M is clearly included in β(G ); in fact it is the smallest subgroup of β(G ) stable under conjugation by a symmetry and which acts transitively on M. Let x0 = π(y0 ) be a point in M and let sx0 = β(By0 ) be the symmetry at this point. Consider the automorphism of G given by conjugation by By0 and denote by σ the

Construction of Ricci-type connections by reduction and induction

55

induced automorphism of the Lie algebra g of G . Let p = {C ∈ g | σ (C) = −C} and k = {C ∈ g | σ (C) = C}. The dimension of p is equal to 2n. Indeed, in a basis {e0 , e0 , e1 , . . . , e2n } of 2n+2 R in which e0 = y0 and e0 = Ae0 and ⎛ ⎞ 0 1 0

= ⎝ −1 0 0 ⎠ , 0 0 one has

⎛ ⎞ ⎞ 0λ 0 10 0 A = ⎝1 0 0 ⎠, By0 = ⎝ 0 1 0 ⎠ , 0 0 A 0 0 −I2n ⎧⎛ ⎞ ⎨ b λc A Z g = ⎝ c −b −Z ⎠ , b, c ∈ R; Z ∈ R2n ; B ∈ sp(R2n , ) ⎩ Z A Z B ⎫ ⎬ such that BA = A B , ⎭ ⎛

and ⎫ ⎧⎛ ⎞ ⎬ ⎨ 0 0 A Z p = ⎝ 0 0 −Z ⎠ , Z ∈ R2n . ⎭ ⎩ Z A Z 0 Hence the Lie algebra of the transvection group is equal to β∗ (p + [p , p ]). In all cases the kernel of β is given by exp tA, and the transvection group is described as follows: Case 1: λ > 0, say λ = k 2 with k > 0. In the basis of R2n+2 in which 0 0 In kIn+1 ,

= A= , 0 −kIn+1 −In 0 we have G =

B 0 0 (t B)−1

, B ∈ Gl(n + 1, R) ,

and β of such an element is the identity iff B = λI with λ > 0. The transvection group G is isomorphic to Sl(n + 1, R) and T S n = Sl(n + 1, R)/ Gl(n, R). Case 2: λ < 0, say λ = −k 2 with k > 0. In the basis of R2n+2 in which 0 Ip,n+1−p 0 −kI , A= ,

= −Ip,n+1−p 0 kI 0

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we have G =

B1 B 2 −B2 B1

, B1 + iB2 ∈ U (p, n + 1 − p) ,

and β of such an element is the identity iff B1 + iB2 = exp −ikt. The transvection group G is isomorphic to SU (p, n + 1 − p) and M = SU (p, n + 1 − p)/U (p − 1, n + 1 − p). Case 3: λ = 0, rank A = k = p + q. In the basis of R2n+2 in which ⎞ ⎛ 00 0 A = ⎝ 0 0 Ipq ⎠ 00 0 and ⎞

1 0 0

= ⎝ 0 0 −I ⎠ , 0 I 0 ⎛

we have

⎧⎛ ⎞ D 0 C ⎨ g = ⎝ −t C 1 −t B F ⎠ , D ∈ sp(R2n+2−2k , ω1 ), B ∈ so(p, q, R), ⎩ 0 0 B ⎫ ⎬ F ∈ gl(k, R), t F = F, C ∈ Mat(2n + 2 − 2k, k, R) . ⎭

Then p is given by the elements of g for which D = 0, C = CJ , F = −J F J , B = −J BJ , where 1 0 J = , 0 −Ik−1 so C = (u 0 . . . 0) for u ∈ R2n+2−2k ,

F = for v ∈ Rk−1 , and

B=

0 tv v 0

0 t w w 0

for w ∈ Rk−1 and w = Ip−1,q w. Hence p ⊕ [p , p ] is the set of all elements in g for which D = 0. The transvection group G has algebra g isomorphic to {(B, F, C)}/(0, RIpq , 0), where B is any element in so(p, q, R), F is any symmetric real k × k matrix, and C is any real (2n + 2 − 2k) × k matrix and the bracket is deﬁned by

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[(B, F, C), (B , F , C )] = ([B, B ], −t C 1 C +t C 1 C −t BF +t B F, CB − C B), so when p+q > 2, the Levi factor is so(p, q, R) and the radical is a two-step nilpotent algebra. If p = 0 and q = 1 the transvection group is R2n and the symmetric space is the standard symplectic vector space. If p = q = 1 or if p = 0 and q = 2, the transvection group is solvable but not nilpotent. The two solvable examples are interesting for building exact quantization.

References [1] P. Baguis and M. Cahen, A construction of symplectic connections through reduction, Lett. Math. Phys., 57 (2001), 149–160. [2] F. Burstall and J. Rawnsley, private communication; see also N. R. O’Brian and J. H. Rawnsley, Twistor spaces, Ann. Global Anal. Geom., 3 (1985), 29–58 and I. Vaisman, Variations on the theme of twistor spaces, Balkan J. Geom. Appl., 3 (1998), 135–156. [3] M. Cahen, S. Gutt, J. Horowitz, and J. Rawnsley, Homogeneous symplectic manifolds with Ricci-type curvature, J. Geom. Phys., 38 (2001), 140–151. [4] M. Cahen, S. Gutt, and J. Rawnsley, Symmetric symplectic spaces with Ricci-type curvature, in G. Dito and D. Sternheimer, eds., Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries, Vol. II, 1999, Mathematical Physics Studies, Vol. 22, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, 81–91. [5] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Wiley, New York, London, 1963. [6] I. Vaisman, Symplectic curvature tensors, Monats. Math., 100 (1985), 299–327; see also M. De Visher, Mémoire de Licence, Bruxelles, 1999.

A mathematical model for geomagnetic reversals J. J. Duistermaat Department of Mathematics Utrecht University Postbus 80 010 3508 TA Utrecht The Netherlands [email protected]

1 Introduction The earth’s magnetic ﬁeld has reversed its polarity many times in history, where the polarity of the magnetic ﬁeld remained the same for very long time intervals, between about one hundred thousand years and many million years, whereas the reversals took place in a relatively short time interval of the order of magnitude of a thousand years. The lengths of time intervals between the subsequent reversals form an irregular sequence with a large variation, which make the reversals look like a (Poisson) stochastic process; see [7]. The direction of the earth’s magnetic ﬁeld has been recorded in the basalt rocks of the ocean ﬂoors and in sedimentary rocks. In the ﬁrst case the alignment of the magnetic particles is frozen in the rock when the upwelling ﬂuid-hot basalt at the mid-ocean ridges solidiﬁes upon cooling and then is pushed aside by the subsequent upwelling. This leads to a fantastic zebra-like stripe pattern in the alignments of the magnetic material in the ocean ﬂoor, the discovery of which actually was one of the most convincing arguments for the concept of sea-ﬂoor spreading. Fixing alignment during sedimentation is another recording mechanism. Compared to the basaltic rocks, the historical record of the earth’s magnetic ﬁeld in sedimentary rocks reveals more details about the shorter time scales, but become harder to interpret in the very old rocks because of the frequent deformations of the sedimentary material. Some years ago Peter Hoyng from SRON (an institute in Utrecht which puts astronomical instruments into satellites) came to me with a question about some partial differential equations, which turned out to be related to a mathematical model which Schmitt, Ossendrijver, and he had formulated for the reversals of the earth’s magnetic ﬁeld in [4] and [5]. In their model, the earth’s magnetic ﬁeld is supposed to be a solution of a stochastically perturbed dynamical system in which the unperturbed deterministic system has two competing stable equilibria. The domains of attraction are separated by the stable manifold of a saddle point at the origin, where the magnetic ﬁeld is equal to zero. A typical feature of their model is that the stochastic perturbation is proportional to the magnetic ﬁeld, and therefore its coefﬁcients vanish at the origin.

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The vanishing of the stochastic term at the origin corresponds to the appearance of the factor B in the term αB in equation (1) in loc. cit., where the factor α contains the stochastic ﬂuctuations. These ﬂuctuations come from the superposition of a turbulent ﬂow superimposed upon the main ﬂow in the liquid outer core. Such turbulent perturbations have no accurate descriptions in terms of simple deterministic models, whereas simple stochastics models lead to reasonable agreement with observations of time averages of turbulent perturbations. In [5] the predictions of the model have been compared in detail with the data on the geomagnetic dipole moment, including the statistics of the reversals. Good, although not complete agreement was obtained and the conclusion in [5] is that the model is consistent with the data. Several groups have made three-dimensional magnetohydrodynamic simulations which looks very much earth-like, leading to a largely dipolar exterior ﬁeld, about 0.5 G at the surface, and sudden reversals. However, the parameter regime of these simulations is still many orders of magnitude removed from reality. For example, the Ekman number (ratio of typical magnitude of viscous to Coriolis forces) is about 10−15 , but the simulations can only get down to 10−6 . The agreement therefore could be just accidental. Apart from this, we believe that there is merit in “simple’’ analytical models which might give insight into what goes on both in the simulations and in the earth. It should also be observed that in other astrophysical objects the behavior of the magnetic ﬁeld can be very different. For instance, the magnetic ﬁeld of the sun has a strong deterministic component which is periodic with a period of about 22 years instead of being stationary. Accordingly, the physical models of the sun are quite different from those of the earth. See, for instance, [9]. Let u(x) denote the expectation value of the ﬁrst time of exit from one of the domains of attraction, for the stochastic process which starts at the point x ∈ . The function u(x) satisﬁes an elliptic inhomogeneous linear partial differential equation in the domain , with the Dirichlet boundary condition that u(x) = 0 when x ∈ ∂ . The problem is singularly perturbed because the assumption of the smallness of the stochastic perturbation of the deterministic system leads to a small positive coefﬁcient in front of the second-order derivatives. Another complication is that the vanishing of the stochastic term at the origin implies that the differential operator is no longer elliptic at the origin, a point on the boundary of . Actually, if we replace the deterministic vector ﬁeld by its linearization at the origin, then the stochastic process is invariant under homotheties, and the mean exit time would be constant on every ray in starting from the origin. This leads to discontinuous behavior at the origin, which we conjecture is inherited by the function u(x) in our nonlinear model. Other quantities, like the invariant probability measure (if it exists) satisfy similar singularly perturbed partial differential equations, which also degenerate at the origin. From my knowledge of partial differential equations, I could not answer Hoyng’s question about the asymptotic behavior of the solutions as the stochastic perturbation tends to zero. Instead, prompted by suggestions of the stochastic experts in our department, I tried to apply the theory of Freidlin and Wentzell [10, 2] concerning small stochastic perturbations of dynamical systems. It turned out that this could not be done directly, because their theory is about compact domains, whereas ours is not

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and, more importantly, our problem degenerates at the origin. Still, the arguments of the theory appear to carry through, although the full proof of this is not yet ﬁnished and will be the subject of a more extensive paper. In this paper I will conﬁne myself to the description of the results which would follow from the properly extended theory of Freidlin and Wentzell. If the stochastic terms are of small order , then the probabilities turn out to be of 2 order e−c/ , in which c is some positive real number. Because ratios of such exponential expressions are again such exponential expressions, the asymptotics for the probabilities depend very sensitively on the value of the coefﬁcient c. Such behavior is called large deviations.

2 The stochastic process The model consists of a stochastic differential equation dx(t) = dt · b(x(t)) +

k

dBj ◦ Xj (x(t)),

j =1

where the process x(t) is a curve in the conﬁguration space M, which is assumed to be a smooth manifold. The b(x) and Xj (x), 1 ≤ j ≤ k, are vector ﬁelds on M, whereas dBj denote independent Brownian motions on the reals. The vector ﬁeld b describes the deterministic dynamical system, whereas dBj ◦ Xj (x) is a Brownian motion proportional to the vector ﬁeld Xj (x). The symbol ◦ indicates that we use the Stratonovich integral, in which the symmetric midpoint rule at subsequent time steps is used in the approximating Riemann sums. It leads to a coordinate invariant calculus on the manifold M; see [1, pp. 74, 75, 111–115], [6, pp. 100, 101, 233–235], or [8, pp. 24, 35–37], which is convenient when we make substitutions of variables. The expectation value u(x, t) of f (x(t)), under the condition that x(0) = x, satisﬁes the partial differential equation ∂u/∂t = Au,

u(x, 0) = f (x),

in which the second-order linear partial differential operator 1 2 Xj 2 k

A=b+

j =1

is called the inﬁnitesimal generator of the process. Here the vector ﬁelds b and Xj are viewed as ﬁrst-order linear partial differential operators. The probability distribution µ, giving a chance to arrive in a given subset at time t if one starts at the point x, satisﬁes the adjoint equation ∂µ/∂t = A∗ µ, where µ at t = 0 is equal to the Dirac measure at x. Here A∗ denotes the adjoint of the operator A acting on distributions.

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Because both the deterministic system and the stochastic perturbation have variable coefﬁcients, one needs a priori estimates which ensure that the solutions of the stochastic differential equation exist with probability one for all time. Similarly, one needs a priori estimates for the existence of an invariant probability measure µ, i.e., a measure which satisﬁes the equation A∗ µ = 0, where the uniqueness of such a measure follows if the inﬁnitesimal generator A is elliptic. The required a priori estimates can be obtained with the use of Lyapunov-like functions W which satisfy inequalities like AW ≤ −1 in the complement of a compact subset of M; see [3].

3 Large deviations If Xj (x) ∼ σj (x) for small , then the principal symbol of A is equal to 2 a(x)(ξ, ξ )/2, in which the inner product a(x) on covectors is deﬁned by a(x)(ξ, η) =

k

σj (x), ξ · σj (x), η,

ξ, η ∈ (Tx M)∗ .

j =1

That is, the partial differential equations are singularly perturbed , which implies, among other things, that they are numerically unstable. The inner product a(x) induces an inner product g(x) = a(x)−1 on the tangent space at x, which deﬁnes a Riemannian structure on the manifold M. The theory of Freidlin and Wentzell [10], [2] states the following. For every smooth curve γ : [0, T ] → M, there exists a real number S(γ ) such that for every δ > 0 and η > 0, there exists an 0 > 0 with the following property. If p denotes the probability that the process x(t) which starts at γ (0) has the property that for every t ∈ [0, T ] the distance from x(t) to γ (t) does not exceed δ, then e−S(γ )+η)/ ≤ p ≤ e−(S(γ )−η)/ 2

2

whenever 0 < || ≤ 0 . The required smoothness is that γ should have a square integrable derivative, and the estimates are uniform when T remains bounded, γ remains in a compact subset of M and S(γ ) remains bounded. Here S(γ ) denotes the “action integral’’ 1 T S(γ ) = g(γ (t))(v(t), v(t))dt, 2 0 where we have written v(t) = γ (t) − b(γ (t)), and η > 0 is arbitrarily small. In other words, the coefﬁcient in the exponent of the small deviations estimate is equal to one half of the integral of the square of the length of the difference between the actual velocity vector and the velocity vector of the deterministic system. Here the length of the tangent vector v at x, with respect to the Riemannian structure g, is equal to g(x)(v, v)1/2 . Of course, one could also include the factor 1/ 2 in the deﬁnition of the functional S, which would amount to replacing the Riemannian structure g by −2 g, or

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replacing the expression a(x)(ξ, ξ )/2 by the principal symbol 2 a(x)(ξ, ξ )/2 of the inﬁnitesimal generator A of the stochastic process. However natural this may be, we follow the custom of using a functional S which does not depend on , and write the factor 1/ 2 in the exponents of the asymptotic estimates of the probabilities. For quite general systems, Freidlin and Wentzell derived similar exponential estimates for the following: • • •

the probability that the process which starts at x leaves a given domain near a given point, the expectation value of the ﬁrst time of exit from the domain, and the invariant probability distribution in the position space.

It is a great challenge for analysts to derive such estimates from the singularly perturbed partial differential equations which are satisﬁed by these quantities. Here “directly’’ would mean “without probabilistic arguments.’’ The results of Freidlin and Wentzell are described in terms of the two-point function V (x, y) = inf {S(γ ) | γ (0) = x, γ (T ) = y}, in which T is arbitrary. It can be proved that this function is Lipschitz continuous in all the variables. Let D be a relatively compact open subset of the domain of attraction of a stable equilibrium s of the deterministic system. Let E be the set of points of the boundary where V (s, z), z ∈ ∂D, is minimal. Then with overwhelming probability the process, which starts at any given point x ∈ D, will leave D near E. Moreover, again with overwhelming probability, the process, after leaving a small neighborhood of s for the last time, stays close to a minimizing curve from s to a point y ∈ E. As we will discuss in the next Section 4, such a curve is equal to the projection to the position space of a solution of the Hamiltonian system on the unstable manifold of (s, 0). Finally, the mean exit time from D is (asymptotically for 2 2 → 0) equal to eV (s,y)/ , modulo factors eη/ with arbitrarily small |η|. Note that because V (s, y) > 0, this means that the mean exit time is exponentially large when 2 is small.

4 A Hamiltonian system If the minimum is attained at a curve γ , then γ satisﬁes the Euler–Lagrange equations [L]i :=

d ∂L ∂L − = 0 in which x = γ (t), v = γ (t), dt ∂vi ∂xi

for the Lagrange function L(x, v) =

1 g(x)(v − b(x), v − b(x)). 2

Moreover, γ (t) and b(γ (t)) have the same g-length, meaning that

(4.1)

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g(x)(v, v)1/2 = g(x)(b(x), b(x))1/2

when x = γ (t), v = γ (t).

(4.2)

Via the Legendre transform, these minimizing curves correspond to the solutions (x(t), ξ(t)) of the Hamiltonian system deﬁned by the function 1 H (x, ξ ) = b(x), ξ + a(x)(ξ, ξ ), 2 where, moreover, H (x(t), ξ(t)) ≡ 0.

(4.3)

Proof. The Legendre transform assigns to an arbitrary function L(x, v) of the position x and the velocity v the function H (x, ξ ) = v, ξ − L(x, v)

(4.4)

of the position x and the momentum vector ξ , in which v is expressed as a function v = v(ξ ) of ξ by means of the equations ∂L(x, v)/∂vi = ξi ,

1 ≤ i ≤ n.

(4.5)

The point is that the Euler–Lagrange equations [L] = 0 for the function L, which express that γ (t) is a stationary curve for the integral T S= L(γ (t), γ (t))dt, 0

are equivalent to the condition that (γ (t), ξ(t)) is a solution curve of the Hamiltonian system deﬁned by the function H , in which ξ(t) = ξ is deﬁned by (4.5). The Hamiltonian system deﬁned by the function H is the system of ordinary differential equations in the (x, ξ )-space, the phase space of classical mechanics, deﬁned by the equations (4.7) and (4.8) below. In our case, 1 g(x)(v − b(x), v − b(x)), (4.6) 2 which yields g(x)(v − b(x)) = ξ or v = b(x) + a(x)ξ , and the formula (4.3) for H (x, ξ ) readily follows. It is proved in [2, Lemma 3.1, p. 120] that (4.2) holds for every t ∈ [0, T ], if γ is a minimizing curve. Because the substitution of ξ = g(x)(v − b(x)) and (4.6) in (4.3) yields that L(x, v) =

1 H = g(x)(v, v − b(x)) − g(x)(v − b(x), v − b(x)) 2 1 1 = g(x)(v, v) − g(x)(b(x), b(x)), 2 2 it follows that the condition (4.2) is equivalent to the condition that H ≡ 0 along the solution curves of the Hamiltonian system.

A well-known general property of the Legendre transform is that the function L can be found back from the function H by means of the formula L(x, v) = x, ξ − H (x, ξ ), where ξ = ξ(v) is the solution of the equation ∂H (x, ξ )/∂ξ = v. In our case we read

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from (4.6) and g(x)(v − b(x)) = ξ , v − b(x) = a(x)ξ that L = a(x)(ξ, ξ )/2. Our recipe for minimizing curves therefore is to look among the solutions (x(t), ξ(t), s(t)) of the system of ordinary differential equations dxi /dt = ∂H (x, ξ )/∂ξi , dξi /dt = −∂H (x, ξ )/∂xi , ds/dt = a(x)(ξ, ξ )/2,

(4.7) (4.8) (4.9)

for those which satisfy x(0) = x, x(T ) = y, H (x(0), ξ(0)) = 0, s(0) = 0, and such that s(T ) is minimal. Note that H (x(0), ξ(0)) = 0 and (4.7) and (4.8) imply that H (x(t), ξ(t)) = 0 for every t. If x(t), 0 ≤ t ≤ T is a minimizing curve from x to y, then V (x, y) = s(T ). For a given point x ∈ M, the solutions of the Hamiltonian system which start at the points (x, ξ ) ∈ T∗ M such that H (x, ξ ) = 0 ﬁll up a so-called Lagrange submanifold of the cotangent bundle T∗ M of M. This is a smoothly immersed n-dimensional submanifold of the 2n-dimensional phase space T∗ M such that the canonical two-form n dξi ∧ dxi i=1

of T∗ M vanishes identically on . If (x(t), ξ(t)) is a solution curve of the Hamiltonian system such that x(0) = x and H (x, ξ(0)) = 0, then for every T > 0 the curve x(t) with 0 ≤ t ≤ T is a candidate for being a minimizing curve from x to y = x(T ), and if this is the case, we have s(T ) = V (x, y). The s(T ) deﬁne a smooth function σ on the open subset + of which consists of the points (x(T ), ξ(T )) with T > 0. The projection from + to the base manifold M (the projection which forgets the ξ -components of the points of the phase space) is locally a diffeomorphism from an open neighborhood 0 in + of the point (y0 , η0 ) onto an open neighborhood M0 of y0 in M if and only if the tangent space of at the point (y0 , η0 ) is transversal to the tangent space of the ﬁber of the projection. If this is the case, then 0 can be written as the set of (y, η(y)) where y varies in M0 and η(y) depends smoothly on y ∈ M0 . In this case s(y) := σ (y, η(y)) deﬁnes a smooth function of y ∈ M0 . Moreover, η(y) = ds(y), which needs an argument. Proof. If γ (t) is a family of solutions of the Euler–Lagrange equations depending smoothly on the parameter , then in the variational equation only the boundary terms survive, viz. T ∂S ∂ ∂γ ∂L ∂γ ∂L = , , Ldt = − . ∂ ∂ 0 ∂ ∂v t=T ∂ ∂v t=0 (This is a classical observation of Hamilton.) If γ (0) = x for all , then the second term on the right-hand side is equal to zero, and recognizing ∂L/∂v as the momentum η(y), we arrive at v, ds(y) = v, η(y) if v = ∂γ (T )/∂ and y = γ (T ). On the other hand,

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∂ ∂T

0

T

∂γ ∂L , , = ∂t ∂v t=T

Ldt = L|t=T

in which the second equality follows from the condition that H = 0 along the corresponding solution curve of the Hamiltonian system; see (4.4) and (4.5). The tangent space of M at y = γ (T ) is spanned by the vectors ∂γ (T )/∂ and ∂γ (T )/∂T , where the γ (t) range over the arbitrary families of solutions of the Euler–Lagrange equations [L] = 0 with γ (0) = x and H = 0 along the corresponding solution of the Hamiltonian system. It follows that v, ds(y) = v, η(y) for every v ∈ Tv M, which means that ds(y) = η(y).

Because H = 0 on , it follows that the function s satisﬁes the nonlinear ﬁrstorder partial differential equation H (y, ds(y)) = 0. If the curves x(t) such that x(0) = x and x(T ) = y ∈ M0 are minimizing, then s(y) = V (x, y) and it follows that the function y → V (x, y) is smooth and satisﬁes the nonlinear ﬁrst-order partial differential equation ∂V (x, y) H x, (4.10) = 0, y ∈ M0 . ∂y Moreover, the description in terms of s(y) leads to a quite explicit local construction of the function y → V (x, y). Remark 4.1. The partial differential equation (4.10) alone is far from sufﬁcient to determine the function y → V (x, y) even if we add the condition that V (x, x) = 0. Remark 4.2. One can also look at the solutions (x(t), ξ(t)), −T ≤ t ≤ 0, of the Hamiltonian system on H = 0 such that x(0) = y, i.e., x(t) ends up at the given point y ∈ M. Here T > 0 and we have shifted the interval [0, T ] to [−T , 0] in order to have a ﬁxed ﬁnal time and a variable initial time (equal to −T ). Under conditions analogous to the conditions leading to (4.10), we have that ∂V (x, y) H x, − (4.11) = 0, x ∈ M0 , ∂x which is the same nonlinear ﬁrst-order partial differential equation as (4.10), but with the vector ﬁeld b(x) replaced by −b(x); see (4.3). If x is a hyperbolic equilibrium point of the unperturbed deterministic system dx/dt = b(x), then the Lagrange manifold has to be replaced by the unstable manifold of the equilibrium point (x, 0) of the Hamiltonian system. Here we note that it is a general fact that the unstable manifold of a hyperbolic equilibrium point of a Hamiltonian system is a Lagrange submanifold of the phase space. We also note that in this case the curve (x(t), ξ(t)) with 0 ≤ t ≤ T is replaced by a curve (x(t), ξ(t)) with −∞ < t ≤ T , and the conditions x(0) = x, H (x(0), ξ(0)) = 0 are replaced by the conditions that x(t) → x and ξ(t) → 0 as t → −∞. The quantity s(T ), which is the candidate for the value of V (x, y), has to be replaced by the limit of s(T ) − s(t) as t → −∞. Under conditions as in the text preceding (4.10), the function y → V (x, y)

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locally satisﬁes the nonlinear ﬁrst-order partial differential equation (4.10) even if x is an equilibrium point of the unperturbed deterministic system. In a similar way, if y is a hyperbolic equilibrium point of the unperturbed deterministic system, then the Lagrange manifold is replaced by the stable manifold of the equilibrium point (y, 0) of the Hamiltonian system, the curve (x(t), ξ(t)) with 0 ≤ t ≤ T is replaced by a curve x(t), ξ(t)) with 0 ≤ t < ∞, and the condition that x(T ) = y is replaced by the conditions that x(t) → y and ξ(t) → 0 as t → ∞. Furthermore, the quantity s(T ), which is the candidate for the value of V (x, y), has to be replaced by the limit of s(t) as t → ∞. Under the natural geometric conditions discussed before, the function x → V (x, y) locally satisﬁes the nonlinear ﬁrst-order partial differential equation (4.11) even if y is an equilibrium point of the unperturbed deterministic system. In general, there may exist x, y ∈ M for which no minimizing curves exist between x and y even if we allow the aforementioned curves which run in inﬁnite time from and to the equilibrium point x and y, respectively. One cause can be that the minimum is attained by a chain of stationary curves from x to y, which runs via a sequence of equilibrium points of the unperturbed deterministic system. It could also happen that the inﬁmum is approached by a sequence of curves which run out of a sequence of compact subsets of M which absorbs M. Often, as in our models for geomagnetic reversals, one assumes conditions which imply that the latter does not occur. Even if minimizing curves exist, it can happen that the Lagrange manifold + is constituted in such a way that for y in a neighborhood of y0 there is a number of different η = η(j ) (y), depending smoothly on y, such that (y, η(j ) (y)) ∈ + , leading to different competing candidates sj (y) = σ (y, η(j ) (y)) for the function y → V (x, y), which is equal to the minimum over j of the functions sj . Let Mj denote the region where V (x, y) is equal to the smooth function sj (y).At the transition between the region Mj and Mk with k = j , the function y → V (x, y) is still continuous, but only piecewise smooth instead of smooth. Such boundaries between the region Mj resemble phase transitions in statistical mechanics. Also the functions sj (y) themselves can develop singularities, for instance, when (y, dsj (y)) approaches points at which is no longer transversal to the ﬁber of the projection onto the base manifold. This phenomenon is analogous to the occurrence of caustics in geometrical optics. This description of the minimizing curves in terms of the Hamiltonian system is not mentioned in [10] and [2]. It is an example of the application of symplectic geometry to variational calculus, and via Freidlin–Wentzell theory to stochastic perturbations of dynamical systems.

5 Hoyng’s model Hoyng’s model for the earth’s magnetic ﬁeld is a stochastic perturbation of a deterministic system which has two stable equilibria (corresponding to the present average magnetic ﬁeld and its opposite), of which the domains of attraction are separated by

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the stable manifold of a saddle point at the origin (corresponding to the zero magnetic ﬁeld). Another important feature of his model is that the stochastic vector ﬁelds are linear. Therefore also the stochastic ﬂuctuations vanish at the origin, and the stochastic process which starts at the origin will remain there with probability one. One of the problems in the theory is to obtain estimates which ensure that the process which gets close to the origin will not remain there for too long a time in such a way that we may consider the origin as stochastically isolated from the complement of the origin. If this holds, then we can take as our conﬁguration space M the complement of the origin in the vector space. (The elements of the vector space V represent the magnetic ﬁelds of the earth.) A heuristic argument that the origin is stochastically isolated from the complement of the origin is that if we replace the unperturbed deterministic vector ﬁeld b(x) by its linear approximation b (o)x at the origin o, then the stochastic differential equation is invariant under the homotheties x → cx for any c > 0. It follows that if we parametrize the complement of the origin by x = es u, in which s ∈ R and u is a point on the unit sphere U , then we obtain a stochastic differential equation on R × U which is invariant under the translations (s, u) → (s +a, u), a ∈ R. Such a stochastic differential equation is complete in the sense that it has a well-deﬁned stochastic process for all positive times, and the completeness for s → −∞ corresponds to the stochastic isolation of the origin in V from the complement M ! R × U of the origin in V . Because the conﬁguration space M is not compact and the domain of attraction of a stable equilibrium point s of b(x) is not relatively compact, we cannot apply the theory of [10], [2] directly to our situation. However, we believe that the arguments can be extended to prove the analogous statements. In particular we think that as a consequence of the fact that the boundary of is equal to the stable manifold of the saddle point at the origin o, the most probable way of leaving one of the domains of attraction is via a neighborhood of o. This corresponds to a heteroclinic connection from the hyperbolic equilibrium point (s, 0) to the equilibrium point (o, 0) of the Hamiltonian system. Moreover, we conjecture that the mean time of exit is asymptotically equal to 2 2 V (s,o)/ e , modulo factors eη/ with arbitrarily small |η|. Here it may be instructive to observe that if we replace the unperturbed deterministic vector ﬁeld b(x) by its linearization b (o)x at the origin, then the invariance of the stochastic differential equation under homotethies implies that V (cx, o) = V (x, o) for every x ∈ V and every c > 0, which means that the function V (x, o) only depends on the direction of the vector x and not on its distance to the origin. Because V (x, o) = 0 when x belongs to the stable manifold of the origin and the function x → V (x, o) is not identically equal to zero, this implies that the function x → V (x, o) is not continuous at the origin o. We conjecture that such a discontinuous behavior of the function x → V (x, o) at the origin is inherited by any vector ﬁelds and the deterministic vector ﬁeld vanishes at the origin. This discontinuity is analogous to the discontinuity at the origin of the mean exit time, which was mentioned in the introduction.

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6 A very simple prototype For the unperturbed deterministic system, we take the system x˙ = x(1 − x 2 ),

y˙ = −2y

in the (x, y)-plane, which has stable equilibria at (1, 0) and (−1, 0), with domains of attraction equal to the right and left half-plane, respectively. The origin (0, 0) is an equilibrium point of saddle point type. Its stable manifold is equal to the y-axis, the common boundary of the two domains of attraction of the stable equilibrium points. (See Figure 1.) For the stochastic vector ﬁelds, we take x∂/∂x, y∂/∂y, x∂/∂y, y∂/∂x, which leads to the Hamiltonian function H = x(1 − x 2 )p − 2yq + (x 2 + y 2 )(p 2 + q 2 )/2. Figure 2 shows the projections γ on the conﬁguration space (the (x, y)-plane) of the solutions on the unstable manifold of the point ((x, y), (p, q)) = ((1, 0), (0, 0)). Note that the minimizing curves starting from the stable equilibrium point (1, 0) are parts of these curves γ . In order to see more clearly what happens with these curves near the origin, we show an enlargement there in Figure 3. An interesting feature of this picture is the appearance of caustics. Using symmetry arguments, one obtains that the minimizing curves are parts of the curves γ which are stopped at the x- and the y-axis. This leads to a much “quieter’’ picture, but still with a caustic near the origin. There is a curve C inside caustic with its end point at the cusp, which is the boundary between the region where s1 (x, y) < s2 (x, y) and V ((1, 0), (x, y)) = s1 (x, y) and the region where s2 (x, y) < s1 (x, y) and V ((1, 0), (x, y)) = s2 (x, y), where s1 (x, y) and s2 (x, y) are two competing smooth functions of (x, y) as described in Section 4. Correspondingly, the function (x, y) → V ((1, 0), (x, y)) is continuous, but has a jump in its gradient along the curve C. Note that in the notation the points x and y in M in Section 4 have been replaced by (1, 0) and (x, y), respectively. The necessity of competing minimal curves can be explained as a consequence of the fact that S(γ ) = ∞ if γ runs on the x-axis from (1, 0) to (0, 0). This implies that the minimizing curve from (1, 0) to (x, 0) with small positive x must make a detour in the upper or lower half-plane. On the other hand, if x is close to 1, the minimizing curve from (1, 0) to (x, 0) runs over the x-axis. The transition appears in the region where the minimizing curves which run along the x-axis compete with the minimizing curves which make a detour. (See Figure 4.) Figure 5 shows the most probable exit paths, both from the right half-plane and from the left half-plane. These exit via the origin. It can be proved that they not only have to make a detour in the upper and lower half-plane, but actually approach the origin tangentially to the y-axis, the stable manifold of the saddle point at the origin. In the higher-dimensional model, they generically approach the saddle point from the direction of the eigenvector of the linearized deterministic system corresponding to the negative eigenvalue which is closest to zero.

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Just for fun, Figure 6 shows the projections on the conﬁguration space on the unstable manifold of the point ((1, 0), (0, 0)) if we take only x∂/∂y and y∂/∂x as the stochastic vector ﬁelds. In this case the Hamiltonian function is equal to

H = x 1 − x 2 p − 2yq + x 2 q 2 + y 2 p 2 /2. Actually, this was the ﬁrst toy example which Peter Hoyng gave me. The ﬁgure eights straddling the x-axis correspond to solutions of the Hamiltonian system which run to inﬁnity in a ﬁnite time, while their projections in the conﬁguration space converge to a limit point on the x-axis, running faster and faster through an inﬁnite sequence of decreasing ﬁgure eights.

References [1] K. D. Elworthy, Stochastic Differential Equations on Manifolds, Cambridge University Press, London, 1982. [2] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, SpringerVerlag, New York, 1998. [3] R. Z. Has’minskiˇı (Khas’minskii), Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. [4] P. Hoyng, M.A. J. H. Ossendrijver, and D. Schmitt, The geodynamo as a bistable oscillator, Geophys. Astrophys. Fluid Dynam., 94 (2001), 263–314. [5] P. Hoyng, D. Schmitt, and M. A. J. H. Ossendrijver, A theoretical analysis of the observed variablity of the geomagnetic dipole ﬁeld, Phys. Earth Planetary Interiors, 130 (2002), 143–157. [6] N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. [7] R. T. Merrill, M. W. McElhinny, and P. L. McFadden, The Magnetic Field of the Earth, Academic Press, New York, 1966. [8] B. Øksendal, Stochastic Differential Equations, 1st ed., Springer-Verlag, Berlin, 1989; 5th ed., 1998. [9] M. A. J. H. Ossendrijver, P. Hoyng, and D. Schmitt, Stochastic excitation and memory of the solar dynamo, Astron. Astrophys., 313 (1996), 938–948. [10] A. D. Wentzell and M. I. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys, 25-1 (1970), 1–55.

Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization∗ Kurt Ehlers1 , Jair Koiller2 , Richard Montgomery3 , and Pedro M. Rios4 1 Department of Mathematics

Truckee Meadows Community College 7000 Dandini Boulevard Reno, NV 89512-3999 USA [email protected] 2 Fundação Getulio Vargas Praia de Botafogo 190, Rio de Janeiro 22253-900 Brazil [email protected] 3 Mathematics Department University of California at Santa Cruz Santa Cruz, CA 95064 USA [email protected] 4 Department of Mathematics University of California at Berkeley Berkeley, CA 94720 USA [email protected] Dedicated to Alan Weinstein on his 60th birthday. Abstract. A nonholonomic system, for short “NH,’’ consists of a conﬁguration space Qn , a Lagrangian L(q, q, ˙ t), a nonintegrable constraint distribution H ⊂ T Q, with dynamics governed by Lagrange–d’Alembert’s principle. We present here two studies, both using adapted moving frames. In the ﬁrst we explore the afﬁne connection viewpoint. For natural Lagrangians L = T −V , where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇N H which mimics Levi-Civita’s. Local geometric invariants are obtained by Cartan’s method of equivalence. As an example, we analyze Engel’s (2–4) distribution. This is the ﬁrst such study for a distribution that is not strongly nonholonomic. In the second part we study ∗ The authors thank the Brazilian funding agencies CNPq and FAPERJ: a CNPq research

fellowship (JK), a CNPq post-doctoral fellowship at Berkeley (PMR), a FAPERJ visiting fellowship to Rio de Janeiro (KE). (JK) thanks the E. Schrödinger Institute, Vienna, for ﬁnancial support during Alanfest and the Poisson Geometry Program, August 2003.

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G-Chaplygin systems; for those, the constraints are given by a connection φ : T Q → Lie(G) on a principal bundle G → Q → S = Q/G and the Lagrangian L is G-equivariant. These systems compress to an almost Hamiltonian system (T ∗ S, H φ , N H ), N H = can +(J.K), with d(J.K) = 0 in general; the momentum map J : T ∗ Q → Lie(G) and the curvature form K : T Q → Lie(G)∗ are matched via the Legendre transform. Under an s ∈ S dependent time reparametrization, a number of compressed systems become Hamiltonian, i.e., N H is sometimes conformally symplectic. A necessary condition is the existence of an invariant volume for the original system. Its density produces a candidate for conformal factor. Assuming an invariant volume, we describe the obstruction to Hamiltonization. An example of a Hamiltonizable system is the “rubber’’ Chaplygin’s sphere, which extends Veselova’s system in T ∗ SO(3). This is a ball with unequal inertia coefﬁcients rolling without slipping on the plane, with vertical rotations forbidden. Finally, we discuss reduction of internal symmetries. Chaplygin’s “marble,’’ where vertical rotations are allowed, is not Hamiltonizable at the compressed T ∗ SO(3) level. We conjecture that it is also not Hamiltonizable when reduced to T ∗ S 2 . “Nonholonomic mechanical systems (such as systems with rolling contraints) provide a very interesting class of systems where the reduction procedure has to be modiﬁed. In fact this provides a class of systems that give rise to an almost Poisson structure, i.e., a bracket which does not necessarily satisfy the Jacobi identity’’ (Marsden and Weinstein [2001]).

1 Introduction and outline Cartan’s moving frames method is a standard tool in Riemannian geometry.1 In analytical mechanics, the method goes back to Poincaré [1901], perhaps earlier, to Euler’s rigid body equations, perhaps much earlier, to the cave person who invented the wheel. Let q ∈ Rn be local coordinates on a conﬁguration space Qn , and consider a local frame, deﬁned by an n × n invertible matrix B(q), ∂ ∂ = bij , ∂πj ∂qi

n

Xj =

π˙ j Xj =

i=1

q˙i

∂ , ∂qi

π˙ = A(q)q, ˙

A = B −1 .

(1.1) In mechanical engineering (Hamel [1949], Papastavridis [2002]), moving frames are disguised under the keyword quasi-coordinates, nonexisting entities π such that ∂f ∂qi ∂f ∂f = = bij = Xj (f ). ∂πj ∂qi ∂πj ∂qi i

i

Let {i }i=1,...,n be the dual coframe to {Xj }, i = “dπi ’’ =

j

aij dqj .

1 Cartan [1926]; there is a recent English translation from the Russian translation (Cartan

[2001]). One of the most important applications was the construction of characteristic classes by Alan’s advisor, S. S. Chern. Our taste for moving frames in mechanics is a small tribute to his inﬂuence.

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1.1 Moving frames: Lagrangian and Hamiltonian mechanics The Euler–Lagrange 1-form may be rewritten as:2 n ∂L d ∂L − − Fr dqr dt ∂ q˙r ∂qr r=1 ⎛ ⎞ n n n ∗ ∗ ∗ ∂L ∂L d ∂L ⎝ = − + γkji π˙ j − Rk ⎠ k = 0, dt ∂ π˙ k ∂πk ∂ π˙ i k=1

i=1

(1.2)

j =1

where L∗ (q, π˙ , t)= L(q, B(q)π˙ , t) is the Lagrangian written in “quasi-coordinates’’ and Rk = s Fs bsk are the covariant components of the total force (external, Fext , and constraint force λ). The so-called Hamel transpositional symbols γkji = γjik = ns,=1 bsk bj (∂ais /∂q − ∂ai /∂qs ) are precisely the moving frame structure coefﬁcients (Koiller [1992]). If the velocities are restricted to a subbundle H ⊂ T Q, a constraint force λ appears. The d’Alembert–Lagrange principle3 implies that λ belongs to the anihilator Ho ⊂ T ∗ Q of H, hence exerting zero work on admissible motions q˙ ∈ H: [L] :=

d ∂L ∂L − − Fext = λ ∈ Ho , dt ∂ q˙ ∂q

q˙ ∈ H.

(1.3)

Using moving frames, constraints can be eliminated directly. If Ho is spanned by the last r forms J , s + 1 ≤ J ≤ n (s = n − r), then equations of motion result from setting the ﬁrst s Euler–Lagrange differentials equal to zero: n n d ∂L∗ ∂L∗ ∂L∗ i − + γkj π˙ j − Fkext = 0 dt ∂ π˙ k ∂πk ∂ π˙ i i=1

(1 ≤ k ≤ s).

(1.4)

j =1

Strikingly, the Hamiltonian counterparts of (1.2) and (1.4) are simpler, although terms of the less known.4 The philosophy is to ﬁght against Darboux’s dictatorship. In local coframe {i }1≤i≤n , any element pq ∈ T ∗ Q can be written as pq = mi i (q). The natural 1-form α on T ∗ Q keeps the familiar confusing expression α := pdq = m. Consequently, the canonical symplectic form := dα may be written as

:= dp ∧ dq = dm ∧ + md.

(1.5)

The second term md, which deviates from Darboux’s format, is not a nuisance; it carries most valuable information. For instance, Kostant–Arnold–Kirillov–Souriau’s bracket in T ∗ G, G a Lie group, can be immediately visualized: take a (left- or right-) 2 Atributed to Hamel, but certainly known by Poincaré. Quasi-coordinates can be found in

Whittaker [1937] and were ﬁrst used in mechanics by Gibbs; see Pars [1965]. 3 According to Sommerfeld [1952], this gives the most natural foundation for mechanics.

4 A “moving frames operational system’’ for Hamiltonian mechanics in T ∗ Q was given in

Koiller, Rios, and Ehlers [2002].

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invariant coframe and apply H. Cartan’s “magic formula’’ on d. So moving frames are ideally suited when a Lie symmetry group G is present.5 Example: Mechanics in SO(3) To ﬁx notation, we now review the standard example. The Lie algebra basis Xi ∈ sO(3) = TI SO(3), i = 1, 2, 3 (inﬁnitesimal rotations around the x, y, z-axis at the identity) can either be right or left transported, producing moving frames on SO(3) denoted {Xir } and {Xi }, respectively. Let {ρi }1≤i≤3 and {λi }1≤i≤3 denote their dual coframes (right- and left-invariant forms in SO(3)). To represent angular momenta, we use Arnold’s notations (Arnold [1989]): capital letters mean objects in the body frame, lowercase objects in the space frame. Thus for instance, = RL, where L is the angular momentum in the body frame and is the angular momentum in space; likewise ω = R relate the angular velocities. The canonical 1-form in T ∗ SO(3) is given by α = 1 ρ1 + 2 ρ2 + 3 ρ3 = L1 λ1 + L2 λ2 + L3 λ3 , so

can = =

di ρi + 1 dρ1 + 2 dρ2 + 3 dρ3 dLi λi + L1 dλ1 + L2 dλ2 + L3 dλ3 ,

where by Cartan’s structure equations, dλ1 = −λ2 ∧ λ3 , . . . and dρ1 = ρ2 ∧ ρ3 , . . . (cyclic). A left-invariant metric is given by an inertia operator L = A . Euler’s rigid body equations follow immediately. Poisson action of S 1 on SO(3) Consider the left S 1 action on SO(3) given by exp(iφ) · R := S(φ)R, where S(φ) is the rotation matrix about the z-axis: ⎛ ⎞ ⎛ ⎞ cos(φ) − sin(φ) 0 0 −1 0 S(−φ)S (φ) = ⎝ 1 0 0 ⎠ = X3 . S(φ) := ⎝ sin(φ) cos(φ) 0 ⎠ , 0 0 1 00 0 Two matrices are in the same equivalence class iff their third rows, which we denote by γ , called the Poisson vector, are the same: R1 ∼ R2 ⇐⇒ R1−1 kˆ = R2−1 kˆ = γ ∈ S 2 . ˆ The So we have a principal bundle π : SO(3) → S 2 , γ = π(R) = R −1 kˆ = R † k. derivative of π is −1 ˙ = −(R −1 RR ˙ −1 )k = −(R −1 R)(R ˙ γ˙ = π∗ (R) )k = −[ ]γ = − × γ = γ × , (1.6) 5 As we learned from Alan at the banquet, the etymology for symplectic is “capable to join,’’

themes and people. The latter is one of the most important aspects of the symplectic “creed.’’ Provocation: taking moving frames, adapted to some other mathematical structure for Q, would the non-Darboux term provide a local symplectic invariant?

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where we used the customary identiﬁcation6 [ ] ∈ sO(3) ↔ ∈ R3 , Arnold [1989]. The lifted action to T ∗ SO(3) has momentum map J = 3 . Connection on S 1 → SO(3) → S 2 Take the usual bi-invariant metric , on SO(3) so that both {Xi } and {Xir } are orthonormal moving frames. The tangent vectors to the ﬁbers are (d/dφ)S(φ) · R = right X3 . Consider the mechanical connection associated to , , namely, that horizonright tal and vertical spaces are orthogonal. The horizontal spaces are generated by X1 right and X2 . The connection form is φ = ρ3 . The horizontal lift of γ˙ to R is the tangent vector R˙ such that (1.7)

hor = R −1 R˙ = [γ˙ × γ ]. Note that hor is the −90 degrees rotation of γ˙ inside Tγ S 2 . The curvature of this connection κ = dρ3 is the area form of the sphere. Reduction of S 1 symmetry It is convenient for reduction to use (a, 3 ), a ∈ R3 , a ⊥ γ , L := a × γ + 3 γ ,

(1.8)

where a is a vector perpendicular to γ . The vector a has an intrinsic meaning: Consider a moving frame e1 , e2 in S 2 , with dual coframe θ1 , θ2 . Then vγ = v1 e1 + v2 e2 parametrizes T S 2 , and pγ = a · dγ = p1 θ1 + p2 θ2 parametrizes T ∗ S 2 , a = γi dγi = 0 denotes both an element of T ∗ S 2 and the p1 e1 + p2 e2 . Here a · dγ , canonical 1-form. Our parametrization for SO(3) is R(φ, γ ) = S(φ) · R(γ ), R(γ ) = rows(e1 , e2 , γ ). Then L = p2 e1 − p1 e2 + 3 γ corresponds to = (p2 , −p1 , 3 ) along the section φ = 0. The right-invariant forms are compactly represented as ρ3 = dφ − (de1 , e2 ),

ρ1 + iρ2 = −i exp(iφ)(θ1 + iθ2 ).

(1.9)

Lifting v ∈ T S 2 to an horizontal vector in T SO(3) is simple:

hor = [(v1 e1 + v2 e2 ) × γ ] = [v2 e1 − v1 e2 ] or

hor(v) = v2 X1r − v1 X2r . (1.10)

Hence any vector R˙ ∈ T SO(3) can be written as R˙ = ω1 X1 + ω2 X2 + ω3 X3 with ω1 = v2 , ω2 = −v1 . Any covector pR ∈ T ∗ SO(3) can be written as pR = p1 π ∗ (θ1 ) + p2 π ∗ (θ2 ) + 3 ρ3 . The reduced symplectic manifold J −1 (3 )/S 1 ≡ T ∗ S 2 can be explicitly constructed, taking the section φ = 0. Let i : T ∗ S 2 → T ∗ SO(3), i(γ , p1 , p2 ) = (R(γ ), ),

= (p2 , −p1 , 3 ).

(1.11)

6 We will drop the [•] and • in what follows and mix all notation, hoping no confusion will arise. Equation (1.6) is one half of every system of ODEs for S 1 -equivariant mechanics in

SO(3). Of course, we also obtain γ˙ = − × γ by differentiating Rγ = k (we could use the notation γ = K, but we won’t).

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Then from (1.9) we get i ∗ ρ2 = −θ1 , i ∗ ρ1 = θ2 , and i ∗ d = di ∗ yields i ∗ dρ1 = dθ2 ,

i ∗ dρ2 = −dθ1 ,

i ∗ dρ3 = i ∗ ρ1 ∧ i ∗ ρ2 = −θ2 θ1 = θ1 θ2 .

We get immediately

red = i ∗ ( T ∗ SO(3) ) = d(p1 θ1 + p2 θ2 ) + 3 area = can + 3 areaS2 . (1.12) T ∗S2 T ∗S2 All references to the moving frame disappear, but the expression can = d(p1 θ1 + T ∗S2 p2 θ2 ) suggests that whenever a natural mechanical system in T ∗ SO(3) reduces to T ∗ S 2 ≡ T S 2 , there is a preferred choice for the moving frame {e1 , e2 }γ : namely, that which diagonalizes the Legendre transform Tγ S 2 → Tγ∗ S 2 ≡ Tγ S 2 of the reduced (Routh) Lagrangian. 1.2 Nonholonomic systems A NH system (Q, L, H) consists of a conﬁguration space Qn , a Lagrangian L : T Q × R → R, and a totally nonholonomic constraint distribution H ⊂ T Q. The dynamics are governed by Lagrange–d’Alembert’s principle.7 Usually L is natural, L = T − V , where T is the kinetic energy associated to a Riemannian metric , , and V = V (q) is a potential. By totally nonholonomic, we mean that the ﬁltration H ⊂ H1 ⊂ H2 ⊂ · · · ends in T Q. Each subbundle Hi+1 is obtained from the previous one by adding to Hi combinations of all possible Lie brackets of vector ﬁelds in Hi . To avoid interesting complications we assume that all have constant rank. Equivalently, let Ho ⊂ T ∗ Q the codistribution of “admissible constraints’’ annihilating H; dually, one has a decreasing ﬁltration of derived ideals ending in zero. Internal symmetries of NH systems: Noether’s theorem An internal symmetry occurs whenever a vector ﬁeld ξQ ∈ H preserves the Lagrangian. For natural systems ξQ is a Killing vector ﬁeld for the metric. Noether’s theorem from unconstrained mechanics remains true. The argument (see Arnold, Kozlov, and Neishtadt [1988]) goes as follows: denote by φξ (s) the one-parameter d group generated by ξ and let φ(s, t) = φξ (s) · q(t), so φ = ds φ = ξQ (φ), where q(t) is chosen as a trajectory of the nonholonomic system. Differentiating with red spect to s the identitly L(φ(s, t), dt φ(s, t)) = const after a standard integration by d ∂L parts we get dt ( ∂ q˙ φ ) = [L]φ . This vanishes precisely when φ = ξQ ∈ H, so Iξ :=

∂L ∂ q˙

· ξ = const.

7 “Vakonomic’’ mechanics uses the same ingredients, but the dynamics are governed by the

variational principle with constraints, and produce different equations; see, e.g., Cortés, de Léon, de Diego, and Martínez [2003]. The equations coincide if and only if the distribution is integrable. In spite of many similarities, there are striking differences between NH and holonomic systems. For instance, NH systems do not have (in general) a smooth invariant measure. Necessary and sufﬁcient conditions for the existence of the invariant measure were ﬁrst given (explicitly in coordinates) by Blackall [1941].

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External symmetries: G-Chaplygin systems External (or transversal) symmetries occur when group G acts on Q, preserving the Lagrangian and the distribution H, meaning that g∗ Hq = Hgq . In the most favorable case one has a principal bundle action Gr → Qn → S m , m + r = n, where H forms the horizontal spaces of a connection with 1-form φ : T Q → Lie(G). These systems are called G-Chaplygin.8 Terminology ´ Since Bates and Sniatycki [1993], and Bloch, Krishnaprasad, Marsden, and Murray [1996], several authors have called attention to these two types of symmetries. Re´ duction of internal symmetries was already described in Sniatycki [1998]. To stress the difference, reduction of external symmetries is called compression here. The word reduction will be used for internal symmetries. LR systems Veselov and Veselova [1986], Veselov and Veselova [1988] considered Lie groups Q = G with left-invariant metrics, with constraint distributions given by right translation of D ⊂ Lie(G), i.e., the constraints are given by right-invariant forms. For a LR-Chaplygin system, in addition there is a decomposition Lie(G) = Lie(H ) ⊕ D, where H is a Lie subgroup such that Ad h−1 D = h−1 Dh = D. Therefore, H → G → S = G/H is a H -Chaplygin system; the base S is the homogeneous space of cosets H g. Fedorov and Jovanovic [2003] considered the case where G is compact and Lie(H ) is orthogonal to D with respect to the bi-invariant metric.9 Compression of G-Chaplygin systems From symmetry, it is clear that the Lagrange–d’Alembert equations compress to the base T S.10 In covariant form, the dynamics take the form [Lφ ] = F (s, s˙ ), where Lφ (s, s˙ ) = L(s, h(˙s )) is the compressed Lagrangian in T S; h(˙s ) is the horizontal 8 A“historical’’remark (by JK). Chaplygin considered the abelian case. During a post-doctoral

year in Berkeley, way back in 1982, I became interested in NH systems with symmetries. Alan directed me to two wonderful books: Hertz [1899] Foundation of Mechanics and Neimark and Fufaev [1972]. In the latter I learned about (abelian) Chaplygin systems, presented in coordinates. I said to Alan that I would like to examine nonabelian group symmetries, and Alan immediately made a diagram on his blackboard, and told me: “well, then, the constraints are given by a connection on a principal bundle.’’ This was the starting point of Koiller [1992]. 9 These conditions are not met in the marble and rubber Chaplygin spheres (see Section 3.2); however, Veselov’s result (Theorem 3.3 below) on invariant volume forms still holds. 10 The full dynamics can be reconstructed from the compressed solutions, horizontal lifting the trajectories via φ, since the admissible paths are horizontal relative to the connection. This last step is not “just’’ a quadrature; in the nonabelian case, a path-ordered integral is in order. For G = SO(3), Levi [1996] found an interesting geometric construction.

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lift to any local section and F is a pseudogyroscopic force.11 In order to write F explicitly, take group quasi-coordinates (s, s˙ , g, π˙ ). Write q = gσ (s), with g ∈ G and a local section σ (s) of Q → S. Fix a basis Xk for the Lie algebra, [XK , XL ] = J X , X(π cKL ˙) = π˙ I XI . Any tangent vector q˙ ∈ Tσ (s) Q can be written as J ˙ ))·σ (s). Horizontal vectors are represented by π˙ = b(s)·˙s , where q˙ = dσ (s)·˙s +X((π b(s) is an r ×m matrix. The connection 1-form may be written as φ(q) ˙ = π˙ −b(s)· s˙ . Then ⎛ ⎞ ∗ m r r b ∂L b K ⎠ ⎝ Ki − Kj + [Lφ ] = F (s, s˙ ), F = s˙j . bU i bVj cU V ∂ π˙ k ∂qj ∂qi K=1

j =1

U,V =1

(1.13) 1.3 Main results Using the moving frames method we present results on two aspects of nonholonomic systems. • •

Cartan’s equivalence, using Cartan’s geometric description of NH systems via afﬁne connections (Cartan [1928]). The objective is to ﬁnd all local invariants. Chaplygin systems: compression of external symmetries, reduction of internal symmetries. The objective is to generalize Chaplygin’s “reducing factor’’ method (Chaplygin [1911]), namely, verify if Hamiltonization is possible (via conformally symplectic structures).

Results on Cartan’s equivalence In Section 2 we analyze NH systems under the afﬁne connection perspective. We pursue the (local) classiﬁcation program proposed by Cartan [1928] using his equivalence method. See Koiller, Rodrigues, and Pitanga [2001] and Tavares [2002] for a rewrite of Cartan’s paper in modern language. Cartan’s method of equivalence is a powerful method for uncovering and interpreting all differential invariants and symmetries in a given geometric structure. In Ehlers [2002] NH systems in a 3-manifold with a contact distribution were classiﬁed. Here we go one step further, looking at Engel’s distribution in 4-manifolds (see deﬁnition below). Our results are summarized in Theorem 2.3. The “role model’’ here is the rolling penny example (no pun intended). This is the ﬁrst such study for a distribution that is not strongly nonholonomic. Next in line is studying the famous Cartan 2–3–5 distribution. Results on G-Chaplygin systems Instead of using (1.13) in TS, we may describe the compressed system in T ∗ S as an 11 This nonholonomic force represents, philosophically, a concealed force in the sense of Hertz

[1899], having a geometric origin. This force vanishes in some special cases, not necessarily requiring the constraints to be holonomic. Equivalently, the dynamics in T S is the geodesic spray of a modiﬁed afﬁne connection. One adds to the induced Levi-Civita connection in T S a certain tensor B(X, Y ). This NH connection in general is nonmetric (Koiller [1992]).

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almost Hamiltonian system12 iX N H = dH,

H = H φ : T ∗ S → R,

∗

T S

N H = can + (J.K),

(1.14)

where H φ is the Legendre transform of the compressed Lagrangian. (J.K) is a semibasic 2-form on T ∗ S which in general is not closed. As one may guess, J is the momentum map, and K is the curvature of the connection. Ambiguities cancel, since J is Ad ∗ -equivariant while K is Ad-equivariant. The construction is independent of the point q on the ﬁber over s. Under an s ∈ S dependent time reparametrization, dτ = f (s)dt, several interesting compressed G-Chaplygin systems become Hamiltonian. A necessary condition is the existence of an invariant volume (Theorem 3.3) whose density F produces a candidate f = F 1/(m−1) , m = dim(S) for a conformal factor. Chaplygin’s “rubber’’ ball (vertical rotations forbidden) is, as far we know, a new example, and generalizes the well-known Veselova system in SO(3) (Proposition 3.6). We describe the obstruction to Hamiltonization as the 2-form iX d(f N H ) (Theorem 3.4) and we discuss further reduction by internal symmetries. An example of the latter situation is Chaplygin’s “marble’’ (a hard ball with unequal inertia coefﬁcients rolling without slipping on the plane). It is non-Hamiltonizable in T ∗ SO(3), and our calculations suggest that it is also non-Hamiltonizable when reduced to T ∗ S 2 (Theorem 3.8). Compare with Borisov and Mamaev [2001]. What does Hamiltonization accomplish? Why do we focus so much on the question of Hamiltonizability? The example of the reduced equations for Chaplygin’s skate (after a two-dimensional Euclidean symmetry is removed) shows that changing time scale in a nonholonomic systems can completely change its character. In this example (see, e.g., Koiller [1992]) the fully reduced equations of motion are not Hamiltonian because every solution is asymptotic in forward and backward time to a point which depends on which solution you choose. However, after rescaling time the fully reduced equations become Hamiltonian, namely, the harmonic oscillator. However, this Hamiltonian vector ﬁeld is incomplete because along one of the coordinate axes, the time rescaling is not deﬁned.13 In light of this example, why is time rescaling interesting? The answer is that it is interesting mostly in the context of integrability, where no singularities are removed in the phase space. See Section 3.

2 Nonholonomic geometry: Cartan equivalence A Cartan nonholonomic structure is a triple (Q, G = ·, ·, H), where Q is an ndimensional manifold endowed with a Riemannian metric G and a rank r totally 12 For details, see Koiller, Rios, and Ehlers [2002], Koiller and Rios [2001]. The Hamiltonian

compression for Chaplygin systems was ﬁrst explored, in the abelian case, by Stanchenko [1985]. The nonclosed term was described as a semibasic 2-form, depending linearly on the ﬁber coordinate in T ∗ S, but its geometric content was not indicated there. 13 We thank one of the referees for this observation.

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nonholonomic distribution H. Our motivation for studying such a structure is a free particle moving in Q, nonholonomically constrained to H, with kinetic energy T = 12 ·, ·. The nonholonomic geodesic equations are obtained by computing accelerations using the Levi-Civita connection associated with G and orthogonally projecting the result onto H. The projected connection is called a nonholonomic connection (Lewis [1998]), and was introduced by Cartan [1928]. A distribution H is strongly nonholonomic if any basis of vector ﬁelds spanning H on U ⊂ Q, together with their Lie brackets, span the entire tangent space over U . The equivalence problem for nonholonomic geometry was revisited in Koiller, Rodrigues, and Pitanga [2001] and the generalization to arbitrary nonholonomic distributions was discussed. Engel manifolds provide the simplest example involving distributions that are not strongly nonholonomic.14 The main question we address is the following. Given two nonholonomic struc¯ H), ¯ is there a (local) diffeomorphism f : U ⊂ Q → ¯ G, tures (Q, G, H) and (Q, ¯ carrying nonholonomic geodesics in Q to nonholonomic geodesics in Q? ¯ U¯ ⊂ Q In Cartan’s approach, this question is recast as an equivalence problem. The nonholonomic structure is encoded into a subbundle of the frame bundle over Q called a G-structure. The diffeomorphism f exists if the two corresponding G-structures are locally equivalent. Necessary and sufﬁcient conditions for the G-structures to be equivalent are given in terms of differential invariants found using the method of equivalence. Outline Our main example is the equivalence problem for nonholonomic geometry on an Engel manifold. Let Q be a four-dimensional manifold and H a rank two distribution. H is an Engel distribution if and only if, for any vector ﬁelds X and Y locally spanning H, and some functions a, b : Q → R, the vector ﬁelds X, Y , Z = [X, Y ], and W = a[X, Z] + b[Y, Z] form a local basis for T Q. By an Engel manifold, we mean a fourdimensional manifold endowed with an Engel distribution. We begin by describing the nonholonomic geodesic equations. In the spirit of Cartan’s program, we express them in terms of connection 1-forms and (co)frames adapted to the distribution. This formulation is particularly well suited to the problem at hand; the nonholonomic geodesic equations are obtained by writing the ordinary geodesic equations in terms of the Levi-Civita connection 1-form and crossing out terms corresponding to directions complementary to H. We then set up the equivalence problem for nonholonomic 14 Historical remarks. Cartan [1928] introduced the equivalence problem for nonholonomic

geometry and studied the case of manifolds endowed with strongly nonholonomic distributions. In his address, Cartan warned against attempts to study other cases because of the “plus compliqués’’ computations involved. In the meantime strides have been made in the equivalence method by Robert Gardner and his students that allow computations to be made at the Lie algebra level rather than at the group level (Gardner [1989]). This together with symbolic computation packages such as MathematicaTM make equivalence problems tractable in many important cases. See Gardner [1989], Bryant [1994], Montgomery [2002], Grossman [2000], Ehlers [2002], Hughen [1995], and Moseley [2001] for some applications.

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geometry and give a brief description of the equivalence method as it is applied to our main example. We conclude this section by applying the method of equivalence to the case of nonholonomic geometry on an Engel manifold. We derive all differential invariants associated with the nonholonomic structure and show that the symmetry group of such a structure has dimension at most four. 2.1 Nonholonomic geodesics: Straightest paths Totally nonholonomic distributions A distribution H is a rank r vector subbundle of the tangent bundle T (Q) over Q. Let H1 = H + [H, H] and Hi = [H, Hi ], and consider the ﬁltration H ⊂ H1 ⊂ · · ·Hi ⊂ · · · ⊂ T Q. H is totally nonholonomic if and only if, for some k, Hk = T Q at all points in Q. For the present discussion we will assume that each Hi has constant rank over Q. As a speciﬁc example, consider the Engel distribution H on R4 with coordinates ∂ ∂ ∂ ∂ , X2 = ∂x + w ∂y + y ∂z }. There are, in fact, local (x, y, z, w), spanned by {X1 = ∂w coordinates on any Engel manifold so that the distribution is given by this normal form, see Montgomery [2002]. Then {X1 , X2 , X3 = [X1 , X2 ]} spans the three-dimensional distribution H1 , and {X1 , X2 , X3 , X4 = [X2 , X3 ]} spans the entire T R4 . A path c : R → Q is horizontal if c(t) ˙ ∈ Hc(t) for all t. Chow’s theorem implies that if H is totally nonholonomic, then any two points in Q can be joined by a horizontal path (see Montgomery [2002]). At the other extreme, the classical theorem of Frobenius implies that H is integrable, which is to say that Q is foliated by submanifolds whose tangent spaces coincide with H at each point, if and only if [Xi , Xj ] ∈ H for all i and j (Warner [1971]). In what follows we will need a description of distributions in terms of differential ideals. Details can be found in Warner [1971] or Montgomery [2002]. Let I = H⊥ be the ideal in ∗ (Q) consisting of the differential forms annihilating H. If H is rank r, then I is generated by n − r independent 1-forms. The ﬁrst derived ideal of I is the ideal (I) := {θ ∈ I|dθ ≡ 0 mod(I)}. (2.1) If we set I (0) = I and I (n+1) = (I (n) ) we obtain a decreasing ﬁltration I = I (0) ⊃ I (1) ⊃ · · · ⊃ 0. The ﬁltration terminating with the 0 ideal is equivalent to the assumption that the distribution is completely nonholonomic. We note that I (j ) = (Hj )⊥ for j = 1, but this is not true in general for j > 1 (see Montgomery [2002]). At the other extreme, the differential ideal version of the Frobenius theorem implies that H is integrable if and only if (I) ⊂ I (Warner [1971]). For the Engel example, the 1-forms η1 = dy − wdx and η2 = dz − ydx generate the ideal I. Notice that dη2 = η1 ∧ dx so η2 ∈ I (1) but dη1 cannot be written in terms of η1 or η2 ; therefore, η1 ∈ / I (1) .

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The nonholonomic geodesic equations There are two different geometries commonly deﬁned on a nonholonomic structure (Q, G = ·, ·, H): sub-Riemannian geometry and nonholonomic geometry. In sub-Riemannian geometry one is interested in shortest √ paths. The length of a path c : [a, b] → Q joining points x and y is (c) = c, ˙ cdt. ˙ The distance from x to y is d(x, y) = inf((c)) taken over all horizontal paths joining x to y. In nonholonomic geometry one is interested in straightest paths, which are solutions to the nonholonomic geodesic equations. Hertz [1899] was the ﬁrst to notice that shortest = straightest unless the constraints are holonomic.15 The nonholonomic geodesic equations are obtained by computing the acceleration of a horizontal path c : R → Q using the Levi-Civita connection associated with G and orthogonally projecting the result onto H. It is convenient to adopt the following indicial conventions: 1 ≤ I, J, K ≤ n, 1 ≤ i, j, k ≤ r (= rank(H)), r + 1 ≤ ν ≤ n.

(2.2)

Let e = {eI } be a local orthonormal frame for which the ei span H, and let η = {ηI } be the dual coframe deﬁned by ηI (eJ ) = δI J , the Kronecker delta function. We note that the ην annihilate H and the metric, restricted to H is g|H = η1 ⊗ η1 + · · · + ηr ⊗ ηr . The Levi-Civita connection can be expressed in terms of local 1-forms ωI J = −ωJ I satisfying Cartan’s structure equation dη = −ω ∧ η (Hicks [1965]). A horizontal path c : R → M is a nonholonomic geodesic if it satisﬁes the nonholonomic geodesic equations ⎡ ⎤ d ⎣ (vi ) + vj ωij (c) ˙ ⎦ ei = 0, (2.3) dt j

where 1 ≤ i, j, ≤ r and vi = ηi (c) ˙ are the quasi-velocities. Example: The vertical rolling penny Astandard example of a mechanical system modeled by a nonholonomic Engel system is that of a coin rolling without sliping on the Euclidean plane. Consider a coin of radius a rolling vertically on the xy-plane. The location of the coin is represented by the coordinates (x, y, θ, φ). The point of contact of the coin with the plane is (x, y), the angle made by the coin with respect to the positive x-axis is θ, and the angle made by the point of contact, the center of the coin, and a point marked on the outer edge of the coin is φ. The state space can be identiﬁed with the Lie group SE(2) × SO(2) where the ﬁrst factor is the group of Euclidean motions locally parametrized by x, 15 The terminology straightest path for a nonholonomic geodesic was, in fact, coined by Hertz

himself.

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y and θ. The mass of the coin is m, the moment of inertia in the θ direction is J and the moment of inertia in the φ direction is I . The kinetic energy, which deﬁnes a Riemannian metric on the state space, is T =

J I m (dx ⊗ dx + dy ⊗ dy) + dθ ⊗ dθ + dφ ⊗ dφ. 2 2 2

(2.4)

The penny rolls without slipping giving rise to the constraints ˙ x˙ = (a cos θ )φ,

˙ y˙ = (a sin θ )φ.

Consider the orthonormal frame (X1 , X2 , X3 , X4 ), where # ∂ ∂ ∂ 2 X1 := + a sin θ + a cos θ , ∂x ∂y dφ ma 2 + I # 2 ∂ , X2 := J ∂θ # ∂ 2 ∂ X3 := + cos θ − sin θ , m ∂x ∂y # ∂ 2 ∂ X4 := + sin θ cos θ . m ∂x ∂y

(2.5)

(2.6)

Note that the constraint subspace is H = span{X1 , X2 }, and H(1) = span{X1 , X2 , X3 }. The dual coframe is (η1 , η2 , η3 , η4 ), where $ # ma 2 + I J 1 2 dφ, η := dθ, η := 2 2 # # m m 3 4 (− sin θ dx + cos θ dy), η := (cos θ dx + sin θ dy − dφ). η := 2 2 (2.7) To compute the Levi-Civita connection form, we determine ω = [ωI J ] such that ωI J = −ωJ I and dη = −ω ∧ η. Using simple linear algebra, we ﬁnd % % ⎛ ⎞ m m 3 √1 2 √1 η η 0 0 2 2 2 J (ma +I ) 2 J (ma +I ) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ % % ⎜ − √1 ⎟ m 1 m 3 1 √ η 0 − η 0 ⎜ ⎟ 2 J (ma 2 +I ) 2 J (ma 2 +I ) ⎜ ⎟ ⎜ ⎟ (2.8) ω=⎜ ⎟ √ % % ⎜ √1 ⎟ 0 − √ 2 η2 ⎟ ⎜ − 2 J (mam2 +I ) η2 √12 J (mam2 +I ) η1 J ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ √ 2 2 √ η 0 0 0 J

so, in particular, ω12 = −ω21 =

1 2

#

m η3 . J (ma 2 + I )

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Let c : R → Q be a nonholonomic geodesic given by c(t) ˙ = v1 (t)X1 + v2 (t)X2 . From the structure equations we see immediately that ω12 (c(t)) ˙ = −ω21 (c(t)) ˙ =0 d d and the nonholonomic geodesic equations reduce to dt (v1 ) = dt (v2 ) = 0. The ˙ θ˙ ) = AX1 + BX2 . In particular, nonholonomic geodesics are solutions to (x, ˙ y, ˙ φ, √ √ 2Aa cos θ (t) 2Aa sin θ (t) x˙ = √ , y˙ = √ , ma 2 + I ma 2 + I (2.9) √ √ 2A 2B ˙ ˙ φ=√ , θ= √ . J ma 2 + I The trajectories are spinning in place (A = 0), rolling along a line (B = 0), or circles (A, B = 0). 2.2 Equivalence problem of nonholonomic geometry Cartan’s method of equivalence starts by encoding a geometric structure in terms of a subbundle of the coframe bundle called a G-structure. We begin this section by describing the G-structure for nonholonomic geometry16 . We then give a brief outline of some of the main ideas behind the method of equivalence as it is applied in our example of nonholonomic geometry on an Engel manifold. Details about the method of equivalence can be found in Gardner [1989], Montgomery [2002], or Bryant [1994]. We then derive the local invariants associated with a nonholonomic structure on a four-dimensional manifold endowed with an Engel distribution. Initial G-structure for nonholonomic geometry A coframe η(x) at x ∈ Qn is a basis for the cotangent space Tx∗ (Q). Alternatively, we can regard a coframe as a linear isomorphism η(x) : Tx (Q) → Rn where Rn is represented by column vectors. A coframe can then be multiplied by a matrix on the left in the usual way. The set of all coframes at x is denoted Fx∗ (Q) and has the projection mapping π : Fx∗ (Q) → x. The coframe bundle F ∗ (Q) is the union of the Fx∗ (Q) as x varies over Q.Acoframe on Q is a smooth (local) section η : Q → F ∗ (Q) and is represented by a column vector of 1-forms (η1 , . . . , ηn )tr , where “tr’’ indicates transpose. F ∗ (Q) is a right Gl(n)-bundle with action Rg η = g −1 η where g is a matrix in Gl(n). Let G be a matrix subgroup of Gl(n). A G-structure is a G-subbundle of F ∗ (Q). We now describe the G-structure encoding the nonholonomic geometry associated with a nonholonomic structure (Q, G, H). Given a nonholonomic structure (Q, G, H) we can choose an orthonormal coframe η = (ηi , ην )tr on U ⊂ Q so that the ην annihilate H and use this coframe to write down the nonholonomic geodesic equations as described above. On the other hand, given acoframe η¯ = (η¯ i , η¯ ν )tr on Q we can ¯ where H¯ is construct a nonholonomic structure (Q, G¯ = η¯ i ⊗ η¯ i + η¯ ν ⊗ η¯ ν , H) 16 This G-structure was ﬁrst presented by Cartan in his 1928 address to the International

Congress of Mathematicians (Cartan [1928])

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annihilated by the η¯ ν . How is η¯ related to η if it is to lead to the same nonholonomic geodesic equations as η? In order to preserve H we must have ην − η¯ ν = 0 (mod I ). In matrix notation, any modiﬁed coframe η¯ must be related to η by i i Ab η¯ η = , (2.10) 0 a η¯ ν ην where A ∈ Gl(r), a ∈ Gl(n − r), and b ∈ M(k, n − r). If we were studying the geometry of distributions, there would be no further restrictions. In order to preserve the metric restricted to H, we must further insist that A ∈ O(r). We would then have the starting point for the study of sub-Riemannian geometry (see Montgomery [2002], Hughen [1995], or Moseley [2001]). It is important to observe that in nonholonomic geometry we need the full metric and not just its restriction to H (as in sub-Riemannian geometry) to obtain the equations of motion. Cartan [1928] showed that in order to preserve the nonholonomic geodesic equations, we can only add covectors that are in the ﬁrst derived ideal to the ηi . Since this fact is central to our analysis, we sketch the argument here (see Koiller, Rodrigues, and Pitanga [2001] for details). Suppose η¯ = gη with connection 1-form deﬁned by d η¯ = −ω¯ ∧ η. ¯ For simplicity, assume that A = id; then ηj ≡ η¯ j (mod I). The geodesic equations are preserved if and only if ωij (T ) = ω¯ ij (T ) for all T ∈ H, in other words ωij ≡ ω¯ ij (mod I). Note also that η¯ ν ≡ 0(mod I). Subtracting the structure equations for dηi and d η¯ i , we get dηi − d η¯ i = −ωij ∧ ηj − ωiν ∧ ην + ω¯ ij ∧ η¯ j + ω¯ iν ∧ η¯ ν ≡ 0

(mod I).

Now η¯ i = ηi + biν ην so we also have dηi − d η¯ i = dηi − (dηi + dbiν ην + biν dην ) ≡ −biν dην

(mod I).

Therefore, biν dην ≡ 0 (mod I) or, equivalently, biν ην ∈ I (1) . This completes the argument. We further subdivide our indicial notation: let r +1≤φ ≤s

(= rank H1 ),

s + 1 ≤ ≤ n.

Adapted coframes A covector η = (ηi , ηφ , η )tr ) arranged so that 1. The ηφ and I, η i generate 2 2. ds |H = η ⊗ ηi , 3. The η generate the ﬁrst derived ideal I (1) ,

is said to be adapted to the nonholonomic structure. In matrix notation, the most general change of coframes that preserves the nonholonomic geodesic equations is of the form η¯ = gη where

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⎞ A 0 b g = ⎝ 0 a1 a2 ⎠ 0 0 a3 ⎛

(2.11)

with A ∈ O(k), b ∈ M(n − s, k), a1 ∈ Gl(s − k), a2 ∈ M(n − s, s − k), and a3 ∈ Gl(n − s). The set of all such block matrices form a matrix subgroup of Gl(n) which we shall denote G0 . The initial G-structure for nonholonomic geometry on (Q, ds 2 , H) is a subbundle B0 (Q) ⊂ F ∗ (Q) (or simply B0 if there is no risk of confusion) with structure group G0 deﬁned above. All local sections of B0 (Q) lead to the same nonholonomic geodesic equations. In this way, the initial G-structure B0 (Q) completely characterizes the nonholonomic geometry. πQ

πN

Two G-structures, B(Q) → Q and B(N ) → N , are said to be equivalent if there is a diffeomorphism f : Q → N for which f1 (B(Q)) = B(N ) where f1 is the induced bundle map. (If we think of b ∈ B(Q) as a linear isomorphism b : TπQ (b) Q → Rn then f1 (b) = b ◦ (f∗ )−1 where f∗ is the differential of f .) Our original question as to whether there is a local diffeomorphism that carries nonholonomic geodesics to nonholonomic geodesics can be answered by determining whether the associated G-structures are locally equivalent. 2.3 A tutorial on the method of equivalence Necessary and sufﬁcient conditions for the equivalence between G-structures are given in terms of differential invariants which are derived using the method of equivalence. In this section, we brieﬂy describe some of the main ideas behind the method of equivalence as it is applied in our example. Details and other facets of the method together with many examples can be found in the excellent text by Robert Gardner (Gardner [1989]). One of the principal objects used in the method of equivalence is π the tautological 1-form. Let B(Q) → Q be a G-structure with structure group G whose Lie algebra is Lie(G). The tautological 1-form on B(Q) is an Rn -valued 1-form deﬁned as follows. Let η : U ⊂ Q → B(Q) be a local section of B(Q) and consider the inverse trivialization U × G0 → B(Q) deﬁned by (x, g) → g −1 η(x). Relative to this section, the tautological 1-form is deﬁned by

(b) = g −1 (π ∗ η),

(2.12)

where b = g −1 η. From (2.12) one can verify that the tautological 1-form is semibasic (i.e., (v) = 0 for all v ∈ ker(π∗ )), has the reproducing property η¯ ∗ = η, ¯ where η¯ is any local section of B(Q), and is equivariant: Rg∗ = g −1 . The components of the tautological 1-form provide a partial coframing for B(Q) and form a basis for the semibasic forms on B(Q). The following proposition reduces the problem of ﬁnding an equivalence between G-structures to ﬁnding a smooth map that preserves the tautological 1-form. (See Gardner [1989] or Bryant [1994] for a proof.)

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Proposition 2.1. Let B(Q) and B(N ) be two G-structures with corresponding tautological 1-forms Q and N , and let F : B(Q) → B(N ) be a smooth map. If G is connected and F ∗ ( N ) = Q , then there exists a local diffeomorphism f : Q → N for which F = f1 , i.e., the two G-structures are equivalent. To ﬁnd the map F in this proposition we would like to apply Cartan’s technique of the graph (see Warner [1971], p. 75): if we could ﬁnd an integral manifold ⊂ B(Q) × B(N ) of the 1-form θ = Q − N that projects diffeomorphically onto each factor, then would be the graph of a function h : Q → N for which h∗1 N = Q . By the above proposition the G-structures would then be equivalent. We generally cannot apply this idea directly because Q and N do not provide full coframes on B(Q) and B(N ) as is required in the technique of the graph. In the example of nonholonomic geometry on Engel manifolds, and indeed in many important examples (see Gardner [1989], Hughen [1995], Moseley [2001], Montgomery [2002], Ehlers [2002]), applying the method of equivalence leads to a new G-structure called an e-structure. An e-structure is a G-structure endowed with a canonical coframe. Differentiating both sides of (2.12) one can verify that d satisﬁes the structure equation d = −α ∧ + T , (2.13) where T is a semibasic 2-form on B(Q) and α is called a pseudoconnection: a Lie(G)valued 1-form on B(Q) that agrees with the Mauer–Cartan form on vertical vector ﬁelds. Here, Lie(G) is the Lie algebra of G. Summarizing, Pseudoconnection:

α = g −1 dg + semibasic Lie(G)-valued 1-form.

(2.14)

The components of the pseudoconnection together with the tautological 1-form do provide a full coframe on the G-structure, but unlike the tautological 1-form, the pseudoconnection is not canonically deﬁned. Understanding how changes in the pseudoconnection affect the torsion is at the heart of the method of equivalence. For any G-structure, that part of the torsion that is left unchanged under all possible changes of pseudoconnection is known as the intrinsic torsion. The intrinsic torsion is the only ﬁrst-order differential invariant of the G-structure (Gardner [1989]). As an example, the intrinsic torsion for the G-structure B of a general distribution (equation 2.10) is the dual curvature of the distribution (Cartan [1910], see also Montgomery [2002]). In the case of a rank two distribution on a four-dimensional manifold, the structure equations for the tautological 1-form are ⎛ ⎞ ⎞ ⎛ 1⎞ ⎛ 1⎞ ⎛

T

1 A11 A12 β13 β14 ⎜ 2 ⎟ ⎜ A21 A22 β23 β24 ⎟ ⎜ 2 ⎟ ⎜ T 2 ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ d⎜ (2.15) ⎝ 3 ⎠ = − ⎝ 0 0 α33 α34 ⎠ ∧ ⎝ 3 ⎠ + ⎝ T 3 ⎠ ,

4 0 0 α34 α44

4 T4 where T I = J
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Reduction and prolongation There are two major steps in the equivalence method: prolongation and reduction (see Gardner [1989] or Montgomery [2002]). In the case of nonholonomic geometry on an Engel manifold a sequence of reductions lead to an e-structure. A brief outline of the reduction procedure is as follows. The ﬁrst step involves writing out the structure equations for the tautological 1-form . A semibasic Lie(G)-valued 1-form is added to the pseudoconnection to make the torsion as simple as possible. Gardner [1989] calls this step absorption of torsion. The action of G on the torsion is deduced by differentiating both sides of the identity Rg∗ ( ) = g −1 . The action of G is used to simplify part of the torsion. The isotropy subgroup of that choice of simpliﬁed torsion is then the structure group of the reduced G-structure. In the case of nonholonomic geometry on an Engel manifold this procedure is repeated until an e-structure is obtained. Suppose that is the canonical coframing on the resulting manifold B. The i form a basis for the 1-forms on B so we can write d I = cJI K J ∧ K . (2.16) J
Relationships between the cJI K are found by differentiating this equation. The resulting torsion functions provide the “complete invariants’’ for the geometric structure (see Gardner [1989], p. 59, Bryant [1994], pp. 9–10, or Cartan [2001]). Many important examples have integrable e-structures. An e-structure is integrable if the cJI K are constant (Gardner [1989]). In this case we can apply the following result from Montgomery [2002]. Lemma 2.2. Let B be an n-dimensional manifold endowed with a coframing . Then the (local) group G of diffeomorphisms of B that preserves this coframing is a ﬁnitedimensional (local) Lie group of dimension at most n. The bound n is achieved if and only if the e-structure is integrable. In this case the cJI K are the structure constants of G, G acts freely and transitively on B, and the coframe can be identiﬁed with the left-invariant 1-forms on G. The Jacobi identities are found by differentiating d = J
Nonholonomic systems via moving frames

⎛

A11 ⎜ A21 ⎜ ⎝ 0 0

A12 A22 0 0

0 0 a33 0

93

⎞

B14 B24 ⎟ ⎟, a34 ⎠ a44

(2.17)

where A = [AI J ] ∈ O(2), a33 a44 = 0, and B14 and B24 are arbitrary. Let = ( 1 , 2 , 3 , 4 )tr be the tautological 1-form on B0 . The structure equations are ⎞ ⎞ ⎛ 1⎞ ⎛ ⎛

1 0 γ 0 β14 ⎜ 2 ⎟ ⎜ −γ 0 0 β24 ⎟ ⎜ 2 ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ d⎜ ⎝ 3 ⎠ = − ⎝ 0 0 α33 α34 ⎠ ∧ ⎝ 3 ⎠

4 0 0 0 α44

4 ⎛ 1 1 ⎞ 1 2 ∧ 3 T13 ∧ 3 + T23 ⎜ T 2 1 ∧ 3 + T 2 2 ∧ 3 ⎟ 13 23 ⎟, +⎜ (2.18) 3 1 ∧ 2 ⎝ ⎠ T12 4 1 ∧ 3 + T 4 2 ∧ 3 T13 23 where we have chosen the pseudoconnection so that the remaining Tjik are zero. 4 = 0. Also,

4 ∈ I (1) so d 4 = 0 mod ( 3 , 4 ) and we must therefore have T12 3 cannot 3 (1) 3 3 4 / I so d = 0 mod ( , ); therefore, the torsion function T12

∈ equal zero. The pseudoconnection for this choice of torsion is not unique. We can, for instance, add arbitrary multiples of 4 to the βi4 and αi4 . Following Cartan’s prescription, we investigate the induced action of G0 on the torsion. Let g ∈ G0 . To simplify notation, functions and forms pulled back by Rg ˆ = ( ˆ 1, ˆ 2, ˆ 3, ˆ 4 )tr and will be indicated by a hat so, for instance, Rg∗ = k k ∗ Rg (Tij ) = Tˆij . We have ⎛ ˆ1⎞ ⎛ ⎞

# ⎜ ⎟ ˆ2⎟ ⎜ # ⎜ ⎟. ⎟=⎜ −1 3 4 3 ⎝ ⎠ ⎝ ˆ det(a )(a44 − a34 ) ⎠ ˆ4 det(a −1 )(a33 4 )

(2.19)

To determine the induced action of G0 on the torsion we differentiate both sides ˆ 3 . For 3 , we compute of the identity Rg∗ 3 = 3 ˆ1 ˆ 3 + αˆ 34 ∧ ˆ 4 + Tˆ12 ˆ 2, Rg∗ (d 3 ) = αˆ 33 ∧

∧ 3 1 = det(A−1 )Tˆ12

∧ 2

(mod 3 , 4 ),

and ˆ 3 = det(a −1 )(a44 d 3 − a34 d 4 ) d 3 1

∧ 2 ) = det(a −1 )(a44 T12

(mod 3 , 4 ) (mod 3 , 4 ).

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3 is therefore The induced action of G0 on T12 3 Rg∗ (T12 )=

det(A) 3 T . a33 12

(2.20)

3 = 0 we can force it to equal 1 using the action of G . The stabilizer Since T12 0 subgroup G1 for this choice of torsion consists of matrices of the form (2.17) with 3 1 ∧ 2 is the (normalized) dual curvature a33 = where = det(A). Note that T12 of the distribution. The structure equations for the G1 -structure B1 are ⎛ ⎞ ⎞ ⎛ 1⎞ ⎛

1 0 γ 0 β14 ⎜ 2 ⎟ ⎜ −γ 0 0 β24 ⎟ ⎜ 2 ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ d⎜ ⎝ 3 ⎠ = − ⎝ 0 0 0 α34 ⎠ ∧ ⎝ 3 ⎠

4 0 0 0 α44

4 ⎛ ⎞ 1 1 ∧ 3 + T 1 2 ∧ 3 T13 23 2 1 ∧ 3 + T 2 2 ∧ 3 ⎜ ⎟ T13 23 ⎟ +⎜ (2.21) ⎝ T 3 1 ∧ 3 + T 3 2 ∧ 3 + 1 ∧ 2 ⎠ . 13 23 4 1 ∧ 3 + T 4 2 ∧ 3 T13 23

Let g ∈ G1 . We write the inverse of g as ⎛ A11 A21 ⎜ A 12 A22 g −1 = ⎜ ⎝ 0 0 0 0

0 0 a¯ 33 0

⎞ B¯ 14 B¯ 24 ⎟ ⎟, a¯ 34 ⎠ a¯ 44

(2.22)

so, in particular, a¯ 33 = , a¯ 34 = −a34 (a44 )−1 , and a¯ 44 = (a44 )−1 . We have ⎛ ˆ1⎞ ⎛ ⎞

A11 1 + A21 2 + B¯ 14 4 ⎜ ˆ 2 ⎟ ⎜ A12 1 + A22 2 + B¯ 24 4 ⎟ ⎟ ⎜ ⎟. Rg∗ = ⎜ (2.23) ⎝ ⎠ ˆ3⎠ = ⎝ a¯ 33 3 + a¯ 34 4 ˆ4 a¯ 44 4

For the next reduction we differentiate both sides of the identity Rg∗ 4 = ˆ4 . We have 4 ˆ1 4 ˆ2 ˆ 4 + Tˆ13 ˆ 3 + Tˆ23 ˆ3

∧

∧ Rg∗ d 4 = αˆ 44 ∧

(mod 4 ),

4 4 = a¯ 33 ((A11 Tˆ13 + A12 Tˆ23 ) 1 ∧ 3 4 4 + (A21 Tˆ13 + A22 Tˆ23 ) 2 ∧ 3 )

(mod 4 ).

On the other hand, 4 1 4 2 ˆ 4 = a¯ 44 d 4 = a¯ 44 (T13 d

∧ 3 + T23

∧ 3 ) (mod 4 ). 4 , T 4 ) is therefore The induced action of G1 on the torsion plane (T13 23

Nonholonomic systems via moving frames

4 Tˆ13 4 ˆ T23

=

−1 A a44

4 T13 4 T23

95

.

(2.24)

4 , T 4 ) = (0, 0) since I (2) = 0 implies that d 4 ∧ 4 = 0 and The torsion plane (T13 23 4 = 0. We can therefore use the action to force we have already established that T12 4 4 4 (T13 , T23 ) = (0, 1). The torsion T23 2 ∧ 3 can be interpreted as the (normalized) 4 , T 4 ) = (0, 0) dual curvature of the rank three distribution H1 . The statement that (T13 23 is equivalent to H not being integrable. To determine the subgroup that stabilizes this choice of torsion, we investigate −1 0 0 0 Rg∗ A = = . (2.25) 1 1 1 a44

As A ∈ O(2), it must be of the form

1 2 0 0 2

,

(2.26)

where 1 , 2 ∈ {−1, 1}. We must also have a44 = 1 2 so that the stabilizer subgroup G2 consists of matrices of the form ⎛ ⎞ 1 2 0 0 B14 ⎜ 0 2 0 B24 ⎟ ⎜ ⎟ (2.27) ⎝ 0 0 1 a34 ⎠ , 0 0 0 1 2 where 1 , 2 ∈ {−1, 1} and B14 , B24 and a34 ∈ R. We compute ⎛ ˆ1⎞ ⎛ ⎞

1 2 1 − B14 4 ⎜ ˆ 2 ⎟ ⎜ 2 2 − B24 4 ⎟ ⎟ ⎜ ⎟ Rg∗ = ⎜ ⎝ ˆ 3 ⎠ = ⎝ 1 3 − 2 a34 4 ⎠ . ˆ4 1 2 4

(2.28)

The structure equations are now ⎛ ⎞ ⎛ ⎞ β14 ∧ 4

1 4⎟ ⎜ 2 ⎟ ⎜ ⎟ = − ⎜ β24 ∧ ⎟ d⎜ (2.29) ⎝ 3 ⎠ ⎝ α34 ∧ 4 ⎠

4 0 ⎞ ⎛ 1 1 ∧ 2 + T 1 1 ∧ 3 + T 1 2 ∧ 3 T12 13 23 2 1 ∧ 2 + T 2 1 ∧ 3 + T 2 2 ∧ 3 ⎟ ⎜ T12 13 23 ⎟. +⎜ 3 3 1 2 1 3 2 3 ⎠ ⎝

∧ + T13 ∧ + T23 ∧ 4 4 4 1 4 2 3 2 4 3 4 T14 ∧ + ∧ + T24 ∧ + T34 ∧ ˆ B2 is not an e-structure, so again we differentiate both sides of the identity Rg∗ = to determine the action of G2 on the torsion. After some computation, we ﬁnd that

96

K. Ehlers, J. Koiller, R. Montgomery, and P. M. Rios 1 1 1 2 1 1 ˆ 1 = 1 2 (T13 d

∧ 3 + T23

∧ 3 + T12

∧ 2 )

− B14 2 ∧ 3

(mod 4 ),

2 1 2 2 2 1 ˆ 2 = 2 (T13 d

∧ 3 + T23

∧ 3 + T12

∧ 2 )

− 1 B24 2 ∧ 3

(mod 4 ),

3 1 3 2 ˆ 3 = 1 (T13 d

∧ 3 + T23

∧ 3 + 1 ∧ 2 )

− 2 a34 2 ∧ 3

(mod 4 ).

Also, 1 1 3 1 1 1 Rg∗ (d 1 ) = 2 Tˆ13

+ 1 2 Tˆ23 + 1 Tˆ12

∧ 2

(mod 4 ),

2 1 3 2 2 1 Rg∗ (d 2 ) = 2 Tˆ13

+ 1 2 Tˆ23 + 1 Tˆ12

∧ 2

(mod 4 ),

3 1 3 3 Rg∗ (d 3 ) = 1 Tˆ13

+ 1 2 Tˆ23 + 1 1 ∧ 2

(mod 4 ).

Matching the 2 ∧ 3 terms, we ﬁnd that 1 1 = T23 − 1 2 B14 , Tˆ23 2 2 Tˆ23 = 1 T23 − 2 B24 ,

(2.30)

3 3 = 2 T23 − 1 a34 . Tˆ23 1 = T 2 = T 3 = 0. The stabilizer We can therefore use the action of G1 to force T23 23 23 subgroup Gﬁnal for this choice of torsion consists of matrices of the form ⎛ ⎞ 1 2 0 0 0 ⎜ 0 2 0 0 ⎟ ⎜ ⎟ (2.31) ⎝ 0 0 1 0 ⎠ . 0 0 0 1 2

The reduced structure group is discrete so we now have an e-structure Bﬁnal . The tautological 1-form ( 1 , 2 , 3 , 4 )tr provides a full coframing for Bﬁnal . The Bﬁnal structure equations are ⎞ ⎛ 1

∧ 2 ⎞ ⎛ 1 1 1 ⎞ ⎛ 1 T1 ⎜ 1 ∧ 3 ⎟ T12 T13 T14 0 T24

1 34 ⎟ ⎜ ⎜ 2 ⎟ ⎜ T 2 T 2 T 2 0 T 2 T 2 ⎟ ⎜ 1 ∧ 4 ⎟ 24 34 ⎟ ⎜ ⎟ = ⎜ 12 13 14 ⎟ d⎜ (2.32) ⎝ 3 ⎠ ⎝ 1 T 3 T 3 0 T 3 T 3 ⎠ ⎜ 2 ∧ 3 ⎟ , ⎟ ⎜ 13 14 24 34 4 1 T4 T4 ⎝ 2 ∧ 4 ⎠

4 0 0 T14 24 34

3 ∧ 4 where the Tijk are functions on Bﬁnal . What remains is to determine any second order relations between the torsion functions. To determine them we use the fact that d 2 = 0. 4 = T 2 + T 3 . We summarize these results After some computation, we ﬁnd that T14 12 13 in the following theorem:

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97

Theorem 2.3. Associated to any nonholonomic Engel structure {Q, G = ·, ·, H} there is a canonical G ∼ = Z2 × Z2 -structure Bﬁnal . The tautological 1-form ( 1 , 2 , 3 , 4 )tr provides a canonical coframing for Bﬁnal . The Bﬁnal structure equations are ⎞ ⎛ 1

∧ 2 ⎞ ⎛ 1 1 ⎞ ⎛ 1 1 T1 ⎜ 1 ∧ 3 ⎟ T12 T13 T14 0 T24

1 34 ⎟ ⎜ 1 2 2 2 4⎟ ⎟ ⎜ 2 ⎟ ⎜ T 2 T 2 T14 0 T24 T34 ⎟ ⎜ ⎟ = ⎜ 12 13 ⎜ ∧ ⎟. d⎜ (2.33) 3 3 T 3 ⎠ ⎜ 2 ∧ 3 ⎟ ⎝ 3 ⎠ ⎝ 1 T 3 T14 0 T24 ⎟ ⎜ 13 34 2 +T3 1 T4 T4 ⎝ 2 ∧ 4 ⎠

4 0 0 T12 13 24 34

3 ∧ 4 According to the framing lemma (Lemma 2.2) the largest Lie group of symmetries of a nonholonomic structure on an Engel manifold is the dimension of Bﬁnal which is four. In this case the TJIK are constants and can be identiﬁed with the structure constants of the four-dimensional Lie algebra of the symmetry group. The Jacobi identities are obtained using the identity d 2 = 0. We have computed them, and it appears that the set of possible symmetry algebras form a rather complicated subvariety of the variety of all four-dimensional Lie algebras. We leave as an open problem the classiﬁcation of all possible four-dimensional symmetry algebras for nonholonomic structures on an Engel manifold. The rolling penny (continued ) An example of a structure with maximal symmetry is given by the rolling penny. A Bﬁnal -adapted coframe for the penny-table system is $ ma 2 + I dφ, η1 = 2 # J 2 dθ, (2.34) η = 2 & J (ma 2 + I ) η3 = (− sin θ dx + cos θ dy), 2 # m η4 = (cos θ dx + sin θ dy − dφ). 2 The structure equations are dη1 = 0, dη2 = 0,

$

ma 2 + I 2 dη3 = η1 ∧ η2 − η ∧ η4 , m # m 2 4 dη = η2 ∧ η3 . J ma 2 + I

(2.35)

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K. Ehlers, J. Koiller, R. Montgomery, and P. M. Rios

The torsion functions are constant, so by the framing lemma (Lemma 2.2) we can identify these constants with the structure constants Lie group of symmetries of this system. We recognize them as the structure constants for the Lie algebra of the group SE(2) × SO(2) which is isomorphic to the conﬁguration space of the penny-table system. Bﬁnal -adapted frames and coframes The e-structure Bﬁnal has a canonical coframing which descends to a coframing and hence a framing, up to signs, on Q. There should be a relationship between this framing and a canonical line ﬁeld possessed by any Engel manifold. In this section we brieﬂy describe this relationship. If Q and H are both oriented, then Q is parallelizable and the following constructions can be made globally (Montgomery [2002]). Let η be a Bﬁnal adapted coframe on U ⊂ Q with dual frame X = {XI } deﬁned ¯ U, by ηI (XJ ) = δI J . If η¯ is any other Bﬁnal -adapted coframe with dual frame Xon then by Theorem 2.3, η¯ is related to η by η¯ 1 = 1 2 η1 , η¯ 2 = 2 η1 , η¯ 3 = 1 η3 , η¯ 4 = 1 2 η4 . The dual frames are related in precisely the same way: X¯ 1 = 1 2 X1 , X¯ 2 = 2 X2 , X¯ 3 = [X¯ 1 , X¯ 2 ] = 1 X3 , and X¯ 4 = [X¯ 2 , X¯ 3 ] = 1 2 X4 . An important feature of an Engel distribution is the presence of a canonical line ﬁeld L ⊂ H (Montgomery [2002], Kazarian, Montgomery, and Shapiro [1997]). L is deﬁned by the condition that [L, H1 ] ⊂ H1 . Here we are abusing notation, using L for the line ﬁeld or a vector ﬁeld spanning L. We have Corollary 2.4. Let η = ηI be a Bﬁnal -adapted coframe. Let X = {XI } be the dual frame deﬁned by ηI (XJ ) = δI J ; then L = span(X1 ). Proof. Suppose L is spanned by the vector ﬁeld Y = aX1 +bX2 . Since η4 annihilates H1 we have η4 ([X3 , Y ]) = 0. Then 0 = η4 ([X3 , Y ]) = X3 η4 (Y ) − Y η4 (X3 ) − dη4 (X3 , Y ) = −dη4 (X3 , Y ). But dη4 ≡ η2 ∧ η3 mod(η4 ), so we must have 0 = η2 ∧ η3 (X3 , Y ) = η2 (X3 )η3 (Y ) − η3 (X3 )η2 (Y ), = −η3 (X3 )η2 (Y ), = −b. L is therefore spanned by X1 . This concludes the arguement. There is a natural metric, associated with Bﬁnal , on Q given by gnat = η˜ 1 ⊗ η˜ 1 + · · ·η˜ 4 ⊗ η˜ 4 where η˜ is any Bﬁnal -adapted coframe. Clearly all Bﬁnal -adapted coframes induce this same metric; using the sub-Riemannian metric gnat |H we form L⊥ within H so that H = L ⊕ L⊥ . By construction, X2 spans L⊥ .

Nonholonomic systems via moving frames

99

3 Nonholonomic dynamics: Chaplygin Hamiltonization Historically, Hamiltonization of nonholonomic systems started with Chaplygin’s last multiplier method . In the new time, the dynamics obeys Euler–Lagrange equations without extra terms; the gyroscopic force (1.13) “magically’’ disappears! When, after a time reparametrization the compressed system can be described as a Hamiltonian system, symplectic techniques can be employed. A number of NH systems have been Hamiltonized, and some interesting ones are Liouville-integrable; see Veselov and Veselova [1988], Kozlov [2002], Fedorov and Jovanovic [2003], Fedorov [1989], Dragovic, Gajic´ , and Jovanovic [1998], Borisov and Mamaev [2002a], Borisov and Mamaev [2002b], Borisov, Mamaev, and Kilin [2002], Borisov and Mamaev [2001], Jovanovic [2003]. 3.1 Compression to T ∗ S, S = Q/G; existence of invariant measures We recall from the introduction that the compressed system has a concise almost Hamiltonian dH φ = iXN H N H ,

∗

T S

N H := can + (J.K),

d N H = 0

(in general),

∗

T S is the canonical 2-form of T ∗ S and the (J.K) term is a semibasic 2where can form, which in general is nonclosed. It combines the momentum J of the G-action on T ∗ Q, and the curvature K of the connection. As this is important for the remaining, we outline the derivation (see Koiller, Rios, and Ehlers [2002] for details). Given the coframe coordinates m, (q) in T ∗ Q (see 1.5) the Poisson bracket matrix relative to I , dmI is 0n I n [] = [ ]−1 = (3.1) −In E

with EJ K = mI dI (eJ , eK ) = −mI I [eJ , eK ].

(3.2)

→ acting Let us consider the case of a principal bundle π : on the left, r = n − s. Recall our convention: capital roman letters I, J, K, etc., run from 1 to n. Lower case roman characters i, j, k run from 1 to s. Greek characters α, β, γ , etc., run from s + 1 to n. Fix a connection λ = λ(q) : Tq Q → Lie(G) deﬁning a G-invariant distribution H of horizontal subspaces. Denote by K(q) = dλ ◦ Hor : Tq Q × Tq Q → Lie(G) the curvature 2-form (which is, as is well known, Ad-equivariant). Choose a local frame ei on S. For simplicity, we may assume that Qn

ei = ∂/∂si

Ss

with Lie group Gr

(3.3)

are the coordinate vector ﬁelds of a chart s : S → Rs . Let ei = h(ei ) be their horizontal lift to Q. We complete to a moving frame of Q with vertical vectors eα which we will specify in a moment. The dual basis will be denoted i , α and we write pq = mi i + mα α . These are in a sense the “least moving’’ among all the moving frames adapted to this structure. We now describe how the n × n matrix E = (EI J ) looks like in this setting.

100

K. Ehlers, J. Koiller, R. Montgomery, and P. M. Rios

(i) The s × s block (Eij ). Decompose [ei , ej ] = h[ei , ej ] + V [ei , ej ] = V [ei , ej ] into vertical and horizontal parts. The choice (3.3) is convenient, since ei and ej commute: [ei , ej ] is vertical. Hence Eij = −pq [ei , ej ] = −mα α [ei , ej ].

(3.4)

Now by Cartan’s rule, K(ei , ej ) = ei λ(ej ) − ej λ(ei ) − λ[ei , ej ] = −λ[ei , ej ] ∈ Lie(G). Thus we showed that [ei , ej ]q = −K(ei , ej ) · q Moreover, let J :

T ∗Q

→

Lie(G)∗

(3.5)

the momentum mapping. We have

(J (pq ), Kq (ei , ej )) = pq (K(ei , ej ).q) = −pq [ei , ej ] (= Eij ). Theorem 3.1 (the J.K formula). Eij = (J (pq ), Kq (ei , ej )).

(3.6)

This gives a nice description for this block, under the choice [ei , ej ] = 0. Notice that the functions Eij depend on s and the components mα , but do not depend on g. This is because the Ad ∗ -ambiguity of the momentum mapping J is cancelled by the Ad-ambiguity of the curvature K. The other blocks are not needed here, but we include for completeness. (ii) The r × r block (Eαβ ). Choose a basis Xα for Lie(G). We take eα (q) = Xα · q as the vertical distribution. Choosing a point qo allows identifying the Lie group G with the ﬁber containing Gqo , so that id → qo . Through the mapping g ∈ G → gqo ∈ Gqo the vector ﬁeld eα is identiﬁed with a right-invariant (not left-invariant!) vector ﬁeld γ in G. The commutation relations for the eα [eα , eβ ] = −cαβ eγ appear with a minus sign. Therefore, γ (3.7) Eαβ = mγ cαβ . (iii) The s × n block (Eiα ). The vectors [ei , eα ] are vertical, but their values depend on the speciﬁc principal bundle one is working with, and there are some noncanonical choices. Given a section σ : US → Q over the coordinate chart s : US → Rm on S, γ we need to know the coefﬁcients biα in the expansion γ

[ei , eα ](σ (s)) = biα (s)eγ . Then

γ

Eiα (σ (s)) = −mγ biα (s).

(3.8)

At another point on the ﬁber, we need the adjoint representation Ad g : Lie(G) → Lie(G), X → g∗−1 Xg, described by a matrix (Aµα (g)) such that Ad g (Xα ) = Aµα (g)Xµ . Then

γ

[ei , eα ](g · σ (s)) = −mγ biµ (s)Aµα (g).

(3.9) (3.10)

Nonholonomic systems via moving frames

101

The clockwise diagram Starting on ps ∈ T ∗ S we go clockwise to Pq ∈ Leg(H) ⊂ T ∗ Q, for some q on the ﬁber π −1 (s) of Q over s. H ⊂ T Q −→ Leg(H) ⊂ T ∗ Q Leg

↑ h | TS

(3.11) ←−

(Legφ )−1

T ∗ S.

Taking differentials of all maps in (3.11) we obtain an induced principal connection φˆ in the bundle G → Leg(H) → T ∗ S. Let v, w, z ∈ Tps (T ∗ S), V , W, Z horizontal lifts at Pq ∈ Leg(H), and denote by Kˆ the curvature of this induced connection. ´ The following proposition is basically a rephrasing of a result in Bates and Sniatycki [1993]. Proposition 3.2. d(J.K)(v, w, z) = cyclic(dJ (V ), K(W, Z)).

(3.12)

Densities of invariant measures and a dimension dependent exponent A necessary and sufﬁcient condition for the existence of an invariant measure for compressed Chaplygin systems was obtained by Cantrijn, Cortés, de Léon, and de Diego [2002] (Theorem 7.5). Since in T ∗ S there is a natural Liouville measure dvol = ds1 · · · dsm dp1 · · · dpm , where (s, p) are coordinates in T ∗ S, the density function F produces an educated guess for a time reparametrization which may Hamiltonize the compressed system. If dim(S) = m and f N H is closed, the time-reparametrized vector ﬁeld XN H /f has the invariant measure f m dvol. XN H will have the invariant measure f m−1 ds1 · · · dsm dp1 · · · dpm . Working backwards, if a measure density F is known so that F (s)dvol is an invariant measure for XN H , then the obvious candidate for conformal factor is 1 f = F (s) m−1 . (3.13) This dimension dependent exponent will be relevant in the Chaplygin marble; see Section 3.2. Invariant measures for LR systems Let Q = G be a unimodular Lie group and identify T G ≡ T ∗ G via the bi-invariant metric. Assume that H ⊂ G is a subgroup acting on the left and preserving the distribution: Dhg = hDg = hDg (which boils down to Ad h−1 D = h−1 Dh = D). The

102

K. Ehlers, J. Koiller, R. Montgomery, and P. M. Rios

Legendre transform Leg : Lie(G) → Lie(G) ≡ Lie∗ (G) of a natural, left-invariant Lagrangian, is represented by a positive symmetric transformation A : Lie(G) → Lie(G), the inertia operator. For each g ∈ G, let Pg1 and Pg2 be, respectively, the projections of Lie(G) relative to the decomposition Lie(G) = Ad g −1 Lie(H ) ⊕ Ad g −1 D. We can also think of Pg2 as a map Pg2 : Tg G → Dg, projection parallel to the vertical spaces Lie(H )g. Let Pg2 o Legg : Dg → Dg. This map descends to the compressed Legendre transform Legs : Ts S → Ts S ≡ Ts∗ S, where S = G/H is the homogeneous space whose metric is induced by the bi-invariant metric on G. Consider the function φ

F (s) = det Legφs .

(3.14)

The following result is a rephrasing of a theorem by Veselov and Veselova [1988]; see also Fedorov and Jovanovic [2003], Theorem 3.3.17 Theorem 3.3. The reduced LR-Chaplygin system in the homogeneous space T ∗ (G/H ) always has the invariant measure ν = F (s)−1/2 ds1 · · · dsm dp1 · · · dpm ,

F (s) = det Legφs .

(3.15)

The density can be also calculated by the “dual’’ formula F (s) = det(A) det(Pg2 oA−1 |g −1 Lie(H )g )

(3.16)

(Pg1 is the projection over g −1 Lie(H )g parallel to g −1 Dg). The second formula may be easier to use if there are few constraints. Almost Hamiltonian systems Let be a nondegenerate (but in general, nonclosed) 2-form on M 2n , and H a function on M. Denote (as usual) by X = XH the skew-gradient vector ﬁeld deﬁned by iX = dH . We say XH is almost Hamiltonian. If α is a closed 1-form, the vector ﬁeld X = Xα deﬁned by iX = α is called locally almost Hamiltonian. Distilling a construction in Stanchenko [1985], we formalize an extension of the notion of a conformally symplectic structure. The 2-form is called H (or α)-afﬁne symplectic if there is a function f > 0 on M and a 2-form o such that (i) iX o ≡ 0; (ii) − o is nondegenerate; and ˜ = f ( − o ) is closed.18 (iii) 17 We do not need to assume D and Lie(H ) to be orthogonal with respect to the bi-invariant

metric. 18 We must admit, however, that we found no example yet where the afﬁne term is really

needed. This notwithstanding, at any point where X = 0, the contraction condition yields d = 2n equations on d(d − 1)/2 unknowns (local coordinate coefﬁcients of o ). This allows additional freedom to Hamiltonize X rather than just requiring conformality of .

Nonholonomic systems via moving frames

103

The ﬁrst condition implies that X does not “see’’ o . Together with the third, we ˜ get (X/f, •) = dH so the vector ﬁeld X/f is (truly) Hamiltonian with respect to ˜ the symplectic form . The closedness condition can be restated as d( − o ) = ( − o ) ∧ θ,

where θ = df/f.

(3.17)

When (3.17) holds with α a closed (but not necessarily exact) 1-form, we say that is locally afﬁne symplectic. The following proposition describes the obstruction to Hamiltonization once f is given. Theorem 3.4. Given a locally almost hamiltonian system ( , α) and an educated guess f > 0, an afﬁne term o exists with d(f − o ) = 0 if and only if iX d(f ) = 0. The proof is quite easy. The vector ﬁeld X satisﬁes iX = α. Since the same equation holds by replacing X by X/f and by f , to expedite notation we may assume f ≡ 1. Let us prove that o exists if iX d = 0. Since d(iX ) = dα = 0, we see that the Lie derivative LX = 0. Consider a regular point of X. By the ﬂow box theorem there are coordinates so that X = ∂/∂x1 . Since LX = 0, the coefﬁcients of this 2-form do not depend on the coordinate x1 (but there may exist terms with a dx1 factor). However, our hypothesis i∂/∂x1 d = 0 ensures that there are no terms containing a dx1 factor in d . Thus d can be thought of as a 3-form in the space of the remaining coordinates. By Poincaré’s theorem d = d o , where o is a 2-form in the space of the remaining coordinates. Hence iX o = 0 and d( − o ) = 0, as desired. The converse is even easier. 3.2 Examples: Veselova’s system and Chaplygin spheres (marble or rubber) Veselov and Veselova [1986, 1988] considered one of the simplest nonholonomic LRChaplygin systems, Q = SO(3) with a left-invariant metric L = T = 12 (A , ), and subject to a right-invariant constraint which, without loss of generality, can be assumed to be ρ3 = 0. Hence the admissible motions satisfy ω3 = 0, where ω is the angular velocity viewed in the space frame. This is a LR Chaplygin system on S 1 → SO(3) → S 2 . Chaplygin’s ball is a sphere of radius r and mass µ, whose center of mass is assumed to be at the geometric center, but the inertia matrix A = diag(I1 , I2 , I3 ) may have unequal entries. Thus its Lagrangian is given by 2L = (A , ) + µ(x˙ 2 + y˙ 2 + z˙ 2 ). The conﬁguration space is the Euclidean group Q = SE(3). In the case of the marble, the ball rolls without slipping on a horizontal plane, with rotations about the z-axis allowed.19 Thus the distribution of admissible velocities is 19 Chaplygin [2002] showed that the 3D problem is integrable using elliptic coordinates in the

sphere; for n > 3 the problem is open. For basic informations, see Fedorov and Kozlov [1995], pp. 147–149, on the 3D case and pp. 153–156 for the general n-dimensional case. For a detailed account of the algebraic integrability of “Chaplygin’s Chaplygin sphere’’; see Duistermaat [2000]. Schneider [2002] analyzed control theoretical aspects.

104

K. Ehlers, J. Koiller, R. Montgomery, and P. M. Rios

deﬁned by D : z˙ = 0, x˙ = rω1 , y˙ = −rω2 . Both Lagrangian and constraints are preserved under the action of the Euclidean motions in the plane, together with the vertical translations. G = SE(2) × R acts on Q via (φ, u, v, w).(R, x, y, z) = (S(φ)R, eiφ (u + iv), z + w). The dynamics could be directly reduced to D/G (see, e.g., Zenkov and Bloch [2003]), but we will proceed in two stages. First, we Chaplygin-compress the dynamics from T Q to T SO(3) using the translation subgroup of SE(3), regarding the constraint distribution as an abelian connection on Q with base space S = SO(3) and ﬁber R3 ; the connection form is given by αmarble := (dx − rρ2 , dy + rρ1 , dz).

(3.18)

There is another S 1 action on Q, this time acting on the ﬁrst factor only: eiφ (R, z) = (S(φ)R, z). This action preserves the Lagrangian but does not preserve the distribuiφ tion: D(S(φ)R,z) = e∗ D(R,z) . However, its inﬁnitesimal action is given by the right vector ﬁeld X3r ∈ D. Noether’s theorem applies, so pφ = 3 is a constant of motion. Therefore, Chaplygin’s marble equations can be reduced, on each level set 3 , to T (SO(3)/S 1 ) = T S 2 . In the case of Chaplygin’s rubber ball,20 rotations about the vertical axis are forbidden (since such rotations would cause energy dissipation). Here the constraints are deﬁned by a subdistribution H ⊂ D with Cartan’s 2-3-5 growth numbers and, in fact, deﬁning a connection on SE(2) × R → Q → S 2 with 1-form ˆ dx − rρ2 , dy + rρ1 , dz). αrubber := (ρ3 k,

(3.19)

The extrinsic viewpoint For clarity we present the classical, direct derivation of the equations of motion, following the “extrinsic viewpoint’’ advocated by the Russian Geometric Mechanics school (Borisov and Mamaev [2002a]). • For the rubber Chaplygin ball (and Veselova’s): in the space frame one has ˙ = τ , where τ = λkˆ is the torque exerted by the constraint force. The torque is vertical because (τ, ω) = 0 for all ω with third component equal to zero. Viewed in the body frame, L˙ + × L = λγ , (3.20) Together (1.6), one gets a closed system of ODEs in the space (L, γ ) ∈ R3 × R3 , provided the relation between and L is obtained. In Veselova’s example, = A−1 L. The multiplier can be eliminated by differentiating the constraint equation ( , γ ) = 0. After a simple computation, one gets 20 This problem was not studied by Chaplygin. For the physical justiﬁcation, see Neimark and

Fufaev [1972] and Cendra, Ibort, de Léon, de Diego [2004]. As far as we know its integrability has not yet been established. Formally, Veselova’s system is the limit of Chaplygin’s rubber ball as r → 0.

Nonholonomic systems via moving frames

λ=

(L, A−1 γ × A−1 L) . (γ , A−1 γ )

105

(3.21)

Besides the standard integrals of motion 2H = (A−1 L, L), (γ , γ ) = 1, (A−1 L, γ ) = 0, Veselov and Veselova [1988] showed that there is a quartic polynomial integral G = (L, L) − (L, γ )2

(3.22)

and an invariant measure21 µ = f (γ )dL1 ∧ dL2 ∧ dL3 ∧ dγ1 ∧ dγ2 ∧ dγ3 ,

f (γ ) = (A−1 γ , γ )−1/2 . (3.23)

• For Chaplygin’s marble: the angular momentum at the contact point in the space frame is constant. An engineer would argue that both gravity and friction produce no torque at that point; a mathematician would use the fact that the admissible vector ﬁelds Vi ∈ H given by right

V1 := −r∂/∂y + X1

,

right

V2 := r∂/∂x + X2

,

right

V3 := X3

(3.24)

preserve the Lagrangian, and would invoke NH-Noether’s theorem. Whichever explanation chosen, differentiating RL = = RL and Rγ = k, one gets Chaplygin’s equations L˙ = − × L, γ˙ = − × γ . (3.25) These two form a coupled system, since again is a linear function of L depending only on γ : ˜ − µr 2 (γ , )γ , L = Lγ ( ) = A + µr 2 γ × ( × γ ) = A

A˜ := A + µr 2 id. (3.26)

A simple way to get this map is to look at the total energy 2T = (ω, ) = ( , L) = (A , ) + µ(x˙ 2 + y˙ 2 ) = (A , ) + µr 2 (ω12 + ω22 ), (3.27) which can be also written as 2T = ( , L) = (A , ) + µr 2 ( , γ × ( × γ )) = ( , A + µr 2 γ × ( × γ ))). (3.28) The expression γ × (• × γ ) represents the projection in the plane perpendicular to γ , and we get (3.26). An ansatz for the inverse of the map (3.26) is (Duistermaat [2000]),

= (L, γ ) = (Lγ )−1 (L) = A˜ −1 L + α(L)A˜ −1 (γ ), (3.29) and one gets the interesting expression for α(L) (which will be used in equation (3.48) and Proposition 3.8): 21 The level sets of the four integrals are 2-tori, since there are no ﬁxed points in the dynam-

ics. The existence of an invariant measure in the tori allows the explicit integration via Jacobi’s theorem. Veselov and Veselova [1988] found a “rather unexpected connection with Neumann’s problem.’’

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α(L) = µr 2 The function

(γ , A˜ −1 L) . 1 − µr 2 (γ , A˜ −1 γ )

(3.30)

f (γ ) := [1 − µr 2 (γ , A˜ −1 γ )]−1/2

was found by Chaplygin to be the density of an invariant measure in

(3.31) R6 :

νR6 = f (γ )dγ1 dγ2 dγ3 dL1 dL2 dL3 .

(3.32)

This follows from Veselova’s theorem, as F (γ ) = 1 − µr 2 (γ , A˜ −1 γ ) is (up to a constant factor) the determinant of the linear map → L = L( ; γ ). For direct proofs of invariance of the measure, see Duistermaat [2000] or Fedorov and Kozlov [1995]. A system of ODE’s for the rubber ball can be derived in a similar fashion. For the angular momentum at the contact point, we get the same equation (3.20) from Veselova’s system, but the relation between and L is (3.26), the same as in Chaplygin’s marble. Differentiating ( , γ ) = 0 the multiplier can be eliminated. Hamiltonization of Veselova’s system The compressed Lagrangian is Lcomp =

1 (A(γ˙ × γ ), γ˙ × γ ), 2

(3.33)

since = γ˙ × γ ; the momentum map corresponding to the S 1 -action is J = 3 = (L, γ ). Thus (J.K) = 3 dρ3 = (A , γ )dρ3 , where dρ3 is the area form of S 2 . The compressed Legendre transform is γ˙ → a =

∂L∗ = γ × A(γ˙ × γ ). ∂ γ˙

The nonholonomic 2-form in T ∗ S 2 is

N H = da ∧ dγ + (A(γ˙ × γ ), γ )dρ3 .

(3.34)

Being a two-degrees of freedom system, a general result from Fedorov and Jovanovic [2003] (Theorem 3.5) guarantees that this system is Hamiltonizable. In order to verify that N H is conformally symplectic, it is simpler to use γ˙ as coordinates, that is, we pull back N H to T S 2 via Leg∗ . We get

N H = d(γ × A(γ˙ × γ ))) ∧ dγ + (γ , A(γ˙ × γ ))dρ3 . Proposition 3.5. Veselova’s system is conformally symplectic, d(f N H ) = 0, with conformal factor f = f (γ ) = (A−1 γ , γ )−1/2 . (3.35) As expected, it is the density of the Veselova invariant measure µ = f (γ )dLdγ obtained via Proposition 3.3. The orthonormal frame in S 2 diagonalizing (3.33) provides explicit coordinates for integration via the Hamilton–Jacobi method.

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Chaplygin’s rubber ball The dynamics compress to T ∗ S 2 , and by the same general result in Fedorov and Jovanovic [2003], we know in advance that the system is Hamiltonizable. Choose a moving frame e1 , e2 in S 2 . The horizontal lift from γ˙ = v1 e1 + v2 e2 to Hor(γ˙ ) ∈ T (SE(3)) is easily done via (1.10): Hor(γ˙ ) = v2 (X1r − r∂/∂y) − v1 (X2r + r∂/∂x). Composing dαrubber = (ρ1 ∧ ρ2, −rρ3 ∧ ρ1 , rρ2 ∧ ρ3 , 0) with Hor, we get Krubber = ˆ 0, 0, 0), where dS is the S 2 area form. Thus for the term (J.K) we need only (dS k, the third component of the angular momentum, m3 = (M, γ ) = (A , γ ), where we insert (1.10) = γ˙ × γ = v2 e1 − v1 e2 . Therefore,

N H = T ∗ S 2 + (A(γ˙ × γ ), γ ) · dS.

(3.36)

Here γ˙ = v1 e1 + v2 e2 ∈ T S 2 corresponds to pγ = p1 θ1 + p2 θ2 via the Legendre map Legcomp of the compressed Lagrangian Lcomp =

1 1 A(v2 e1 − v1 e2 , v2 e1 − v1 e2 ) + µr 2 (v12 + v22 ). 2 2

(3.37)

Clearly, this system becomes Veselova for r = 0. Using Proposition 3.3 and Fedorov’s result for two degrees of system, we get Proposition 3.6. Proposition 3.6. The compressed rubber ball system is Hamiltonizable. The conformal factor is f = [det Legcomp ]−1/2 '

= (I1 I2 I3 )−1/2 (A−1 γ , γ ) + µr 2

(

γ22 + γ33 I2 I3

)

γ 2 + γ33 γ 2 + γ23 µ2 r 4 + + 1 + 1 I1 I3 I1 I2 I1 I2 I3

(3.38)

*−1/2

.

Proof. We checked using spherical coordinates and MathematicaTM .22 3.3 Chaplygin’s marble is not Hamiltonizable at the T ∗ SO(3) level The homogeneous sphere In a nutshell, the dynamics in the homogeneous case are embarrassingly simple. The angular velocity in space is constant, so the attitude matrix R evolves as a oneparameter group R = exp([ω]t), so and ω are constant. The vector γ (t) describes a circle in the sphere perpendicular to ω, and L(t) the curve given by L(t) = (I + µr 2 )ω − ω3 γ (t). Provided is not vertical, L and γ are never parallel. The invariant tori are always foliated by closed curves and the two frequencies coincide. From 22 We can provide the (short) notebook under request. It should be investigated if the rubber

ball problem is integrable. Does a (quartic) integral still exist?

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the constraint equations we see that the motion of the contact point in the plane is a straight line. Shooting pool with a perfect Chaplygin ball is very dull.23 Let us use these simple results as template for our operational system. In terms of the right coframe, we have

N H = d1 ρ1 + d2 ρ2 + d3 ρ3 + 1 ρ2 ρ3 + 2 ρ3 ρ1 + 3 ρ1 ρ2 − µr 2 (ω2 ρ3 ρ1 + ω1 ρ2 ρ3 ).

(3.39)

This formula holds in general. In the nonhomogeneous case one must write ω1 and ω2 in terms of and R ∈ SO(3): ω = R = R γ (R −1 ) which seems to be a quite involved expression, a haunting monster we will avoid, until a ﬁnal confrontation in Proposition 3.8. In the homogeneous case, life is much easier: ω = R κ1 I R −1 m = I1 m, so the dependence of ω on R disappears. The Hamiltonian is given by * ' 2 1 21 + 22 H = + 3, 2 2 I + µr I where

µr 2 1 = 1 + m1 , I m1 ω1 = , I

µr 2 2 = 1 + I m2 ω2 = , I

To obtain the equations of motion we solve ⎛ ⎞ ⎛ ω1 0 0 0 1 ⎜ ω2 ⎟ ⎜ 0 0 0 0 ⎜ ⎟ ⎜ ⎜ ω3 ⎟ ⎜ 0 0 0 0 ⎜ ⎟=⎜ ⎜ ˙1 ⎟ ⎜ −1 0 0 0 ⎜ ⎟ ⎜ ⎝ ˙2 ⎠ ⎝ 0 −1 0 −3 0 0 −1 I ω2 ˙3

0 1 0 3 0 −I ω1

m2 ,

3 = m3 , ω3 =

m3 . I

⎞ ⎛ ⎞ 0 0 ⎟ ⎜0 ⎟ 0 ⎟ ⎜ ⎟ ⎟ ⎜0 ⎟ 1 ⎟ · ⎜ ⎟, ⎜ ⎟ −I ω2 ⎟ ⎟ ⎜ ω1 ⎟ ⎠ ⎝ ω2 ⎠ −I ω1 0 ω3

(3.40)

where we have used H1 = 1 /(I + µr 2 ) = m1 /I = ω1 , and similarly, H2 = ω2 , H2 = ω2 . This gives, as expected ˙1 = (I ω3 )H2 − I ω2 H3 = 0,

˙2 = · · · = 0,

˙3 = · · · = 0. right

Thus ωi = mi /I = const, i = 1, 2, 3, and the vector ﬁeld is simply X = ω1 X1

+

23 We found the following relevant information in www.ot.com/skew/ﬁve/myths. html (Top

Ten Myths in Pool or the Laws of Physics Do Apply): “4. If the cue is kept level, contacting the cueball purely left or right of its center will make it curve as it rolls. (No! The rolling cue ball can have two completely independent components to its angular momentum. Basically, this means that it can rotate in the manner of a top while rolling slowly forward along a straight line. In general, spin on a cue ball is of two types; follow/draw is the spin like tires on a car, while English is the spin like a child’s toy ‘top.’ Separately, neither one will make a ball curve! If they are combined—e.g., strike low-left giving left English and draw—then the spin is called masse (“mass-ay’’), and the ball will curve as it travels.)’’

Nonholonomic systems via moving frames right

109

right

ω2 X2 + ω3 X3 (no components in the ﬁber directions ∂/∂mi ). We now use Theorem 3.4. Using m as coordinates, the nonholonomic 2-form is given by µr 2

N H = 1 + (dm1 ρ1 + dm2 ρ2 ) I + dm3 ρ3 + (m1 ρ2 ρ3 + m2 ρ3 ρ1 + m3 ρ1 ρ2 ) 2

so that d N H = − µrI (dm1 ρ2 ρ3 + dm2 ρ3 ρ1 ). It is easy to see that the equation d N H = N H ∧ α has no solution. Indeed, suppose α = A1 dm1 + A2 dm2 + A3 dm3 + B1 ρ1 + B2 ρ2 + B3 ρ3 . Taking the exterior product, and looking at terms like dm1 dm2 ρ2 we see that all the A’s must be zero. Examining the coefﬁcient of ρ1 ρ2 ρ3 we get B1 m1 + B2 m2 + B3 m3 ≡ 0 so all the B’s are also zero. Hence the homogeneous Chaplygin sphere, as simple at it can be, has no conformal symplectic structure! In fact, it does not have an afﬁne symplectic structure either. A short calculation shows that µr 2 (−dm1 m2 ρ3 + dm1 ρ2 m3 − dm2 m3 ρ1 + dm2 ρ3 m1 ) = 0. I2 By continuity, for sufﬁciently close but different inertia coefﬁcients the inequalities persist. We have also done the calculation for the nonhomogeneous case and things only get worse. But, it still remains a possibility: is the reduced system to T ∗ S 2 Hamiltonizable? The impatient reader can go directly to Theorem 3.8. iX d N H =

3.4 Chaplygin’s marble: Reduction to T ∗ S 2 Using (1.8), L = a × γ + 3 γ , Chaplygin’s marble equations in (L, γ )-space directly reduce to T ∗ S 2 : γ˙ = γ × ,

a˙ = −2H γ + (γ , ) · (a × γ + 3 γ )

with

= (a, γ ; 3 ) = A˜ −1 L + µr 2

(3.41)

(γ , A˜ −1 L) A˜ −1 (γ ). 2 −1 ˜ 1 − µr (γ , A γ )

S 1 reduction of the homogeneous sphere to T ∗ S 2 The homogeneous Chaplygin sphere when reduced to T ∗ S 2 produces a more interesting system. Equations (3.41) become γ˙ =

1 a, I + µr 2

a˙ = ω3 a × γ −

1 |a|2 γ . I + µr 2

(3.42)

One observes that (a, γ ) = 0 and that |a|2 is conserved. So at each level set, we get an isotropic 3D oscillator with a Lorentz force.24 24 Alan Weinstein commented on more than one occasion that “unreduction’’ sometimes is

even nicer than reduction: unreducing a nontrivial system may lead to a trivial one. Alan credits this to Guillemin and Sternberg; one reference could be Guillemin and Sternberg [1980].

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Dimension count argument In hindsight, we can give two simple arguments why the Chaplygin marble could not be Hamiltonizable at the T ∗ SO(3) level. First, if (T ∗ SO(3), N H , H ) were Hamiltonizable, the system would be Liouville integrable by “mere’’symmetries, due to the existence of three independent ﬁrst integrals H, 3 , 21 + 22 , 23 . But it is known that integrability of Chaplygin’s marble stems not from symmetries, but from a special choice of separating coordinates (Duistermaat [2000]), namely, elliptic coordinates on the sphere. Second, Stanchenko [1985] veriﬁed that Chaplygin’s density function F (3.31) of the system in R6 also gives an invariant measure on T ∗ SO(3) (see also Duistermaat [2000], Section 7), F = [1 − µr 2 (γ , A˜ −1 γ )]−1/2 . (3.43) Were the compressed system Hamiltonizable in T ∗ SO(3), the conformal factor (time 1 reparametrization) would be F (γ ) m−1 , with m = 3; see (3.13). But the correct time reparametrization holds with m = 2 instead of m = 3. This strongly suggests that Hamiltonization should be attempted after reduction of the internal S 1 symmetry. νT ∗ SO(3) = F (γ )dλ1 dλ2 dλ3 dL1 dL2 dL3 ,

Phase locking The fact that Chaplygin’s sphere is integrable implies an interesting phase locking property. For simplicity, consider a resonant torus and a periodic solution, γ (T ) = γ (0), L(T ) = L(0). We may assume that R(0) = identity, so R(T ) preserves both k and . If we assume = ±k, then R(T ) must also be the identity (there is only one orthogonal matrix with two different eigenvectors with equal eigenvalues 1). Since the rotational conditions are reproduced after time T , there is a “planar geometric phase’’ (meaning a translation), z = ( x, y). From Duistermaat [2000], Section 11, one knows this direction. Proposition 3.7. On average,

z moves in the direction of × k.

In the normal direction k ×(×k) there is a “swaying motion,’’ with zero average, see Duistermaat [2000], (11.71), and Remark 11.11. This result depends on the explicit solution in terms of elliptic coordinates, but the zero average can be proved in a more elementary way, see Duistermaat, Section 8.2. In the direction × k one has d (z(t), × k) = r(ω × k, × k) = r(ω, − 3 k) = r(2T − 3 ω3 ) > 0. dt Duistermaat [2000] shows (in Section 9.2) that by a suitable change of coordinates, one may assume that 3 = 0, so in this equivalent problem, the velocity in this direction is simply 2rT . Chaplygin’s marble via the almost Hamiltonian structure After this detour, we hope the reader will appreciate a concise way of describing this system. The clockwise map is

Nonholonomic systems via moving frames T (SO(3) × R2 )

−→ Leg

T ∗ (SO(3) × R2 ) ( , x, ˙ y) ˙ → (M = A , Px = µx, ˙ Py = µy) ˙

↑ h | T SO(3)

111

↑ h | ←−

(Legφ )−1

T ∗ SO(3)

←

L,

(3.44) where (x, ˙ y) ˙ = rω × k, and (Px , Py ) = µrω × k. We now compute the “gyroscopic’’ 2-form (J.K) = r(−Px dρ2 + Py dρ1 ) = µr(−xdρ ˙ 2 + ydρ ˙ 1 ) = −µr 2 (ω2 dρ2 + ω1 dρ1 ). (3.45) To obtain ω1 and ω2 as functions in T ∗ SO(3), we use the Legendre transformation: ω = R = RA−1 M, so (J.K) is a combination of the basic forms ρ3 ∧ ρ1 , ρ2 ∧ ρ3 (coefﬁcients linear in M and functions of R). S 1 invariance We claim that N H is S 1 -invariant (S 1 acting only in the ﬁrst factor of Q = SO(3) × R3 ). For the canonical term this is a standard symplectic fact. The (J.K) term is invariant as well: (3.45), written in terms of the left-invariant forms, depends only on the Poisson vector γ : (J.K) = µr 2 (γ × ( (L, γ ) × γ ), dλ) .

(3.46) right

In fact, the S 1 action generated by the right-invariant vector ﬁeld X3 maintains the projection γ ﬁxed. We know (general nonsense) that the right-invariant vector ﬁelds ˙ = λi (Rφ R[ ]) = preserve the left-invariant forms: Rφ∗ λi = λi . (Proof: (Rφ∗ λi )(R) 1

i .) Since under the left S action (actually under the left action of SO(3) on SO(3)) the value of remains unchanged, the (J.K) term is preserved. The twisted action generator and S 1 reduction MW reduction method works ﬁne, although X3r is the Hamiltonian vector ﬁeld of J = 3 , relative to the canonical symplectic form, but not relative to N H . We just change to the twisted S 1 -action generator X˜ 3 , deﬁned by iX˜ 3 N H = −d3 . A simple computation gives X˜ 3 = Xr − m2 ∂ + m1 ∂ , where m1 = 1 − µr 2 ω1 , 3

∂1

∂2

m2 = 2 − µr 2 ω2 . The reduced manifold is the quotient of a level 3 in T ∗ SO(3), ˜ A concrete realization is achieved using (1.11). Taking identifying the ﬂow lines φ. ∗ the pullback via i , the reduced form is then

red = can + 3 areaS 2 − µr 2 (ω1 dθ2 − ω2 dθ1 ), T ∗S2

(3.47)

where we recall the parametrizations p1 θ1 + p2 θ2 ∈ T ∗ S 2 , R(γ ) = rows(e1 , e2 , γ ).

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In (3.47) we must write ω1 , ω2 explicitly in terms of p1 , p2 , 3 . To write this explicitly, there is no other option than to confront the monster (which actually is not that terrible): from = R −1 ω = ω1 e1 + ω2 e2 + ω3 γ and (3.26), we get ω = R(γ ) γ [R(γ )]† ,

= (p2 , −p1 , 3 ),

(3.48)

where γ is explicitly given by (3.30). Theorem 3.8. iX d(f red ) = 0, f (γ ) = [1 − µr 2 (γ , A˜ −1 γ )]−1/2 . Proof. We used spherical coordinates (fâute de mieux) and a MathematicaTM notebook. It misses being conformally symplectic by very little (even in the homogeneous case).25 Our calculation shows that Chaplygin’s sphere is not afﬁne symplectic even at the T ∗ S 2 level, so Chaplygin’s sphere integrability is due to a speciﬁc nonholonomic phenomenon. This observation is in accordance with the opening statement in Duistermaat [2000]: “Although the system is integrable in every sense of the word, it neither arises as a Hamiltonian system, nor is the integrability an immediate consequence of the symmetries.’’

4 Recent developments and ﬁnal comments NH systems have a reputation of having peculiar (even rebellious) dynamic behavior (Arnold, Kozlov, and Neishtadt [1988]). In spite of good progress, the general theory for NH systems is way behind the theory of Hamiltonian systems. For instance, although the groundwork for a Hamilton–Jacobi theory for NH systems has been set up in Weber [1986], not much has been achieved since. We have no intention (or competence) to make a survey of recent developments in NH systems, especially regarding reduction of symmetries; nevertheless it may be worth registering the intense activity going on. Recent books of interest are Cushman and Bates [1997], Cortés [2002], Oliva [2002], Bloch [2003] and a treatise in the mechanical engineering tradition is Papastavridis [2002].26 Reports on Mathematical Physics has been publishing NH papers regularly, and Regular and Chaotic Dynamics devoted large parts of vols. 1/2 (2002) to NH systems. For older eastern European literature, see P. M. M. USSR, J. Appl. Math. Mechanics, which has strongly inﬂuenced Chinese mechanics as well. For a historical account of NH systems, from a somewhat “antireductionist’’ perspective, see Borisov and Mamaev [2002a]. 25 Borisov and Mamaev [2002] showed by a subtle numerical evidence that, in the original time,

Chaplygin’s marble is not Hamiltonizable at any level of reduction. The question whether Chaplygin’s marble is Hamiltonizable in the new time dt/dτ = f (γ ) was addressed in Borisov and Mamaev [2001]. They provide a bracket structure in terms of the coordinates ˜ γ ), with L˜ = L/f (γ ). Using a computer algebra program we (L, γ ) or the coordinates (L, checked that the second brackets satisfy the Jacobi identity. However, we could not recover Chaplygin’s equations for the L coordinates, even in the homogenous case. 26 Reviewed in Koiller [2003].

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4.1 Invariant measures and integrability Kupka and Oliva [2001] and Kobayashi and Oliva [2003] ﬁnd conditions ensuring a special, but very interesting situation, where the Riemann measure in T Q induced by the metric in Q is an invariant measure for the NH system. Invariant measures for systems with distributional symmetries were characterized in Zenkov and Bloch [2003]. Curiously, although a number of interesting NH systems have been solved using Abelian functions, a precise deﬁnition for integrability of a NH system is still lacking (Bates and Cushman [1999]). These examples suggest that the presence of an invariant measure must be imposed as a necessary (although not sufﬁcient) condition for integrability (whatever it may be), see Kozlov [2002]. Most of them have enough integrals of motion that the dynamics occur on invariant two-dimensional tori. Due to the invariant measure, the ﬂow becomes linear in these tori after a time rescaling. This follows from Jacobi’s multiplier method and Kolmogorov’s theorem (Arnold [1989]). Time reparametrization indicates the possibility of an afﬁne symplectic structure. We believe that characterizing NH systems possessing an afﬁne symplectic structure (if needed, after some reduction stage) could be an interesting project. As a ﬁrst step, one may examine the existing literature to see which examples ﬁt. We list a few papers for that purpose: Veselov and Veselova [1988], Veselov and Veselova [1986], Fedorov [1989], Cushman, Hermans, and Kemppainen [1995], Zenkov [1995], Zenkov and Bloch [2000], Dragovic, Gajic´ , and Jovanovic [1998], Jovanovic [2003], Fedorov and Jovanovic [2003]. One can hope that the manifestly geometric character of (1.14) can be instrumental to understand when, where and why Hamiltonization is possible. Moreover, a prior geometric understanding of the invariant volume form conditions is a more general question. It would be also interesting to tie the “Hamiltonizable’’ question with the invariants from the Cartan equivalence viewpoint, see below. 4.2 Nonholonomic reduction ´ The difﬁculties in reduction for general NH systems are explained in Sniatycki [2002]. There are four current theories of reduction of symmety for nonholonomic systems27 : (i) projection methods; see Marle [1995], Dazord [1994]; (ii) the distributional Hamiltonian approach, initiated by Bocharov and Vinogradov ´ [1977] and developed in Bates and Sniatycki [1993], Cushman, Kemppainen, ´ ´ ´ Sniatycki, and Bates [1995], Sniatycki [1998], and Cushman and Sniatycki [2002]. (iii) bracket methods, initiated by Mashke and van der Schaft [1994], and developed by Koon and Marsden [1998] and Sn´ iatycki [2001]; (iv) Lagrangian reduction; see Cendra, Marsden, and Ratiu [2001]. A few other references in this rapidly developing theme, besides those already mentioned are Bates [2002], Koon and Marsden [1997], Cantrijn, de Léon, Marrero, and de Diego [1998], Cortés and de Léon [1999], Marle [1998], Marle [2003]. 27 We thank one of the referees for this information.

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Almost Poisson, almost Dirac approaches Mashke and van der Schaft [1994] were the ﬁrst to describe a NH system using an almost-Poisson structure,28 x˙i = {xi , H }MS . This bracket, deﬁned on the manifold P = Leg(H) ⊂ T ∗ Q, where Leg : T Q → T ∗ Q is the Legendre transformation, in general does not satisfy the Jacobi identity. They proved that the Jacobi identity holds if and only if the constraints are integrable. In Koiller, Rios, and Ehlers [2002] we gave a moving frames based derivation of the bracket structure. For some recent work on the MS-bracket and also Dirac estructures (the latter introduced in Courant [1990]), see Cantrijn, de Léon, de Diego [1999], Koon and Marsden [1998], Ibort, de Léon, Marrero, and de Diego [1999], Clemente-Gallardo, Maschke, and van der Schaft [2001]. In spite of these advances, a complete understanding of the NH bracket geometry is still in order.29 4.3 G-Chaplygin systems via afﬁne connections Trajectories of the compressed system can be described as geodesics of an afﬁne connection ∇ N H in S (Vershik and Fadeev [1981], Koiller [1992]). For background in this approach, see Lewis [1998] and references therein. Consider the parallel transport operator along closed curves; if the holonomy group is always conjugate to a subgroup of S0(m), then the connection is metrizable. This means that there is a metric such that ∇ N H is precisely the Levi-Civita connection of this metric. More generally, one may want to know when the geodesics of ∇ N H are, up to time reparametrization, the geodesics of a Riemannian metric. This is a traditional area in differential geometry, whose roots go back to the 19th century, and goes under the name of projectively equivalent connections (Cartan [1937], Eisenhart [1925], Kobayashi and Nomizu [1963], Sharpe [1997]). Grossman [2000] studies integrability of geodesics equations via the equivalence method. Our problem, then, is to ﬁnd conditions for the NH connection to be projectively equivalent to a Riemannian connection. It would be also interesting to tie the Hamiltonization question with the canonical system and invariants of the Cartan equivalence method. When an internal symmetry group is present, it would be desirable to construct a projected connection in S for each set of conserved momenta, and address these issues in the reduced level. Acknowledgments Our thanks to Hans Duistermaat for information on Chaplygin’s sphere and the referees for very good criticism and suggestions. This is also a special occasion to thank our mathematical family: Alan, of course; mathematical brothers and sisters from all continents (especially Yilmaz Akyildyz and Henrique Bursztyn) and cousins (especially Tudor Ratiu and 28 Physicists are never shy to use the word “super’’ in their endeavors; on the other hand we,

mathematicians, prefer to use low-key terminology, like “almost-quasi-twisted-(freakaz-)‘oid’s’’; this certainly does not help our image problem with applied people, see Papastavridis [2002] and Koiller [2003]. 29 Observations by J. Marsden (joint work with H. Yoshimura), and by C. Marle in their Alanfest talks are important steps in this direction.

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Debra Lewis), “uncle’’ Jerry Marsden, and, last but not least, “grandfather’’ Chern (with his gentle voice, commanding us to keep interested in Math, up to his age). We shall toast in many more Alanfests: Madadayo!

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Thompson’s conjecture for real semisimple Lie groups∗ Sam Evens1 and Jiang-Hua Lu2 1 Department of Mathematics

The University of Notre Dame Notre Dame, IN 46556 USA [email protected] 2 Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong [email protected] Dedicated to Professor Alan Weinstein for his 60th birthday. Abstract. A proof of Thompson’s conjecture for real semisimple Lie groups has been given by Kapovich, Millson, and Leeb. In this paper, we give another proof of the conjecture by using a theorem of Alekseev, Meinrenken, and Woodward from symplectic geometry.

1 Introduction Thompson’s conjecture [T] for the group GL(n, C), which relates eigenvalues of matrix sums and singular values of matrix products, was ﬁrst proved by Klyachko in [Kl]. In [Al-Me-W], by applying a Moser argument to certain symplectic structures, Alekseev, Meinrenken, and Woodward gave a proof of Thompson’s conjecture for all quasisplit real semisimple Lie groups. In [Ka-Le-M1], Kapovich, Millson, and Leeb have, among other things, proved Thompson’s conjecture for an arbitrary semisimple Lie group G0 . A direct geometric argument can also be found in [Ka-Le-M3, Section 4.2.4]. In this note, we give a different proof of Thompson’s conjecture for arbitrary semisimple real groups by extending the proof of Alekseev, Meinrenken, and Woodward for quasisplit groups. In fact, we prove a stronger result, Theorem 2.2, which implies Thompson’s conjecture. Let G0 = K0 A0 N0 be an Iwasawa decomposition of G0 and let g0 = k0 + p0 be a compatible Cartan decomposition of the Lie algebra of g0 . Theorem 2.2 asserts that for each l ≥ 1, there is a diffeomorphism ∗ The research of the ﬁrst author was partially supported by NSF grant DMS-9970102. The

research of the second author was partially supported by NSF grant DMS-0105195, by HHY Physical Sciences Fund, and by the New Staff Seeding Fund at HKU.

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L : (A0 N0 )l → (p0 )l which relates the addition on p0 with the multiplication on A0 N0 and intertwines naturally deﬁned K0 -actions. When G0 is quasisplit, Theorem 2.2 follows from results in [Al-Me-W]. The key step in our proof of Theorem 2.2 for an arbitrary G0 is to relate an arbitrary real semisimple Lie algebra g0 to a quasisplit real form in its complexiﬁcation. In Section 2, we state Theorem 2.2 and show that it implies Thompson’s conjecture. Inner classes of real forms and quasisplit real forms are reviewed in Section 3. The proof of Theorem 2.2 is given in Section 4. Since the version of the Alekseev– Meinrenken–Woodward theorem we present in this paper is not explicitly stated in [Al-Me-W], we give an outline of its proof in Section 5, the appendix.

2 Thompson’s conjecture Let G be a complex connected reductive algebraic group with an antiholomorphic involution τ . Let G0 be a subgroup of the ﬁxed point set Gτ of τ containing the identity connected component. Then G0 is a real reductive Lie group in the sense of [Wa, pp. 42–45], which implies that G0 has Cartan and Iwasawa decompositions. Let g0 be the Lie algebra of G0 . Fix a Cartan decomposition g0 = k0 + p0 of g0 , and let G0 = P0 K0 be the corresponding Cartan decomposition of G0 . Let a0 ⊂ p0 be a maximal abelian subspace of p0 . Fix a choice + res of positive roots in the restricted root system res for (g0 , a0 ), and let n0 be the subspace of g0 spanned by the root vectors for roots in + res . Then g0 = k0 + a0 + n0 is an Iwasawa decomposition for g0 . Let A0 = exp(a0 ) and N0 = exp(n0 ). Then we have the Iwasawa decomposition G0 = K0 A0 N0 for G0 . Consider now the space X0 := G0 /K0 with the left G0 -action given by G0 × (G0 /K0 ) −→ G0 /K0 : (g1 , gK0 ) −→ g1 gK0 ,

g1 , g ∈ G0 .

(2.1)

Thompson’s conjecture is concerned with K0 -orbits in X0 . Identify exp

p0 ∼ = P0 ∼ = G0 /K0 = X0 via the Cartan decomposition G0 = P0 K0 . The K0 -action on X0 in (2.1) becomes the adjoint action of K0 on p0 . Orbits of K0 in p0 are called (real) ﬂag manifolds. Let + a+ 0 ⊂ a0 be the closed Weyl chamber determined by res . It is well known that every + K0 -orbit in p0 goes through a unique element λ ∈ a0 . On the other hand, we can also identify A0 N0 ∼ = G0 /K0 = X0 via the Iwasawa decomposition G0 = A0 N0 K0 . Then the K0 -action on X0 becomes the following action of K0 on A0 N0 : k · b := p(kb) = p(kbk −1 )

for k ∈ K0 , b ∈ A0 N0 ,

(2.2)

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123

where p : G0 → A0 N0 is the projection b1 k1 → b1 for k1 ∈ K0 and b1 ∈ A0 N0 . Let E0 : p0 → A0 N0 be the composition of the identiﬁcations: exp

E0 : p0 ∼ = P0 ∼ = G0 /K0 ∼ = A0 N0 .

(2.3)

Then E0 is K0 -equivariant, and E0 (a0 ) = A0 . Thus every K0 -orbit in A0 N0 goes + through a unique point a = exp λ ∈ A+ 0 := exp a0 . Thompson’s conjecture for G0 is concerned with the sum of K0 -orbits in p0 and the product of K0 -orbits in A0 N0 . To further prepare for the statement of the conjecture, let l ≥ 1 be an integer, and consider the two maps a : p0 × p0 × · · · × p0 −→ p0 : (ξ1 , ξ2 , . . . , ξl ) −→ ξ1 + ξ2 + · · · + ξl , m : A0 N0 × A0 N0 × · · · × A0 N0 −→ A0 N0 : (b1 , b2 , . . . , bl ) −→ b1 b2 · · · bl . Clearly, a is K0 -equivariant for the diagonal action of K0 on (p0 )l . On the other hand, deﬁne the twisted diagonal action T of K0 on (A0 N0 )l by k −→ Tk := ν −1 ◦ δk ◦ ν : (A0 N0 )l −→ (A0 N0 )l ,

(2.4)

where δk is the diagonal action of k ∈ K0 on (A0 N0 )l , and ν : (A0 N0 )l −→ (A0 N0 )l is the diffeomorphism given by ν(b1 , b2 , . . . , bl ) −→ (b1 , b1 b2 , . . . , b1 b2 · · · bl ).

(2.5)

Them m is K0 -equivariant. See Remark 2.4 for motivation of the twisted diagonal action. Let e be the identity element of A0 N0 and identify Te (A0 N0 ) ∼ = = a0 + n0 ∼ g0 /k0 ∼ . We will regard the map a, respectively, the diagonal K -action on (p0 )l , p = 0 0 as the linearization of the map m, respectively, the twisted diagonal K0 -action on (A0 N0 )l , at the point (e, e, . . . , e). Notation 2.1. For λ ∈ a0 , we will use Oλ to denote the K0 -orbit in p0 through λ. For a ∈ A0 , we will use Da to denote the K0 -orbit in A0 N0 through the point a. If = (λ1 , λ2 , . . . , λl ) ∈ (a0 )l , we set aj = exp(λj ) for 1 ≤ j ≤ l and O = Oλ1 × Oλ2 × · · · × Oλl

and

D = Da1 × Da2 × · · · × Dal .

In this paper, we will prove the following theorem. Theorem 2.2. For every integer l ≥ 1, there is a K0 -equivariant diffeomorphism L : (A0 N0 )l → (p0 )l such that m = E0 ◦ a ◦ L and L(D ) = O for every ∈ (a0 )l . Theorem 2.2 now readily implies the following Thompson’s conjecture for G0 . Corollary 2.3 (Thompson’s conjecture). For each = (λ1 , λ2 , . . . , λl ) ∈ (a0 )l , the two spaces (m−1 (e) ∩ D )/K0 = {(b1 , b2 , . . . , bl ) ∈ D : b1 b2 · · · bl = e}/K0

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and (a−1 (0) ∩ O )/K0 = {(ξ1 , ξ2 , . . . , ξl ) ∈ O : ξ1 + ξ2 + · · · + ξl = 0}/K0 are homeomorphic. In particular, one is nonempty if and only if the other is. Proof. Let L : (A0 N0 )l → (p0 )l be the diffeomorphism in Theorem 2.2. Then L induces a homeomorphism L : m−1 (e) → a−1 (0). Since L(D ) = O and L is K0 -equivariant, it induces a homeomorphism from (m−1 (e) ∩ D )/K0 to (a−1 (0) ∩ O )/K0 .

Remark 2.4. Equip X0 = G0 /K0 with the G0 -invariant Riemannian structure deﬁned by the restriction of the Killing form of g0 on p0 . For x1 , x2 ∈ X0 , let x1 x2 be the unique geodesic in X0 connecting x1 and x2 . Then there is a unique λ ∈ a+ 0 such that g · x1 = ∗ and g · x2 = (exp λ) · ∗ for some g ∈ G0 , where ∗ = eK0 ∈ X0 is the base + point. The element λ ∈ a+ 0 is called [Ku-Le-M] the a0 -length of x1 x2 . Representing the vertices of an ∗-based l-gon in X0 by (∗, b1 · ∗, b1 b2 · ∗, . . . , b1 b2 · · · bl · ∗) for some b1 , b2 , . . . , bl ∈ A0 N0 , we can regard (b1 , b2 , . . . , bl ) as the set of edges of l the l-gon. Then for ∈ (a+ 0 ) , the space {(b1 , b2 , . . . , bl ) ∈ D : b1 b2 · · · bl = e}/K0 can be identiﬁed with the space of G0 -equivalence classes of closed l-gons in X0 with ﬁxed “side length’’ . Similarly, the space {(ξ1 , ξ2 , . . . , ξl ) ∈ O : ξ1 + ξ2 + · · · + ξl = 0}/K0 can be identiﬁed with the space of equivalent l-gons with ﬁxed side length in the “inﬁnitesimal symmetric space’’ p0 . Using the right A0 N0 -action on K0 given by k b := q(kb),

b ∈ A0 N0 ,

k ∈ K0 ,

(2.6)

where q : G0 → K0 is the projection q(b1 k1 ) = k1 for b1 ∈ A0 N0 and k1 ∈ K0 , it is easy to see that Tk as deﬁned in 2.4 is also given by Tk (b1 , b2 , . . . , bl ) := (k1 · b1 , k2 · b2 , . . . , kl · bl ),

(2.7)

k b1 b2 ···bj −1

where k1 = k, kj = for 2 ≤ j ≤ l. In the appendix (Section 5), we will see that this formula naturally arises in the context of Poisson Lie group actions. Remark 2.5. When G0 = GL(n,& R), recall that the singular values of g ∈ G0 are by deﬁnition the eigenvalues of gg t . Thompson’s conjecture for GL(n, R) says that, for any collection (λ1 , λ2 , . . . , λl ) of real diagonal matrices, the following two statements are equivalent (see [Al-Me-W, Section 4.2]): 1. there exist matrices gj ∈ GL(n, R) whose singular values are entries of aj = exp(λj ) and g1 g2 · · · gl = e; 2. there exist symmetric matrices ξj whose eigenvalues are entries of λj and such that ξ1 + ξ2 + · · · + ξl = 0.

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125

Remark 2.6. By a theorem of O’Shea and Sjamaar [O-S], the set a−1 (0) ∩ O is nonempty if and only if = (λ1 , λ2 , . . . , λl ) lies in a certain polyhedral cone P in + l + (a+ 0 ) . For G0 = SL(2, R), we have a0 = R = {λ ∈ R : λ ≥ 0}. In this case 2 p0 may be identiﬁed with R and the K0 -action may be identiﬁed with rotation. We know from classical triangle inequalities that P for l = 3 is given by the inequalities |λ1 − λ2 | ≤ λ3 ≤ λ1 + λ2 .

(2.8)

The fact that these triangle inequalities are also the necessary and sufﬁcient conditions for λ1 , λ2 and λ3 to be the sides of a geodesic triangle in the hyperbolic 2-space SL(2, R)/SO(2) follows from the law of cosines in this space, which says that cos θ =

cosh λ1 cosh λ2 − cosh λ3 , sinh λ1 sinh λ2

where θ is the angle between the two sides with lengths λ1 and λ2 . For any arbitrary real semisimple Lie group G0 , there have been intensive research activities on the inequalities that describe the polyhedral cone P. We refer to [Ku-Le-M] for explicit examples of these polyhedral cones when X0 has rank 3 and to [Fu, Ka-Le-M1, Ka-Le-M2, Ka-Le-M3] for an account of the history and connections between this problem and others ﬁelds such as Schubert calculus, representation theory, symmetric spaces, geometric invariant theory, and integrable systems. See also Remark 4.9. Remark 2.7. Finally, we remark that it is enough to prove Theorem 2.2 for G with trivial center. Indeed, let Z be the center of G, let Z0 = G0 ∩ Z, and let G 0 = G0 /Z0 with Lie algebra g 0 = g0 /z0 , where the Lie algebra z0 of Z0 is the center of g0 . Let j : G0 → G 0 and g0 → g 0 be the natural projections. Let k 0 = j (k0 ) and p 0 = j (p0 ), and let K0 = j (K0 ), A 0 = j (A0 ), and N0 = j (N0 ). By [Kn1, Corollary 1.3], Z0 = (K0 ∩ Z0 )(A0 ∩ Z0 ), and exp : z0 ∩ a0 → Z0 ∩ A0 is an isomorphism. Thus G 0 = K0 A 0 N0 is an Iwasawa decomposition for G 0 , and g 0 = k 0 + p 0 is a Cartan decomposition for g 0 . Moreover, p0 ∼ = p 0 ⊕ (a0 ∩ z0 ) and A0 ∼ = A 0 × (A0 ∩ Z0 ) and ∼ N0 = N0 . Let a : (p 0 )l −→ p 0 ,

m : (A 0 N0 )l −→ A0 N0 ,

a : (a0 ∩ z0 )l −→ a0 ∩ z0 ,

m : (A0 ∩ Z0 )l −→ A0 ∩ Z0

be, respectively, the addition and multiplication maps. If L : (A 0 N0 )l → (p 0 )l is a diffeomorphism satisfying the requirements in Theorem 2.2 for the group G 0 , then L = (L , (log)l ) will be a diffeomorphism from (A0 N0 )l to (p0 )l satisfying the requirements in Theorem 2.2 for the group G0 , where we use the obvious identiﬁcations between (A0 N0 )l ∼ = (A 0 N0 )l × (A0 ∩ Z0 )l and (p0 )l ∼ = (p 0 )l × (a0 ∩ z0 )l .

3 Inner classes of real forms and quasisplit real forms Let g be a semisimple complex Lie algebra. Recall that real forms of g are in one-toone correspondence with complex conjugate linear involutive automorphisms of g.

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For such an involution τ , the corresponding real form is the ﬁxed point set gτ of τ . We will refer to both gτ and τ as the real form. Throughout this paper, if V is a set and σ in an involution on V , we will use V σ to denote the set of σ -ﬁxed points in V . Let G be the adjoint group of g. Deﬁnition 3.1 ([A-B-V, Deﬁnitions 2.4 and 2.6]). Two real forms τ1 and τ2 of g are said to be inner to each other if there exists g ∈ G such that τ1 = Ad g τ2 . A real form τ of g is said to be quasisplit if there exists a Borel subalgebra b of g such that τ (b) = b. Inner classes of real forms are classiﬁed by involutive automorphisms of the Dynkin diagram D(g) of g. Indeed, let Aut g be the group of complex linear automorphisms of g. Its identity component is the adjoint group G. Let Aut D(g) be the automorphism group of the Dynkin diagram D(g) of g. There is a split short exact sequence [A-B-V, Proposition 2.11], !

1 −→ G −→ Aut g −→ Aut D(g) −→ 1.

(3.1)

Denote by R the set of all real forms of g. Let θ be any compact real form of g. Deﬁne " : R −→ Aut D(g) : " (τ ) = !(τ θ).

(3.2)

Then " (τ )2 = 1 for each τ , and τ1 and τ2 are inner to each other if and only if " (τ1 ) = " (τ2 ). Conversely, for every involutive d ∈ Aut D(g) , we can construct γd ∈ Aut g such that τ := γd θ is a real form with " (τ ) = d (see (3.4) below). Thus the map " gives a bijection between inner classes of real forms of g and involutive elements in Aut D(g) [A-B-V, Proposition 2.12]). Deﬁnition 3.2. Let d be an involutive automorphism of the Dynkin diagram D(g) of g. We say that a real form τ of g is of inner class d or in the d-inner class if " (τ ) = d. By [A-B-V, Proposition 2.7], every inner class of real forms of g contains a quasisplit real form that is unique up to G-conjugacy. In the following, for each involutive d ∈ Aut D(g) , we will construct an explicit quasisplit real form τd in the d-inner class. We will then show that, up to G-conjugacy, every real form in the d-inner class is of the form τ = Ad w˙ 0 τd , where w0 is a certain Weyl group element and w˙ 0 a representative of w0 in G. We ﬁrst ﬁx once and for all the following data for g: Let h be a Cartan subalgebra of g, and let be the corresponding root system. Fix a choice of positive roots + in , and let be the basis of simple roots. Let ·, · be the Killing form on g. For each α ∈ , let E±α be root vectors such that [Eα , E−α ] = Hα for all α ∈ + , where Hα is the unique element of h deﬁned by H, Hα = α(H ) for all H ∈ h, and the numbers mα,β for α, β ∈ deﬁned for α = −β by [Eα , Eβ ] = mα,β Eα+β =0

if α + β ∈ otherwise

Thompson’s conjecture for real semisimple Lie groups

127

satisfy m−α,−β = −mα,β . We will refer to the set {Eα , E−α : α ∈ + } as (part of) a Weyl basis. Using a Weyl basis {Eα , E−α : α ∈ + }, we can deﬁne a compact real form k of g as √ √ k = spanR { −1Hα , Xα := Eα − E−α , Yα := −1(Eα + E−α ) : α ∈ + }. (3.3) Let θ be the complex conjugation of g deﬁning k. We also deﬁne a split real form η0 of g by setting η0 |a = id, and η0 (Eα ) = Eα for every α ∈ . Clearly θ η0 = η0 θ. The inner class for η0 is easily seen to be the automorphism of the simple roots given by −w 0 , where w 0 is the longest element in the Weyl group W of (g, h). An explicit splitting of the short exact sequence (3.1) can be constructed using the Weyl basis. Indeed, for any d ∈ Aut D(g) , deﬁne γd ∈ Aut g by requiring γd (Hα ) = Hdα

and

γd (Eα ) = Edα

(3.4)

for each simple root α. Then d → γd is a group homomorphism from Aut D(g) to Aut g and is a section of ! in (3.1). Moreover, every γd commutes with both θ and η0 because they commute on a set of generators of g. Lemma 3.3. For an involutive element d ∈ Aut D(g) , let γ−w0 d ∈ Aut g be the lifting of −w0 d ∈ Aut D(g) as deﬁned in (3.4). Deﬁne τd = η0 γ−w0 d .

(3.5)

Then τd is a quasisplit real form of g in the d-inner class. Proof. We know that (τd )2 = 1 because −w0 ∈ Aut D(g) is in the center. Since τd maps every positive root vector to another positive root vector, it is a quasisplit real form. Finally, since " (τd ) = !((γ−w0 )(γ−w0 d )) = d, we see that τd is in the d-inner class.

To relate an arbitrary real form in the d-inner class with the quasisplit real form = k + a + n is an Iwasawa τd , we recall some deﬁnitions from [Ar]. Note ﬁrst that g decomposition, where a = spanR {Hα : α ∈ } and n = α∈ + gα . Deﬁnition 3.4. A real form τ of g is said to be normally related to (k, a) and compatible with + if 1. τ θ = θ√ τ , and τ (h) = h; √ 2. aτ ⊂ ( −1k)τ is maximal abelian in ( −1k)τ ; 3. if α ∈ + is such that α|aτ = 0, then τ (α) ∈ + , where τ (α) ∈ a∗ is deﬁned by τ (α)(λ) = α(τ (λ)) for λ ∈ a. We will call a real form with properties 1–3 an Iwasawa real form relative to (k, a, n). Remark 3.5. Once properties 1 and 2 in Deﬁnition 3.4 are satisﬁed, property 3 is equivalent to the set r( + )\{0} ⊂ res being a choice of positive roots for the restricted root system res of (gτ , aτ ), where r is the map dual to the inclusion aτ → a. If τ is an Iwasawa real form relative to (k, a, n), then so is Ad t τ Ad −1 t for any t ∈ exp(ia).

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Proposition 3.6. 1. Every real form of g is conjugate by an element in G to a real form that is Iwasawa relative to (k, a, n). 2. Suppose that τ is an Iwasawa real form relative to (k, a, n) and suppose that τ is in the d-inner class. Let w0 be the longest element of the subgroup of W generated by the reﬂections corresponding to roots in the set 0

= {α ∈

: α|aτ = 0} = {α ∈

: τ (α) = −α}.

Then there is a representative w˙ 0 of w0 in G such that τ = Ad w˙ 0 τd .

(3.6)

Proof. Statement 1 follows from [Ar, Proposition 1.2, Section 2.8, and Corollary 2.5]. Statement 2: Assume now that τ is an Iwasawa real form relative to (k, a, n) and that τ is in the d-inner class. Consider τd τ . Since both τ and τd are in the d-inner class, τd τ = Ad g for some g ∈ G. Since both τ and τd leave h invariant, the element g represents an element in the Weyl group W . Let 0 = ∩ 0 . By [Ar, Section 2.8], α ∈ 0 if and only if α is in the linear span of 0 . For every α ∈ 0 , we have τd τ (α) = −τd (α) ∈ − + , and for every α ∈ + − 0 , since τ (α) ∈ + , we have

(τd τ )(α) ∈ + . Thus g represents the element w0 = (w0 )−1 . Remark 3.7. Recall from [Ar] that the Satake diagram of τ is the Dynkin diagram of g with simple roots in 0 painted black, simple roots in − 0 painted white, and a two-sided arrow drawn between two white simple roots α and α if τ (α) = α + β for some β ∈ 0 . From (3.6) we see that α = −w0 d(α) if α is a white simple root. Conversely, given a Satake diagram for a real form of g, let c ∈ Aut D(g) be deﬁned by + −w0 α if α is black, (3.7) c(α) = if α is white, α where w0 is the longest element in the subgroup of the Weyl group of (g, h) generated by the black dots in the Satake diagram, and α → α is the order-2 involution on the set of white dots in the diagram. Then c is involutive, and the inner class of the real form is d = −w0 c. We now return to the real form τ in Proposition 3.6. Set + 1

= {α ∈

Then τ (

+ 1)

⊂

+

: α|aτ = 0},

+ 1.

(

+ 1)

=

+

∩

0

= {α ∈

+

: α|aτ = 0}.

Set n1 =

α∈

+ 1

gα ,

n 1 =

α∈(

gα .

(3.8)

+ 1)

Then n1 is τ -invariant, and, since there are no noncompact imaginary roots for the Cartan subalgebra hτ of gτ , we have τ |n 1 = θ|n 1 [Ar, Proposition 1.1]. Since the

Thompson’s conjecture for real semisimple Lie groups

restriction of know that

+

129

to aτ gives a choice of positive restricted roots for (gτ , aτ ), we gτ = kτ + aτ + (n1 )τ

is an Iwasawa decomposition of

(3.9)

gτ .

4 Proof of Theorem 2.2 By Remark 2.7, it is enough to prove Theorem 2.2 when G of adjoint type. Let g0 be a real form of g, the Lie algebra of G. Then by Proposition 3.6, we can assume that g0 = gτ , where τ is the involution on g given by (3.6), and d is the inner class of g0 . The lifting of τ to G will also be denoted by τ . Let G0 contain the connected component of the identity of the subgroup Gτ . In this section, we will prove Theorem 2.2 for G0 . The ﬁrst step in our proof of Theorem 2.2 for G0 is to realize the various objects associated to G0 as ﬁxed point sets of involutions on the corresponding objects related to G. We will then apply a theorem of Alekseev–Meinrenken–Woodward, stated as Theorem 4.7 below, whose proof using Poisson geometry will be outlined in Section 5, the appendix. We will keep all the notation from Section 3. In particular, set √ k0 = kτ , p0 = ( −1k)τ , a0 = aτ , and n0 = (n1 )τ . Then g0 = k0 +p0 is a Cartan decomposition of g0 , and g0 = k0 +a0 +n0 an Iwasawa decomposition of g0 . Let K be the connected subgroup of G with Lie algebra k. Let K0 = K ∩ G0 , K τ = K ∩ Gτ , and let N1 = exp(n1 ),

A0 = exp(a0 ),

and

N0 = N1 ∩ Gτ = exp(n0 ).

Lemma 4.1. G0 = K0 A0 N0 is an Iwasawa decomposition of G0 , and Gτ = K τ A0 N0 is an Iwasawa decomposition of Gτ . Proof. The statements follow from [Wa, Lemma 2.1.7] and the facts that K0 and K τ are maximally compact subgroups of G0 and of Gτ, respectively.

Let A = exp a and N = exp n so that G = KAN is an Iwasawa decomposition of G. We will now identify A0 N0 with the ﬁxed point set of an involution on AN, although we note that the involution τ does not leave AN invariant unless τ = τd . Deﬁne σ : AN −→ AN : σ (b) = p(τ (b)), (4.1) where p : G = KAN → AN is the projection b1 k1 → b1 for k1 ∈ K and b1 ∈ AN. Recall from (3.6) that τ = Ad w˙ 0 τd on g. Since both τ and τd commute with θ, the element w˙ 0 is in K. As for the case of G0 , we can use the Iwasawa decomposition G = KAN to deﬁne a left action of K on AN ∼ = G/K by k · b := p(kb),

k ∈ K,

b ∈ AN.

(4.2)

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Then σ : AN → AN is also given by σ (b) = w˙ 0 · τd (b)

for b ∈ AN .

(4.3)

Lemma 4.2. σ : AN → AN is an involution, and A0 N0 = (AN )σ , the ﬁxed point set of σ in AN. Proof. The fact that σ 2 = 1 follows from the fact that w˙ 0 τd (w˙ 0 ) = 1. Recall that A0 N0 is the ﬁxed point set of τ in AN1 , where N1 = exp(n1 ) and n1 ⊂ n is given in (3.8). Since σ coincides with τ on AN1 , we have A0 N0 ⊂ (AN )σ . Suppose now that b ∈ AN is such that σ (b) = b. Then there exists k ∈ K such that τ (b) = bk. By [Sl, Proposition 7.1.3], we can decompose b as b = gak1 for some k1 ∈ K, a ∈ A and g ∈ Gτ . Then τ (b) = gτ (a)τ (k1 ), and thus gτ (a)τ (k1 ) = gak1 k, from which it follows that τ (a) = a and τ (k1 ) = k1 k. Thus k = k1−1 τ (k1 ), and hence τ (bk1−1 ) = bk1−1 . Write bk1−1 = b2 k2 with k2 ∈ K τ and b2 ∈ A0 N0 using the Iwasawa decomposition of Gτ . It follows then that k2 = k1−1 and b2 = b, so b ∈ A0 N0 .

Now let l ≥ 1 be an integer. As in Section 2, we have the twisted diagonal action k → Tk of K on (AN )l given by Tk = ν −1 ◦ δk ◦ ν : (AN )l −→ (AN )l ,

(4.4)

where ν : (AN )l → (AN )l is as in (2.5) with A0 N0 replaced by AN, and δk denotes the diagonal action of k ∈ K on AN. Set (τd )l = (τd , τd , . . . , τd ) : (AN )l → (AN )l . Lemma 4.3. For an integer l ≥ 1, deﬁne σ (l) = Tw˙ 0 ◦ (τd )l : (AN )l −→ (AN )l . Then σ (l) is an involution, and the ﬁxed point set of σ (l) is (A0 N0 )l . Proof. Let σ l = (σ, σ, . . . , σ ) : (AN )l → (AN )l , where σ : AN → AN is as in Lemma 4.2. Since τd is a group automorphism of AN, we have σ (l) = ν −1 ◦ δw˙ 0 ◦ ν ◦ (τd )l = ν −1 ◦ (δw˙ 0 ◦ (τd )l ) ◦ ν = ν −1 ◦ σ l ◦ ν. Thus (σ (l) )2 = 1. Moreover, let b = (b1 , b2 , . . . , bl ) ∈ (AN )l , and let b = ν(b). Then σ (l) (b) = b if and only if σ l (b ) = b , which in turn is equivalent to bj ∈ A0 N0 for each 1 ≤ j ≤ l because of Lemma 4.2 and because of the fact that A0 N0 is a subgroup of AN.

√ Let p = −1k, so g = k + p is a Cartan decomposition of g. Let E : p → AN be the composition of the identiﬁcations exp

E:p ∼ = exp(p) ∼ = G/K ∼ = AN.

(4.5)

Then E is K-equivariant with respect to the action of K on p by conjugation and the action of K on AN given in (4.2).

Thompson’s conjecture for real semisimple Lie groups

131

Lemma 4.4. E ◦ (τ |p ) = σ ◦ E, and E0 = E|p0 : p0 → A0 N0 . Proof. Consider E −1 : AN → p. For g ∈ G, deﬁne g ∗ = θ (g −1 ). Then for every b ∈ AN, E −1 (b) = 12 log(bb∗ ) for all b ∈ AN, and thus 1 log(w˙ 0 τd (b)τd (b)∗ w˙ 0−1 ) 2 1 log(τd (b)τd (b)∗ ) = Ad w˙ 0 τd (E −1 (b)) = τ (E −1 (b)). = Ad w˙ 0 2

E −1 (σ (b)) =

Thus E ◦ (τ |p ) = σ ◦ E, and E(p0 ) = A0 N0 by Lemma 4.2. It also follows that E|p0 = E0 .

Notation 4.5. For λ ∈ a ⊂ p, let Oλ be the K-orbit in p through λ, and let Da be the K-orbit in AN through a = exp λ ∈ A. For = (λ1 , λ2 , . . . , λl ) ∈ (a)l , set O = Oλ1 × Oλ2 × · · · × Oλl

and

D = Da1 × Da2 × · · · × Dal .

(4.6)

Recall from Section 2 that, for λ ∈ a0 , Oλ denotes the K0 -orbit in p0 through λ, and Da denotes the K0 -orbit in A0 N0 through a = exp λ ∈ A0 . Lemma 4.6. Let λ ∈ a0 and a = exp λ ∈ A0 . Then 1. τd (λ) = λ, so both Oλ ⊂ p and Da ⊂ AN are τd -invariant; 2. Oλ is τ -invariant, and (Oλ )τ = Oλ ; 3. Da is σ -invariant and (Da )σ = Da . Proof. 1. Let α be a simple root such that τ (α) = −α. Then for any λ ∈ a0 , α(λ) = α(τ (λ)) = (τ (α))(λ) = −α(λ) = 0. Thus rα (λ) = λ, where rα is the reﬂection in a deﬁned by α. Since w0 is a product of such reﬂections (see Proposition 3.6), we see that w0 acts trivially on a0 . Thus every λ ∈ a0 is also ﬁxed by τd . 2. This is a standard fact. See, for example, [O-S, Example 2.9]. We remark that the key point is to show that (Oλ )τ is connected. 3. Statement 3 follows from statement 2 and Lemma 4.4.

We can now state the Alekseev–Meinrenken–Woodward theorem. Set a : p × p × · · · × p −→ p : (x1 , x2 , . . . , xl ) −→ x1 + · · · + xl , m : AN × AN × · · · × AN −→ AN : (b1 , b2 , . . . , bl ) −→ b1 b2 · · · bl . As for the case of G0 , we will equip pl with the diagonal K-action by conjugation, and we will equip (AN )l with the twisted diagonal action T given by (4.4).

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Theorem 4.7 (Alekseev–Meinrenken–Woodward [Al-Me-W]). For every quasisplit real form τd given in (3.5) and for every integer l ≥ 1, there is a K-equivariant diffeomorphism L : (AN )l → pl such that m = E ◦ a ◦ L,

and (τd )l ◦ L = L ◦ (τd )l .

Moreover, L(D ) = O for every ∈

(4.7)

al .

Remark 4.8. Theorem 4.7, whose proof will be outlined in the appendix (Section 5), is a consequence of the Moser isotopy lemma for Hamiltonian K-actions on Poisson manifolds, a rigidity theorem for such spaces. More precisely, we will see in the appendix (Section 5) that the map E −1 ◦ m : (AN )l → p is a moment map for the twisted diagonal K-action T on (AN )l with respect to a Poisson structure π1 on (AN)l , and (τd )l is an anti-Poisson involution for π1 . Moreover, the symplectic leaves of π1 are precisely all the orbits O for ∈ al . The quintuple Q1 = ((AN )l , π1 , T , E −1 ◦ m, (τd )l ) will be referred to as a Hamiltonian Poisson K-space with anti-Poisson involution. In fact, Q1 belongs to a smooth family Qs = ((AN )l , πs , Ts , E −1 ◦ ms , (τd )l ) as the case for s = 1, and when s = 0, Ts is the diagonal K-action on (AN )l and E −1 ◦ m0 = a ◦ (E −1 )l . The Moser isotopy lemma, Proposition 5.1 in Section 5, implies that Qs is isomorphic to Q0 by a diffeomorphism ψs of (AN )l for every s ∈ R. The map L in Theorem 4.7 is then taken to be (E −1 )l ◦ ψ1 . We will assume Theorem 4.7 for now and prove Theorem 2.2 for G0 . Proof of Theorem 2.2. Let L : (AN )l → pl be as in Theorem 4.7. Since L is Kequivariant and intertwines (τd )l : (AN )l → (AN )l and (τd )l : pl → pl , it also intertwines σ (l) = Tw˙ 0 ◦ (τd )l : (AN )l −→ (AN )l

and

τ l = δw˙ 0 ◦ (τd )l : pl −→ pl .

Thus by Lemma 4.3, we know that L((A0 N0 )l ) = (p0 )l . Denote L|(A0 N0 )l : (A0 N0 )l → (p0 )l also by L. Then clearly L is K0 -equivariant, and since E0 : p0 → A0 N0 coincides with the restriction of E : p → AN to p0 , we see that m = E0 ◦a ◦L. Finally, let = (λ1 , λ2 , . . . , λl ) ∈ (a0 )l . Then by Lemma 4.6, we see that L(D ) = L(D ∩ (A0 N0 )l ) = O ∩ (p0 )l = O .

+ Remark 4.9. Let a+ = {x ∈ a : α(x) ≥ 0 ∀α ∈ + } so that a+ 0 = a0 ∩ a . By + l Kirwan’s convexity theorem, there exists a polyhedral cone P ⊂ (a ) such that ∈ P if and only if a−1 (0) ∩ O is nonempty. The cone P is explicitly computed l in [B-S]. Set P0 = P ∩ (a0 )l ⊂ (a+ 0 ) . Then by a theorem of O’Shea–Sjamaar [O-S], −1 the set a (0) ∩ O is nonempty if and only if ∈ P0 . If we use Pd ⊂ (aτd )l to denote the polyhedral cone P ∩ (aτd )l for the quasisplit form τd , it follows from a0 ⊂ aτd that P0 = Pd ∩ (a0 )l . (4.8)

A statement related to this fact is given in [Fo].

Thompson’s conjecture for real semisimple Lie groups

133

5 Appendix: Alekseev–Meinrenken–Woodward theorem In this appendix, we give an outline of the proof of the Alekseev–Meinrenken– Woodward theorem, stated as Theorem 4.7 in this paper. [Al-Me-W, Theorem 3.5] shows the existence of a diffeomorphism L : D → O satisfying (4.7) for each ∈ al . To show that all the L ’s come from a globally deﬁned L on all of (AN)l , one uses the Moser isotopy lemma for Hamiltonian Poisson Kspaces proved in [Al-Me]. What we present here is a collection of arguments from [Al, Al-Me-W, Ka-M-T, Al-Me]. 5.1 Gauge transformation for Poisson structures Recall that a Poisson structure on a manifold M is a smooth section π of ∧2 T M such that [π, π] = 0, where [ , ] is the Schouten bracket on the space of multi-vector ﬁelds on M. For a smooth section π of ∧2 T M, we will use π # to denote the bundle map π # : T ∗ M → T M : π # (α) = π(·, α) for all cotangent vectors α. Similarly, for a 2-form γ on M, we will set γ # : T M → T ∗ M : γ # (v) = γ (·, v) for all tangent vectors v. Suppose now that π is a Poisson structure on M and that γ is a closed 2-form on M. If the bundle map 1 + γ # π # : T ∗ M → T ∗ M is invertible, the section π of ∧2 T M given by (π )# = π # (1 + γ # π # )−1 : T ∗ M −→ T M

(5.1)

is then a Poisson structure on M. The Poisson structure π will be called the gauge transformation of π by the closed 2-from γ , and we write π = Gγ (π ). It is clear from (5.1) that π and π have the same symplectic leaves. If S is a common symplectic leaf, then the symplectic 2-forms ω and ω coming from π and π differ by iS∗ γ , where iS : S → M is the inclusion map. See [Se-W] for more details. 5.2 The Poisson Lie groups (K, sπK ) and (AN, •s , πAN,s ) The group AN carries a distinguished Poisson structure πAN . Indeed, let , be the imaginary part of the Killing form of g and identify k with (a + n)∗ via , . For x ∈ k, denote by x¯ the right-invariant 1-form on AN deﬁned by x. Let xAN be the generator of the action of exp(tx) on AN according to (4.2). Then the unique section πAN of ∧2 T (AN) such that # (x) ¯ = xAN ∀x ∈ k πAN (5.2) is a Poisson bivector ﬁeld on AN. The Poisson structure πAN makes AN into a Poisson Lie group in the sense that the group multiplication map AN × AN → AN : (b1 , b2 ) → b1 b2 is a Poisson map, where AN × AN is equipped with the product Poisson structure πAN × πAN . We refer to [L-W] and [L] for details on Poisson Lie groups and Poisson Lie group actions and to [L-Ra]) for details on πAN . In particular, the dual Poisson Lie group of (AN, πAN ) is K together with the Poisson structure πK , explicitly given by πK = r0 − l0 , where

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0 =

1 Xα ∧ Yα ∈ ∧2 k 2 + α∈

with Xα , Yα ∈ k given in (3.3) and r0 and l0 being, respectively, the right- and left-invariant bivector ﬁelds on K determined by 0 . It follows from (5.2) that the symplectic leaves of πAN are precisely the orbits of the K-action on AN given in (4.2). Now let d be an involutive automorphism of the Dynkin diagram of g, and let τd be the quasisplit real form of g given in (3.5). Recall that τd leaves AN invariant and deﬁnes a group isomorphism on AN. It is easy to check (see also [Al-Me-W, Section 2.3]) that τd : (AN, πAN ) → (AN, πAN ) is anti-Poisson, i.e., τd ∗ πAN = −πAN . Similarly, τd : K → K is anti-Poisson for πK . We will denote the restrictions of τd to K and to AN both by τd , and we will refer to (K, πK , τd ) and (AN, πAN , τd ) as a dual pair of Poisson Lie groups with anti-Poisson involutions. In the context of Poisson Lie groups, the K-action on AN given in (4.2) and the AN-action on K given in (2.6) (with AN replacing A0 N0 and K replacing K0 ) are, respectively, called the dressing actions. As noticed in [Al] and [Ka-M-T], we in fact have a smooth family of Poisson Lie groups (AN, •s , πAN,s ) for s ∈ R. Indeed, for s ∈ R − {0}, let Fs : p → p be the diffeomorphism x → sx, and let Is = E ◦ Fs ◦ E −1 : AN → AN. Let πAN,s be the Poisson bivector ﬁeld on AN such that (Is )∗ πAN,s = sπAN , or πAN,s (b) = s(Is−1 )∗ (πAN (Is (b))),

b ∈ AN.

(5.3)

Deﬁne the group structure •s : AN × AN → AN by b1 •s b2 := Is−1 (Is (b1 )Is (b2 )),

b1 , b2 ∈ AN.

Then since s = 0, the map (AN ×AN, πAN,s ×πAN,s ) → (AN, πAN,s ) : (b1 , b2 ) → b1 b2 is a Poisson map, so (AN, •s , πAN,s ) is a Poisson Lie group for each s ∈ R, s = 0. On the other hand, p ∼ = k∗ has the linear Poisson structure πp,0 deﬁned by the Lie algebra k. Let •0 and let πAN,0 be the pullbacks to AN by E −1 : AN → p of the abelian group structure on p and the Poisson structure πp,0 on p. Then we get a smooth family of Poisson Lie group structures (•s , πAN,s ) on AN for every s ∈ R (see [Al] and [Ka-M-T]). The dual Poisson Lie group of (AN, •s , πAN,s ) is again the Lie group K (with group structure independent on s) but with the Poisson structure sπK , if we identify again k ∼ = (a + n)∗ via the imaginary part of the Killing form of g. It is also clear that τd is a group isomorphism for •s and is anti-Poisson for πAN,s . Thus we get a dual pair of Poisson Lie groups (K, sπK , τd ) and (AN, •s , πAN,s , τd ) with anti-Poisson involutions for each s ∈ R. 5.3 The Poisson structure πs on (AN )l As noted in [Al], for each s ∈ R, the Poisson structure πAN,s on AN is related to πAN by a gauge transformation. Recall that for x ∈ k ∼ = (a + n)∗ , x¯ is the right-invariant 1-form on AN deﬁned by x. Let lx be the differential of the linear function ξ → x, ξ

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on p, and let xp be the vector ﬁeld on p generating the adjoint action of exp(tx) ∈ K on p. By [Al-Me-W, Proposition 3.1], there is a 1-form β on p such that β(0) = 0 and −(dβ)# (xp ) = E ∗ x¯ − lx

∀x ∈ k.

Moreover, (τd )∗ β = −β for every quasisplit real form τd given in (3.5). For s ∈ R − {0}, let βs = 1s (Fs∗ β). Since β(0) = 0, the family βs extends smoothly to β0 = 0. Let 1 α = (E −1 )∗ β and αs = (E −1 )∗ βs = (Is∗ α) s for all s ∈ R. Then it is easy to show that, for every s ∈ R, πAN,s = Gd(α−αs ) (πAN ) = G−dαs (πAN,0 ). Assume now that l ≥ 1 is an integer. For each s ∈ R, set ms : (AN )l −→ AN : (b1 , b2 , . . . , bl ) −→ b1 •s b2 •s · · · •s bl . By generalizing the “linearization" procedure of Hamiltonian symplectic (K, sπK )spaces described in [Al-Me-W] to the case of Poisson manifolds, one can show that πs := Gdms∗ αs (πAN,s × πAN,s × · · · × πAN,s )

(5.4)

is a well-deﬁned Poisson structure on (AN )l for each s ∈ R. Deﬁne the twisted diagonal action Ts of K on (AN )l by k −→ Ts,k := νs−1 ◦ δk ◦ νs ,

(5.5)

where, again, δk denotes the diagonal action of k on (AN )l for the action of K on AN given in (4.2), and νs ∈ Diff ((AN )l ) is given by νs (b1 , b2 , . . . , bl ) = (b1 , b1 •s b2 , . . . , b1 •s b2 •s · · · •s bl ). Note that Ts,k is Tk when s = 1 and is δk when s = 0. Then again it follows from [Al-Me-W] that for each s ∈ R, the action Ts of K on (AN )l is Hamiltonian with respect to the Poisson structure πs with the map E −1 ◦ ms : (AN )l → p ∼ = k∗ as a moment map. Moreover, for every quasisplit real form τd deﬁned in (3.5), the Cartesian product (τd )l = τd × τd × · · · × τd is anti-Poisson for πs for every s ∈ R. 5.4 Moser isotopy lemma Let U be a connected Lie group with Lie algebra u. Suppose that σU is an involutive automorphism of U with the corresponding involution on u denoted by σu . Deﬁne σu∗ = −(σu )∗ . If (M, πM , ) is a Hamiltonian Poisson U -space, an anti-Poisson involution σM of (M, πM ) is said to be compatible with σU if ◦ σM = σu∗ ◦ . The following Moser isotopy lemma for Hamiltonian Poisson U -spaces with anti-Poisson involutions is proved in [Al-Me]. See [Al-Me-W] for the symplectic case.

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Proposition 5.1. Let U be a connected compact semisimple Lie group with Lie algebra u, and let (M, πs , s ), s ∈ R, be a smooth family of Hamiltonian Poisson U -spaces. Suppose that there exists a smooth family of 1-forms s on M with 0 = 0 such that πs = Gds π0 for every s ∈ R. Assume also that π0 has compact symplectic leaves. Then (M, πs , s ) is isomorphic to (M, π0 , 0 ) for every s ∈ R as Hamiltonian Poisson U -spaces, i.e., there exists ψs ∈ Diff (M) for s ∈ R with ψ0 = id, such that for every s ∈ R, (1) πs = ψs ∗ π0 ; (2) s ◦ ψs = 0 . If σU is an involutive automorphism on U , and if for each s ∈ R, σM,s is an anti∗ ˙ = −˙ s , where Poisson involution for πs compactible with σU and such that σM,s s d ˙s = ds s for s ∈ R, then ψs can be chosen such that ψs ◦ σM,0 = σM,s ◦ ψs for all s ∈ R. 5.5 Proof of Theorem 4.7 Consider the Hamiltonian Poisson K-space (M = (AN )l , πs , s ) with s = E −1 ◦ ms . The action of K on (AN )l induced by (πs , s ) is the twisted diagonal action Ts given in (5.4). From the deﬁnition of πs , we know that πs = Gds π0 , where s = ms∗ αs −

l

pj∗ αs

j =1

with pj : (AN )l → AN denoting the projection to the j ’th factor. For every quasisplit real form τd given in (3.5), since τd is a group isomorphism for (AN, •s ), we have τd∗ s = −s , and thus τd∗ ˙s = −˙s for every s ∈ R. Let σM,s = (τd )l and let ψs ∈ Diff ((AN )l ) be as in Proposition 5.1. Then L := (E −1 )l ◦ψ1−1 : (AN )l → (p)l is the diffeomorphism in Theorem 4.7. Indeed, E ◦ a ◦ L = E ◦ a ◦ (E −1 )l ◦ ψ1−1 = E ◦ 0 ◦ ψ1−1 = E ◦ 1 = m, where the second equality follows from the identity a ◦ (E −1 )l = E −1 ◦ m0 , which is a trivial consequence of the fact that m0 is the pullback of addition by the map E.

Acknowledgments Although we do not explicitly use any results from [Fo], our paper is very much inspired by [Fo]. We thank P. Foth for showing us a preliminary version of [Fo] and for helpful discussions. We also thank J. Millson and M. Kapovich for sending us the preprint [Ka-Le-M1] and R. Sjamaar for answering some questions. Thanks also go to the referee for helpful comments.

References [A-B-V]

Adams, J., Barbasch, D., and Vogan, D., The Langlands Classiﬁcation and Irreducible Characters for Real Reductive Groups, Birkhäuser, Boston, 1992.

Thompson’s conjecture for real semisimple Lie groups [Al] [Al-Me-W]

[Al-Me] [Ar] [B-S] [Fo] [Fu] [Ka-Le-M1] [Ka-Le-M2] [Ka-Le-M3]

[Ka-M-T] [Kl] [Kn1] [Ku-Le-M]

[L-W] [L]

[L-Ra] [O-S] [Sl] [Se-W]

[T] [Wa]

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Alekseev, A., On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom., 45 (1997), 241–256. Alekseev, A., Meinrenken, E., and Woodward, C., Linearization of Poisson actions and singular values of matrix products, Ann. Inst. Fourier Grenoble, 51-6 (2001), 1691–1717. Alekseev, A., and Meinrenken, E., Poisson geometry and the Kashiwara–Vergne conjecture, C. R. Math. Acad. Sci. Paris, 335 (2002), 723–728. Araki, S., On root systems and an inﬁnitesimal classiﬁcation of irreducible symmetric spaces, J. Math. Osaka City Univ., 13-1 (1962), 1–34. Berenstein, A., and Sjamaar, R., Coadjoint orbits, moment polytopes, and the Hilbert–Mumford criterion, J. Amer. Math. Soc., 13-2 (2000), 433–466. Foth, P., A note on Lagrangian loci of quotients, Canad. Math. Bull., to appear; math.SG/0303322. Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.), 37-3 (2000), 209–249. Kapovich, M., Leeb, B., and Millson, J., Polygons in symmetric spaces and buildings, preprint, 2002. Kapovich, M., Leeb, B., and Millson, J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, math.RT/0210256. Kapovich, M., Leeb, B., and Millson, J., Convex functions on symmetric spaces and geometric invariant theory for weighted conﬁgurations on ﬂag manifolds, math.DG/0311486. Kapovich, M., Millson, J., and Treloar, T., The symplectic geometry of polygons in hyperbolic 3-space, Asian J. Math., 4-1 (2000) (Kodaira’s issue), 123–164. Klyachko, A., Random walks on symmetric spaces and inequalities for matrix spectra, Linear Algebra Appl., 319-1–3 (2000), 37–59. Knapp, A., Representation Theory of Semi-Simple Groups, Princeton University Press, Princeton, NJ, 1986. Kumar, S., Leeb, B., and Millson, J., The generalized triangular inequalities for rank 3 symmetric spaces of non-compact type, in Explorations in Complex and Riemannian Geometry: A Volume Dedicated to Robert Greene, Contemporary Mathematics, Vol. 332, American Mathematical Society, Providence, RI, 2003, 171–195. Lu, J.-H., and Weinstein, A., Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom., 31 (1990), 501–526. Lu, J.-H., Moment mappings and reductions of Poisson Lie groups, in Proc. Seminaire Sud-Rhodanien de Géométrie, Mathematical Sciences Research Institute Publications Series, Springer-Verlag, New York, 1991, 209–226. Lu, J.-H., and Ratiu, T., On the non-linear convexity theorem of Kostant, J. Amer. Math. Soc., 4-2 (1991), 349–361. O’Shea, L., and Sjamaar, R., Moment maps and Riemannian symmetric pairs, Math. Ann., 317-3 (2000), 415–457. Schlichtkrull, H., Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Birkhäuser, Basel, 1984. Severa, P., and Weinstein, A., Poisson geometry with a 3-form background, in Proceedings of the International Workshop on Non-Commutative Geometry and String Theory, Keio University, Tokyo, 2001; also available at math.SG/0107133. Thompson, R., Matrix Spectral Inequalities, John Hopkins University Press, Baltimore, 1988. Wallach, N., Real Reductive Groups I, Academic Press, New York, 1988.

The Weinstein conjecture and theorems of nearby and almost existence∗ Viktor L. Ginzburg Department of Mathematics University of California at Santa Cruz Santa Cruz, CA 95064 USA [email protected] Dedicated to Alan Weinstein on the occasion of his 60th birthday. Abstract. The Weinstein conjecture, as the general existence problem for periodic orbits of Hamiltonian or Reeb ﬂows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding some fundamental properties of Hamiltonian ﬂows. In this paper we survey some recently developed and well-known methods of proving various particular cases of this conjecture and the closely related almost existence theorem. We also examine differentiability and continuity properties of the Hofer–Zehnder capacity function and relate these properties to the features of the underlying Hamiltonian dynamics, e.g., to the period growth.

1 Introduction Without exaggeration, one can say that Arnold’s conjecture and the Weinstein conjecture have been the two problems determining the development of symplectic topology over the past twenty years. The Weinstein conjecture [We4], the problem we are interested in here, concerns the existence of closed characteristics on a compact hypersurface of contact type. To be more precise, consider a regular compact contact type level of a Hamiltonian on a symplectic manifold. Then, the Weinstein conjecture as we understand it now asserts that the level must carry at least one periodic orbit of the Hamiltonian ﬂow. This paper is a survey of some recent and well-known results concerning a particular perspective on the Weinstein conjecture, originating from the so-called almost existence theorem. The main focus of this survey is on the action selector and symplectic capacity methods. Arguing along the lines of these methods, we reprove a number ∗ The work is partially supported by the NSF and by the faculty research funds of the University

of California at Santa Cruz.

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of particular cases of the conjecture and also reﬁne the capacity–displacement energy inequality for noncompact convex symplectically aspherical manifolds (see Sections 2.5.2 and 3.3) and analyze in Section 4 the problem of continuity of the Hofer–Zehnder capacity. Some aspects of the conjecture are just touched upon or not considered at all in this survey. Among the latter is, for instance, the Weinstein conjecture for general contact manifolds. The contents of the survey are outlined in more detail at the end of the introduction. The Weinstein conjecture was motivated by numerous results establishing the existence of periodic orbits under various, often rather restrictive, conditions on the level (e.g., convex or bounding a star-shaped domain); see, e.g., [Mo, Ra, We1, We2, We3]. The feature distinguishing the Weinstein conjecture from these results is the requirement that the level have contact type is invariant under symplectomorphisms (as is the assertion) while the hypotheses of the earlier theorems are not. In its original form for Hamiltonians on a linear symplectic space, the Weinstein conjecture was proved by Viterbo [Vi1]. Since then, the conjecture has been established for many other symplectic manifolds (sometimes under additional requirements on the level); see, e.g., [FHV, HV1, HV2, LT, Lu1, Lu2, Lu4, Vi4], to mention several results. Among these manifolds are products of complex projective spaces, manifolds of the form P × Cn (where P is compact symplectic and the product is given a split symplectic structure), and subcritical Stein manifolds. However, in general, the conjecture is still neither proved nor disproved. For example, the Weinstein conjecture in its full generality is open for cotangent bundles and T4 . Starting from the Weinstein conjecture, one can go in two different directions. One direction is to dispose of the Hamiltonian and the ambient symplectic manifold and focus exclusively on the level of contact type. This naturally leads to the question of whether a Reeb ﬂow on a compact contact manifold necessarily has a periodic orbit; see, e.g., [Ho3]. Along these lines, Hofer [Ho2] proved the existence of periodic orbits for the Reeb ﬂow of a contact form on S 3 or on a closed three-dimensional contact manifold M with π2 (M) = 0, and also for the Reeb ﬂow of an overtwisted contact form. This approach interprets the conjecture as a question about the dynamics of Reeb ﬂows and leads to the notions of contact homology and symplectic ﬁeld theory [EGH, BEHWZ]. Another direction is to view the conjecture as a question about the dynamics of Hamiltonian ﬂows on the ambient symplectic manifold. This is the perspective with which we are concerned here. More speciﬁcally, we focus on such problems as whether and how the assumption that the level has contact type can be relaxed, whether the existence of periodic orbits is typical, and so on. The contact type requirement cannot be dropped entirely: there exists a proper function on R2n (C ∞ -smooth if 2n ≥ 6 and C 2 -smooth if 2n = 4) with a regular level carrying no periodic orbits; see [Gi1, Gi3, GG1, GG2, He1, He2, Ke2]. (Constructions of such functions are known as counterexamples to the Hamiltonian Seifert conjecture.) Nevertheless, the existence of periodic orbits on the level sets of a ﬁxed Hamiltonian is a generic phenomenon: almost all (in the sense of measure theory) regular levels of a C 2 -smooth proper function on R2n carry periodic orbits of the Hamiltonian

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ﬂow. This result, due to Hofer and Zehnder and to Struwe, is known as the almost existence theorem [HZ3, St]. The almost existence theorem holds for many but not all symplectic manifolds (sometimes under some additional restrictions on the Hamiltonian; see [GG3]). For example, the theorem holds for CPn , products P × Cn with split symplectic structure (where P is geometrically bounded) [HV1, Lu1, MDSl, Schl], subcritical Stein manifolds [Ci, Ke3, Lu1], and small neighborhoods of certain non-Lagrangian submanifolds [Ke3, Sc3, Schl]. However, as has been observed by Zehnder [Ze], the almost existence theorem fails for certain Hamiltonians on T2n equipped with an irrational symplectic structure (Zehnder’s torus). In fact, for this Hamiltonian system, there is an interval of energy values without periodic orbits. Moreover, this phenomenon is stable under C k -small (k > 2n) perturbations of the Hamiltonian [He3, He4]. A sibling of the almost existence theorem is the theorem of dense or nearby existence which guarantees the existence of periodic orbits for a dense set of energy levels or, equivalently, near a ﬁxed level; see, e.g., [CGK, FH, FS, HZ2, HZ3]. Both of these theorems imply the Weinstein conjecture and below we will discuss the relation between these theorems. Another reason to consider the almost existence and nearby existence theorems is that a broad class of Hamiltonian systems for which energy levels fail to have contact type naturally arises in classical mechanics. In this class are, for example, the systems describing the motion of a charge in a magnetic ﬁeld, which we will discuss shortly; see also [Gi2]. For such systems one can expect the almost existence theorem to hold and periodic orbits to exist for all low-energy values. However, periodic orbits need not exist on all energy levels even when the symplectic form is exact near a level (the horocycle ﬂow; see Example 4.7). Furthermore, as has been pointed out by Kerman [Ke1], the analysis of such systems is closely related to a generalization of the Weinstein–Moser theorem [Mo, We2]. The Weinstein–Moser theorem asserts that a smooth function H on R2n attaining a nondegenerate minimum at the origin must have at least n distinct periodic orbits on every level near the origin [Mo, We2]. Let us now replace R2n by an arbitrary symplectic manifold W and the nondegenerate minimum at one point by a Morse–Bott nondegenerate minimum along a closed symplectic submanifold M ⊂ W . Then, conjecturally, every level of H near M carries at least one periodic orbit or even a number of periodic orbits. We will refer to this conjecture as the generalized Weinstein–Moser conjecture. As a particular example motivating our interest in this question, consider π the standard kinetic energy H on the cotangent bundle W = T ∗ M → M, equipped with a twisted symplectic structure ω0 + π ∗ , where ω0 is the standard symplectic structure dp ∧ dq and is a closed two-form on M. This system describes the motion of a charge on M in the magnetic ﬁeld ; see [Gi2]. When is nondegenerate M turns into a symplectic submanifold of W . The generalized Weinstein–Moser conjecture has been proved in a number of particular cases [GK1, Ke1], but in general the question is still open. Recently, however, some progress has been made along the lines of the almost existence theorem. Namely, it has been shown that almost all levels close to M of a function H attaining a minimum along M carry contractible periodic orbits [CGK, GG3, Ma2]—this is the

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relative (with respect to M) almost existence theorem for small energy values. (Note that unless M is a Morse–Bott nondegenerate minimum of H , we cannot expect such periodic orbits to exist on all levels near M [GG3].) This implies the almost existence theorem for low-energy periodic orbits of a charge in a nondegenerate magnetic ﬁeld. Moreover, under suitable additional hypothesis, the genuine, nonrelative, almost existence theorem holds near M [Ke3]. In the setting of magnetic ﬁelds, these results can be further reﬁned: periodic orbits must exist whenever = 0 [Schl]. (We will brieﬂy discuss the proof of this result in Section 3.5.) We also refer the reader to [GK2, Ma1, Pol2] for related results. The nearby existence theorem is weaker and often easier to prove than the almost existence theorem. The pattern has been that, in many cases, the nearby existence theorem was proved ﬁrst and then followed by the almost existence theorem; see, e.g., [HZ1], [HZ3, St], [CGK], and [GG3]. As we have pointed out above, both theorems imply the Weinstein conjecture. Conversely, essentially every proof of a particular case of the Weinstein conjecture in the Hamiltonian setting translates into a proof of either the nearby existence or almost existence theorem, although it is not always easy to establish which of these theorems is proved. As of today, almost existence is veriﬁed in virtually all the cases where the nearby existence has been proved. Probable exceptions are some of the results from [Vi4] on periodic orbits in cotangent bundles and, perhaps, the results of [LT] and hence of [Lu2, Lu3, Lu4]. Organization and contents of the paper In this paper we focus on the aspects of the Weinstein conjecture related to global Hamiltonian dynamics, and hence our treatment of the conjecture in general is by no means comprehensive. For example, we do not even touch upon the Weinstein conjecture for contact manifolds. (The reader interested in this conjecture is referred to the surveys [Ho3, Ho4, Ho5] in addition to the references given above.) Furthermore, we do not mention the fruitful connection between the Weinstein conjecture and Gromov–Witten invariants, although we do discuss the holomorphic curve approach to the proof of the conjecture. In Sections 2 and 3 we outline methods of proving the nearby existence and almost existence theorems and, hence, the Weinstein conjecture. These sections can be viewed as a brief introduction to certain concepts of symplectic topology (action selectors, constructions of symplectic capacities, Hofer’s metric, symplectic homology, etc.), albeit strictly focused on a speciﬁc task and not even mentioning many aspects of the subject. However, these sections should not be taken as an introduction to symplectic topology in general. For example, we assume the reader’s familiarity with Floer homology. Section 4 concerns some simple, but apparently not present in the literature, properties of the Hofer–Zehnder capacity function. Although the paper is complemented by an extensive bibliography, the list of references contains only the papers immediately relevant to our discussion. Inevitably, this list is incomplete and omits many important contributions to the subject, and our exposition emphasizes the publications that have most inﬂuenced the author’s thinking.

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Conventions In this paper, all manifolds are assumed to be without boundary.Asymplectic manifold W will be called convex if W is either closed (i.e., compact) or open and convex at inﬁnity. Here W is said to be convex at inﬁnity if there exists: a hypersurface ⊂ W which separates W into one set with compact closure and another, U , with noncompact closure; and a ﬂow ϕt (for t ≥ 0) of symplectic dilations on U , which is transversal to ∂U = . Recall also that (W, ω) is symplectically aspherical if c1 |π2 (W ) = ω|π2 (W ) = 0. (In some instances, this condition can be replaced by ω|π2 (W ) = 0.) We refer the reader to, e.g., [AL, CGK] for a discussion of geometrically bounded symplectic manifolds and to [HZ3, MDSa1, MDSa2, Pol3] for a general introduction to symplectic topology. Let us now ﬁx the sign conventions in the deﬁnition of the action functional. Let H ∈ C ∞ (S 1 × W ) and let x : S 1 → W be a contractible loop in W . Here we will use the action functional AH deﬁned as AH (x) = − x¯ ∗ ω + H (t, x)dt, D2

S1

where x¯ : → W is a map of a disk, bounded by x. By the least action principle, contractible one-periodic orbits of H are precisely the critical points of AH . The action spectrum S(H ) of H is the set of critical values of AH , i.e., the collection of action values of AH on the contractible periodic orbits. (Since we are assuming that ω|π2 (W ) = 0, the action AH is single valued. Otherwise, one has to pass to a suitable covering of the loop space and deal with numerous other technical difﬁculties.) We set Ht = H (t, ·), where t ∈ S 1 , and denote by XH the Hamiltonian vector ﬁeld of H . D2

2 Nearby existence theorems Nearby existence theorems are usually proved by Floer homological methods. Below we discuss some of these methods, focusing on the key ideas and omitting many (often quite nontrivial) technical details. Throughout this section, all ambient symplectic manifolds (W, ω) are assumed to be convex and symplectically aspherical, unless explicitly stated otherwise as is, for example, in Section 2.4. The argument is particularly transparent for open manifolds which are convex at inﬁnity. Hence we consider this case ﬁrst. 2.1 The action selector method: Convex at inﬁnity open manifolds In this section we focus on convex at inﬁnity open manifolds and outline the approach to the proof of the nearby existence theorem utilizing the notion of an action selector. In our proof of the existence of an action selector and its properties, we mainly follow [FS, Sc3]. However, many elements of our argument are already contained in [HZ2, HZ3], where the action selector is deﬁned differently; see also, e.g., [Oh2, Vi3].

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2.1.1 Action selectors and nearby existence theorem Let W be a convex at inﬁnity open symplectic manifold. We denote by HU , where U is an open subset of W , the space of compactly supported smooth Hamiltonians H : S 1 × U → R. For our purposes, it is convenient to adopt the following deﬁnition. An action selector is a C 0 -continuous function σ : HW → R such that1 (AS1) σ (H ) ∈ S(H ), (AS2) σ is monotone, i.e., H1 ≤ H2 implies that σ (H1 ) ≤ σ (H2 ), and (AS3) σ (H ) = max H , when H is a C 2 -small function on W with a unique maximum. Let us point out some consequences of (AS1)–(AS3) which are essential for what follows. First, recall that the action spectrum S(H ) is compact and nowhere dense; see, e.g., [HZ3, Sc3]. (This elementary, but not entirely trivial, fact can be thought of as a version of Sard’s theorem for AH ; cf. [Zv].) Furthermore, as is easy to see, S(H ) depends only on the time-one ﬂow ϕH of H viewed as an element of the universal cover D of the group of Hamiltonian symplectomorphisms; the homotopy to identity is given by H . (In fact, when W is symplectically aspherical, S(H ) depends only on the time-one ﬂow of H regarded as an element of the group of symplectomorphisms.) Then it follows from (AS1) and the continuity of σ that σ (H ) also depends only on ϕH ∈ D. Furthermore, σ (H ) > 0

when H is nonnegative and nonconstant,

(2.1)

by (AS2) and (AS3) and, by (AS1) and (AS2), σ (H ) = 0 when H ≤ 0. The nearby existence theorem for functions on U ⊂ W can be (and often is) easily derived from the existence of an action selector σ which meets one additional requirement that σ be a priori bounded on HU . To state the result, it is convenient to introduce the notion of a shell or thickening of a hypersurface. A shell {s } in U is an embedding × (−, ) → U , where is a closed manifold such that dim = dim U − 1. We identify × (−, ) with its image, set s = × s and assume, in addition, that = 0 divides U into two components: one bounded (i.e., such that its closure in U is compact) and the other bounded or unbounded. (In many instances, this assumption can be omitted.) The following, nearly trivial, proposition sums up a number of versions of the nearby existence theorem. Proposition 2.1. Assume that there exists an action selector σ : HW → R such that σ (K) ≤ CU for every function K ∈ HU and some constant CU independent of K. Then the nearby existence theorem holds for proper functions H on U : contractible 1 The properties (AS1)–(AS3) should not be taken as an attempt to axiomatize the notion

of an action selector (see [FGS])—such axioms would almost certainly be different from (AS1)–(AS3) and would include a version of the inequality (2.4) below. Our objective is to list the properties of the action selector that are most essential for the proof of the nearby existence theorem.

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in W periodic orbits exist for a dense set of values of H . Equivalently, for every shell {s } there exists a value s0 such that s0 carries a contractible in W closed characteristic.2 As we will soon see, there exists an action selector on R2n which is a priori bounded on bounded domains [HZ3] and hence the nearby existence theorem holds for R2n . Remark 2.2. In this proposition and in Proposition 3.1, the assumption that the shell divides W is not essential; see [MS]. Proof of Proposition 2.1. Given a shell {s }, consider a smooth function F on W such that • •

F is equal to some positive constant C > CU on the bounded component of the complement to the shell and is identically zero on the unbounded component, within the shell, F depends only on s and is a monotone decreasing function with range [0, C] and such that 0 and C are the only critical values of F .

Note that the only critical values of F on W are again 0 and C. Furthermore, 0 < σ (F ) < C by (2.1) and since σ (F ) ≤ CU < C. Hence, any contractible one-periodic orbit of F with action σ (F ) must be nontrivial and lie on one of the regular levels of F , i.e., on some hypersurface s0 . Hence, at least one of these hypersurfaces carries a contractible in W closed characteristic.

2.1.2 The existence of an action selector Let us turn now to the problem of the existence of an action selector. There are numerous constructions of action selectors for different classes of manifolds W : in [HZ2, HZ3] an action selector has been constructed for W = R2n using a direct variational method on the space of H 1/2 -loops; in [Vi3] an action selector has been deﬁned for cotangent bundles using generating functions; this construction has been extended to Lagrangian submanifolds in [Oh1, Oh2] using Floer homology; in [Sc3, Oh2] an action selector has been introduced for symplectically aspherical closed manifolds W , again by utilizing the Floer homology methods; these results have been extended to convex at inﬁnity manifolds in [FS]. Finally, in [Oh3], an action selector was deﬁned for closed symplectic manifolds which are not necessarily convex or symplectically aspherical. Let us now outline the construction of an action selector, following mainly [FS, Sc3], for open symplectically aspherical manifolds that are convex at inﬁnity. Let W be such a manifold. Recall that the Floer homology HF(H ) for H ∈ Cc∞ (S 1 ×W ) is deﬁned and independent of H , i.e., for any two functions these groups are canonically isomorphic [Fl1, Fl2]; see also [PSS, Sa, Sc1]. Since W is not closed but convex at inﬁnity, one has to extend H to a function with a suitable growth at 2 Depending on whether this result is thought of in terms of H or a shell, it is referred to as a

dense existence or nearby existence theorem.

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inﬁnity, without creating new periodic orbits, and then work with the Floer homology of the resulting function; see [FS, Oh1, Vi4]. Moreover, HF(H ) is isomorphic to H∗ (W ) up to a shift of degrees, but this fact and the speciﬁc nature of the isomorphism are not very essential for us until later stages of the argument. At the moment, one may interpret H∗ (W ) as a common notation for the groups HF(H ) identiﬁed with each other for different functions H . Furthermore, the ﬁltered Floer homology HF[a,b) (H ) is deﬁned for any interval [a, b) with endpoints outside of S(H ). This homology is constant under deformations of a, b and H as long as a and b are outside of the action spectrum of H , see, e.g., [Vi4]. There is a natural map j a : H∗ (W ) = HF(H ) → HF[a,∞) (H ) induced by the quotient of complexes. Fix a nonzero element u ∈ H∗ (W ) and set σu (H ) = inf {a | j a u = 0}. It is easy to see by using the invariance and monotonicity properties of Floer homology that σu meets the requirements (AS1) and (AS2). Let [max] ∈ H∗ (W ) be the class of the maximum of a C 2 -small bump function. By calculating the Floer homology of such a bump function, one can show that [max] = 0. The selector σ = σ[max] obviously satisﬁes (AS1)–(AS3). However, the C 0 -continuity of σ , or in general of σu , requires a proof (see, e.g., [Oh2, Sc3]) and this proof relies on the explicit construction of Floer’s continuation map. The key to the proof is the following observation. Let H0 ≥ H1 be two Hamiltonians whose one-periodic orbits are nondegenerate. Consider periodic orbits x0 of H0 and x1 of H1 such that there exists a connecting homotopy trajectory joining x0 and x1 for a certain “linear’’ monotone homotopy from H0 to H1 . Then AH0 (x0 ) − AH1 (x1 ) ≤

1

max H0 (t, p) − H1 (t, p) dt.

0 p∈W

(2.2)

This inequality can be obtained by a direct calculation and the C 0 -continuity of σ readily follows from (2.2); see [Sc3] for details. In a similar vein, (2.2) implies the inequality σ (H ) ≤

1

max H (t, p)dt,

0 p∈W

(2.3)

which can be established by setting H = H0 in (2.2), taking a C 2 -small function as H1 and then letting H1 go to zero. 2.1.3 An a priori bound for σ One class of domains U for which σ is a priori bounded is the class of displaceable domains, i.e., U such that there exists a Hamiltonian symplectomorphism ϕH with

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H ∈ HW moving U away from itself: ϕH (U ) ∩ U = ∅. For example, bounded open subsets of R2n are displaceable. To show that σ is a priori bounded on a displaceable domain U , one argues as follows. t and ϕ t be time-dependent Hamiltonian ﬂows generated by H and K in Let ϕH K t ◦ ϕ t , i.e., HW . Denote by H #K ∈ HW the Hamiltonian generating ϕH K t −1 ) (p)). H #K(t, p) = H (t, p) + K(t, (ϕH

The crucial feature of the selector σ deﬁned in Section 2.1.2 is that σ is subadditive: σ (H #K) ≤ σ (H ) + σ (K)

(2.4)

or more generally σu∩v (H #K) ≤ σu (H ) + σv (K), where u ∩ v is the intersection of homology classes u and v. The subadditivity of σ is proved in [Oh2, Sc3] by making use of the pair-of-pants product introduced in [Sc2]; see also [PSS]. This is a product HF[a,∞) (H ) ⊗ HF[b,∞) (K) → HF[a+b,∞) (H #K) which is intersection of cycles on the full Floer homology H∗ (W ) and such that the corresponding diagram commutes. (At this stage one speciﬁcally utilizes the isomorphism HF(H ) → H∗ (W ) deﬁned in [PSS], for this isomorphism sends the pair-of-pants product to the intersection of cycles. However, it might also be possible to prove the subadditivity of σ[max] by using the Floer continuation map.) Once the existence of such a product is established, (2.4) follows from the deﬁnition of σ ; see [Sc3, Section 4]. The subadditivity of σ implies an a priori bound for σ . Let K be a Hamiltonian whose time-one ﬂow displaces supp H , where, by deﬁnition, supp H = , t )−1 so that supp Ht . Denote by K − the Hamiltonian generating the ﬂow (ϕK 1 t∈S − K#K = 0. Then σ (H ) ≤ σ (K) + σ (K − ). (2.5) To prove (2.5), we argue as in [FS]. First, observe that since ϕK displaces supp H , oneperiodic orbits of H #K are exactly one-periodic orbits of K, and, as a consequence, S(H #K) = S(K). The same is, of course, true when H is replaced by the Hamiltonian sH with s ∈ R, i.e., S((sH )#K) = S(K). By continuity of σ and “discontinuity’’ of S(K), we conclude that σ ((sH )#K) ∈ S((sH )#K) = S(K) is independent of s. Setting s = 0 and s = 1 yields σ (K) = σ (H #K). Therefore, σ (H ) = σ (H #K#K − ) ≤ σ (H #K) + σ (K − ) = σ (K) + σ (K − ), which proves (2.5). Among other important examples of displaceable domains U are small neighborhoods of closed non-Lagrangian submanifolds of middle dimension whose normal bundles have nonvanishing sections [LS, Pol1]. This (combined with Macarini’s stabilization trick, [Ma1]), leads to a local version of the nearby existence theorem for twisted cotangent bundles over surfaces (other than the torus) or, in higher dimensions, for exact magnetic ﬁelds; see [FS].

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2.2 The action selector method: closed manifolds In this section we will brieﬂy point out changes required in the action selector proof of the nearby existence theorem when W is a closed manifold. First, let us observe that the construction of an action selector outlined in Section 2.1.2 carries over to Hamiltonians on closed manifolds W. (Here, as everywhere in this section, W is assumed to be symplectically aspherical.) Such an action selector has the property σ (H + const) = σ (H ) + const and hence is never a priori bounded. To circumvent this problem, one chooses a suitable normalization of Hamiltonians. (This is also necessary to ensure that σ (H ) depends only on ϕH .) However, in contrast with open manifolds where compactly supported Hamiltonians are automatically normalized, there appears to be no natural choice of normalization for closed manifolds. One possible choice is to restrict the action selector to Hamiltonians vanishing on a neighborhood of a ﬁxed point. This should allow one to extend word-for-word the results and constructions of Section 2.1 to closed manifolds. However, certain details of this approach are still to be worked out and we leave its discussion for a later occasion. Here, we use the traditional normalization requiring the mean value of Hamiltonians to be zero. Morespeciﬁcally, let HW be the class of smooth functions on S 1 × W normalized so that W Ht ωn = 0 and let HU be formed by Hamiltonians H ∈ HW such that supp XHt ⊂ U for all t ∈ S 1 . Proposition 2.1 still holds when the class HU is deﬁned in such a fashion. However, when W is closed, obtaining an a priori bound for a selector on HU is more difﬁcult than in the case of an open manifold, even though this still might be possible. We refer the reader to [FGS] for a detailed analysis of this question. (The proof of the a priori bound from Section 2.1.3 does not go through because inequality (2.5) need not hold. The reason is that even when W U is connected, S(H #K) will differ from S(K) by the constant equal to H |W U .) Here, following [Sc3], we choose a somewhat different approach. Let, as in Section 2.1.2, σ[max] be the action selector associated with the Floer homology class [max] ∈ H∗ (W ) corresponding to the maximum of a C 2 -small bump function (shifted to have zero mean). Set γ (H ) = σ[max] (H ) + σ[max] (H − ). Equivalently, γ (H ) = σ[max] (H ) − σ[min] (H ), where σ[min] is the action selector associated with the Floer homology class [min] ∈ H∗ (W ) corresponding to the minimum of a negative C 2 -small shifted bump function. Similar to action selectors for normalized functions, γ (H ) depends only on ϕH ∈ D. 1 When H is C 2 -small, we have γ (H ) = 0 (max H − min H )dt. Furthermore, γ in H , subadditive, but not necessarily monotone. Then, since is still C 0 -continuous t −1 Ht− (p) = −Ht (ϕH ) (p) , inequality (2.3) translates to γ (H ) ≤ 0

1

max H (t, p) − min H (t, p) dt. p∈W

p∈W

(2.6)

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Note also that γ (K − ) = γ (K). As we have pointed out above, the proof of (2.5) given in Section 2.1.3 does not go through for the normalization we use here. However, a similar but more involved argument proves the upper bound γ (H ) ≤ 2γ (K),

(2.7)

where the time-one ﬂow of K displaces U ⊃ supp(XH ). (Here, by deﬁnition, , supp(XH ) = t∈S 1 supp XHt .) We refer the reader to [Sc3] for detailed proofs of these facts. Observe now that Proposition 2.1 holds when an a priori bounded selector σ is replaced by γ which satisﬁes (2.6) and is a priori bounded on U . (To see this, only minor modiﬁcations in the proof are needed.) As a consequence of (2.7), we immediately obtain the nearby existence theorem for displaceable domains in compact symplectically aspherical manifolds. 2.3 Limitations of the action selector method The class of manifolds W and domains U to which the action selector method applies is rather limited. For example, a bounded selector or a bounded function γ do not exist for U = W = T2 (nor for T4 or T2 ×S 2 with split symplectic structure). Indeed, it is easy to see that there is a function H ∈ HT2 with S(H ) = {min H, max H } and such that σ[max] (H ) = max H and σ[min] (H ) = max H are both arbitrarily large. The same is true for tubular neighborhoods of the zero section in cotangent bundles of some compact manifolds (e.g., of surfaces with genus g ≥ 1). As a consequence, Proposition 2.1 cannot be applied to prove nearby existence on these manifolds as long as the standard Floer homology (even accounting for noncontractible orbits) is employed. This is the case already for W = T2 even though every regular level of H is comprised of (not necessarily contractible) periodic orbits. The problem is that the periodic orbit can “migrate’’ from one homotopy class to another depending on the function in question. Another class of manifolds to which this method does not apply is formed by geometrically bounded, but not convex manifolds. Among such manifolds are many twisted cotangent bundles and also universal coverings of some manifolds, which justiﬁes the interest in this class. The problem is that there is no known satisfactory deﬁnition of an action selector for general geometrically bounded manifolds. For example, the obstacle in the homological approach is that HF[a,b) (H ) for H ∈ HW has been deﬁned only for intervals [a, b] which do not contain zero, while the homological deﬁnition of σ requires HF[a,b) (H ) to be deﬁned for all intervals. One possible solution is to consider an action selector only on the class of nonnegative (autonomous or time-dependent) Hamiltonians or a yet more narrow class of functions (containing the functions F from the proof of Proposition 2.1) and to extend the homological deﬁnition to this class. The proof of the a priori bound outlined in this section will not carry over to such a narrow class, but a different argument (e.g., akin to the symplectic homology method from Section 2.4) may.

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Furthermore, when a geometrically bounded manifold is “asymptotically convex,’’ as are for instance twisted cotangent bundles, one should be able to deﬁne Floer homology for all intervals of action by suitably extending the function at inﬁnity as in the convex case. When this is done, many of the results that hold for convex manifolds should remain valid, although certain technical difﬁculties have to be dealt with. However, no work in either of these directions has yet been carried out and it is not clear whether or not this would lead to new results. 2.4 Symplectic homology A slightly different approach to proving the nearby existence theorem, which also utilizes Floer homology, relies on symplectic homology introduced and investigated in [CFH, CFHW, FH, FHW]. Let, as above, U be a domain in a symplectic manifold W . The symplectic homology of U is deﬁned as SH[a,b) (U ) = lim HF[a,b) (H ), ← − H

where the inverse limit is taken over all H ∈ Cc∞ (S 1 × U ). The symplectic homology of U detects closed characteristics in an arbitrarily narrow thickening {s } of = ∂U , provided that is smooth. In particular, when SH[a,b) (U ) = 0, in any thickening {s } there exists a hypersurface s0 carrying a closed characteristic. (This fact is an essentially immediate consequence of the deﬁnition; see [FH].) Therefore, the nearby existence theorem holds for a function H whenever the symplectic homology of the sublevels {H < c} is nontrivial. The problem is thus reduced to verifying that symplectic homology does not vanish. Let us show how this approach works, for example, when W = R2n . Let U be a bounded domain in R2n which we assume to contain the origin. Thus, there exist two open balls Br and BR centered at the origin and such that Br ⊂ U ⊂ BR . The symplectic homology has a natural monotonicity property: an inclusion U ⊂ V induces a map SH[a,b) (V ) → SH[a,b) (U ). The map # : SH[a,b) (BR ) → SH[a,b) (Br ) factors as SH[a,b) (BR ) → SH[a,b) (U ) → SH[a,b) (Br ), and hence SH[a,b) (U ) = 0 when # = 0. The symplectic homology of a ball and the map # can be calculated explicitly (see [FHW, Her1]) and, indeed, it turns out that # = 0 for some b > a > 0 and the nearby existence in R2n follows. Today, this calculation can be carried out particularly easily (see [BPS, CGK]) if one makes use of Poz´ niak’s theorem giving Floer homology in the Morse–Bott nondegenerate case [Poz]. This method also applies in the setting of the generalized Weinstein–Moser theorem discussed in the introduction. Let U be a neighborhood of a closed symplectic submanifold M of a geometrically bounded symplectically aspherical manifold W . Then, by taking suitably deﬁned symplectic tubular neighborhoods of M as Br and

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BR , one can show that SH[a,b) (U ) = 0 [CGK]. This leads to a proof of a nearby existence theorem for narrow shells enclosing M or small values of functions having a minimum (say, equal to zero) along M. Note that since here we utilize Floer homology only for intervals that do not contain zero, the method can be used for geometrically bounded manifolds which are not necessarily convex. Furthermore, this method can also be cast in the framework of energy selectors deﬁned on a suitable class of functions (see [Her1]). However, the beneﬁts of this approach are unclear and we omit the details here. Remark 2.3 (stability of the area spectrum [CFHW]). As a side remark, let us mention one more application of symplectic homology and Poz´ niak’s theorem. Here, again, we focus on the main idea of the argument rather than on a complete proof. Let be a smooth hypersurface in W . The area (or action) spectrum A() is the collection of symplectic areas bounded by contractible closed characteristics on , including iterated closed characteristics. It is not hard to see using Poz´ niak’s theorem that when is the boundary of an open domain U , A() = {a ∈ R | SH[a−,a+) (U ) = 0 for any small > 0}, provided that has contact type and all (iterated) closed characteristics on are nondegenerate. (An additional, more subtle, argument is needed here when different closed characteristics bound equal areas.) Observing that the right-hand side of this equality depends solely on U , we conclude that under these hypotheses A() is determined by U . This is the stability of the area spectrum theorem [CFHW]. More speciﬁcally, let and be smooth contact type hypersurfaces in W bounding open domains U and U , respectively. Assume that all (iterated) closed characteristics on and are nondegenerate and that there exists a symplectomorphism of W sending U to U . Then A() = A( ). 2.5 Other applications In this section we brieﬂy discuss some other applications of the action selector, which are important for what follows. 2.5.1 Homological capacity An action selector σ can be used to introduce an invariant, the homological capacity chom (U ), of a domain U in a symplectic manifold W . Assume ﬁrst that W is open and convex at inﬁnity. To an action selector σ on W , we associate a function chom on open subsets of W by setting chom (U ) = sup{σ (H ) | H ∈ HU } ∈ (0, ∞]. By deﬁnition, chom (U ) < ∞ if and only if σ is a priori bounded on U , and hence the nearby existence theorem holds for U . When W is closed, we set

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chom (U ) = sup{γ (H ) | H ∈ HU } ∈ (0, ∞]. The homological capacity is invariant under symplectomorphisms of W , monotone with respect to inclusions of domains, and homogeneous of degree one with respect to scaling of ω, i.e., chom (U, λω) = |λ| chom (U, ω) for any nonzero λ ∈ R. Here we do not touch upon the general deﬁnition and properties of symplectic capacities and refer the reader to, e.g., [Ha, Her1, Her2, Ho1, HZ2, HZ3] for a detailed discussion of this subject. In what follows we will always assume that chom is associated to σ = σ[max] or γ = σ[max] − σ[min] . Then, it is not hard to show that chom (BR ) = π R 2 by calculating the Floer homology of bump functions on BR (see [FHW] and Section 2.4). Similarly, chom (S 2 ) = area(S 2 ). However, chom (W ) = ∞, when W is a closed orientable surface other than S 2 or W = T2 × S 2 or W = T4 . In Section 3.3 we will utilize the homological capacity chom as an upper bound for the Hofer–Zehnder capacity. 2.5.2 Hofer’s geometry Denote by D the universal cover of the group of Hamiltonian symplectomorphisms of W . To be more precise, D is formed by time-one ﬂows ϕK of Hamiltonians K ∈ HW , where each ϕK comes together with the homotopy class (with ﬁxed end points) of t for t ∈ [0, 1]. Hofer’s norm · is deﬁned as the path ϕK H

1

K H =

(max Kt − min Kt )dt

0

for K ∈ HW . For ψ ∈ D, set ρ(ψ) = inf { K H | K generates ψ, i.e., ψ = ϕK }. It is easy to see that ρ(ψ, ϕ) := ρ(ψϕ −1 ) is a bi-invariant metric on D, provided that ρ is nondegenerate, i.e., ρ(ψ) > 0 iff ψ = id . It turns out that ρ, known as Hofer’s metric, is indeed nondegenerate for any symplectic manifold W . Nondegeneracy of ρ has been established through a series of more and more general results starting with W = R2n (see, e.g., [HZ3]) and ending with the proof for an arbitrary W in [LMcD1]. We refer the reader to [Pol3] for an introduction to Hofer’s geometry and further references. Here we only outline the proof of nondegeneracy for symplectically aspherical convex manifolds. Assume ﬁrst that W is open and convex at inﬁnity. Let ϕK = id. Pick an open set U displaced by ϕK , i.e., such that ϕK (U ) ∩ U = ∅, and let H ∈ HU . The idea is to again utilize (2.5), but this time to obtain a lower bound for the right-hand side. Namely,

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1 1 (2.3) yields the inequality σ (K) ≤ 0 max Kt dt and also σ (K − ) ≤ − 0 min Kt dt, as can be easily seen from the deﬁnition of K − . Therefore, by (2.5), σ (H ) ≤ K H , and hence, once supp H ⊂ U and σ (H ) > 0, 0 < σ (H ) ≤ ρ(ϕK ).

(2.8)

This proves nondegeneracy of ρ for convex at inﬁnity symplectically aspherical open manifolds. Furthermore, let us deﬁne the displacement energy of U as e(U ) = inf ρ(ψ), where ψ ∈ D displaces U . Then (2.8) translates into the upper bound chom (U ) ≤ e(U ), (2.9) which is a minor improvement over the upper bound chom (U ) ≤ 2e(U ) established in [FS]. When W is closed and symplectically aspherical, we argue in a similar fashion. Let, as above, ϕK displace U . Note that γ (K) ≤ ρ(ϕK ), by (2.6). Hence, we conclude from (2.7) that γ (H ) ≤ 2ρ(ϕK ) 1 for any H ∈ HU . When H is a C 2 -small nonzero function, γ (H ) = 0 (max Ht − min Ht )dt > 0. Therefore, ρ(ϕK ) > 0, which proves nondegeneracy of ρ. Furthermore, we also obtain the upper bound [Sc3]: chom (U ) ≤ 2e(U ).

(2.10)

In fact, the more accurate upper bound (2.9) still holds in this case; see [FGS]. Nondegeneracy of ρ for an arbitrary symplectic manifold has been established in [LMcD1] by showing, using entirely different methods, that cGr (U ) ≤ 2ρ(ϕK ), where cGr (U ) is the Gromov capacity of U , i.e., cGr (U ) = sup{π R 2 | BR symplectically embeds into U }. Note also that since cGr (U ) ≤ chom (U ), (2.8) yields that in fact cGr (U ) ≤ ρ(ϕK ), whenever W is symplectically aspherical, open, and convex at inﬁnity.

3 Almost existence theorems Virtually all known proofs of the almost existence theorems are based on the notion of the Hofer–Zehnder capacity [HZ3], with the exception of [St] which precedes this notion.

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3.1 Hofer–Zehnder capacity and almost existence Let V be a symplectic manifold without boundary. Denote by HHZ (V ) the class of smooth nonnegative functions K on V such that • •

K is compactly supported if V is not closed or K vanishes on some open set if V is compact, K is constant near its maximum.

We will refer to such functions as Hofer–Zehnder functions. Also, let us call a nontrivial periodic orbit of K with period T ≤ 1 fast. Otherwise, an orbit will be called slow. A Hofer–Zehnder function without nontrivial fast periodic orbits will be said to be admissible. Following [HZ2, HZ3], recall that the Hofer–Zehnder capacity of V is deﬁned as cHZ (V ) = sup{max K | K ∈ HHZ (V ) and K is admissible} ∈ (0, ∞]. The capacity cHZ (V ) does not change when the assumptions that K is nonnegative and/or that K is constant near its maximum are dropped [GG3]. The Hofer–Zehnder capacity has the same general properties as the homological capacity. One should think of cHZ as a higher dimensional analogue of the area. We will elaborate on this point in Section 4 and here we only mention that by using the area–period relation (see Section 4) one can show that for closed orientable surfaces cHZ is exactly equal to the area [Sib], in contrast with chom . The following result asserts that to prove almost existence in V , it sufﬁces to establish that cHZ (V ) is ﬁnite. Proposition 3.1 ([HZ3]). Assume that cHZ (V ) < ∞. Then the almost existence theorem holds for proper C 2 -functions H on V : periodic orbits of H exist on almost all, in the sense of measure theory, regular levels of H . Equivalently, for every shell {s } which bounds a domain in V , the hypersurfaces s carry closed characteristics for almost all s. This proposition is a rather simple consequence of the deﬁnition of cHZ and the Arzela–Ascoli theorem; see [HZ3]. The assumption that {s } bounds a domain is superﬂuous [MS]. In Section 4 we will prove a more precise version of Proposition 3.1. One can also incorporate the homotopy class of an orbit in the almost existence theorem and in the deﬁnition of cHZ [Sc3]. The simplest way to do this is as follows. Fix an ambient symplectic manifold W . Let us now modify the deﬁnition of the Hofer–Zehnder capacity by requiring V to be an open subset of W and requiring K ∈ HHZ (V ) to have no nontrivial contractible in W fast periodic orbits. We denote the resulting capacity by coHZ (V ). Then Proposition 3.1 holds for contractible in W periodic orbits provided that coHZ (V ) < ∞. Clearly, cHZ (V ) = coHZ (V ), when W is simply connected, and cHZ (V ) ≤ coHZ (V ) (3.1) in general. The strict inequality is possible: for example, cHZ (S 1 ×(0, 1)) = area(S 1 × (0, 1)) < coHZ (S 1 × (0, 1)) = ∞, where we view the annulus S 1 × (0, 1) as a subset

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of itself. This also shows that the choice of the ambient manifold, even though it is not included in the notation, affects the value of coHZ . (Replace W = S 1 × (0, 1) by W = R2 .) In this connection, we also note that cHZ (U ) < ∞ for any bounded open subset U of T ∗ Tn [Ji], while coHZ (U ) = ∞ when U contains the zero section. Indeed, U can be symplectically embedded into a bounded subset of R2n since Tn admits a Lagrangian embedding in R2n . In particular, the almost existence theorem holds in T ∗ Tn . Beyond dimension two, little is known about the capacity cHZ in contrast with the capacity coHZ . For example, all results discussed in what follows deal with contractible periodic orbits and thus concern the capacity coHZ . Of course, by (3.1), an upper bound for coHZ implies an upper bound for cHZ . 3.2 Finiteness of the Hofer–Zehnder capacity In view of Proposition 3.1, to prove the almost existence theorem in V it sufﬁces to show that cHZ (V ) < ∞. Let us start by proving that the Hofer–Zehnder capacity of a bounded domain in R2n is ﬁnite. Theorem 3.2 ([HZ2, HZ3]). Let U be a bounded domain in R2n . Then cHZ (U ) < ∞. Moreover, cHZ (BR ) = π R 2 , where BR is the ball of radius R. Proof. By monotonicity, the ﬁrst assertion is an immediate consequence of the inequality cHZ (BR ) < ∞. To show that cHZ (BR ) = π R 2 , it sufﬁces to prove that cHZ (BR ) ≤ π R 2 ; the opposite inequality is easy. (It is usually the case that establishing an upper bound for the capacity is hard while a lower bound can be easily obtained by deﬁnition.) The key is the following result. Proposition 3.3 ([HZ3]). Let H be a Hofer–Zehnder function on BR such that max H > πR 2 . Then the Hamiltonian ﬂow of H has a nontrivial one-periodic orbit. Following [GG3], let us outline a proof of the proposition which relies on the calculation of the Floer homology and generalizes to some other situations. Let H be as in Proposition 3.3. There exist nonnegative functions K − ≤ H ≤ K + , supported in BR and depending only on the distance to the origin, and such that the action of K ± on the nearest to the origin nontrivial one-periodic orbits is greater than max K + . (These orbits are Hopf circles on a sphere enclosing the origin.) To ﬁnd such functions, we make use of the assumption that H is constant near its maximum and take, as K ± , functions that squeeze H from above and below as tightly as possible and that depend only on the distance to the origin. Now, applying Poz´ niak’s theorem [Poz], one can show that HF[a,∞) (K ± ) = Z2 m for some m and a > max K + ≥ max H . This homology group is “generated’’ by the nontrivial one-periodic orbits closest to the origin. Moreover, by examining the natural homotopy from K + to K − , one can prove that the monotonicity map, which factors as

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Z2 = HF[a,∞) (K + ) → HF[a,∞) (H ) → HF[a,∞) (K − ) = Z2 , m m m [a,∞)

is an isomorphism. Hence, HFm (H ) = 0. Every trivial periodic orbit of H has action in the interval [0, max H ]. Thus, since a > max H , the ﬂow of H must have a nontrivial one-periodic orbit. This concludes the proof of Proposition 3.3 and Theorem 3.2.

Observe that Proposition 3.3 is stronger than what is needed: the proposition guarantees the existence of a nontrivial orbit with period T = 1 while T ≤ 1 is sufﬁcient to prove Theorem 3.2. Yet, this fact, somewhat surprisingly, appears to have no interesting applications. Also note that there are compact aspherical symplectic manifolds with inﬁnite Hofer–Zehnder capacity. The basic example of such a manifold is Zehnder’s torus [Ze], i.e., a torus T2n with an irrational symplectic structure. All other examples of such manifolds known to the author are derivatives of Zehnder’s torus. As has been pointed out in the introduction, the nearby/almost existence theorem also fails for Zehnder’s torus. Let us now elaborate on some general principles concerning the almost existence or nearby existence theorem and the Weinstein conjecture. The existence problem for periodic orbits on a given energy level is dual in a certain, rather vague, sense to the existence problem for periodic orbits of a ﬁxed period. The orbits of a ﬁxed period are known to exist (Arnold’s conjecture) and a variety of methods (e.g., Floer homology) of proving this fact have been developed during the last two decades. By contrast, unless the function is assumed to be convex (and then the duality acquires a precise meaning), the problem for a ﬁxed energy level may fail to have a solution, as counterexamples to the Hamiltonian Seifert conjecture show. Moreover, there seems to be no direct method to tackle the problem of existence of periodic orbits for a sufﬁciently large set of energy values (e.g., dense or full measure). All known methods, including those outlined above, ﬁrst reduce the problem to proving that the ﬂow of a function H with sufﬁciently large variation max H − min H possesses (nontrivial, fast, or period one) periodic orbits. Thus, all proofs of these theorems hinge on a general principle that a compactly supported function with sufﬁciently large variation must have fast nontrivial periodic orbits or even one-periodic orbits if the function is constant near its maximum. This principle, which is often true but fails in general (Zehnder’s torus!), can already be proved for some manifolds by utilizing, for example, Floer homology. Coming back to the discussion of the Hofer–Zehnder capacity, let us mention just one more of its properties. Namely, cHZ (ZR ) = π R 2 , where ZR = DR2 ×R2n−2 ⊂ R2n is a symplectic cylinder [HZ2, HZ3]. (Thus, the Hofer–Zehnder capacity of ZR is ﬁnite even though its volume is inﬁnite.) This can be easily proved as follows [GG3]. First note that our proof of Proposition 3.3 readily extends to ellipsoids and shows that the capacity of an ellipsoid is equal to its minimal cross-section area (by a plane through the origin). Then exhausting ZR by more and more elongated ellipsoids we see that cHZ (ZR ) = π R 2 . The next application of our method concerns the generalized Weinstein–Moser conjecture, the motion of a charge in a magnetic ﬁeld, and the relative almost exis-

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tence theorem. Let U be a neighborhood of a closed symplectic submanifold M of a geometrically bounded symplectically aspherical manifold W . Then, we can emulate the proof of Proposition 3.3 by taking, as K ± , functions that depend only on the distance to M in a suitable metric and that are supported near M. As a consequence, we obtain the relative almost existence theorem: near M, almost all levels of a function H attaining a minimum along M carry contractible periodic orbits [GG3]. (Note that this, in turn, requires replacing the ordinary Hofer–Zehnder capacity in Theorem 3.2 by its relative counterpart. The relative Hofer–Zehnder capacity cHZ (W, M), where M ⊂ W is a compact subset, is deﬁned just as the ordinary capacity, but the function K is required to attain its maximum along M, see, e.g., [GG3, HZ2, La] for more details.) Furthermore, a more reﬁned version of the Floer homology calculation outlined here can be used to show that coHZ (U ) < ∞ when U is a small neighborhood of a closed symplectic submanifold M with homologically trivial normal bundle in a closed symplectically aspherical manifold [Ke3]. This implies the almost existence theorem in U . o with c 3.3 Comparison of cHZ hom

In this section we will outline a different approach to obtaining an upper bound for cHZ . To be more speciﬁc, we prove that coHZ (U ) ≤ chom (U ). In our proof of this inequality we draw heavily on [HZ3, FS, Sc3], but some details appear to be new. Let W be an open manifold (with ω|π2 (W ) = 0) on which a continuous selector σ satisfying (AS1) and (AS2) is deﬁned. Assume also that σ (H ) = max H

when H is C 2 -small and independent of time.

(3.2)

This requirement strengthens (AS3) and holds for the energy selector deﬁned in Section 2.1.2 because the Floer complex of a C 2 -small autonomous Hamiltonian is equal to its Morse complex; see [FHS]. Let chom be the associated homological capacity. Proposition 3.4 ([FS, Sc3]). coHZ (U ) ≤ chom (U ) for any U ⊂ W . This, by (3.1), implies that cHZ (U ) ≤ chom (U ). Proof. The proposition is an immediate consequence of the following. Lemma 3.5. Let H ∈ HW be an autonomous Hamiltonian whose ﬂow has no fast nontrivial contractible periodic orbits. Then σ (H ) = max H . To derive the proposition from the lemma, we just use the deﬁnition of the capacities: chom (U ) = sup{σ (H ) | H ∈ HU } ≥ sup{max H | H as in the lemma} = coHZ (U ).

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The assertion of the lemma is very plausible. When W is convex at inﬁnity and σ = σ[max] , one can argue as follows. By continuity, it sufﬁces to prove the lemma for a generic H . Then, since H does not have nontrivial contractible periodic orbits, the Floer complex of H is generated by the critical points of H and hence σ (H ) is a critical value of H . (Moreover, without loss of generality we may assume that the eigenvalues of d 2 H at the critical points of H are small. Indeed, this can be achieved by replacing H by f ◦ H , where f : R → R is a suitable diffeomorphism which is C 0 -close to identity, see [GG3, Schl]. Then σ (H ) must be the value of H at one of its local maxima, for maxima are the only critical points of H with correct index. Hence, max H = σ (H ) if H has a unique maximum.) However, to show that σ (H ) is delivered by the global maximum, one has to analyze the Floer differential of H and this indeed can be done; see, e.g., [FS, Sc3]. At this point one considers the homotopy λH with λ ∈ (0, 1] and fully uses the assumptions on H . Note also that it seems to be unknown whether in the hypotheses of the lemma fast orbits can be replaced by one-periodic orbits. One can also prove the lemma in a formal set-theoretic way, using only the continuity of σ , (AS1), (AS2) and (3.2). Denote by S the action spectrum S(H ). Observe that by the hypothesis of the lemma, every contractible one-periodic orbit of λH is trivial for all λ ∈ (0, 1], and hence the action spectrum S(λH ) is again comprised entirely of the critical values of λH . Therefore, S(λH ) = λS. Set σ (λ) = σ (λH ) ∈ λS. Then σ (λ)/λ is a continuous function of λ with values in S. Since the action spectrum S is a nowhere dense, σ (λ)/λ = const. For a small λ > 0, the function λH is C 2 small and hence σ (λH ) = max(λH ) = λ max H . It follows that σ (λ) = λ max H for all λ ∈ (0, 1] which concludes the proof of the lemma and proposition.

Proposition 3.4 also holds (see [Sc3]) when W is closed and symplectically aspherical and chom is the homological capacity deﬁned in Section 2.5.1 for γ = σ[max] − σ[min] . This can be proved essentially in the same way as the result for open manifolds. Combining Proposition 3.4 with (2.9) and (2.10), we infer that a displaceable open set has a ﬁnite Hofer–Zehnder capacity: coHZ (U ) ≤ e(U )

if W is open,

cHZ (U ) ≤ 2e(U ) if W is closed, o

(3.3) (3.4)

whenever W is convex and symplectically aspherical. These are versions of the socalled “capacity–displacement energy inequality.’’ In particular, the almost existence theorem holds for displaceable sets. For example, let L ⊂ W be a closed nonLagrangian submanifold of middle dimension whose normal bundle has a nonvanishing section. Assume also that on W there exists a selector σ satisfying (3.2). Then, by [LS, Pol1], a small neighborhood U of L is displaceable and hence cHZ (U ) < ∞. As in Section 2.1.3, this leads to a local version of the almost existence theorem for twisted cotangent bundles over surfaces (other than the torus) or, in higher dimensions, for exact magnetic ﬁelds; see [FS, Ma1].

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Proposition 3.4 explains why the almost existence theorem can often be proved in the same setting as that of the weaker nearby existence theorem. Indeed, most of the proofs of the former, with some notable exceptions, rely on ﬁrst establishing that a version of the homological capacity is ﬁnite. However, in this case the Hofer–Zehnder capacity is also ﬁnite, which implies the almost existence theorem. Remark 3.6. Proposition 3.3 suggests that in the deﬁnition of the Hofer–Zehnder capacity one may replace the condition that K has no fast periodic orbits by a weaker requirement that it has no one-periodic orbits (see [Ke3]). The resulting invariant is still a capacity. For this capacity the upper bound from Proposition 3.4 may fail to hold, unless an additional requirement on the critical set of K is imposed; see [FGS]. However, this capacity is a priori bounded on R2n and sufﬁcient for the proof of the almost existence theorem. Furthermore, the inequality T ≤ D cH (h) in Proposition 4.2 is then replaced by the equality. (In fact, when cHZ is modiﬁed in this way, for every converging sequence cH (h) − cH (hi ) /(h − hi ) → A there exists an A-periodic orbit (not necessarily simple) on the level H = h.) As a consequence, this capacity is represented on hypersurfaces of restricted contact type in R2n (see Proposition 4.4). 3.4 Holomorphic curves and almost existence The holomorphic curve approach to the nearby existence and almost existence theorems goes back to [FHV, HV2]. Here we outline only the main idea of the method, following essentially [HV2] and omitting technical details which are in some instances quite involved. Let (W, ω) be a closed symplectic manifold and let L− and L+ be two disjoint closed submanifolds of W . Fix a generic almost complex structure J compatible with the symplectic structure and consider the space of J -holomorphic spheres u : S 2 → W in a given free homotopy class A ∈ [S 2 , W ] and with the north pole on L+ and the south pole on L− . The resulting metric space of J -holomorphic curves is not compact: the noncompact group of conformal transformations of S 2 ﬁxing the poles acts properly on it. To make this space compact, we require that u∗ ω = ω, A /2, (3.5) D−

where D− is the southern hemisphere in S 2 . This condition eliminates the conformal ﬂow from one pole to another, parametrized by R, and forces the resulting space M0 to be compact. This space depends on L± , A, and J . For a generic J , the space M0 is a smooth manifold and its free S 1 -equivariant cobordism class [M0 ] is independent of J . (Here the action of S 1 on M0 arises from the S 1 -action on S 2 by rotations about the vertical axis.) Note that in reality the manifold W and the class A must satisfy some additional conditions, e.g., A must be minimal, i.e., ω, A = m(W, ω); the deﬁnition of m(W, ω) is recalled below. Now let H be a smooth nonnegative function on W such that H ≡ 0 near L− and H ≡ max H near L+ . Consider the perturbed Cauchy–Riemann equation for the maps u : S 2 → W :

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∂¯J u + λ∇H = 0,

(3.6)

where λ ≥ 0 is a real parameter. Note that since ∇H = 0 in some neighborhoods of L± , this equation simply means that u is holomorphic near the poles. To make sense of this equation away from the poles, we view S 2 as the cylinder S 1 × R with the two poles attached. Then the coordinates on the cylinder are used to deﬁne ∂¯J u as a vector ﬁeld along u. Let Mλ be the space of solutions of (3.6) which are in the class A, send the north and south poles to L± , and satisfy (3.5). When λ = 0, we obtain the space M0 introduced above. In general, the solutions of (3.6) can be thought of as gradient trajectories for AλH connecting trivial periodic orbits on L+ with those on L− . Then, a calculation shows that the difference of actions on these periodic orbits, which is λ max H , is bounded from above by the symplectic area of u, i.e., λ max H ≤ ω, A . As a consequence, Mλ = ∅ when λ is large. Let us examine now the disjoint union of the spaces Mλ . Under suitable genericity hypotheses, this is a smooth manifold. If this manifold is compact, it gives a cobordism from M0 to the empty set, i.e., [M0 ] = 0. Just as in Floer’s theory, compactness can fail only when a family of solutions uλ ∈ Mλ converges as λ → λ0 ≤ ω, A / max H to a broken solution which “hangs up’’ on a contractible one-periodic orbit of λ0 H , i.e., a contractible 1/λ0 -periodic orbit of H . We conclude that H must have contractible one-periodic orbits, provided that max H > ω, A and [M0 ] = 0. (Note that at this point one still has to show that the orbits found are nontrivial.) This approach leads to a few versions of the almost existence theorem, all relying on the same basic requirement that the space M0 is not cobordant to zero. In other words, the space of holomorphic curves “connecting’’ L+ and L− should be sufﬁciently large for the method to apply. In particular, it may be helpful to start with larger submanifolds L± or with a function H such that H ≡ 0 and H ≡ max H on large subsets. (The tradeoff is that this may result in the almost existence theorem for a restricted class of functions H .) Let us illustrate these considerations by some examples. When L± are points and the method is applicable to W and A, we conclude that cHZ (W ) ≤ ω, A and hence prove the almost existence theorem in W . There are, however, rather few manifolds W for which [M0 ] = 0 when L± are points. One example is CPn with the standard symplectic form normalized as the reduction of the unit sphere in Cn+1 . In this case the standard J is already generic, M0 = S 1 with S 1 acting by translations, and we see that cHZ (CPn ) = π [HV2]. It is also possible that this reasoning can be utilized to show that other coadjoint orbits of compact semisimple Lie groups have ﬁnite Hofer–Zehnder capacity. Now let W = P ×S 2 , where (P , β) is a compact symplectic manifold, as in [HV2]. Then taking the class of the ﬁber S 2 as A and applying, with some modiﬁcations, this method to L− = P × {0} and L+ = P × {∞}, one can show that the relative capacity

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cHZ (P × DR2 , P × {0}) = π R 2 as long as area(DR2 ) = π R 2 < m(P , β, J ), where u∗ β > 0 | u is a nonconstant J -holomorphic sphere . m(P , β, J ) := inf S2

In particular, m(P , β, J ) = ∞ when π2 (P ) = 0 [FHV]. Note also that m(P , β, J ) is always positive by Gromov’s compactness theorem. Furthermore, assume that ∗ 2 u β > 0 | u : S → P ≥ 0. m(P , β) := inf S2

In contrast with m(P , β, J ) ≥ m(P , β), this constant can be zero. (For instance, this is the case for P = S 2 × S 2 where the area of the ﬁrst component is 1 and the second component has an irrational area.) By taking a point in P × {0} as L− and L+ as before, one can show that cHZ (P ×S 2 , L+ ) = area(S 2 ) as long as area(S 2 ) < m(P , β) [HV2]. As a consequence, cHZ (P × DR2 ) = π R 2 if π R 2 < m(P , β). (In fact, there are strong indications that cHZ (P × DR2 ) = π R 2 for any geometrically bounded symplectic manifold P and any R > 0; see [MDSl, Schl].) Symplectic manifolds P for which m(P , β) > 0 are called rational. The method also extends to the setting where P is not closed but is geometrically bounded [Lu1]. (These results have further applications to the existence problem for periodic orbits of a charge in a magnetic ﬁeld, see [Ma1, Ma2].) For some other incarnations and applications of the holomorphic curve method, we refer the reader to [GL, Ke3, LT, Lu2, Lu3, Lu4]. In particular, for uniruled symplectic manifolds (such as T2 × S 2 ) and symplectic toric manifolds the Weinstein conjecture was established in [Lu2, Lu4]. In the context of the generalized Weinstein–Moser conjecture, it was proved in [Ke3] that a small neighborhood of a rational symplectic submanifold M ⊂ W has ﬁnite Hofer–Zehnder capacity coHZ . Finally, one may replace holomorphic spheres by holomorphic curves of higher genus to obtain a sufﬁciently large space M0 as in [LT, Lu3]. Extra care in interpreting (3.6) is then needed. For instance, imposing some additional conditions on H may be necessary, e.g., requiring H to be locally constant outside of a shell separating L− from L+ . This, depending on the details of the approach, leads to a version of either the nearby existence or almost existence theorem; see [LT, Lu2, Lu3, Lu4]. The holomorphic curve approach in the form outlined above does not apply to compact manifolds that generically have too few holomorphic curves. Among such manifolds is, for example, T4 with the standard symplectic structure. It is not known whether or not the Hofer–Zehnder capacity of T4 is ﬁnite. 3.5 Hofer’s geometry and almost existence Applications of Hofer’s geometry to nearby and almost existence theorems are based on a principle relating fast periodic orbits to minimazing properties of geodesics in t which is a one-parameter Hofer’s metric. Namely, consider the Hamiltonian ﬂow ϕH subgroup in D and hence can be viewed as a geodesic in D. Then, conjecturally, t is length minimizing for t ∈ [0, 1), provided that H has no fast periodic orbits; ϕH

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see [MDSl] and references therein. Various particular cases of this conjecture have been proved. We refer the reader to [Pol3] for a detailed discussion and additional references and to [MDSl], following [LMcD2], for more recent results relevant to our discussion. The application of Hofer’s geometry to the circle of questions considered in this paper was pioneered in [Pol2]. Below we will just brieﬂy indicate the logic of the argument following closely the most recent work [Schl] and in fact suppressing the connection with minimizing properties of geodesics in Hofer’s metric. Let W be a geometrically bounded symplectic manifold, which we do not require to be symplectically aspherical or convex, and let H be an autonomous Hamiltonian supported in U ⊂ W . The cornerstone of the method is the inequality coHZ (U ) ≤ 4e(U ),

(3.7)

which implies almost existence in U , provided that e(U ) < ∞. (Note that stronger inequalities (3.3) and (3.4) hold when W is symplectically aspherical and convex. Hence, the emphasis here is on extending the class of manifolds for which a version of the capacity–displacement energy inequality is proved.) The proof of (3.7) is based on the following two results: (i) Assume that supp H ⊂ U ⊂ W and H H > 4e(U ). Then ρ(ϕH ) < H H . (ii) H has contractible fast periodic orbits, provided that ρ(ϕH ) < H H . The ﬁrst assertion (i) is proved by the curve shortening method, ubiquitous in Hofer’s geometry and going back to [Sik]; see the references above for other incarnations of this method. As stated, the second assertion (ii) is obtained in [Schl] as a consequence of the results of [MDSl]. (This is the point where length minimizing geodesics enter the picture.) The inequality (3.7) is sufﬁcient to establish the almost existence theorem for only a very limited class of domains U . However, (3.7) can be combined with Macarini’s stabilization trick [Ma1]. This leads to an upper bound similar to (3.7), but with e(U ) replaced by the displacement energy of U × S 1 in W × T ∗ S 1 [Schl], which has a somewhat broader range of applications. In particular, by modifying (3.7) in this way, one can prove the almost existence theorem for low-energy periodic orbits of a charge in a nonvanishing magnetic ﬁeld and a conservative force ﬁeld [Schl]. We conclude this section by pointing out that it may be possible to prove (3.7) directly, without invoking minimizing properties of geodesics in Hofer’s geometry, by adapting the argument from [MDSl].

4 Hofer–Zehnder capacity function What we see as a central open problem concerning the almost existence theorem is the question whether or not this theorem is sharp. Consider, for example, a smooth proper function H : R2n → R which we assume to be bounded from below. By the almost existence theorem, almost all regular levels in [min H, ∞) carry periodic

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orbits. The counterexamples to the Hamiltonian Seifert conjecture show that H may have a discrete collection of aperiodic levels, i.e., regular levels without periodic orbits [Gi1, Gi3, Gi4, GG1, GG2, He1, He2, Ke2]. Moreover, such levels can accumulate to a degenerate critical level of H [GG3]. However, it is still unknown if aperiodic levels can accumulate to a regular level either with or without periodic orbits. In particular, it is not known whether the set of aperiodic energy values can be dense or be a Cantor set. 4.1 Deﬁnition of the capacity function Throughout this section, we will focus on functions on R2n although most of our discussion carries over to functions on any manifold of bounded capacity. To concentrate on the essential part of the problem, let us assume that the function has only one critical point, the minimum. The key to the proof of the almost existence theorem is the Hofer–Zehnder capacity function associated with H : cH (h) = cHZ ({H < h}) < ∞,

where h > min H .

This is a monotone increasing function on (min H, ∞) and the levels H = h, where cH is Lipschitz, carry periodic orbits [HZ3]. (This is a simple consequence of the Arzela–Ascoli theorem and shortly we will recall the argument.) Since cH is monotone increasing, it is differentiable almost everywhere and the almost existence theorem follows. (Note that there is no reason to expect every non-Lipschitz value to be aperiodic. In particular, aperiodic points are not entirely visible from the properties of cH .) Any zero measure set is the set of non-Lipschitz points of some monotone increasing function [Na, p. 214]. Hence, one possible approach to the problem (but not the only one) is to investigate additional properties of the capacity function which distinguish it from an arbitrary monotone increasing function. We will soon see that some of these readily arise from the Arzela–Ascoli theorem. The following elementary observation illustrates our point [Gi5]. Denote by Z the collection of aperiodic values of H . Furthermore, let ZT be the collection of levels where all periodic orbits have period greater than T . It is clear that ZT Z= T ∈Z+

and that, by theArzela–Ascoli theorem, ZT is open. Hence, Z is a Gδ set. Furthermore, if Z is dense, every set ZT is also dense. From this we infer that Z must be a residual set when Z is dense. In particular, the set Z of aperiodic values cannot be a countable dense set. 4.2 Area–period relation The role of the capacity function in the proof of the almost existence theorem can be best understood in terms of the classical area–period relation; see, e.g., [Ar, p. 282]. Let us recall this result.

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Let H be a function on R2 . Denote by area(h) the area bounded by a regular level H = h and by T (h) the period of the periodic orbit on this level. (Here we are assuming that the levels are connected.) Proposition 4.1 (area–period relation). d area(h)/dh = T (h). Note that the same is true for any proper function on a symplectic surface, provided that the level H = h is regular. When this level is comprised of more than one connected component, the right-hand side is the sum of their periods. To prove Proposition 4.1, denote by ω the area form (i.e., the symplectic form) on R2 . Dividing ω by dH near the level, we can write ω = dH ∧ α, where α is a one-form such that α(XH ) = 1. Let $ be the annulus h ≤ H ≤ h+. The following calculation concludes the proof: 1 1 d area(h) ω = lim dH ∧ α = α = T (h). = lim →0 $ →0 $ dh {H =h} This result generalizes to higher dimensions when H is convex. Namely, in this case, on any level H = h there exist periodic orbits γl and γr such that the periods of these orbits are equal to the left and, respectively, right derivatives of cH at h and the symplectic areas bounded by γl and γr are equal to cH (h) [Ne]. When the convexity assumption is dropped, the derivative of the capacity function gives only an upper bound on the period. Proposition 4.2. Assume that the lower derivative D cH (h) = lim inf δ→0

cH (h + δ) − cH (h) δ

is ﬁnite. Then the level H = h carries a periodic orbit with period T ≤ D cH (h) and, as a consequence, D cH (h) > 0. The proposition shows, in particular, that the Hofer–Zehnder capacity is strictly increasing. More speciﬁcally, we have Corollary 4.3. (i) The capacity function cH is strictly increasing. (ii) Let U and V be open bounded subsets of R2n such that U ⊂ V . Then cHZ (U ) < cHZ (V ). It is easy to see that the assumption that the sets are bounded is essential and the requirement U ⊂ V cannot be replaced by U V . Proof of Proposition 4.2. The proof of the proposition is a modiﬁcation of an argument from [HZ3]. Let hi → h be a sequence such that Ai :=

cH (hi ) − cH (h) → A, hi − h

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where 0 ≤ A < ∞. By passing if necessary to a subsequence, we may assume that hi converges to h either from the right or left. In what follows we will assume that hi > h. The other case can be handled in a similar fashion. Observe that either Ai > 0 for all (sufﬁciently large) indexes i or cH is constant on some interval [h, h + δ], with δ > 0, since cH is monotone. Assume ﬁrst that Ai > 0 and pick a sequence bi > 1 converging to one and a sequence 0 < i < (bi − 1)(hi − h)Ai . By the deﬁnition of the Hofer–Zehnder capacity, on the domain {H < h} there exists an admissible function Ki without fast nontrivial periodic orbits and such that max Ki = cH (h)−i . Let fi be a monotone decreasing function on the interval [h, hi ] identically equal to bi (hi − h)Ai near h and zero near hi and such that |fi | ≤ ai bi Ai for some sequence ai → 1+. Set Fi = fi ◦ H on the shell h ≤ H ≤ hi and smoothly extend this function to R2n by requiring it to be constant inside and outside of the shell. Then max(Ki + Fi ) = cH (hi ) + [(bi − 1)(hi − h)Ai − i ] > cH (hi ). Hence, Ki + Fi must have a nontrivial fast periodic orbit which can only be a periodic orbit of Fi . Therefore, H has a periodic orbit with period less than or equal to ai bi Ai in the shell h ≤ H ≤ hi . By passing to the limit and applying the Arzela–Ascoli theorem, we conclude that H has a periodic orbit of period less than or equal to A = lim ai bi Ai on the level H = h. To ﬁnish the proof it sufﬁces to show that cH cannot be constant on any interval [h, h+δ]. (This will also be a direct proof of the corollary.) Assume the contrary. Then cH (hi ) = cH (h) for all large enough i. In this case we again choose a sequence of decreasing functions fi on [h, hi ] such that fi is zero near hi and a positive constant near h. (Thus fi is equal to max fi > 0 near h.) Furthermore, we can make the slope of fi so small that the function Fi has no fast nontrivial periodic orbits. Let 0 < i < max fi and let Ki be as above. It is clear that the function Ki + Fi has no fast nontrivial periodic orbits. On the other hand, we have max(Ki + Fi ) = cH (h) − i + max fi > cH (h) = cH (hi ), and hence Ki + Fi must have fast nontrivial periodic orbits. This contradiction completes the proof.

A question closely related to the area–period relation is that of representability of a capacity. Let U be the domain bounded by a compact smooth hypersurface in R2n . We say that a capacity c is represented on if there is a closed characteristic on that bounds a disc of symplectic area c(U ). For example, as is easy to see from the deﬁnition, chom is represented on smooth hypersurfaces of contact type. Since there are hypersurfaces without closed characteristics, no capacity is represented on every hypersurface. The Hofer–Zehnder capacity is represented on convex hypersurfaces [HZ2]. In fact, when is convex, cHZ (U ) is the minimal symplectic area of a closed characteristic on [HZ2]. However, it is not known if cHZ is represented on every

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contact type, or even restricted contact type, hypersurface. Arguing as in the proof of Proposition 4.2 and utilizing Lemma 3.5, one can prove the following. Proposition 4.4. Let U be the domain bounded by a smooth hypersurface of restricted contact type. Then, cHZ is subrepresented on , i.e., there exists a closed characteristic on with area less than or equal to cHZ (U ). Recall in this connection that even when U is star-shaped there can be a closed characteristic on with area strictly less than cHZ (U ); the “Bordeaux bottle’’ (see [HZ3, p. 99]) is an example. We refer the reader to, e.g., [FS, Her2] for other representability results. 4.3 Period growth and the Hofer–Zehnder capacity function As is immediately clear from the deﬁnition, the Hofer–Zehnder capacity function is necessarily continuous from the left. Beyond this trivial observation, little is known about continuity or differentiability properties of the capacity function. For example, it is not known if this function can be discontinuous at regular values. One, admittedly very naive, approach to this question is based on the estimates of the period growth. To make this more precise, ﬁx a regular level of H , say H = 0. Let τ (h) be the inﬁmum of all periods of periodic orbits in the open shell 0 < H < h or h < H < 0, depending on whether h is positive or negative. Then the growth of the function τ (h) is related to continuity and smoothness of cH at zero as the next proposition shows. Proposition 4.5. (i) τ (h) = O(1/ h) as h → 0+ and τ (h) = o(1/|h|) as h → 0−. (ii) Assume that τ (hk ) ≥ C/ hk for some sequence hk → 0+. Then cH (0+) − cH (0) ≥ C. (iii) Assume that τ (hk ) ≥ C/|hk |α for some sequence hk → 0 and 0 ≤ α ≤ 1. Then | cH (h) − cH (0)| ≥ C|h|1−α . The proofs of these facts follow the same line as the proof of the almost existence theorem in [HZ3]; see the proof of Proposition 4.2. Omitting a detailed argument here, we only mention that (i) is a consequence of the fact that the capacity is bounded. In fact, cHZ ({H < 0}) = ∞ if (i) fails for the left limit and cHZ (V ) = ∞ for any open set V ⊃ {H ≤ 0} if (i) fails for the right limit. Remark 4.6. The analogues of (ii) and (iii) hold when the difference of capacities is replaced by the relative capacity. Proposition 4.5 is difﬁcult to apply to examine discontinuity or nonsmoothness of the capacity function. Indeed, let, for example, H be a Hamiltonian on R2n constructed in [Gi1, Gi3, GG1, GG2, He1, He2, Ke2] such that H = 0 carries no periodic orbits. Then cH is not Lipschitz at zero as is clear already from the results of [HZ3]. To utilize Proposition 4.5, we would need to bound from below minimal periods on the nearby levels H = h. However, the constructions of H afford little insight into the

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dynamics on these levels and obtaining such lower bounds appears to be an extremely difﬁcult problem. Preliminary estimates indicate that cH can probably be Hölder with any α ∈ (0, 1) at H = 0 or even fail to be Hölder. However, there is no convincing evidence that cH can be discontinuous. (Note in this connection that the property of cH to be Hölder with a speciﬁc α or to be continuous, just as to be Lipschitz (see [HZ3]), is determined by the level H = 0 and is independent of the choice of H .) Consider, however, the following example. Example 4.7 (the horocycle ﬂow). Let M be a closed surface equipped with a metric of constant negative curvature −1 and let be the area form on M. Consider the twisted symplectic structure ω = ω0 + π ∗ on W = T ∗ M, where π is the natural projection T ∗ M → M and ω0 is the standard symplectic structure. Set H = p 2 −1. The Hamiltonian ﬂow on the level H = 0 is the horocycle ﬂow and hence has no periodic orbits; see, e.g., [Gi2, Gi4] for details. Assume that the Hofer–Zehnder capacity of the sets {H < h} for h near zero is ﬁnite. (This is not known, although is likely to be true, even for h < 0 close to zero. For small neighborhoods of the zero section (i.e., h ≈ −1), ﬁniteness has been recently proved in [FS]; however, for h > 0, ﬁniteness of cHZ appears to be beyond reach of the methods considered in this paper.) Then we claim that √ cH (h) − cH (0) ≥ lmin h for h > 0, (4.1) where lmin is the minimal length of a closed geodesic on M. Therefore, cH is not smoother at h = 0 than 1/2-Hölder on the right. To prove this, let us ﬁrst note that √ τ (h) = lmin / h for h > 0, (4.2) √ and hence, by (iii), cH (h) − cH (0) ≥ lmin h. Establishing (4.2) directly appears to be rather difﬁcult. Instead, let us observe that there exists a symplectomorphism (T ∗ M {H ≤ 0}, ω) → (T ∗ M M, ω0 ) which sends p 2 to p 2 −1. (Such a symplectomorphism can be constructed, for instance, by judiciously applying Moser’s method.) This allows one to translate the period growth on (T ∗ M, ω0 ) as p → 0, which is lmin / p for the Hamiltonian p 2 /2, to the period growth on (T ∗ M, ω) for H > 0. Note also that by applying the results of [BPS] and using√the above symplectomorphism, one can prove that cHZ ({H < h}, {H ≤ 0}) ≤ lmin h. This, together with the analogue of (4.1) for the relative capacity, proves that cHZ ({H < h}, {H ≤ 0}) = √ lmin h. One can also estimate the period growth on the left. All orbits on the level H = h < 0 are closed and project to geodesic circles on M with √ geodesic curvature kg > 1. A straightforward calculation shows that τ (h) = 2π/ |h| for h < 0 and hence, by √ (iii), cH (0) − cH (h) ≥ 2π |h|. Thus, on the left, cH is again not smoother than 1/2-Hölder. Acknowledgments The author is deeply grateful to Paul Biran, Urs Frauenfelder, Ba¸sak Gürel, Ely Kerman, Debra Lewis, Leonid Polterovich, Felix Schlenk, and the referees for their useful remarks and suggestions.

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Simple singularities and integrable hierarchies∗ Alexander B. Givental and Todor E. Milanov Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected], [email protected] To Alan Weinstein on his 60th birthday. Abstract. The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the An−1 -singularity satisﬁes the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy which is satisﬁed by the total descendent potential of a simple singularity of ADE type. Our description of the hierarchy is parallel to the vertex operator construction of Kac–Wakimoto [17] except that we give both some general integral formulas and explicit numerical values for certain coefﬁcients which in the Kac–Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown.

1 ADE-hierarchies According to Date–Jimbo–Kashiwara–Miwa [6] and I. Frenkel [10], the KdVhierarchy of integrable systems can be placed under the name A1 into the list of more general integrable hierarchies corresponding to the ADE Dynkin diagrams. These hierarchies are usually constructed (see [16]) using the representation theory of the corresponding loop groups. V. Kac and M. Wakimoto [17] describe the hierarchies even more explicitly in the form of the so-called Hirota quadratic equations expressed in terms of suitable vertex operators. One of the goals of this paper is to show how the vertex operator description of the Hirota quadratic equations (certainly the same ones, even though we don’t quite prove this) emerges from the theory of vanishing cycles associated with ADE-singularities. Let f be a weighted-homogeneous polynomial in C3 with a simple critical point at the origin. According to V. Arnold [1] simple singularities of holomorphic functions are classiﬁed by simply-laced Dynkin diagrams: ∗ This material is based on work supported by National Science Foundation grants DMS-

0072658 and DMS-0306316.

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x2 x2 x1N +1 + 2 + 3, N +1 2 2 N −1 2 + x32 , DN , N ≥ 4: f = x1 x2 − x2 AN , N ≥ 1: f =

E6 : f = x13 + x24 + x32 , E7 :

f = x13 + x1 x23 + x32 ,

E8 : f = x13 + x25 + x32 . Let H = C[x1 , x2 , x3 ]/(fx1 , fx2 , fx3 ) denote the local algebra of the critical point. We equip H with a nondegenerate symmetric bilinear form (·, ·) by picking a weightedhomogeneous holomorphic volume ω = dx1 ∧ dx2 ∧ dx3 and using the residue pairing ϕ(x)ψ(x)ω (ϕ, ψ) := Res0 . fx1 fx2 fx3 Let H = H ((z−1 )) be the space of Laurent series f (z) = k∈Z fk zk in one indeterminate z−1 (i.e., ﬁnite in the direction of positive k) with vector coefﬁcients fk ∈ H . We endow H with the symplectic form . 1

(f , g) := (f (−z), g(z))dz. 2π i The polarization H = H+ ⊕ H− , where H+ = H [z] and H− = z−1 H [[z−1 ]], is Lagrangian and identiﬁes H with the cotangent bundle space T ∗ H+ . The Hirota quadratic equations are imposed on asymptotical functions of q = q0 + q1 z + q2 z2 + · · · ∈ H+ . By an asymptotical function we mean an expression of the form = exp

∞

g−1 F (g) (q),

g=0

where usually F (g) will be formal functions on H+ . By deﬁnition, vertex operators are elements of the Heisenberg group acting on such functions. Given a sum f = fk zk (possibly inﬁnite in both directions) one deﬁnes the corresponding vertex operator of the form √

√

e (f− ,q)/ e ∂f+ ⎫ ⎧ ⎫ ⎧ ⎬ ⎨ ⎬ ⎨√ √ a = exp (−1)k+1 f−1−k qka / exp fka ∂/∂qka . ⎭ ⎩ ⎭ ⎩ k≥0

a

k≥0 a

Here fka , qka are components of the vectors fk , qk in an orthonormal basis. We will make use of the vertex operators φ (λ) corresponding to two-dimensional homology classes φ ∈ H2 (f −1 (1)) ! ZN and deﬁned as follows. Take

Simple singularities and integrable hierarchies

f :=

(k)

Iφ (λ)(−z)k ,

(k)

175

(k+1)

where dIφ /dλ = Iφ

k∈Z (−1)

and Iφ

(λ) ∈ H is the following period vector: (−1) (Iφ (λ), [ψa ])

1 := 2π

φ⊂f −1 (λ)

ψa (x)

ω . df

The cycle φ is transported from the level surface f −1 (1) to f −1 (λ), and ψa are weighted-homogeneous functions representing a basis in the local algebra H .1 The (k) functions (Iφ , [ψa ]) are proportional to the fractional powers λma / h−k−1 where h is the Coxeter number and ma = 1 + h deg ψa are the exponents of the appropriate reﬂection group AN , DN , or EN . The lattice H2 (f −1 (1)) carries the action of the monodromy group (deﬁned via morsiﬁcation of the function f ) which is the reﬂection group with respect to the intersection form of cycles. The form is negative deﬁnite, and we will denote ·, · the positive deﬁnite form opposite to it. Let A denote the set of vanishing cycles, i.e., the set of classes α ∈ H2 (f −1 (1)) with α, α = 2 such that the reﬂections φ → φ − α, φα belong to the monodromy group. The Hirota quadratic equation of the ADE type takes on the form ( ) dλ N (h + 1) α −α Resλ=∞ ( ⊗ ) (1) aα (λ) ⊗ (λ) ( ⊗ ) = λ 12h α∈A

ma ∂ ∂ + + k (qka ⊗ 1 − 1 ⊗ qka ) ⊗ 1 − 1 ⊗ ( ⊗ ). h ∂qka ∂qka a k≥0

(2) The tensor product sign means that the functions depend on two copies q and q of the variable q, and the objects on the left of ⊗ refer to q = q while those on the right to q = q . The equation can be interpreted as follows. Set q = x + y, q = x − y and expand (1), (2) as a power series in y. Namely, rewrite the vertex operators: / 0 / 0 a α (λ) ⊗ −α (λ) = exp −1/2 yka exp 2(−1)k+1 f−1−k fka 1/2 ∂yka , a where the coefﬁcients fka (respectively, f−1−k ) are proportional to negative (respectively, positive) fractional powers of λ. The residue sum (which should be understood here as the coefﬁcient at λ0 ) can therefore be written as a power series ym Pm (∂y ) in y with coefﬁcients Pm which are differential polynomials. Also, ⊗ = (x + y)(x − y) can be expanded into the Taylor power series in y with coefﬁcients which are quadratic expressions in partial derivatives of (x). Finally, the operator in (2) assumes the form 2 a,k (ma / h + k)yka ∂yka . Equating coefﬁcients 1 As it follows, for instance, from [13], the integral on the right-hand side depends only on

the class [ψa ] ∈ H .

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in (1), (2) at the same monomials ym , we obtain a hierarchy of quadratic relations between partial derivatives of (x). In particular, the equation corresponding to y0 shows that

aα =

α∈A

N (h + 1) 12h

(3)

is a necessary condition for consistency of the hierarchy (i.e., for existence of a nonzero solution ). According to C. Hertling (see the last chapter in [15]), for any weightedhomogeneous singularity the expressions N (h+1)/(12h) and h−2 a ma (h−ma )/2 coincide. Therefore, the operator on the right-hand side of the Hirota equation is twice the Virasoro operator2

ma (h − ma ) + k yka ∂yka + . h 4h2 a

ma a,k

The coefﬁcients aα actually depend only on the orbit of the vanishing cycle α under the action of the classical monodromy operator deﬁned by transporting the cycles in f −1 (λ) around λ = 0 and acting as one of the Coxeter elements in the reﬂection group. In fact, the root system A consists of N such orbits with h elements each. Summing the vertex operators within the same orbit acts as taking the average over all h branches of the function λ1/ h . Thus the total sum does not contain fractional powers of λ when expanded near λ = ∞. The exact values of the coefﬁcients aα can be described as follows. To a vector β ∈ H2 (f −1 (1), C) ! CN , associate the meromorphic 1-form on CN Wβ := −

dγ , x 1 . β, γ 2 2 γ , x

(4)

γ ∈A

Let w be an element of the reﬂection group and α and β = wα be two roots. Then aβ /aα = exp

w−1 κ

κ

Wα =

1

κ, γ α,γ

2 /2−β,γ 2 /2

,

(5)

γ ∈A

where κ ∈ CN denotes an eigenvector of the classical monodromy operator M with the eigenvalue exp(2π i/ h). The right-hand side does not depend on the path connecting κ with w −1 κ since Wα is closed with logarithmic poles on some mirrors and with periods that are integer multiples of 2π i. It does not depend on the normalization of κ since Wα is homogeneous of degree 0. Also, the identity (see, e.g., [5, Section V.6.2]) γ , x2 = 2hx, x γ ∈A 2 In a sense it corresponds to the vector ﬁeld λ∂ in the Lie algebra of vector ﬁelds on the λ

line—see Section 7 for further information about this.

Simple singularities and integrable hierarchies

177

M −h κ M −1 κ implies that i xa ∂/∂xa Wα = −2h and shows that κ Wα = h−1 κ Wα = −4π i so that aMα = aα as expected. While the ratios of aα are determined by (5), the normalization of aα is found from (3) which says that the average value of aα is (h + 1)/12h2 . Later we give two other descriptions of the coefﬁcients aα —as certain limits and as explicit case-by-case values. Conjecture. The Hirota quadratic equation (1)–(5) coincides (up to certain rescaling of the variables qka ) with the corresponding ADE-hierarchy of Kac–Wakimoto [17]. In Section 8, we conﬁrm this conjecture in the cases AN , D4 , and E6 .3

2 The total descendent potential The second goal of this paper is to generalize to the ADE-singularities the result of [12] that the total descendent potential associated to the An−1 -singularity in the axiomatic theory of topological gravity is a tau-function of the nKdV-hierarchy (or Gelfand–Dickey-hierarchy). According to E. Witten’s conjecture [22] proved by M. Kontsevich [18], the following generating function for intersection indices on the Deligne–Mumford spaces satisﬁes the equation of the KdV-hierarchy:4 ' * m ∞ 1 g−1 k ψi + (6) DA1 = exp qk ψ i . m! Mg,m g,m i=1

k=0

In the axiomatic theory, the total descendent potential is, by deﬁnition, an asymptotical function of the form D = exp g−1 F (g) (q), g≥0

where F (g) are formal functions on H+ near the point q = −1z. (Here 1 is the unit element in the local algebra H .) This convention called the dilaton shift is already explicitly present in (6). The formal functions F (g) called the genus g descendent potentials are supposed to satisfy certain axioms dictated by Gromov–Witten theory. The axioms (while not entirely known) are to include the so-called string equation (SE), dilaton equation (DE), topological recursion relations (TRR or 3g − 2-jet property), and Virasoro constraints. According to [14], the genus-0 axioms SE + DE + TRR for F (0) are equivalent to the following geometrical property (∗) of the Lagrangian submanifold L ⊂ H = T ∗ H+ deﬁned as the graph of dF (0) : 3 There has been a new development in the subject which leads, in particular, to the proof of

the conjecture: motivated partly by this paper, E. Frenkel found a simple formula for the analogues in the Kac–Wakimoto theory of the coefﬁcients aα . 4 Here ψ is the ﬁrst Chern class of the line bundle over M g,m formed by the cotangent lines i to the curves at the ith marked points.

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(∗) L is a Lagrangian cone with the vertex at the origin and such that tangent spaces L to L are tangent to L exactly along zL. In other words, the cone L is swept by the family τ ∈ H → zLτ of isotropic subspaces which form a variation of semiinﬁnite Hodge structures in the sense of S. Barannikov [3]. According to his results, this deﬁnes a Frobenius structure on the space of parameters τ . In the case of ADE-singularities (and, more generally, ﬁnite reﬂection groups) the Frobenius structures have been constructed by K. Saito [20]. Consider the miniversal deformation fτ (x) = f (x) + τ 1 ψ1 (x) + · · · + τ N ψN (x), where {ψa } form a weighted-homogeneous basis in the local algebra H , and ψN = 1. The tangent spaces Tτ T to the parameter space T ! CN are canonically identiﬁed with the algebras of functions on the critical schemes crit (fτ ): ∂τ a → ∂fτ /∂τ a mod (∂fτ /∂x). The multiplication • on the tangent spaces is Frobenius with respect to the following residue metric: . . . 1 3 ψa (x)ψb (x)ω . (∂τa , ∂τb )τ := ∂fτ ∂fτ ∂fτ 2π i ∂x1 ∂x2 ∂x3

The residue metric is known to be ﬂat and together with the Frobenius multiplication, the unit vectors ∂τ N and the Euler vector ﬁeld E :=

N

(deg τ a )τ a ∂τ a ,

deg τ a = 1 − (ma − 1)/ h,

a=1

forms a conformal Frobenius structure on T (see [7]). On the other hand, the condition (∗) involves only the symplectic structure on H and the operator of multiplication by z and thus admits the following twisted loop group of symmetries: L(2) GL(H ) = {M ∈ End(H )((1/z)) | M(−z)∗ M(z) = 1}. According to a result from [14], when the Frobenius structure associated with the cone L is semisimple, one can identify L with the Cartesian product LA1 × · · · × LA1 of N = dim H copies of the cone LA1 deﬁned by the genus 0 descendent potential (0) FA1 = lim→0 ln DA1 . The identiﬁcation is provided by a certain transformation Mτ from (a completed version of) L(2) GL(H ) whose construction depends on the choice of a semisimple point τ . A number of results in Gromov–Witten theory suggests that the higher genus theory inherits the symmetry group L(2) GL(H ) (see [11, 14]). This motivates the following construction of the total descendent potential of a semisimple Frobenius manifold. Adopt the following rules of quantization ˆ of quadratic Hamiltonians. Let {. . . , pa , . . . , qb , . . . } be a Darboux coordinate system on the symplectic space (H, ) compatible with the polarization H = H+ ⊕ H− . Then

Simple singularities and integrable hierarchies

(qa qb )ˆ = qa qb /,

(qa pb )ˆ = qa ∂/∂qb ,

179

(pa pb )ˆ = ∂ 2 /∂qa ∂qb .

This gives a projective representation of the Lie algebra L(2) gl(H ) in the Fock space. The central extension is due to ˆ = {F, G}ˆ + C(F, G), [Fˆ , G] where C is the cocycle satisfying + C(pα pβ , qα qβ ) =

1 if α = β, 2 if α = β

and equal to 0 for any other pair of quadratic Darboux monomials. Introduce the total descendent potential as an asymptotical function: D := C(τ )Mˆ τ [DA1 ⊗ · · · ⊗ DA1 ], where Mˆ := exp(ln M)ˆ, and C(τ ) is a normalizing constant possibly needed to keep the right-hand side independent of the choice of a semisimple point τ . This deﬁnition has been tested in [11, 12] and is known to agree with the TRR, SE, DE, and Virasoro constraints. Here is a more explicit description of Mτ and C(τ ) in the form applicable to Frobenius manifolds of simple singularities. Consider the complex oscillating integral efτ (x)/z ω. JB (τ ) = (−2π z)−3/2 B

Here B is a noncompact cycle from the relative homology group lim Hm (Cm , {x : Re(fτ /z) ≤ −c}) ! ZN .

c→∞

We will use the notation ∂1 , . . . , ∂N for partial derivative with respect to a ﬂat (and weighted-homogeneous) coordinate system (t 1 , . . . , t N ) of the residue metric. We treat the derivatives z∂a JB as components of a covector ﬁeld z ∂a JB dt a ∈ T ∗ T which can be identiﬁed with a vector ﬁeld via the residue metric and—via its LeviCivita connection—with an H -valued function JB (z, τ ). According to K. Saito’s theory, these functions satisfy in ﬂat coordinates the differential equations z∂a J = (∂a •)J

(7)

together with the homogeneity condition (z∂z − µ + z−1 E•)J = 0,

(8)

where µ = −µ∗ is the diagonal operator with the eigenvalues 1/2 − ma / h. The latter equation yields an isomonodromic family of connection operators ∇τ = ∂z − µ/z + (E•)/z2 regular at z = ∞ and turning into ∂z − µ/z at τ = 0.

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According to [8], there exists a (unique in the ADE case) gauge transformation of the form Sτ (z) = 1 + S1 (τ )z−1 + S2 (τ )z−2 + · · · (i.e., near z = ∞) conjugating ∇τ to ∇0 and such that Sτ∗ (−z)Sτ (z) = 1. It satisﬁes the homogeneity condition (z∂z + LE )Sτ = [µ, Sτ ]. On the other hand, let τ be semisimple. Then the functions fτ have N nondegenerate critical points x (a) (τ ) with the critical values ua (τ ) and the Hessians a (τ ). The local coordinate system {ua } (called canonical) diagonalizes the product • and the residue metric ∂/∂ua • ∂/∂ub = δab ∂/∂ub ,

(∂/∂ua , ∂/∂ub )τ = δab

Deﬁne an orthonormal coordinate system #(τ ) : CN → Tτ T = H,

#(q 1 , . . . , q N ) =

qa

&

−1 b a ∂/∂u .

a ∂/∂u

a

,

and put Uτ = diag[u1 (τ ), . . . , uN (τ )]. Stationary phase asymptotics of the oscillating integrals JBa , a = 1, . . . , N, near the corresponding critical points x (a) yield a fundamental solutions to the system (7), (8) in the form #(τ )Rτ (z)eUτ /z ,

Rτ = 1 + R1 (τ )z + R2 (τ )z2 + · · · +,

Rτt (−z)Rτ (z) = 1.

The matrix series Rτ satisﬁes the homogeneity condition (z∂z + LE )Rτ = 0 and, according to [11], an asymptotical solution with this property is unique up to reordering or reversing the basis vectors in CN . Deﬁne 1 τ aa c(τ ) := R1 (τ )dua 2 a aa a as the local potential of the 1-form R1 du /2 (which is known to be closed [7]). In the above notations, the total descendent potential of the ADE-singularity assumes the form Uˆτ ⊗N D = ec(τ ) Sˆτ−1 #(τ )Rˆ τ e z DA . (9) 1 The right-hand side is known to be independent of τ (see [11]) and deﬁnes D (up to a constant factor) as an asymptotical function of q = q0 + q1 z + q2 z2 + · · · in the formal neighborhood of q = τ − z with semisimple τ . Our main result is the following theorem. Theorem 1. The total descendent potential (9) of a simple singularity satisﬁes the corresponding Hirota quadratic equation (1)–(5). In Section 4, we discuss Hirota quadratic equations of the KdV-hierarchy. The plan for the proof of Theorem 1 is to reduce the Hirota quadratic equations for D to those for DA1 by conjugating the vertex operators in (1), (2) past the quantized symplectic transformations from (9). In Section 5, we describe the results of such conjugations by quoting corresponding theorems from [12]. The residue in (1) is computed in Section 6 and is compared with (2) in Section 7. The case-by-case tables for the coefﬁcients aα are presented in Section 8. A key to all our computations is the phase form and its properties discussed in next section.

Simple singularities and integrable hierarchies

181

3 Phase forms and root systems Consider a ﬂat family of cycles φ ∈ H2 (fτ−1 (λ)) in the nonsingular Milnor ﬁbers (0) and deﬁne the period vector Iφ (λ, τ ) ∈ H by (0)

(Iφ (λ, τ ), ∂a ) := ∂a

(−1) 2π

φ⊂fτ−1 (λ)

ω . dfτ

(10)

It is a multiple-valued vector function on the complement to the discriminant which (0) turns into Iφ (λ) from Section 1 at τ = 0. ˜ α,β (deﬁned in [12, Section 7]) is given by the formula The phase form W ˜ α,β (λ, τ ) := W

N

(0)

(Iα(0) (λ, τ ) • Iβ (λ, τ ), ∂τ a )dτ a .

i=1

This is a multiple-valued 1-form on the complement to the discriminant and depends bilinearly on the cycles α, β (to be chosen in (f0−1 (1) and transported to fτ−1 (λ)). According to [12], the phase forms have the following properties: ˜ α,β = 0. 1. d W ˜ α,β = 0, i.e., W ˜ is determined by the restriction 2. L∂λ +∂N W ˜ α,β (0, τ ), Wα,β (τ ) := W

˜ α,β (λ, τ ) = Wα,β (τ − λ1). W

3. LE Wα,β = 0. 4. Near a generic point of the discriminant ⊂ T the form Wα,β becomes single(0) valued on the double cover and has a pole of order ≤ 1 on D (since Iα have a pole of order ≤ 1/2). 2 5. δγ Wα,β = −2π iα, γ β, γ , where γ is the cycle vanishing over a generic point of the discriminant, and δγ is a small loop going twice (in the positive direction deﬁned by complex orientations) around the discriminant near this point. Proposition 1. Wα,β = −

1 dγ , x . α, γ β, γ 2 γ , x γ ∈A

Proof. The phase form Wα,β becomes single-valued on the Chevalley cover representing T as the quotient of CN = H 2 (f0−1 (1), C) by the monodromy group. The properties (1) and (4) show that it has at most a logarithmic pole on the mirrors γ , x = 0. Property 5 controls the residues on the mirrors. The difference of the leftand right-hand sides has to be a holomorphic 1-form, homogeneous of degree 0 by property 3 and therefore vanishes identically.

Corollary 1. iE Wα,β = −α, β.

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A. B. Givental and T. E. Milanov

Indeed, the Euler vector ﬁeldbecomes h−1 xa ∂xa on the Chevalley cover, so that the equality follows from γ ∈A α, γ β, γ = 2hα, β. This is one more general property of phase forms established in [12]. Corollary 2. The phase form Wβ of Section 1 coincides with Wβ,β . Remark. The inverse to the Chevalley quotient map is given by the period map τ → [ω/dfτ ] ∈ H 2 (fτ−1 (0), C) ; H 2 (f0−1 (1), C) ! CN . (0)

The periods Iα are deﬁned via the differential of the inverse Chevalley map and therefore represent parallel translations of the cycles α considered as covectors in CN . The value of phase form Wα,β , which is also a covector, is constructed as the Frobenius product α • β of covectors (deﬁned by the isomorphisms Tτ T ! Tτ∗ T based on the residue metric). Thus the formula 1 α, γ β, γ γ α • β := 2 γ , x γ ∈A

deﬁnes on (CN )∗ a family of commutative associative multiplications depending on the parameter x.5 It would be interesting to ﬁnd a representation-theoretic interpretation of this structure deﬁned entirely in terms of the root system A. We prove several further properties of phase forms needed in our computations. Proposition 2. In the case of ADE-singularities, suppose that β has integer intersection indices with all α ∈ A and2 is invariant under the monodromy around a discriminant-avoiding loop γ . Then γ Wβ,β ∈ 2π iZ. Proof. This is [12, Section 7, Proposition 1].

It would be interesting to ﬁnd out if the property remains valid for nonsimple singularities. We will see in Section 7 that the coefﬁcients aα introduced in Section 1 can be equivalently deﬁned via the following limits bα . Start by choosing (τ1 , . . . , τN ) = −1 = (0, . . . , 0, −1) in the role of the base point in T and identify A with the set of −1 vanishing cycles in H2 (f−1 (0)) = H2 (f0−1 (1)). Let us also ﬁx τ ∈ T such that fτ is a Morse function, and let u be one of the critical values of fτ so that τ − u1 ∈ . We may assume that τ − (u + 1)1 ∈ / and that the straight segment connecting τ − (u + 1)1 with τ − u1 does not intersect . For each α ∈ A, pick a discriminantavoiding path γα connecting −1 with τ − (u + 1)1 and further with τ − u1 along the straight segment and such that α becomes the vanishing cycle when transported along γα from 1 to τ − u1. Assuming that integration of the phase form is performed along this path, we put 3 + −ε τ −(u+ε)1 2dt . (11) Wα,α − bα := lim exp − ε→0 t −1 −1 5 We thank V. A. Ginzburg, who explained to us that this is a special case of a family of

Frobenius structures constructed by A. P. Veselov [21].

Simple singularities and integrable hierarchies

183

Proposition 3. The limit exists and does not depend on the choice of the path of integration provided that the path terminates at a generic point of the discriminant and that the cycle α transported along the path vanishes over this point. Proof. We may assume that u = u1 is the ﬁrst of the canonical coordinates U = (u1 , . . . , uN ), and therefore u1 = 0 is the local equation of the discriminant branch. (0) Since α is vanishing at the end of the path, the period vector Iα has the following N expansion (here 1i stand for the standard basis vectors in C ): #τ−1 Iα(0) (λ, τ ) = & Since #(1i ) =

√

±2 2(λ − u1 )

i ∂/∂u

i,

11 + (λ − u1 )

a i (U )1i + o(λ − u1 ) .

we have (1i • 1j , ∂/∂uk ) = δij δik , and therefore

2du1 (Iα(0) • Iα(0) , ∂/∂uk )duk |λ=0 = + 4a 1 (U )du1 + O(−u1 ). −u1 0 u =0 We see that the integral u11=−1 Wα,α diverges the same way as − −1 2dt/t so that the difference converges. This proves the existence of the limit. Removing this singular term we ﬁnd that the integral [4a1 (U )du1 +O(−u1 )] vanishes along any path inside the discriminant branch u1 = 0. This shows that the limit bα is locally constant as a function of the path’s endpoint on the discriminant and therefore is globally constant due to the irreducibility of the discriminant. Finally, precomposing a path with a discriminant-avoiding loop γ with trivial monodromy of the cycle α does not change bα thanks to Proposition 2.

Wα,α =

Corollary. aα /aβ = bα /bβ for all α, β ∈ A. Proposition 4. Let δε be a small loop of radius ε around the discriminant near a generic point τ − u1, 2 and let α, β = ±1, where β is the cycle vanishing at this point. Then limε→0 δε Wα,α = −π i. Proof. We have α = ±β/2 + α , where α is invariant under the monodromy around (0) (0) (0) δε . Expanding Iα = Iα ± Iβ/2 near λ = u as in the proof of Proposition 3, we ﬁnd .

. Wα,α = δε

δε

du + −2u

.

√ √ O( u)du = −π i + O( ε) → −π i.

δε

In fact, this property has been already used in [12].

4 Two forms of the KdV-hierarchy Consider the miniversal deformation of the A1 -singularity in the form fu (x) := (x12 + x22 + x32 )/2 + u. The vanishing cycle α can be identiﬁed with the real sphere (x12 + x22 + x32 ) = 2(λ − u). The period is

184

A. B. Givental and T. E. Milanov

& dx1 ∧ dx2 ∧ dx3 d 4 π(2(λ − u))3/2 = 4π 2(λ − u). = dfu dλ 3 α √ (−1) Since (1, 1) = Res dx1 ∧dx2 ∧dx3 /x1 x2 x3 = 1, we have Iα (λ, u) = 2 2(λ − u), (k) and more generally, I±α (λ, u) = ±2(d/dλ)k (2(λ − u))−1/2 , k ∈ Z. The Coxeter transformation swaps α and −α and so aα = a−α = (h + 1)/12h2 = 1/16. The equation (1), (2) in this example assumes the form ( ) dλ A1 ±α 1 A1 ∓α (λ) ⊗ (λ) ( ⊗ ) = 16 l + ( ⊗ ), (12) Resλ=∞ λ 8 ± where l=

2k + 1 2

k≥0

(qk ⊗ 1 − 1 ⊗ qk )(∂qk ⊗ 1 − 1 ⊗ ∂qk ).

(13)

Here we use the notation A1 φ (λ) to single out the vertex operators φ (λ) of the A1 -singularity. In order to identify the condition (12) for with √ the KdV-hierarchy in [16, 17] corresponding to the root system A1 , we denote 2λ by ζ , rescale the + t )/2, y = (t − t )/2, variables by qk = (2k + 1)!! t2k+1 , and put xm = (tm m m m m where m = 1, 3, 5, . . . . In this notation l = mym ∂ym , and (12), (13) becomes 5 4 dζ 4 ζ m √ym −2 ζ −m √∂ym m e e −1−8 mym ∂ym (x + y)(x − y) = 0. Res ζ This coincides with [16, equation (14.13.1)] characterizing tau-functions of the KdV-hierarchy. Another form of the Hirota quadratic equation for is based on the representation of the KdV-hierarchy as the mod 2 reduction of the KP-hierarchy (see [16, Section 14.11]). It can be rephrased (see [12]) as the condition ( ) dλ A1 ∓α/2 A1 ±α/2 (λ) ⊗ (λ) √ ( ⊗ ) has no pole in λ. (14) ± λ ± Indeed, in the previous notations this can be rewritten as the property e

2

ym ζm √ −

e

ζ −m m

√

∂ym

(x + y)(x − y) contains no ζ −m for odd m > 0.

This coincides with the mod 2 reduction of the KP-hierarchy of the Hirota equation [16, equation (14.11.5)]. According to a result from [16, Section 14.13], this condition is actually equivalent to (12). In Section 6, we will use the fact that (according to Kontsevich’s theorem) the function = DA1 satisﬁes both forms (12) and (14) of the KdV-hierarchy.

5 Symplectic transformations of vertex operators φ

Generalizing the construction of Section 1, introduce the vertex operator τ (λ) corresponding to the vector f ∈ H [[z, z−1 ]] of the form

Simple singularities and integrable hierarchies

fτφ (λ) :=

185

(k)

Iφ (λ, τ )(−z)k .

k∈Z (0)

Here Iφ

(k)

is the period vector introduced in Section 3, and Iφ

(0)

:= d k Iφ /dλk as (k)

before. For k < 0, the integration constants are taken “equal to 0’’ so that Iφ satisfy the homogeneity conditions 1 (k) (k) (λ∂λ + LE )Iφ (λ, τ ) = µ − − k Iφ (λ, τ ). 2 φ

In particular, 0 coincides with the vertex operator φ from Section 1. We state below several results about behavior of the vertex operators under conjugation by some symplectic transformations and refer to [12, Sections 5, 6, and 7] for the proofs. Theorem A (see [12, Proposition 2]). φ Sˆτ 0 (λ)Sˆτ−1

+ 3 1 τ −λ1 = exp Wφ,φ τφ (λ). 2 −λ1 φ

We have to stress here that in order to compare the vertex operators 0 (λ) and φ τ (λ), one needs to transport the cycle φ from f0−1 (λ) to fτ−1 (λ) along a path in T connecting −λ1 = (0, . . . , −λ) with τ − λ1 = (τ1 , . . . , τN − λ) and avoiding the discriminant corresponding to singular levels fτ−1 (0). It is assumed in the formulation of the theorem that the integral of the phase form is taken along this very path. Similar conventions apply to other formulas of this and following sections involving integration of phase forms. Now let the cycle φ ∈ H2 (fτ−1 (λ) be written as the sum φ = φ, ββ/2 + φ , where φ , β = 0. Here β is the cycle vanishing at a nondegenerate critical point of the function fτ with the critical value u and transported to fτ−1 (λ) along a discriminantavoiding path connecting τ − λ1 and τ − u1. Theorem B (see [12, Proposition 4]). 3 + τ −u1 φ,β φ, β β Wβ,φ τφ (λ)τ 2 (λ). τφ (λ) = exp 2 τ −λ1 The integral here is taken along the path terminating on the discriminant where the phase form is singular. However the singularity is proportional to (λ − u)−1/2 and is therefore integrable. Let us recall that the columns of the matrix Rτ in the asymptotical expansion #(τ )Rτ (z) exp(U/z) correspond to nondegenerate critical points of the Morse function fτ with the critical values ui (τ ). Let βi be the cycle vanishing over ui . Theorem C (see [12, Proposition 3]). 2 cβ (#(τ )Rˆ τ )−1 τcβi (λ)(#(τ )Rˆ τ ) = ec Wi /2 [· · · 1 ⊗ (A1 ui (λ))(i) ⊗ 1 · · · ],

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where

Wi :=

τ −ui 1 τ −λ1

Wβi ,βi −

τN

2dt N − ui − t N

,

cβ

and A1 u (λ) is the vertex operator of the A1 -singularity with the miniversal deformation

x12 2

+

x22 2

+

x32 2

+ u corresponding to the c-multiple of the vanishing cycle. (0)

The behavior of Iβi near λ = ui is described by the asymptotics 2 (0) # −1 (τ )Iβi (λ, τ ) = & (1i + · · · ), 2(λ − ui ) where 1i is the ith basis vector in CN , and the dots mean higher order powers of λ−ui . Respectively, the vertex operator of the A1 -singularity is more explicitly deﬁned by the series f ∈ C[[z, z−1 ]] of the form f=

dk 2c (−z)k , √ k dλ 2(λ − u) k∈Z

where the branch of the square root should be the same as in the above asymptotics. The subscript (i) indicates the position of the vertex operator in the tensor product operator acting on the Fock space of functions of (q(1) , . . . , q(N ) ) = # −1 (τ )q. The integrand in the formula for Wi considered as a 1-form in the space with coordinates (t 1 , . . . , t N ) identical to parameters of the miniversal deformation, while the notation τ = (τ 1 , . . . , τ N ) is reserved for expressing the limits of integration. The phase form W has a nonintegrable singularity at t = τ − ui 1 which happens to cancel out with that of the subtracted term so that the difference is integrable. Finally, the following result is the special case of Theorem A corresponding to the A1 -singularity. Theorem D (see [12, Proposition 3]). 2 λ 2dt A1 cβ c 0 (λ). e−(u/z)ˆA1 ucβ (λ) e(u/z)ˆ = exp − 2 λ−ui t In fact, this result can be obtained more directly using Taylor’s formula. Indeed, for any analytic function I (0) , we have ( ) e−u/z I (k) (λ)(−z)k eu/z = I (k) (λ + u)(−z)k , k∈Z

k∈Z

provided that |u| does not exceed the convergence radius of I (0) √ at λ. Thus √ the transformation in the theorem effectively consists in the translation λ − u ; λ along an origin-avoiding path. The integral in the exponent should be taken along this path.

Simple singularities and integrable hierarchies

187

6 Residue sum In this section, we compute the residue sum ( ) dλ −α α Resλ=∞ bα 0 (λ) ⊗ 0 (λ) D⊗2 λ

(15)

α∈A

assuming that the coefﬁcients bα are deﬁned as in Proposition 3. Introduce the total ancestor potential ⊗N Aτ := Sˆτ D = ec(τ ) #(τ )Rˆ τ e(Uτ /z)ˆ DA . 1

Applying Theorem A of the previous section, we ﬁnd that (15) can be rewritten as ( ) Resλ=∞ λdλ cα τα (λ) ⊗ τ−α (λ) A⊗2 (16) τ , α∈A

+ cα = lim exp −

where

ε→0

τ −(u+ε)1 τ −λ1

Wα,α −

−ε −1

2dt t

3 ,

assuming that α ∈ H2 (fτ−1 (λ)) vanishes at λ = u when transported along the path of integration of the phase form. Note that the factor λ−1 dλ in (15) is replaced by λdλ in (16) due to Corollary 1 from Section 3, which shows that + 3 −λ −λ1 dt exp − Wα,α = exp α, α = λ2 . −1 −1 t (g) The ancestor potential Aτ = exp (g−1) Fτ is a tame asymptotical function (g) in the following sense: Fτ considered as a formal function of tka = qka + δk1 δaN satisﬁes (g)

∂ r Fτ |t=0 = 0 whenever k1 + · · · + kr > 3g − 3 + r. · · · ∂tkarr

∂tka11

⊗N This follows from the analogous property of DA , from the invariance of DA1 under 1 the string ﬂow exp(u/z)ˆ and from the “upper-triangular’’ property of Rˆ τ . We refer to Proposition 5 in [12] for the proof. It is also shown in [12, Section 8] that for tame φ −φ asymptotical functions the vertex operator expressions τ (λ) ⊗ τ (λ)⊗2 can be considered not only as series expansions in fractional powers of λ near λ = ∞, but also as multiple-valued analytical functions deﬁned over the entire range of λ and ramiﬁed only on the discriminant. Moreover, the sum in (16) is manifestly invariant under the entire monodromy group (= the ADE-reﬂection group). Therefore the sum is actually a single-valued differential 1-form on the complement to D. Thus the

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residue (16) at λ = ∞ coincides with the sum of residues at the critical values λ = ui of the function fτ . Our next goal is to take u = ui and compute the residue. In a neighborhood of λ = u, the monodromy group reduces to Z2 generated by the reﬂection σ in the hyperplane orthogonal to two vanishing cycles which we denote ±β. First, consider the summand in (16) corresponding to a σ -invariant cycle α ∈ A. (k) The period vectors Iα (λ, τ ) are therefore single-valued analytic functions near λ = τ −λ1 u. In particular, ln cα , which differs from a constant by τ −(u+1)1 Wα,α , is analytic, too. We conclude that λcα τα (λ) ⊗ τ−α (λ)A⊗2 has no pole at λ = u. Next, consider a pair of cycles α± ∈ A transposed by σ and having intersection indices ±1 with β. We have α± = α ± β/2 where σ α = α . We use Theorem B ±β/2 α ±β/2 and then commute τ across # Rˆ exp(U/z)ˆ using to replace τ ± with τα τ Theorems C and D. The terms from (16) corresponding to α = α± turn into

7 6 λdλ τα (λ) ⊗ τ−α (λ) #(τ )Rˆ τ e(Uτ /z)ˆ ⊗ #(τ )Rˆ τ e(Uτ /z)ˆ ⎡ ⎤ ' *(i)

±β/2 ∓β/2 × ⎣· · · 1 ⊗ d± A1 (λ) ⊗ A1 (λ) ⊗ 1 · · · ⎦ D⊗N ⊗ D⊗N . 0

±

A1

0

A1

(17) The coefﬁcients d± here are + . d± = lim exp − ε→0

τ −(u+ε)1 τ −λ1

+

τ −(u+ε)1 τ −λ1

Wα± ,α± −

−ε −1

−ε

Wβ/2,β/2 +

2dt ± t

u−λ

τ −u1 τ −λ1

dt − 2t

−λ u−λ

Wα ,β

dt 2t

3 .

(18)

We have to emphasize that all integrals here except the ﬁrst one are taken along a short path near λ = u making β vanish, while in the ﬁrst integral this path is precomposed with a loop transforming α± to β. Let us take λ = u + 1 for the base point for such a loop γ± and rearrange the ﬁrst integral as − γ±

Wα± ,α± −

τ −(u+ε)1 τ −(u+1)1

Wβ,β +

τ −λ1 τ −(u+1)1

Wα± ,α± .

Combining this with Wα± ,α± = Wα ,α ± Wα ,β + Wβ/2,β/2 , we can rewrite the exponent in (18) as − γ±

Wα± ,α± −

τ −(u+ε)1

τ −(u+1)1 τ −λ1 τ −(u+1)1

Wβ,β −

Wα ,α +

−ε

2dt ± t

−1 τ −(u+ε)1 τ −(u+1)1

τ −(u+ε)1 τ −(u+1)1

Wβ/2,β/2 +

Wα ,β

−ε −1

dt 2t

(19) (20)

Simple singularities and integrable hierarchies

−

−ε −1

dt − 2t

u−λ −ε

dt − 2t

−λ u−λ

dt . 2t

189

(21)

−λ The integrals in (21) add up to − −1 dt/2t and contribute λ−1/2 to the coefﬁcients d± . The sum in (20) is a function of λ analytic near λ = u (since α is σ -invariant) and is the same for both cycles α± . The values of (19) may depend on the cycle α± but are independent of λ. We claim that in the limit ε → 0 the difference is an odd multiple of π i. Indeed, transporting α− along the composition γ− γ+−1 yields α+ . On the other hand, 2Wα ,β = Wα2+ ,α+ − Wα− ,α− . Thus the difference of the two values of (19) can be interpreted as Wα− ,α− along a loop γε starting and terminating at τ − (u + ε)1 and transporting α− to α+ . Let us compose it with a small loop δε of radius ε around λ = u. Since α+ transports along this loop back to α− , the composite integral γε δε Wα− ,α− ∈ 2π iZ due to Proposition 2 and does not depend on ε. Our claim therefore follows from Proposition 4. We conclude that d± = ±d0 (λ)λ−1/2 where d0 is a nonvanishing analytic function near λ = u. Now we use the fact that DA1 is a tau-function of the KdV-hierarchy (14) to conclude that the factor in (17) of the form ±

7 dλ 6 ±β/2 ∓β/2 ± √ A1 0 (λ) ⊗ A1 0 (λ) (DA1 ⊗ DA1 ) λ

is everywhere analytic in λ. The same remains true after application of the operator ˆ (U/z)ˆ )⊗2 . The vertex operator τα ⊗ τ−α is analytic near λ = u since α is (# Re σ -invariant. Thus (17) has no pole at λ = u and contributes 0 to the residue sum. Finally, consider the summands in (15) with α = ±β. Applying Theorems C and D, we transform the corresponding summands from (16) to the form ⎡ ⎤ *(i) ' ±β ∓β ⊗2 ⊗2 ⊗2 (#(τ )Rˆ τ e(U/z)ˆ )⊗2 ⎣· · · DA ⊗ eλdλ(A1 0 ⊗ A1 0 )DA ⊗ DA · · ·⎦, 1 1 1 ±

where e = exp −

−ε

−1

2dt − t

u−λ −ε

2dt − t

−λ u−λ

2dt t

= exp −

−λ

−1

2dt t

= λ−2 .

The contribution of these terms to the residue sum (16) at λ = ui can be calculated using the form (12) of the KdV-hierarchy for DA1 and is equal to ˆ (U/z)ˆ )⊗2 l (i) (D⊗N )⊗2 , 16(# Re A1

where l (i) = (· · · 1 ⊗ l ⊗ 1 · · · ).

In order to justify this conclusion, recall from the end of Section 5 that conjugation by exp(ui /z) act as translation λ − ui → λ. Also, since DA1 is tame, the vertex operator expression in (12) yields a meromorphic 1-form in λ with a singularity only at λ = 0. Thus the residue in (12) at λ = ∞ is the same as at λ = 0. Let us summarize our computation.

190

A. B. Givental and T. E. Milanov

Proposition 5. The residue sum (15) is equal to ' * N N ⊗N ⊗2 c ˆ −1 (U/z)ˆ ⊗2 (i) ˆ (DA 16(e S # Re + ) l ) . 1 8

(22)

i=1

7 The Virasoro operator Functions of the form ⊗ belong to a Fock space which is the quantization of the symplectic space H ⊕ H, the direct sum of two copies of (H, ). Respectively, the operator

ma ∂ ∂ a a + k (qk ⊗ 1 − 1 ⊗ qk ) (23) ⊗1−1⊗ a h ∂qka ∂qk a k≥0

in (2) is the quantization of a certain quadratic Hamiltonian (Df , f )/2 on H ⊕ H. Let us describe the inﬁnitesimal symplectic transformation D explicitly. Introduce the Virasoro operator l0 := z∂z + 1/2 − µ.6 Since µ∗ = −µ, the operator l0 : H → H is antisymmetric with respect to , and the corresponding quadratic Hamiltonian reads as . 1 1 k + − µ fk , (−1)k f−1−k (l0 f (−z), f (z))dz = 4π i 2 k≥0

=−

N ma k≥0 a=1

h

+ k qka pka .

Comparing this with (23), we conclude that 5 4 −l0 l0 ∈ End(H ⊕ H). D= l0 −l0

(24)

The expression (2) on the right-hand side of the Hirota equation is proportional to ˆ Mˆ ⊗2 )(D×N )⊗2 , ˆ ⊗2 = Mˆ ⊗2 (Mˆ ⊗2 )−1 D( DD A1 where Mˆ = ec(τ ) Sˆτ−1 #(τ )Rˆ τ e(Uτ /z)ˆ . Note that Mˆ ⊗2 is the quantization of a blockdiagonal operator 5 4 4 5 −M −1 l0 M M −1 l0 M M 0 −1 . B := and B DB = 0 M M −1 l0 M −M −1 l0 M 6 The name comes from the property of the operators l := l zl z · · · zl (z repeated m times, m 0 0 0

m = −1, 0, 1, 2, . . . ) to form a Lie algebra isomorphic to the algebra of formal vector ﬁelds x m+1 ∂/∂x on the line and participating in the formulation of the Virasoro constraints (see [11, 14]).

Simple singularities and integrable hierarchies

Proposition 6. M −1 l0 M =

N

i=1 (

A1 l

0)

191

(i) .

Proof. We have E• 1 1 S z∂z + − µ S −1 = z∂z + − µ + , 2 2 z since (z∂z +LE )S = µS−Sµ and ∂a S = z−1 ∂a •S. Next, in the canonical coordinates, E = ui ∂/∂ui , and therefore E• U 1 1 −1 # z∂z + − µ + # = z∂z + − V + , 2 z 2 z where V := # −1 µ# = # −1 LE #. Furthermore, the differential equations ∂a (#ReU/z ) = z−1 (∂a •)(#ReU/z ) translate into (d + # −1 d#)R = z−1 (dU R − RdU ). This implies (LE + V )R = z−1 (U R − RU ), which together with the homogeneity condition (z∂z + LE )R = 0 shows that U 1 1 U R −1 z∂z + − V + R = z∂z + + . 2 z 2 z Finally, e

−U/z

1 U z∂z + + 2 z

1 eU/z = z∂z + . 2

Proposition 7. Mˆ −1 lˆ0 Mˆ = (M −1 l0 M)ˆ+ tr µµ∗ /4. Proof. The quadratic Hamiltonians for z∂z + 1/2, µ, V , ln S, (E•)/z, U/z contain no p 2 -terms, and the quadratic Hamiltonians for z∂z + 1/2, µ, V , ln R contain no q 2 -terms. Therefore, in the quantized version of the previous computation, the only point where the cocycle C makes a nontrivial contribution is U U Rˆ −1 ˆRˆ = R −1 R ˆ + C. z z Let A = ln R, B = U/z. Then the quadratic Hamiltonian of BA − AB contains no q 2 -terms (since R|z=0 = 1). We therefore have d −t Aˆ ˆ t Aˆ ˆ ˆ t Aˆ = e−t Aˆ (BA − AB)ˆet Aˆ + C(B, A) e Be = e−t A (Bˆ Aˆ − Aˆ B)e dt 6 7 d 6 −tA tA 7 = e−tA (BA − AB)etA ˆ + C(B, A) = e Be ˆ + C(B, A). dt Integrating in t from 0 to 1 weﬁnd C = C(B, A). Since A = R1 z + o(z), we compute explicitly C = tr(BA)/2 = i R1ii ui /2.

192

A. B. Givental and T. E. Milanov

This expression, which seems to be a function of τ , has to be a constant, and the value of this constant is well known to be tr µµ∗ /4 (see, for instance, the last chapter in [15]). For the sake of completeness, we include the computation. Namely, comparing the z0 - and z1 -terms in the equation (LE + V )R = z−1 (U R − RU ) we ij ﬁnd V ij = (ui − uj )R1 and, respectively, R1ii = −LE R1ii =

V ij V j i ij j i (ui − uj )R1 R1 = . uj − u i j

j

Thus we have uj V ij V j i 1 i ii ui V ij V j i 1 ij j i 1 = =− u R1 = V V = tr µµ∗ , j i i j 2 2(u − u ) 2(u − u ) 4 4 i

ij

ij

ij

since V = # −1 µ# and V t = # −1 µ∗ #.

Remark. Slightly generalizing Propositions 6 and 7, one obtains the following trans (i) formation formula (see Theorem 8.1 in [11]) Mˆ −1 lˆm Mˆ = i A1 lˆm for the Virasoro ˆ ⊗N operators with m = 0. Since (lˆm − δm,0 /16)DA1 = 0, this implies that D = MD A1

satisﬁes the Virasoro constraints [lˆm − δm,0 tr(µµ∗ /4 + 1/16)]D = 0. In fact, this is [11, Corollary 8.2] specialized to Frobenius structures of weighted-homogeneous singularities. Note that the conjugation l0 → M −1 l0 M of the off -diagonal blocks in the matrix D yields after quantization Mˆ −1 lˆ0 Mˆ = (M −1 l0 M)ˆ (since the cocycle C vanishes on pairs of quadratic Hamiltonians corresponding to block-diagonal and block-off diagonal operators.) Thus Bˆ −1 Dˆ Bˆ = i l (i) − tr µµ∗ /2. Taking into account that 1 1 1 N (h + 1) ma (h − ma ) = + µ − µ = a a 12h 2 a 2 2 2h2 a 1 1 = tr + µµ∗ , 2 4 we conclude that the right-hand side of the Hirota equation (1), (2) can be written as ' * N ⊗N ⊗2 ⊗2 (i) (DA + l ) . Mˆ 1 8 i

Comparing this with Proposition 5, we arrive at the following result. Proposition 8. The function D satisﬁes the Hirota quadratic equation (1), (2) with aα = bα /16. Since D = 0, the Hirota equation is thus rendered consistent, and the following corollary completes the proof of Theorem 1.

Simple singularities and integrable hierarchies

193

Corollary. The average value 4(h + 1) 1 bα = . Nh 3h2 α∈A

Note that in the proof of Theorem 1, we use neither the Virasoro constraints for ⊗N DA1 nor the fact that the N factors in DA are the same. The only relevant conditions 1 for DA1 were both forms of the KdV-hierarchy and the tame property of exp(u/z)ˆDA1 . Thus we have actually proved the following generalization of Theorem 1. Theorem 2. Suppose that tame asymptotical functions 1 , . . . , N are tau-functions of the KdV-hierarchy and remain tame under the string ﬂow i → exp(u/z)ˆi for all u. Then := ec(τ ) Sˆτ−1 #(τ )Rˆ τ e(Uτ /z)ˆ (1 ⊗ · · · ⊗ N ) satisﬁes the corresponding Hirota quadratic equation (1)–(5). Remark. Although the condition for i to remain tame under the string ﬂow is quite restrictive, DA1 is not the only tau-function satisfying it. A large class of examples consists of the shifts DA1 (q + a) where a(z) = a0 + a1 z + a2 z2 + · · · is a series with coefﬁcients ak which are arbitrary series in such that a0 and a1 are smaller than 1 in the -adic norm and ak → 0 in this norm as k → ∞.

8 Kac–Wakimoto hierarchies Let us compare theADE-hierarchies (1)–(5) with the principal hierarchies of the types (1) (1) (1) AN , DN , EN described in [17, Theorem 1.1]. The corresponding Hirota equation [17, equation (1.14)] has the form dζ m∈E 2βi,m −1/2 ym ζ m −m∈E βi,−m 1/2 ∂ym ζ −m /m + + gi e e (x + y)(x − y) ζ i=1 ⎞ ⎛ (25) mym + ρ, ρ⎠ (x + y)(x − y). = ⎝2h N

Res

m∈E+

Here ρ is the sum of the fundamental weights of the root system A, and the value ρ, ρ = Nh(h+1)/12 can be found, for instance, from the tables in [5]. The index set E+ = {ma +kh|a = 1, . . . , N, k = 0, 1, 2, . . . }, and m denote the remainder modulo h. The vertex operators in the sum correspond to a set of roots αi , i = 1, . . . , N, chosen one from each orbit of some Coxeter element M on the root system A. The coefﬁcients βi,m are coordinates of αi with respect to a basis√of eigenvectors Hm of the Coxeter transformation M with the eigenvalues exp(2π −1m/ h) satisfying

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A. B. Givental and T. E. Milanov

the additional normalization condition Hm , H−m = 1.7 The coefﬁcients gi are deﬁned via representation theory of afﬁne Lie algebras. The numerical values of gi are computed in [17] in the cases AN , D4 , and E6 . In order to identify the vertex operators in (25) with those in (1), (2) let us start (−1) by taking ζ = (hλ)1/ h . Then the components of the period vector Iαi with respect to a suitable basis [ψa ] ∈ H will have the form ma / h (λ), [ψa ]) = βi,ma m−1 . (Iα(−1) a (hλ) i

(26)

Indeed, the weighted-homogeneous forms ψa ω/df represent a basis of eigenvectors for the classical monodromy operator in H 2 (f −1 (1), C). Then it is straightforward to check that the relation qka =

k 1

(ma + rh)tma +kh

(27)

r=0

(together with the standard change x + y = t , x − y = t as in Section 4) identiﬁes the vertex operators in (25) with αi ⊗ −αi . Note that replacing αi with any of the h roots from the same M-orbit does not change the corresponding residue in (25) since the new vertex operator would differ from the old one only by the choice of the branch of ζ = (hλ)1/ h . Thus we arrive at the following conclusion. Proposition 9. The choice of the basis {[ψa ] ∈ H } such that (26) holds true and the change of variables (27) identify the Hirota equation (1)–(5) with the corresponding hierarchy of the form (25) provided that gi = h3 aαi = h3 bαi /16. Let us now compute the coefﬁcients bα . First, rewrite the deﬁnition (11) as bα = lim e

−

τ −(u+ε)1 −ε1

ε→0

Wα,α

= lim

ε→0

1 y(ε), γ γ ∈A

α,γ 2 2

ε 1/ h κ, γ

α,γ 2 2

=

v, α4 κ, α4

1 γ ,α=1

x, γ , κ, γ

where ε1/ h κ, y(ε) and x are inverse images under the Chevalley map of −ε1, τ − (u + ε)1 and τ − u1, respectively, x is a generic point on the mirror α, x = 0, and v is determined from the expansion y(ε) = x + ε1/2 v + o(ε1/2 ). We will use this formula in the case of A and D series. Case AN . The root system consists of the vectors γij := ei − ej in the space CN +1 with the standard orthonormal basis e0 , . . . , eN and coordinates z0 , . . . , zN . Take zN +1 1 1 + t1 zN −1 + · · · + tN = (z − zi ). N +1 N +1 N

F (z, τ ) =

i=0 7 As usual, there is a caveat in the case D with odd l when the eigenvalue −1 of the monl

odromy operator has multiplicity 2. The involution of the Dynkin diagram induces an automorphism of the root system which allows one in this case to single out one invariant and (i) (a) one antiinvariant eigenvector, Hl−1 and Hl−1 , orthogonal to each other and normalized by Hl−1 , Hl−1 = 1 each.

Simple singularities and integrable hierarchies

195

Let α = ea − eb and let t = τ − u1 be a generic point on the discriminant. Then the components yi (ε) = xi +ε 1/2 vi +εwi +o(ε) (where xa = xb ) satisfy F (yi , t −ε1) = 0, and therefore ε = F (xi , t) + F (xi , t)ε 1/2 yi + F (xi , t)εwi + F (xi , t)ε

vi2 + o(ε). 2

We have&F (xi , t) = 0 for all i and F (xi , t) = 0 for i = a,& b. This implies that vi = ± 2/F (xa , t) for i = a, b and hence α, v = ±2 2/F (xa , t). Thus α, v4 = 64/F (xa , t)2 . On the other hand, 2 1 1 N −1 2 N −1 (N + 1)F (xa , t) x, γ = (−1) (xi − xa ) = (−1) . 2 i=a,b

γ ,α=1

The eigenvector κ = (N + 1)1/(N +1) (1, η, η2 , . . . , ηN −1 ) of the Coxeter transformation (z0 , . . . , zN ) → (z1 , . . . , zN , z0 ) with the eigenvalue η = exp 2π i/(N + 1) is a preimage of t = −1 under the Chevalley map. We ﬁnd8 1 1 1 κ, α4 κ, γ = (N + 1)2 (ηa − ηb )2 (ηa − ηj ) (ηi − ηb ) j =a

α,γ =1

i=b

= (−1) (N + 1) (η − η ) η 4

N

b 2 N (a+b)

a

= (−1)N −1 (N + 1)4 (2 − ηa−b − ηb−a ). Collecting the results, we ﬁnd bα =

1 16 . a−b 2 − ηb−a ) (N + 1) (2 − η

This agrees with [17, Theorem 1.2], where gi = (N + 1)/(2 − ηi − η−i ) corresponds to αi = e0 − ei . In particular,

N + 1 −2 sin gk = 4 N

k=1

πk N +1

=

N (N + 1)(N + 2) . 12

The middle expression is a special case of Dedekind sums, and the second equality, which follows from our results, is well known in number theory (see, e.g., [4]). Case DN . The root system consists of the vectors ±ei ± ej , i = j , where e1 , . . . , eN is the standard orthonormal basis and (z1 , . . . , zN ) are the corresponding coordinates in CN . The parameters (t1 , . . . , tN ) in the following family of polynomials F (z, t) = z2N + t2 z2N−2 + t3 z2N −4 + · · · + tN z2 + t12 =

N 1 (z2 − zi2 )

i=1 8 We use here the facts that the product 8 2π ik/n ) over all nth roots of unity except (ζ − e k=a ζ = e2π ia/n is equal to the derivative of zn − 1 at z = ζ , i.e., to n/ζ .

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are identiﬁed with coordinates on the Chevalley quotient CN /W . Note that the invariant tN of degree h = 2N − 2 is the coefﬁcient at z2 . Let us assume that x is a generic point on the mirror za ± zb orthogonal to the root α = ea ∓ eb , and that t is the corresponding point on the discriminant, so that xa = ±xb and F (±xa , t) = F (±xa , t) = 0. Taking yi (ε) = xi + ε 1/2 vi + εwi + o(ε) and expanding F (y(ε), t1 , . . . , tN − 1, tN − ε) = 0 in ε, we ﬁnd εxi2 = F (xi , t) + F (xi , t)(ε1/2 vi + εwi ) + F (xi , t)ε

vi2 + o(ε). 2

& & Thus va = 2xa2 /F (xa , t), vb = ∓ 2xa2 /F (xa , t) and α, v4 = 64xa4 /F (xa , t)2 . Furthermore, ⎡ ⎤ 1 1 F (z, t)|z=xa = ⎣2 (z2 − xi2 ) + 4z2 (z2 − xi2 )⎦ a i=a

=

8xa2

1

(xa2

a=b i=a,b

z=xa

− xi2 ).

i=a

Using this we ﬁnd 1

x, α =

1 j =a,b

α,γ =1

(xa2 − xj2 )

1

(∓1)(xi2 − xb2 ) = (±1)N −2

i=a,b

F (xa , t)2 . 64xa4

Next, the eigenvector κ = (1, η, . . . , ηN −2 , 0) of the Coxeter transformation (z1 , . . . , zN ) → (z2 , . . . , zN −1 , −z1 , zN ) with the eigenvalue η = exp πi/(N − 1) is mapped to (t1 , . . . , tN ) = (0, . . . , 0, 1) under the Chevalley map. Assuming ﬁrst that α = ea ∓ eb with a, b < N, we ﬁnd 1 1 κ, γ = (ηa ∓ ηb )4 (ηa )2 (∓ηb )2 (η2a − η2i )(±1)(η2b − η2i ) κ, α4 i=a,b,N

α,γ =1

∓ η b )2 (ηa ± ηb )2 (2 ∓ ηa−b ∓ ηb−a ) . = (±1)N−2 (N − 1)2 (2 ± ηa−b ± ηb−a ) = −(±1)N −2 (N − 1)2

(ηa

Combining with the previous formulas, we compute bα =

(2 ± ηa−b ± ηb−a ) 1 2 (N − 1) (2 ∓ ηa−b ∓ ηb−a )

for α = ea ∓ eb .

Now let α = ea ∓ eN . Then 1 1 κ, γ = η4a (η2a − η2i )(−η2i ) = (−1)N −2 (N − 1) κ, α4 α,γ =1

i=a,N

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and therefore bα = 1/(N − 1). Taking the representatives α1 = eN −1 − e1 ,

. . . , αN −2 = eN −1 − eN −2 ,

αN −1 = eN −1 − eN ,

αN = eN −1 + eN

in the orbits of the Coxeter transformation on A, we ﬁnd gi =

(N − 1) (2 − ηi − η−i ) 2 (2 + ηi + η−i )

gi =

(N − 1)2 2

for i = 1, . . . , N − 2

and for i = N − 1, N.

The identity gk = (N − 1)N (2N − 1)/6, which follows from our general theory, agrees with the value of the Dedekind sum9 N −2 k=1

tan

2

πk 2N − 2

=

(N − 2)(2N − 3) . 3

In the case N = 4, the values gi = 1/2, 9/2, 9/2, 9/2 agree with the values of gi found in [17, Proposition 1.3(a)]. Cases EN . We ﬁnd gi using the packages LiE and MAPLE to compute the ratios via (5) and then apply the normalizing relation (3). In each case EN , let α1 , . . . , αN be the simple roots and M be the Coxeter transformation described by the following diagrams: α 1 α3 α 4 α 5 α 6 • − • − • − • −• | • M = σ1 σ4 σ6 σ2 σ3 σ5 , α2 α1 α 3 α 4 α 5 α 6 α 7 • − • − • − • − • −• | • M = σ1 σ4 σ6 σ2 σ3 σ5 σ7 , α2 α1 α 3 α 4 α 5 α 6 α 7 α8 • − • − • − • − • − • −• | • M = σ 1 σ 4 σ6 σ8 σ 2 σ 3 σ 5 σ 7 . α2 9 It is essentially the same one as in the A-case since sin −2 x − 1 = cot 2 x = tan2 (π/2 − x).

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One can check (e.g., using LiE) that all simple roots α1 , . . . , αN belong to different M-orbits. The following tables represent the values of the corresponding coefﬁcients gi , while the values of bαi can be obtained from them as in Proposition 9. Case E6 . We have bαi = gi /108, where √ √ g1 = g6 = 16+8 3, g3 = g5 = 16−8 3,

√ g2 = 7+4 3,

√ g4 = 7−4 3.

This agrees with the values of gi found in [17, Proposition 1.3(b)]. Case E7 . We have bαi = 2gi /729. Put u = cos(π/9). Then 27 + 36u + 24u2 , 2 147 g4 = + 12u − 96u2 , 2 21 g6 = − − 48u + 72u2 , 2

g1 =

225 + 36u − 144u2 , 2 9 g5 = − 72u + 72u2 , 2 9 g7 = + 36u + 72u2 . 2 g2 =

g3 =

3 , 2

Case E8 . We have bαi = 2gi /3375. Let u = cos(π/15). Then 33 + 80u + 72u2 − 16u3 , 2 123 g3 = − + 568u + 376u2 − 912u3 , 2 745 g5 = + 584u − 376u2 − 624u3 , 2 35 g7 = − + 156u + 136u2 − 256u3 , 2

g1 =

273 + 132u − 136u2 − 128u3 , 2 109 − 368u − 72u2 + 400u3 , g4 = 2 257 − 1220u − 232u2 + 1376u3 , g6 = 2 19 g8 = − + 68u + 232u2 + 160u3 . 2 g2 =

9 Open questions (a) Formula (9) deﬁnes the total descendent potential D as an asymptotical function of q = q0 + q1 z + · · · with semisimple q0 . As shown in [12, Theorem 5], the function DAN extends to arbitrary values of q0 without singularities. We expect the same for DDN and DEN but leave this issue open. (b) B. Dubrovin [7] associates to a Frobenius manifold a dispersionless integrable hierarchy. In particular, the hierarchy (1), (2) for asymptotical functions = exp(F (0) / + F (1) + · · · ) admits the dispersionless limit as → 0 which is an inﬁnite system of equations for F (0) . It is not hard to show that F satisﬁes the dispersionless hierarchy if and only if the Gaussian distributions := exp{dx2 F(q)/2} (where dx2 F is the quadratic differential of F at x) satisfy the original hierarchy (1), (2) for all x. An elegant explicit characterization in terms of the semiinﬁnite Grassmannian of those Gaussian distributions which satisfy the hierarchy of the type AN is given in the appendix in [12]. It would be interesting to generalize the characterization to Cases DN , EN .

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(c) Theorem 1 implies that the genus 0 descendent potential F (0) = lim→0 ln D satisﬁes the corresponding dispersionless hierarchy. The quadratic forms dx2 F (0) (q) depend only on N parameters τ = τ (x) (due to the property (∗) of the cone L = graph dF (0) ) and have the following explicit description (see the appendix in [12]): τ ([St (z)q(z)]0 • [St (z)q(z)]0 , ∂t a )dt a , 0

a

where [S(z)q(z)]0 = S0 q0 + S1 q1 + · · · denotes the z0 -mode. The corresponding Gaussian distributions therefore satisfy the hierarchy (1)–(5). Taking τ = u1 so that [Sτ q]0 = qk uk /k! we conclude that in particular the hierarchy has the 1-parametric family of Gaussian solutions ⎧ ⎛ ⎞ ⎫ ⎨ 1 u vk vl ⎬ ⎝ = exp qk , ql ⎠ dv . ⎩ 2 0 ⎭ k! l! k≥0

l≥0

This imposes nontrivial constraints on the coefﬁcients aα in the Hirota equation (1). It would be interesting to ﬁnd out if these constraints are sufﬁcient in order to determine the coefﬁcients unambiguously. (d) Our computations in Section 8 conﬁrm the Conjecture from Section 1 in Cases AN , D4 , E6 and leave it open in Cases DN with N > 4, E7 , and E8 —mostly because the values of the coefﬁcients gi in the Kac–Wakimoto theory remain unknown. A more conceptual approach to the identiﬁcation of the Hirota equations should rely on the deﬁnition of the coefﬁcients gi given in [17] in terms of representation theory. Namely, the vertex operators Ci± ±αi participate in the so-called principal construction of the basis representation of the afﬁne Lie algebra Aˆ N , Dˆ N , or Eˆ N , and gi = Ci+ Ci− . Here Ci± are certain structure constants whose values remain generally speaking unknown. Our successful description of the products Ci+ Ci− via the phase forms suggests that one should look for the intrinsic role of the phase forms in representation theory and for a description of the individual coefﬁcients Ci± in terms of the phase forms or their generalizations. (e) In representation theory, the hierarchies of ADE type form only a part of a larger list of examples including twisted versions of the afﬁne Lie algebras and nonsimply laced Dynkin diagrams. It would be interesting to ﬁnd the corresponding constructions in singularity theory and, in particular, to associate the Hirota equations with the boundary singularities BN , CN , F4 . (f) B. Dubrovin and Y. Zhang [9] associate an integrable hierarchy to any semisimple Frobenius manifold. In a sense their construction is parallel to the deﬁnition (9) of the total descendent potential D (see [11]) and, in particular, yields objects deﬁned in the complement to the caustic. In this regard the vertex operator description of the hierarchies seems more attractive as it is free of this defect. Of course, theADE-hierarchies (1)–(5) are expected to be equivalent to the hierarchies of Dubrovin–Zhang. It would be interesting to conﬁrm this expectation. (g) Conjecturally, the total descendent potential D extends analytically across the caustic values of q0 in the case of K. Saito’s (semisimple) Frobenius structure

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corresponding to any isolated singularity. (By the way, this is known to be false for boundary singularities or for ﬁnite reﬂection groups other than AN , DN , EN .) Respectively, one should expect the same for the hierarchies of Dubrovin–Zhang. It would be very interesting to give a vertex operator description of the hierarchies together with Theorem 1 for arbitrary (or at least weighted-homogeneous) isolated singularities of functions. The most obvious difﬁculty is that the vertex operator sum (1) over the set of all vanishing cycles (or even orbits of the classical monodromy operator on this set) becomes inﬁnite beyond the ADE list. Nevertheless we believe that the obstructions can be removed by an appropriate generalization of the concepts involved. The ﬁrst examples to study here would be the unimodal singularities P8 , X9 , J10 (see [1]). Their miniversal deformations are closely related to the complex crystallographic reﬂection groups E˜ 6 , E˜ 7 , E˜ 8 (see [19]). Moreover, the question can be extrapolated to the complex crystallographic groups A˜ N , D˜ N , and the threedimensional Frobenius manifold to be called A˜ 1 represents the ﬁrst challenge. Acknowledgments The authors thank V. Kac and M. Wakimoto for very helpful consultations on the contents of their paper [17], E. Frenkel and P. Pribik for their interest and stimulating discussions, and B. Sturmfels for his recommendation of the LiE package.

References [1] V. I. Arnold, Normal forms of functions near degenerate critical points, Weyl groups Ak , Dk , Ek and Lagrangian singularities, Functional Anal.] Appl., 6-2 (1972), 3–25. [2] V. I. Arnold, S. M. Gousein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. II: Monodromy and Asymptotics of Integrals, Monographs in Mathematics, Vol. 83, Birkhäuser, Basel, 1988. [3] S. Barannikov, Quantum periods I: Semi-inﬁnite variations of Hodge structures, Internat. Math. Res. Notices, 2001-23 (2001), 1243–1264. [4] M. Beck, The reciprocity law for Dedekind sums via the constant Ehrhart coefﬁcient, Amer. Math. Monthly, 106-5 (1999), 459–462. [5] N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1968, Chapters IV–VI. [6] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Operator approach to the Kadomtsev– Petviashvili equation: Transformation groups for soliton equations III, J. Phys. Soc. Japan, 50 (1981), 3806–3812. [7] B. Dubrovin, Geometry of 2D topological ﬁled theories, in Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, Vol. 1620, Springer-Verlag, New York, 1996, 120–348. [8] B. Dubrovin. Painlevé transcendents in two-dimensional topological ﬁeld theory, in The Painlevé Property, CRM Series in Mathematics and Physics, Springer-Verlag, New York, 1999, 287–412. [9] B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, arXiv: math.DG/0108160. [10] I. B. Frenkel, Representations of afﬁne Lie algebras, Hecke modular forms and Korteweg–de Vries type equations, in Lie Algebras and Related Topics (New Brunswick, NJ, 1981), Lecture Notes in Mathematics, Vol. 933, Springer-Verlag, Berlin, New York, 1982, 71–110.

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[11] A. Givental, Gromov–Witten invariants and quantization of quadratic hamiltonians, Moscow Math. J., 1-4 (2001), 551–568. [12] A. Givental, An−1 -singularities and nKdV hierarchies, Moscow Math. J., 3 (2003). [13] A. Givental, Asymptotics of the intersection form of a quasi-homogeneous function singularity, Functional Anal. Appl., 16-4 (1982), 294–297. [14] A. Givental, Symplectic geometry of Frobenius structures, Proceedings of the Workshop on Frobenius Structures (MPIM Bonn, July 2002), to appear; arXiv: math.AG/0305409. [15] C. Hertling, Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2002. [16] V. Kac, Inﬁnite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, UK, 1990. [17] V. Kac and M. Wakimoto, Exceptional hierarchies of soliton equations, Proc. Sympos. Pure Math., 49 (1989), 138–177. [18] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., 147 (1992), 1–23. [19] E. Looijenga, On the semiuniversal deformation of a simple-elliptic hypersurface singularity, Topology, 17 (1977), 23–40. [20] K. Saito, On a linear structure of the quotient variety by a ﬁnite reﬂection group, Publ. Res. Inst. Math. Sci., 29-4 (1993), 535–579. [21] A. P. Veselov, On generalizations of Calogero–Moser–Sutherland quantum problem and WDVV equations, J. Math. Phys., 43-11 (2002), 5675–5682. [22] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surv. Differential Geom., 1 (1991), 243–310.

Momentum maps and measure-valued solutions (peakons, ﬁlaments, and sheets) for the EPDiff equation∗ Darryl D. Holm1,2 and Jerrold E. Marsden3 1 Computer and Computational Science Division

Los Alamos National Laboratory MS D413 Los Alamos, NM 87545 USA [email protected] 2 Mathematics Department Imperial College London SW7 2AZ UK [email protected] 3 Control and Dynamical Systems 107-81 California Institute of Technology Pasadena, CA 91125 USA [email protected] To Alan Weinstein on the occasion of his 60th birthday. Abstract. This paper is concerned with the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler–Poincaré equations associated with the diffeomorphism group (of Rn or of an n-dimensional manifold M). It focuses on Lagrangians that are quadratic in the velocity ﬁelds and their ﬁrst derivatives, that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler–Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa–Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume-preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible ﬂuids. ∗ The ﬁrst author is grateful for support by US DOE under contract W-7405-ENG-36 for Los

Alamos National Laboratory, and Ofﬁce of Science ASCAR/AMS/MICS. The research of the second author was partially supported by the California Institute of Technology, by the National Science Foundation through NSF grant DMS-0204474, and by Air Force contract F49620-02-1-0176.

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It is shown that the EPDiff equations are generated by a smooth vector ﬁeld on the diffeomorphism group for sufﬁciently smooth solutions. This is analogous to known results for incompressible ﬂuids—both the Euler equations and the LAE equations—and it shows that for sufﬁciently smooth solutions, the equations are well posed for a short time. Numerical evidence suggests that as time progresses these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior. These nonsmooth, or measure-valued, solutions are higher-dimensional generalizations of the peakon solutions of the CH equation in one dimension. One of the main purposes of the paper is to show that many of the properties of these measure-valued solutions can be understood by the fact that their solution Ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one part of a dual pair.

1 Introduction This paper is concerned with solutions of the EPDiff equations, that is, with the Euler–Poincaré equations associated with the diffeomorphism group in n-dimensions. In particular, we are concerned with singular solutions that generalize the peakon solutions of the Camassa–Holm (CH) equation from one dimension to more spatial dimensions. The CH equation (see Camassa and Holm [1993]) for the dynamics of shallow water in a certain asymptotic regime,1 is ut + 3uux = α 2 (uxxt + 2ux uxx + uuxxx ),

(1.1)

where u(x, t) is the ﬂuid velocity, subscripts denote partial derivatives in position x and time t, α 2 is a positive constant, and the linear dispersion terms normally in CH have been omitted. Equivalently, in Hamiltonian form, this dispersionless CH equation reads as mt = −umx − 2ux m = {m, h(m)}, (1.2) where m = u − α 2 uxx and α 2 is a positive constant. As Camassa and Holm [1993] show, the CH equation is expressed in Hamiltonian form by using the Lie–Poisson bracket {·, ·} deﬁned on the dual Lie algebra of the one-dimensional vector ﬁelds and using the Hamiltonian 1 h(m) = umdx. (1.3) 2 The CH equation may be equivalently expressed in Euler–Poincaré form by using the Lagrangian associated with the H 1 metric for the ﬂuid velocity. This Lagrangian is given as a function of the ﬂuid velocity by the quadratic form 1 (1.4) l(u) = (u2 + α 2 u2x )dx. 2 It follows from Euler–Poincaré theory (see Marsden and Ratiu [1999] and Holm, Marsden, and Ratiu [1998a]) that the one-parameter curve of diffeomorphisms η(x0 , t) depending on parameter t and deﬁned implicitly by 1 See Dullin, Gottwald, and Holm [2001, 2003, 2004] for recent discussions of the derivation

and asymptotic validity of the CH equation for shallow water waves, at one order beyond the Korteweg–de Vries equation.

Measure-valued momentum map

205

∂ η(x0 , t) = u(η(x0 , t), t) ∂t is a geodesic in the group of diffeomorphisms of R (or, with periodic boundary conditions, of the circle S 1 ) equipped with the right invariant metric equal to the H 1 metric at the identity. A remarkable analytical property of the CH equation, conjectured by keeping track of derivative losses in Holm, Marsden, and Ratiu [1998a] and proved in Shkoller [1998] is that the geodesic equations literally deﬁne a smooth vector ﬁeld in the Sobolev H s topology for s > 3/2. That is, in the material representation, the equations have no derivative loss. This property is analogous to the corresponding results for the Euler equations for ideal incompressible ﬂuid ﬂow (discovered by Ebin and Marsden [1970]) and the Lagrangian averaged Euler equations (again conjectured by Holm, Marsden, and Ratiu [1998a] and proved by Shkoller [1998]). As we will explain in Section 3, a similar statement holds for the n-dimensional EPDiff equation if we use the H 1 metric. This is all the more remarkable because smoothness of the geodesic ﬂow is probably not true for the L2 metric. Smoothness of volume-preserving geodesic ﬂow with respect to the L2 metric does hold for the incompressible ﬂow of an ideal Euler ﬂuid, a result proved in Ebin and Marsden [1970]. Before proceeding with a discussion of the general case of the n-dimensional EPDiff equations, we shall quickly review, mostly to establish notation, a few facts about the Euler–Poincaré and Lie–Poisson equations, whose basic theory is explained, for example, in Marsden and Ratiu [1999]. Review of Euler–Poincaré and Lie–Poisson equations Let G be a Lie group and g its associated Lie algebra (identiﬁed with the tangent space to G at the identity element), with Lie bracket denoted [ξ, η] for ξ, η ∈ g. Let : g → R be a given Lagrangian and let L : T G → R be the right invariant Lagrangian on G obtained by translating from the identity element to other points of G via the right action of G on T G. A basic result of Euler–Poincaré theory is that the Euler–Lagrange equations for L on G are equivalent to the (right) Euler–Poincaré equations for on g, namely, to d δ δ = − ad ∗ξ . dt δξ δξ

(1.5)

Here ad ξ : g → g is the adjoint operator, that is, the linear map given by the Lie bracket η → [ξ, η] and ad ∗ξ : g∗ → g∗ is its dual, that is, ad ∗ξ (µ), η = µ, [ξ, η], where , denotes the natural pairing between g∗ and g. Also, δ/δξ denotes the functional derivative of with respect to ξ ∈ g; it is deﬁned to be the element of g∗ such that Dl(ξ ) · η = δl/δξ, η for all η ∈ g, where D denotes the (Frechet) derivative.2 For left invariant systems, we change the sign of the right-hand side in 2 Of course in the ﬁnite-dimensional case there is no real difference between Dl and δl/δξ , but in the inﬁnite-dimensional case one normally does not choose g∗ to be the naive, that

is, the literal functional analysis dual, but rather a convenient space for PDE needs.

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(1.5). The Euler–Poincaré equations can be written in the variational form δ dt = 0,

(1.6)

for all variations of the form δξ = η˙ − [ξ, η] for some curve η in g that vanishes at the endpoints. If the reduced Legendre transformation ξ → µ = δ/δξ is invertible, then the Euler–Poincaré equations are equivalent to the (right) Lie–Poisson equations µ˙ = − ad ∗δh/δµ µ,

(1.7)

where the reduced Hamiltonian is given by h(µ) = µ, ξ − (ξ ). These equations are equivalent (via Lie–Poisson reduction and reconstruction) to Hamilton’s equations on T ∗ G relative to the Hamiltonian H : T ∗ G → R, obtained by right translating h from the identity element to other points via the right action of G on T ∗ G. The Lie–Poisson equations may be written in the Poisson bracket form F˙ = {F, h},

(1.8)

where F : g∗ → R is an arbitrary smooth function and the bracket is the (right) Lie–Poisson bracket given by 4 5 δF δG , {F, G}(µ) = µ, . (1.9) δµ δµ In the important case when is quadratic, the Lagrangian L is the quadratic form associated to a right invariant Riemannian metric on G. In this case, the Euler– Lagrange equations for L on G describe geodesic motion relative to this metric and these geodesics are then equivalently described by either the Euler–Poincaré, or the Lie–Poisson equations. Outline of the paper The main contents of this paper are as follows: 1. In Section 2 we review some basic facts about the EPDiff equations, and in particular we recall a singular solution ansatz of Holm and Staley [2003, 2004] (see equation (2.8) below) which introduces a class of singular solutions that generalize the peakon solutions of the CH equation to higher spatial dimensions. 2. In Section 3 we show that the EPDiff equations possess an interesting smoothness property, namely, they deﬁne a smooth vector ﬁeld (that is, they deﬁne ODEs with no derivative loss) in the Lagrangian representation. There are a number of interesting consequences of this; in particular, it implies that the EPDiff equations are locally well posed for sufﬁciently smooth initial data and that the H 1 diameter of

Measure-valued momentum map

207

Diff (M) is strictly positive. Because of the development of singularities in ﬁnite time, which the numerics suggests, the smooth solutions may not exist globally in time. This smoothness property is similar to the corresponding smoothness property of the Euler equations for ideal incompressible ﬂuid mechanics shown in Ebin and Marsden [1970]. 3. In Section 4, we show that the singular solution ansatz—again see (2.8) below— deﬁnes an equivariant momentum map. We do this in a natural way by identifying the singular solutions with certain curves in the space of embeddings Emb(S, Rn ) of a generally lower-dimensional manifold S into the ambient space Rn (or an n-manifold M) and letting the diffeomorphism group act on this space. The right action of Diff (S) corresponds to the right invariance of the EPDiff equations, while the left action of Diff (Rn ) has a momentum map that gives the desired solution ansatz. This is the main result of the paper. 4. In Section 5 we brieﬂy explore some of the geometry of the singular solution momentum map, in parallel with the corresponding work on singular solutions (vortices, ﬁlaments, etc.) for the Euler equations of an ideal ﬂuid that was developed in Marsden and Weinstein [1983]. 5. Finally, in Section 6, we discuss some of the remaining challenges and speculate on some of the many possible future directions for this work. Historical note This paper is dedicated to our friend and collaborator Alan Weinstein and, for us, this work parallels some of our earlier collaborations with him. Alan’s basic works on reduction, Poisson geometry, semidirect product theory, and stability in mechanics— just to name a few areas—have been, and remain, incredibly inﬂuential and important to the ﬁeld of geometric mechanics. See, for instance, Marsden and Weinstein [1974], Weinstein [1983b], Marsden, Ratiu, and Weinstein [1984], Weinstein [1984], and Holm et al. [1985]. Mechanics on Lie groups was pioneered by Arnold [1966], a reference that is a key foundation for the subject and in particular for this paper. However, this theory was in a relatively primitive state, even by 1980, and it has beneﬁted greatly from Alan’s insights. In fact, the clear distinction between the Euler–Poincaré and Lie– Poisson equations, the former equations on g and possessing a variational structure (with constraints on the variations) and the latter on g∗ with its Lie–Poisson structure took until the 1980s to crystallize, and for the place of Lie and Poincaré in the history of the subject to be clariﬁed. This development and clariﬁcation was greatly aided by Alan’s work, historical researches, and deep insight. Alan has made key contributions to many fundamental concepts in geometric mechanics, such as Lagrangian submanifolds and related structures (Weinstein [1971, 1977]), symplectic reduction (Marsden and Weinstein [1974]), normal modes and periodic orbits (Weinstein [1973, 1978]), Poisson manifolds (Weinstein [1983b]), geometric phases (Weinstein [1990]), Dirac structures (Courant and Weinstein [1988]) groupoids and Lagrangian reduction (Weinstein [1996]) and the plethora of related “oid’’ structures he has been working on during the last decade (just look over the

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151+ papers on MathSciNet he has written!) that will surely play as an important role in the next generation of people working in the area of geometric mechanics as it is with the current new generation. Of Alan’s papers, the one that is most directly relevant to the topics discussed in this paper is Marsden and Weinstein [1983]. Alan himself is still developing the mathematics associated with this area, as in Weinstein [2002].

2 The EPDiff equation This section reviews the EPDiff equation; that is, the Euler–Poincaré (EP) equation associated with the diffeomorphism group of an n-manifold M (which, for simplicity, will be taken primarily as Rn ). This equation coincides with the dispersionless case of the CH equation for shallow water waves in one and two dimensions, discussed in Camassa and Holm [1993]; Kruse, Scheurle, and Du [2001]. It also coincides with the ATME equation (the averaged template matching equation) in two dimensions. The latter equation arises in computer vision; see, for instance, Mumford [1998], Hirani, Marsden, and Arvo [2001], or Miller, Trouvé, and Younes [2002] for a description and further references. We have chosen to call this by a generic name, the EPDiff equation, because it has these various interpretations in different applications. Of course these different interpretations also provide opportunities: for example, this point of view may enable one to see to what extent the singular solutions found in the EPDiff equations are applicable, either for shallow water wave interactions, or for computer vision applications. A recent combination of these ideas in which image processing is informed by concepts of momentum originating in soliton theory appears in Holm, Trouvé, Ratnanather, and Younes [2004]. Statement of the EPDiff equations Treating analytical issues formally at this point, let X denote the Lie algebra of vector ﬁelds on an n-dimensional manifold M (such as Rn ). The space X is the algebra of the diffeomorphism group of M, but the usual Jacobi–Lie bracket is the negative of the (standard) Lie algebra bracket. (See Marsden and Ratiu [1999] for a discussion.) Let : X → R be a given Lagrangian and let M denote the space of oneform densities on M, that is, the momentum densities. The corresponding momentum density of the ﬂuid is deﬁned as m=

δ ∈ M, δu

which is the functional derivative of the Lagrangian with respect to the ﬂuid velocity u ∈ X. If u is the basic dynamical variable, the EPDiff equations are simply the Euler– Poincaré equations associated with this Lagrangian. Equivalently, if m is taken to be the basic dynamical variable, the Legendre transformation allows one to identify the EPDiff equations as the Lie–Poisson equations associated with the resulting

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Hamiltonian. For the case of Rn , we will use vector notation for the momentum density m(x, t) : Rn × R → Rn (a bold m instead of a lightface m). The EPDiff equations are as follows (see Holm, Marsden, and Ratiu [1998a,b]; Holm et al. [2002] for additional background and for techniques for computing the Euler–Poincaré equations for ﬁeld theories), ∂ T m + 9u ·:; ∇m< + 9∇u:; · m< + m(div u) = 0. (2.1) 9 :; < ∂t convection

stretching

expansion

In coordinates i = 1, 2, . . . , n, using the summation convention, and writing m = mi dx i ⊗ d n x (regarding m as a one-form density) and u = ui ∂/∂x i (regarding u as a vector ﬁeld), the EPDiff equations read xi ,

∂mi ∂uj ∂uj ∂ mi + uj j + mj i + mi j = 0. ∂t ∂x ∂x ∂x The EPDiff equations can also be written concisely as

(2.2)

∂m + £u m = 0, (2.3) ∂t where £u m denotes the Lie derivative of the momentum one-form density m with respect to the velocity vector ﬁeld u. As mentioned earlier, if is a quadratic function of u, then the EPDiff equation (2.1) or, equivalently, (2.3), is the Eulerian description of geodesic motion on the diffeomorphism group of the underlying space (in this case Rn ). The corresponding metric is the right invariant metric on the group, whose value on the Lie algebra (the group’s tangent space at the identity—the space of vector ﬁelds) is deﬁned by . Since the Lagrangian is positive and quadratic in u, the momentum density is linear in u and so deﬁnes a positive symmetric operator Qop by δ = Qop u. δu Likewise, for quadratic Lagrangians the velocity u is determined from the momentum m by u = G ∗ m, where G∗ represents convolution with the Green’s function G for the linear operator Qop . m=

Variational formulation Following the variational formulation of EP theory, the particular EP equation (2.1) may be derived from the following constrained variational principle: δ (u)dt = 0. The variations are constrained to have the form ˙ + w · ∇u − u · ∇w. δu = w This assertion may of course be veriﬁed directly. These constraints are analogous to the so-called “Lin constraints’’ used for a similar variational principle for ﬂuid mechanics. (See Marsden and Ratiu [1999] for a discussion and references.)

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Hamiltonian formulation The Legendre transformation yields the Hamiltonian H (m) = m, u − (u), where , is the natural pairing between one-form densities and vector ﬁelds given by integration (L2 pairing). This Hamiltonian is the corresponding quadratic form for the momentum, namely, H (m) = (Q−1 (2.4) op (m)). Of course it often happens that Qop is a differential operator and in this case the inverse is usually given in terms of the convolution with the Green’s function G, corresponding to the appropriate solution domain and boundary conditions: u=

δH (m) = G ∗ m. δm

According to the general theory, the EP equation (2.1) may be expressed in Hamiltonian form by using the Lie–Poisson bracket on M as ∂ m = {m, H }LP = − ad ∗δH /δm m. ∂t

(2.5)

One-dimensional CH peakon solutions We return now to the CH equation (1.2), which, as we have noted, is the same as the EPDiff equation (2.1) for the case of one spatial dimension when the momentum velocity relationship is deﬁned by the Helmholtz equation, m = u − α 2 uxx . In one dimension, the CH equation has solutions whose momentum is supported at points on the real line via the following sum over Dirac delta measures: m(x, t) =

N

pi (t)δ(x − qi (t)).

(2.6)

i=1

The velocity corresponding to this measure-valued momentum is obtained by convolution u = G ∗ m with the Green’s function, G(|x − y|) = 12 e−|x−y|/α , for the one-dimensional Helmholtz operator, Qop = (1 − α 2 ∂x2 ), appearing in the CH momentum velocity relationship m = Qop u. Consequently, the CH velocity corresponding to this momentum is given by a superposition of peaked traveling wave pulses, N 1 u(x, t) = pi (t)e−|x−qi (t)|/α . (2.7) 2 i=1

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Thus the superposition of “peakons’’in velocity arises from the delta function solution ansatz (2.6) for the momentum. Remarkably, the isospectral eigenvalue problem for the CH equation implies that only these singular solutions emerge asymptotically in the solution of the initial value problem in one dimension, as is shown in Camassa and Holm [1993]. Figure 2.1 shows the emergence of peakons from an initially Gaussian velocity distribution and their subsequent elastic collisions in a periodic one-dimensional domain.3 This ﬁgure demonstrates that singular solutions dominate the initial value problem. Thus it is imperative to go beyond smooth solutions for the CH equation; as we shall see, there is numerical evidence that the situation is similar for the EPDiff equation.

Fig. 2.1. This ﬁgure shows a smooth localized (Gaussian) initial condition for the CH equation breaking up into an ordered train of peakons as time evolves (the time direction being vertical, which then eventually wrap around the periodic domain and interacting with other slower emergent peakons and causing a phase shift (see Alber and Marsden [1992]).

Remarkably, the dynamical equations for pi (t) and qi (t), i = 1, . . . , N, that arise from solution ansatz (2.6)–(2.7) comprise an integrable system for any N . This system is studied in (Alber, Camassa, Fedorov, Holm, and Marsden [2001]) and references therein. See also Vaninsky [2002, 2003] for discussions of how the integrable dynamical system for N peakons is related to the Toda chain with open ends. Generalizing the CH peakon solutions to n dimensions Building on the peakon solutions for the CH equation and the pulsons for its generalization to other traveling-wave shapes (see Fringer and Holm [2001]), the papers of Holm and Staley [2003, 2004] introduced the following measure-valued (that is, density valued) ansatz for the n-dimensional solutions of the EPDiff equation (2.1): m(x, t) =

N

Pa (s, t)δ(x − Qa (s, t))ds.

a=1 3 Figure 2.1 was kindly supplied by Martin Staley.

(2.8)

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These solutions are vector-valued functions supported in Rn on a set of N surfaces (or curves) of codimension (n − k) for s ∈ Rk with k < n. They may, for example, be supported on sets of points (vector peakons, k = 0), one-dimensional ﬁlaments (strings, k = 1), or two-dimensional surfaces (sheets, k = 2) in three dimensions. One of the main results of this paper is the theorem stating that the singular solution ansatz (2.8) is an equivariant momentum map. This result helps to organize the theory and to suggest new avenues of exploration, as we shall explain. Substitution of the solution ansatz (2.8) into the EPDiff equations (2.1) implies the following integro-partial-differential equations (IPDEs) for the evolution of such strings and sheets, ∂ a Q (s, t) = ∂t N

b=1

∂ a P (s, t) = − ∂t N

Pb (s , t)G(Qa (s, t) − Qb (s , t))ds ,

(Pa (s, t) · Pb (s , t))

b=1

∂ ∂Qa (s, t)

(2.9)

G(Qa (s, t) − Qb (s , t))ds .

Important for the interpretation of these solutions, the coordinates s ∈ Rk turn out to be Lagrangian coordinates. The velocity ﬁeld corresponding to the momentum solution ansatz (2.8) is given by u(x, t) = G ∗ m =

N

Pb (s , t)G(x − Qb (s , t))ds ,

u ∈ Rn .

(2.10)

b=1

When evaluated along the curve x = Qa (s, t), the velocity satisﬁes N u(x, t)x=Qa (s,t) =

Pb (s , t)G(Qa (s, t) − Qb (s , t))ds =

b=1

∂Qa (s, t) . (2.11) ∂t

Thus the lower-dimensional support sets deﬁned on x = Qa (s, t) and parameterized by coordinates s ∈ Rk move with the ﬂuid velocity. Moreover, equations (2.9) for the evolution of these support sets are canonical Hamiltonian equations, δHN ∂ a Q (s, t) = , ∂t δPa

∂ a δHN P (s, t) = − a . ∂t δQ

(2.12)

The Hamiltonian function HN : (Rn × Rn )N → R is 1 HN = 2

N

(Pa (s, t) · Pb (s , t))G(Qa (s, t) − Qb (s , t))dsds .

(2.13)

a,b=1

This is the Hamiltonian for geodesic motion on the cotangent bundle of a set of curves Qa (s, t) with respect to the metric given by G. This dynamic was investigated numerically in Holm and Staley [2003, 2004] to which we refer for more details of the solution properties.

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As we have mentioned, one of our main goals is to show that the solution ansatz (2.8) can be recast in terms of an equivariant momentum map that naturally arises in this problem. This geometric feature underlies the remarkable reduction properties of the EPDiff equation and “explains’’ why the preceding equations must be Hamiltonian—it is because it is a general fact that equivariant momentum maps are Poisson maps. As explained in general terms in Marsden and Weinstein [1983], the way one implements a coadjoint orbit reduction is through a momentum map, and this holds even for the case of singular orbits (again ignoring functional analytic details). Thus, in summary, the Ansatz (2.12) is the EPDiff analog of the corresponding Anzatz for incompressible ﬂuid mechanics (that is, the EPDiffVol equations) that gives point (or blob) vortex dynamics, vortex ﬁlaments, or sheets. There are, however, some important differences between vortex dynamics for incompressible ﬂows and the dynamics of the measure-valued EPDiff momentum solutions. For example, the Lagrangian representations of the equations of motion show that EPDiff solutions for momentum have inertia, while the corresponding solutions for point (or blob) vortices of the EPDiffVol dynamics have no inertia. What this means is the following: the equations of motion for the spatial vectors specifying the measure valued vorticity solutions on EPDiffVol are ﬁrst order in time, while the dynamical equations for the corresponding spatial vectors Qa (s, t) for measurevalued momentum solutions on EPDiff are second order in time. This difference has profound effects on the properties of the solutions, especially on their stability properties. Numerical investigations of Holm and Staley [2003, 2004] show, for example, that the codimension-one singular momentum solutions of EPDiff are stable, while its higher-codimension singular momentum solutions are very unstable to codimensionone perturbations. In contrast, the codimension-two singular vorticity solutions of EPDiffVol (point vortices in the plane, and vortex ﬁlaments in space) are known to be stable to such perturbations. Comments on the physical meaning of the equations The EPDiff equations with the Helmholtz relation between velocity and momentum are not quite the CH equations for surface waves in 2D. Those would take precisely the same form, but the shallow water wave relation in the 2D CH approximation would be m = u − α 2 Grad Div u,

that is,

mi = ui − α 2 uj,j i ,

that is,

mi = ui − α 2 ui,jj .

rather than the Helmholtz operator form, m = u − α 2 Div Grad u,

The corresponding Lagrangians are, respectively, 1 lCH (u) = (|u|2 + α 2 (Div u)2 )dxdy. 2

(2.14)

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and lEPDiff (u) =

1 2

(|u|2 + α 2 | Grad u|2 )dxdy.

(2.15)

This difference was noted in Kruse, Scheurle, and Du [2001], which identiﬁed (2.14) as the generalization of (1.4) for water waves in two dimensions. One may also verify this by considering the limit of the Green–Nagdhi equations for small potential energy. (The CH equation arises in this limit. The Lake and Great Lake equations of Camassa, Holm, and Levermore [1996, 1997] also arise in a variant of this limit.) Remarkably, the numerics in Holm and Staley [2003, 2004] show that the solutions for a variety of initial conditions are indistinguishable in these two cases. The initial conditions in Holm and Staley [2003, 2004] were all spatially conﬁned velocity distributions. Notice that this difference affects the choice of Hamiltonian, but the equations are still Euler–Poincaré equations for the diffeomorphism group and the description of the Ansatz (2.8) as a momentum map is independent of this difference in the equations. Figure 2.2 shows the striking reconnection phenomenon seen in the nonlinear interaction between wave-trains, as simulated by numerical solutions of the EPDiff equation and observed for internal waves in the ocean. Figure 2.2(a) shows a frame taken from simulations of the initial value problem for the EPDiff equation in two dimensions, excerpted from Holm and Staley [2003, 2004]. (See also Holm et al. [2003].) Figure 2.2(b) shows the interaction of two internal wave trains propagating at the interface of different density levels in the South China Sea, as observed from the Space Shuttle using synthetic aperture radar, courtesy of A. Liu (2002). Most important, both Figures 2.2(a) and 2.2(b) show nonlinear reconnection occurring in the wave-train interaction as their characteristic feature. Figure 2.3 shows a sequence of snapshots illustrating the reconnection phenomenon for singular solutions of EPDiff in two dimensions.4 Interactions among internal waves are generally thought to be described by the KP equation, and so any relations among the KP equation, the EPDiff equation and the 2D CH equation would be of great interest to explore; see Liu et al. [1998]. The derivations of the KP equation and the CH equation differ in the way the gradients of their transverse motions are treated in the asymptotics—order O() for KP and order O(1) for CH; thus some difference in their solution behavior is to be expected.

3 Smoothness of the Lagrangian equations The one-dimensional case Based on a formal argument given in Holm, Marsden, and Ratiu [1998a], it was shown in Shkoller [1998] that the CH equation (1.2) in Lagrangian variables deﬁnes 4 Figures 2.2(a) and 2.3 were kindly supplied by Martin Staley.

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(a)

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(b)

Fig. 2.2. Comparison of evolutionary EPDiff solutions in two dimensions (a) and Synthetic Aperture Radar observations by the Space Shuttle of internal waves in the South China Sea (b). Both ﬁgures show nonlinear reconnection occurring in the wave-train interaction as their characteristic feature.

Fig. 2.3. A single collision is shown to produce reconnection as the faster wave front segment initially moving southwest along the diagonal expands, curves and obliquely overtakes the slower one, which was initially moving rightward (east). This reconnection illustrates one of the collision rules for singular solutions of the two-dimensional EPDiff ﬂow. See Holm and Staley [2004] for a complete treatment.

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a smooth vector ﬁeld. (That is, one obtains an evolution equation with no derivative loss.) This means one can show using ODE methods that the initial value problem is well posed, and one may also establish other important properties of the equations, provided the data is sufﬁciently smooth. As above, we write the relation between m and u as m = Qop u, so that in the one-dimensional case, Qop is the operator Qop = Id −α 2 ∂xx . We ﬁrst recall how the equations are transformed into Lagrangian variables. Introduce the one-parameter curve of diffeomorphisms η(x0 , t) deﬁned implicitly by ∂ η(x0 , t) = u(η(x0 , t), t), ∂t

(3.1)

so that η is a geodesic in the group of diffeomorphisms of R (or, with periodic boundary conditions, of the circle S 1 ) equipped with the right invariant metric equaling the H 1 metric at the identity. We compute the second time derivative of η in a straightforward way by differentiating (3.1) using the chain rule ∂ 2η ∂u . = uux + 2 ∂t ∂t Acting on this equation with Qop and using the deﬁnition m = Qop u yields Qop

∂ 2η ∂m = Qop (uux ) − u∂x (Qop u) + umx + ∂t ∂t 2 = [Qop , u∂x ]u − 2mux = −3α 2 ux uxx − 2mux = −α 2 ux uxx − 2uux ,

where the third step uses the commutator relation calculated from the product rule, [Qop , u∂x ]u = −3α 2 ux uxx . Hence, the preceding equation becomes ∂ 2η 1 = − Q−1 ∂x (α 2 u2x + 2u2 ). 2 op ∂t 2

(3.2)

The important point about this equation is that the right-hand side has no derivative loss for α > 0. That is, if u is in the Sobolev space H s for s > 5/2, then the right-hand side is also in the same space. Regarding the right-hand side as a function of η and ∂η/∂t, we see that it is plausible that the second-order evolution equation (3.2) for η deﬁnes a smooth ODE on the group of H s diffeomorphisms. (This argument requires the use of, for example, weighted Sobolev spaces in the case x ∈ R.) The above is the essence of the argument given in Shkoller [1998], Remark 3.5, which in turn makes use of the type of arguments found in Ebin and Marsden [1970] for the incompressible case and which shows, by a more careful argument, that the

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spray is smooth if s > 3/2. However, one should note that the complete argument is not quite so simple (as in the case of incompressible ﬂuids). A subtlety arises because smoothness means as a function of η, η. ˙ Hence, one must express u in terms of η, namely, through the relation ut = η˙ t ◦ ηt−1 , where the subscript t here denotes that this argument is held ﬁxed, and is not a partial derivative. Doing this, one sees that, while there is clearly no derivative loss, the right-hand side of (3.2) does involve ηt−1 and the map ηt → ηt−1 is known to not be smooth. Nevertheless, the combination that appears in (3.2) is, quite remarkably, a smooth function of η, η˙ as is shown by arguments in Ebin and Marsden [1970]. The n-dimensional case The above argument readily generalizes to n-dimensions, which we shall present in the case of Rn or the ﬂat n-torus Tn for simplicity. Namely, we still have the relation ∂ η(x0 , t) = u(η(x0 , t), t), ∂t between η and u. Consequently, we may compute the second partial time derivative of η in the usual fashion using the chain rule ∂u ∂ 2η . = u · ∇u + ∂t ∂t 2 Therefore, as in the one-dimensional case, we get ∂ 2η ∂m . = [Qop , (u · ∇)]u + u · ∇m + 2 ∂t ∂t Calculating the commutator relation in n-dimensions gives Qop

[Qop , (u · ∇)]u = −α 2 div(∇u · ∇u + ∇u · ∇uT ) + α 2 (∇u) · ∇ div u or, in components, ([Qop , (u · ∇)]u)i = −α 2 ∂k (ui,j uj,k + ui,j uk,j ) + α 2 (ui,j )∂j div u,

(3.3)

with a sum on repeated indices. Upon substituting the preceding commutator relation, the EPDiff equation (2.1) and the vector calculus identity 1 2 α2 T 2 T 2 |u| + |∇u| , (3.4) −∇u · m = α div(∇u · ∇u) − ∇ 2 2 then imply the n-dimensional result Qop

∂ 2η ∂t 2

1 T T T 2 = −α div ∇u · ∇u + ∇u · ∇u − ∇u · ∇u − ∇u (div u) + Id |∇u| 2 1 − u(div u) − ∇|u|2 . 2 A similar computation holds on a Riemannian manifold (or in curvilinear coordinates) 2

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in terms of covariant derivatives. This form of the EPDiff equation is useful for interpreting some of its solution behavior. As in the one-dimensional case, the crucial point is that the right-hand side involves at most second derivatives of u; so there is no derivative loss in the expression for ∂ 2 η/∂t 2 . This is the main idea behind the proof of smoothness. The technical details for the n-dimensional case can be provided following the arguments for the one-dimensional or incompressible cases, as indicated above. This leads to the following result. Proposition 3.1. For s > (n/2) + 1, and M a compact Riemannian manifold, the EPDiff equations as second-order equations on the H s -Diff (M) group, deﬁne a smooth spray. In particular, this implies the following: 1. The EPDiff equations are locally well posed (for short time) for initial data in H s . 2. Two nearby diffeomorphisms can be joined by a unique curve (lying in the neighborhood of these diffeomorphisms) which is the ﬂow of a solution of the EPDiff equations for time ranging from 0 to 1. 3. With initial data in H s , the solutions of the EPDiff equation are C ∞ in time. For example, if M is the three torus, this corresponds to solutions in R3 with spatially periodic boundary conditions. By using methods such as those of Cantor [1975], we one may reasonably expect to establish a similar result for Rn in weighted (Nirenberg–Walker) Sobolev spaces. While local well posedness of the EPDiff equations is a fairly routine matter from the PDE point of view, the other properties are not so simple to obtain by classical PDE methods. In the sections that follow, we will be interested in nonsmooth data. This is in stark contrast to the preceding discussion, which requires initial data that is at least C 1 . Remarkably, the same smoothness results hold for the case of the LAE-α (Lagrangian averaged Euler) equations, a set of incompressible equations in which small scale ﬂuctuations whose size is of order α are averaged. One can view the LAE-α equations as the incompressible version of the EPDiff equations. This smoothness property for the LAE-α equations was shown by Shkoller [1998] for regions with no boundary and for regions with boundary (for various boundary conditions), it was shown in Marsden et al. [2000]. However, unlike the incompressible case, the results apparently do not hold if α is zero (as was also noted in Shkoller [1998]). This sort of smoothness result also appears not to hold for many other equations, such as the KdV equation, even though it too can be realized as Euler–Poincaré equations on a Lie algebra, or as geodesics on a group, in this case the Bott–Virasoro group, as explained in Marsden and Ratiu [1999] and references therein. Development of singularities The smoothness property just discussed does not preclude the development in ﬁnite time of singular solutions from smooth localized initial data, as was indicated in

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Figure 2.1. To capture the local singularities in the EPDiff solution (either verticality in slope, or discontinuities in its spatial derivative) that develop in ﬁnite time from arbitrarily smooth initial conditions, one must enlarge the solution class of interest by considering weak solutions. There are a number of papers on weak solutions of the CH equation (such as, for instance, Xin and Zhang [2000]) which we will not survey here. We just mention that the theory is not yet complete, as it is still unknown in what sense one may deﬁne global unique weak solutions to the CH equations in H 1 —only results that have an energy conservation theorem and allow for the head-on collision of two peakons would be considered interesting. As discussed in Alber, Camassa, Fedorov, Holm, and Marsden [2001] for the CH equation, one most likely must consider weak solutions in the spacetime sense. The steepening lemma of Camassa and Holm [1993] proves that in one dimension any initial velocity distribution whose spatial proﬁle has an inﬂection point with negative slope (for example, any antisymmetric smooth initial distribution of velocity on the real line) will develop a vertical slope in ﬁnite time. Note that the peakon solution (2.7) has no inﬂection points, so it is not subject to the steepening lemma. However, the steepening lemma underlies the mechanism for forming these singular solutions, which are continuous but have discontinuous spatial derivatives; they also lie in H 1 and have ﬁnite energy. We conclude that solutions with initial conditions in H s with s > (n/2) + 1 go to inﬁnity in the H s norm in ﬁnite time, but remain in H 1 and presumably continue to exist in a weak spacetime sense for all time in H 1 . Numerical evidence in higher dimensions and the inverse scattering solution for the CH equation in one dimension (the latter has only discrete eigenvalues, corresponding to peakons) both suggest that the singular solutions completely dominate the time-asymptotic dynamics of the initial value problem (IVP). This singular IVP behavior is one of the main discoveries of Camassa and Holm [1993]. This singular behavior has drawn a great deal of mathematical interest to the CH equation and its relatives, such as EPDiff. The other properties of CH—its complete integrability, inverse scattering transform, connections to algebraic geometry and elliptical billiards, bi-Hamiltonian structure—are, of course, all interesting too. However, the requirement of dealing with singularity as its main solution phenomenon is the primary aspect of CH (and EPDiff). We aim to show that many of the properties of these singular solutions of CH and EPDiff are captured by recognizing that the singular solution ansatz itself is a momentum map. This momentum map property explains, for example, why the singular solutions (2.8) form an invariant manifold for any value of N and why their dynamics form a Hamiltonian system. In one dimension, the complete integrability of the CH equation as a Hamiltonian system and its soliton paradigm explain the emergence of peakons in the CH dynamics. Namely, their emergence reveals the initial condition’s soliton (peakon) content. However, beyond one dimension, we do not have an explicit mechanism for explaining why only singular solution behavior emerges in numerical simulations. One hopes that eventually a theory will be developed for explaining this observed singular solution phenomenon in higher dimensions. Such a theory might, for example, parallel the well-known explanation of the formation of shocks for hyperbolic

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partial differential equations. (Note, however, that EPDiff is not hyperbolic, because the relation u = G ∗ m between its velocity and momentum is nonlocal.) In the remainder of this work, we shall focus on the momentum map properties of the invariant manifold of singular solutions (2.8) of the EPDiff equation.

4 Singular solution momentum map The momentum ansatz (2.8) is a momentum map The purpose of this section is to show that the solution ansatz (2.8) for the momentum vector in the EPDiff equation (2.1) deﬁnes a momentum map for the action of the group of diffeomorphisms on the support sets of the Dirac delta functions. These support sets are the analogs of points on the real line for the CH equation in one dimension. They are points, curves, or surfaces in Rn for the EPDiff equation in n dimensions. This result, as we shall discuss in greater detail later, shows that the solution ansatz (2.8) ﬁts naturally into the scheme of Clebsch, or canonical variables in the sense advocated by Marsden and Weinstein [1983] as well as showing that these singular solutions evolve on special coadjoint orbits for the diffeomorphism group. One can summarize by saying that the map that implements the canonical (Q, P) variables in terms of singular solutions is a (cotangent bundle) momentum map. Such momentum maps are Poisson maps; so the canonical Hamiltonian nature of the dynamical equations for (Q, P) ﬁts into a general theory which also provides a framework for suggesting other avenues of investigation. Theorem 4.1. The momentum ansatz (2.8) for measure-valued solutions of the EPDiff equation (2.1), deﬁnes an equivariant momentum map JSing : T ∗ Emb(S, Rn ) → X(Rn )∗ that we will call the singular solution momentum map. We shall explain the notation used in this statement in the course of the proof. Right away, however, we note that the sense of “deﬁnes’’ is quite simple, namely, expressing m in terms of Q, P (which are, in turn, functions of s) can be regarded as a map from the space of (Q(s), P(s)) to the space of m’s. We shall give two proofs of this result from two rather different points of view. The ﬁrst proof below uses the formula for a momentum map for a cotangent lifted action, while the second focuses on a Poisson bracket computation. Each proof also explains the context in which one has a momentum map. (See Marsden and Ratiu [1999] for general background on momentum maps.) First proof . For simplicity and without loss of generality, let us take N = 1 and so suppress the index a. That is, we shall take the case of an isolated singular solution. As the proof will show, this is not a real restriction.

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To set the notation, ﬁx a k-dimensional manifold S with a given volume element and whose points are denoted s ∈ S. Let Emb(S, Rn ) denote the set of smooth embeddings Q : S → Rn . (If the EPDiff equations are taken on a manifold M, replace Rn with M.) Under appropriate technical conditions, which we shall just treat formally here, Emb(S, Rn ) is a smooth manifold. (See, for example, Ebin and Marsden [1970] and Marsden and Hughes [1983] for a discussion and references.) The tangent space TQ Emb(S, Rn ) to Emb(S, Rn ) at the point Q ∈ Emb(S, Rn ) is given by the space of material velocity ﬁelds, namely, the linear space of maps V : S → Rn that are vector ﬁelds over the map Q. The dual space to this space will be identiﬁed with the space of one-form densities over Q, which we shall regard as maps P : S → (Rn )∗ . In summary, the cotangent bundle T ∗ Emb(S, Rn ) is identiﬁed with the space of pairs of maps (Q, P). These give us the domain space for the singular solution momentum map. Now we consider the action of the symmetry group. Consider the group G = Diff of diffeomorphisms of the space M in which the EPDiff equations are operating, concretely in our case Rn . Let it act on M by composition on the left. Namely, for η ∈ Diff (Rn ), we let η · Q = η ◦ Q. (4.1) Now lift this action to the cotangent bundle T ∗ Emb(S, Rn ) in the standard way (see, for instance, Marsden and Ratiu [1999] for this construction). This lifted action is a symplectic (and hence Poisson) action and has an equivariant momentum map. We claim that this momentum map is precisely given by the ansatz (2.8). To see this, we will recall and then apply the general formula for the momentum map associated with an action of a general Lie group G on a conﬁguration manifold Q and cotangent lifted to T ∗ Q. First, let us recall the general formula. Namely, the momentum map is the map J : T ∗ Q → g∗ (g∗ denotes the dual of the Lie algebra g of G) deﬁned by J(αq ) · ξ = αq , ξQ (q),

(4.2)

where αq ∈ Tq∗ Q and ξ ∈ g, where ξQ is the inﬁnitesimal generator of the action of G on Q associated to the Lie algebra element ξ , and where αq , ξQ (q) is the natural pairing of an element of Tq∗ Q with an element of Tq Q. Now we apply formula (4.2) to the special case in which the group G is the diffeomorphism group Diff (Rn ), the manifold Q is Emb(S, Rn ) and where the action of the group on Emb(S, Rn ) is given by (4.1). The sense in which the Lie algebra of G = Diff is the space g = X of vector ﬁelds is well understood. Hence, its dual is naturally regarded as the space of one-form densities. The momentum map is thus a map J : T ∗ Emb(S, Rn ) → X∗ . To calculate J given by (4.2), we ﬁrst work out the inﬁnitesimal generators. Let X ∈ X be a Lie algebra element. By differentiating the action (4.1) with respect to η in the direction of X at the identity element we ﬁnd that the inﬁnitesimal generator is given by XEmb(S,Rn ) (Q) = X ◦ Q. Thus, taking αq to be the cotangent vector (Q, P), equation (4.2) gives

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J(Q, P), X = (Q, P), X ◦ Q = Pi (s)X i (Q(s))d k s. S

On the other hand, note that the right-hand side of (2.8) (again with the index a suppressed, and with t suppressed as well), when paired with the Lie algebra element X is P(s)δ(x − Q(s))d k s, X = (Pi (s)δ(x − Q(s))d k s)X i (x)d n x n S R S = Pi (s)X i (Q(s)d k s. S

This shows that the expression given by (2.8) is equal to J and so the result is proved.

Second proof . As is standard (see, for example, Marsden and Ratiu [1999]), one can characterize momentum maps by means of the following relation, required to hold for all functions F on T ∗ Emb(S, Rn ), that is, functions of Q and P: {F, J, ξ } = ξP [F ].

(4.3)

In our case, we shall take J to be given by the solution ansatz and verify that it satisﬁes this relation. To do so, let ξ ∈ X so that the left side of (4.3) becomes 4 5 δF i δF δ i F, Pi (s)ξ i (Q(s))d k s = ξ (Q(s)) − P (s) ξ (Q(s)) d k s. i i δPj δQj S S δQ On the other hand, one can directly compute from the deﬁnitions that the inﬁnitesimal generator of the action on the space T ∗ Emb(S, Rn ) corresponding to the vector ﬁeld ∂ ξ i (x) ∂Q i (a Lie algebra element), is given by (see Marsden and Ratiu [1999], formula (12.1.14)): ∂ i ξ (Q(s)), δQ = ξ ◦ Q, δP = −Pi (s) ∂Q which veriﬁes that (4.3) holds. An important element left out in this proof so far is that it does not make clear that the momentum map is equivariant, a condition needed for the momentum map to be Poisson. The ﬁrst proof took care of this automatically since momentum maps for cotangent lifted actions are always equivariant and hence Poisson. Thus, to complete the second proof, we need to check directly that the momentum map is equivariant. Actually, we shall only check that it is inﬁnitesimally invariant by showing that it is a Poisson map from T ∗ Emb(S, Rn ) to the space of m’s (the dual of the Lie algebra of X) with its Lie–Poisson bracket. This sort of approach to characterize equivariant momentum maps is discussed in an interesting way in Weinstein [2002]. The following computation accomplishes this methodology by showing directly that the singular solution momentum map is Poisson. To do this we use the canonical

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Poisson brackets for {P}, {Q} and apply the chain rule to compute {mi (x), mj (y)}. Using the notation δk (y) ≡ ∂δ(y)/∂y k , we get {mi (x), mj (y)} +N 3 N a a b b = dsPi (s, t)δ(x − Q (s, t)), ds Pj (s , t)δ(y − Q (s , t)) a=1

=

N

b=1

dsds {Pia (s), Pjb (s )}δ(x − Qa (s))δ(y − Qb (s ))

a,b=1

− {Pia (s), Qbk (s )}Pjb (s )δ(x − Qa (s))δk (y − Qb (s )) − {Qak (s), Pjb (s )}Pia (s)δk (x − Qa (s))δ(y − Qb (s ))

+ {Qak (s), Qb (s )}Pia (s)Pjb (s )δk (x − Qa (s))δ (y − Qb (s )) .

Substituting the canonical Poisson bracket relations {Pia (s), Pjb (s )} = 0, {Qak (s), Qb (s )} = 0, and {Qak (s), Pjb (s )} = δ ab δkj δ(s − s ) into the preceding computation yields {mi (x), mj (y)} +N 3 N = dsPia (s, t)δ(x − Qa (s, t)), ds Pjb (s , t)δ(y − Qb (s , t)) a=1

=

N

b=1

dsPja (s)δ(x − Qa (s))δi (y − Qa (s))

a=1

−

N

dsPia (s)δj (x − Qa (s))δ(y − Qa (s))

a=1

∂ ∂ = − mj (x) i + j mi (x) δ(x − y). ∂x ∂x ∂ ∂ {mi (x), mj (y)} = − mj (x) i + j mi (x) δ(x − y), ∂x ∂x which is readily checked to be the Lie–Poisson bracket on the space of m’s. This completes the second proof of theorem.

Thus

Each of these proofs has shown the following basic fact. Corollary 4.2. The singular solution momentum map deﬁned by the singular solution ansatz, namely,

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JSing : T ∗ Emb(S, Rn ) → X(Rn )∗ is a Poisson map from the canonical Poisson structure on T ∗ Emb(S, Rn ) to the Lie–Poisson structure on X(Rn )∗ . This is perhaps the most basic property of the singular solution momentum map. Some of its more sophisticated properties are outlined in the following section. Pulling back the equations Since the solution ansatz (2.8) has been shown in the preceding corollary to be a Poisson map, the pullback of the Hamiltonian from X∗ to T ∗ Emb(S, Rn ) gives equations of motion on the latter space that project to the equations on X∗ . Thus the basic fact that the momentum map JSing is Poisson explains why the functions Qa (s, t) and Pa (s, t) satisfy canonical Hamiltonian equations. Note that the coordinate s ∈ Rk that labels these functions is a “Lagrangian coordinate’’ in the sense that it does not evolve in time but rather labels the solution. In terms of the pairing ·, · : g∗ × g → R, (4.4) between the Lie algebra g (vector ﬁelds in Rn ) and its dual g∗ (one-form densities in Rn ), the following relation holds for measure-valued solutions under the momentum map (2.8), m, u = m · ud n x, L2 pairing for m and u ∈ Rn , =

N

(Pa (s, t) · Pb (s , t))G(Qa (s, t) − Qb (s , t))dsds

a,b=1

=

N

Pa (s, t) ·

a=1

˙ ≡ P, Q,

∂Qa (s, t) ds ∂t (4.5)

˙ ∈ which is the natural pairing between the points (Q, P) ∈ T ∗ Emb(S, Rn ) and (Q, Q) T Emb(S, Rn ). The pullback of the Hamiltonian H [m] deﬁned on the dual of the Lie algebra g∗ , to T ∗ Emb(S, Rn ) is easily seen to be consistent with what we had before: 1 1 m, G ∗ m = P, G ∗ P ≡ HN [P, Q]. (4.6) 2 2 In summary, in concert with the Poisson nature of the singular solution momentum map, we see that the singular solutions in terms of Q and P satisfy Hamiltonian equations and also deﬁne an invariant solution set for the EPDiff equations. In fact, H [m] ≡

this invariant solution set is a special coadjoint orbit for the diffeomorphism group, as we shall discuss in the next section.

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Remark. It would be extremely interesting if the smoothness properties explored in Section 3 were also valid on the space T ∗ Emb(S, Rn ). This is obviously valid for the CH equation.

5 Geometry of the momentum map In this section we explore the geometry of the singular solution momentum map discussed in Section 4 in a little more detail. The approach may be stated as follows: simply apply all of the ideas given in Marsden and Weinstein [1983] in a systematic way to the current setting. As in that paper, the treatment is formal, in the sense that there are a number of technical issues in the inﬁnite-dimensional case that are left open. We will mention a few of these as we proceed. Coadjoint orbits We claim that the image of the singular solution momentum map is a coadjoint orbit in X∗ . This means that (modulo some issues of connectedness and smoothness, which we do not consider here) the solution ansatz given by (2.8) deﬁnes a coadjoint orbit in the space of all one-form densities, regarded as the dual of the Lie algebra of the diffeomorphism group. These coadjoint orbits should be thought of as singular orbits—that is, due to their special nature, they are not generic. Recognizing them as coadjoint orbits is one way of gaining further insight into why the singular solutions form dynamically invariant sets—it is a general fact that coadjoint orbits in g∗ are symplectic submanifolds of the Lie–Poisson manifold g∗ (in our case X(Rn )∗ ) and, correspondingly, are dynamically invariant for any Hamiltonian system on g∗ . The idea of the proof of our claim is simply this: whenever one has an equivariant momentum map J : P → g∗ for the action of a group G on a symplectic or Poisson manifold P , and that action is transitive, then the image of J is an orbit (or at least a piece of an orbit). This general result, due to Kostant, is stated more precisely in Marsden and Ratiu [1999], Theorem 14.4.5. Roughly speaking, the reason that transitivity holds in our case is because one can “move the images of the manifolds S around at will with arbitrary velocity ﬁelds’’ using diffeomorphisms of Rn . Symplectic structure on orbits Recall (from, for example, Marsden and Ratiu [1999]), the general formula for the symplectic structure on coadjoint orbits:

µ (ξg∗ (µ), ηg∗ (µ)) = µ, [ξ, η],

(5.1)

where µ ∈ g∗ is a chosen point on an orbit and where ξ, η are elements of g. We use a plus sign in this formula since we are dealing with orbits for the right action.

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As in Marsden and Weinstein [1983], this line of investigation leads to an explicit formula for the coadjoint orbit symplectic structure in the case of Diff . In the present case, it is a particularly simple and transparent formula. Recall that in the case of incompressible ﬂuid mechanics, this procedure leads naturally to the symplectic (and Poisson) structure for many interesting singular coadjoint orbits, such as point vortices in the plane, vortex patches, vortex blobs (closely related to the planar LAE-α equations) and vortex ﬁlaments. An important point is that this structure is independent of how these solutions are parametrized. For the case of the diffeomorphism group, let Om denote the coadjoint orbit through the point m ∈ X∗ (Rn ). Theorem 5.1. The symplectic structure m on Tm Om is given by

m (£u1 m, £u2 m) = − m, [u1 , u2 ]d n x. Proof. We substitute into the general Kirillov–Kostant–Souriau formula (5.1) for the symplectic structure on coadjoint orbits. (As noted above, there is a + sign, since we are dealing with a right invariant system). The only thing needing explanation is that our Lie algebra convention always uses the left Lie bracket. For Diff , this is the negative of the usual Lie bracket, as explained in Marsden and Ratiu [1999].

The momentum map JS and the Kelvin circulation theorem The momentum map JSing involves Diff (Rn ), the left action of the diffeomorphism group on the space of embeddings Emb(S, Rn ) by smooth maps of the target space Rn , namely, Diff (Rn ) : Q · η = η ◦ Q, (5.2) where we recall that Q : S → Rn . As above, the cotangent bundle T ∗ Emb(S, Rn ) is identiﬁed with the space of pairs of maps (Q, P), with Q : S → Rn and P : S → T ∗ Rn . However, there is another momentum map JS associated with the right action of the diffeomorphism group of S on the embeddings Emb(S, Rn ) by smooth maps of the “Lagrangian labels’’ S (ﬂuid particle relabeling by η : S → S). This action is given by Diff (S) : Q · η = Q ◦ η. (5.3) The inﬁnitesimal generator of this right action is d XEmb(S,Rn ) (Q) = Q ◦ ηt = T Q ◦ X, dt t=0

(5.4)

where X ∈ X is tangent to the curve ηt at t = 0. Thus, again taking N = 1 (so we suppress the index a) and also letting αq in the momentum map formula (4.2) be the cotangent vector (Q, P), one computes JS : JS (Q, P), X = (Q, P), T Q · X

Measure-valued momentum map

=

Pi (s) S

227

∂Qi (s) m X (s)d k s ∂s m

=

X(P(s) · dQ(s))d k s k P(s) · dQ(s) ⊗ d s, X(s) = S

S

= P · dQ, X. Consequently, the momentum map formula (4.2) yields JS (Q, P) = P · dQ,

(5.5)

with the indicated pairing of the one-form density P · dQ with the vector ﬁeld X. We have set things up so that the following is true. Proposition 5.2. The momentum map JS is preserved by the evolution equations (2.12) for Q and P. Proof. It is enough to notice that the Hamiltonian HN in equation (2.13) is invariant under the cotangent lift of the action of Diff (S); it merely amounts to the invariance of the integral over S under reparametrization, that is, the change of variables formula; keep in mind that P includes a density factor.

This result is similar to the Kelvin–Noether theorem for circulation of an ideal 2 ﬂuid, which may be written as = c(s) D(s)−1 P(s) · dQ(s) for each Lagrangian circuit c(s), where D is the mass density and P is again the canonical momentum density. This similarity should come as no surprise, because the Kelvin–Noether theorem for ideal ﬂuids arises from invariance of Hamilton’s principle under ﬂuid parcel relabeling by the same right action of the diffeomorphism group, as in (5.3). Note that, being an equivariant momentum map, the map JS , as with JSing , is also a Poisson map. That is, substituting the canonical Poisson bracket into relation (5.5); that is, the relation M(x) = i Pi (x)∇Qi (x) yields the Lie–Poisson bracket on the space of M’s. We use the different notations m and M because these quantitites are analogous to the body and spatial angular momentum for rigid body mechanics. In fact, the quantity m is given by the solution Ansatz; speciﬁcally, m = JSing (Q, P) gives the singular solutions of the EPDiff equations, while M(x) = JS (Q, P) = i i Pi (x)∇Q (x) is a conserved quantity. In the language of ﬂuid mechanics, the expression of m in terms of (Q, P) is an example of a “Clebsch representation,’’ which expresses the solution of the EPDiff equations in terms of canonical variables that evolve by standard canonical Hamilton equations. This has been known in the case of ﬂuid mechanics for more than 100 years. For modern discussions of the Clebsch representation for ideal ﬂuids, see, for example, Holm and Kupershmidt [1983]; Marsden and Weinstein [1983]. One more remark is in order, namely, that the special case in which S = M is of course allowed. In this case, Q corresponds to the map η itself and P just corresponds to its conjugate momentum. The quantity m corresponds to the spatial

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(dynamic) momentum density (that is, right translation of P to the identity), while M corresponds to the conserved “body’’ momentum density (that is, left translation of P to the identity). Dual pairs For reasons that are similar to those for incompressible ﬂuids presented in Marsden and Weinstein [1983], the singular solution momentum map JSing : T ∗ Emb(S, Rn ) → X(Rn )∗ forms one leg of a formal dual pair. We use the words formal dual pair since, in the inﬁnite-dimensional case, the proper mathematical underpinnings for the theory of dual pairs has not yet been developed. Even in the ﬁnite-dimensional case, there are nontrivial issues to be aware of since dual pairs were studied in the basic paper of Weinstein [1983b]; we refer to Ortega and Ratiu [2004], Chapter 11, as well as Blaom [2001] and Ortega [2003] for background, references and a summary of the current state-of-the-art in this topic. These works show the subtlety of the dual pairs notion, even in ﬁnite dimensions, let alone for the inﬁnite-dimensional problem we are dealing with here. The point is that, as we have seen, there is another group that acts on Emb(S, Rn ), namely, the group Diff (S) of diffeomorphisms of S, which acts on the right, while Diff (Rn ) acted by composition on the left (and this gave rise to our singular solution momentum map JSing ). As explained above, the action of Diff (S) from the right gives us the momentum map JS : T ∗ Emb(S, Rn ) → X(S)∗ . We now assemble both momentum maps into one ﬁgure as follows: T ∗ Emb(S, M) JSing X(M)∗

@

@ JS @ R @ X(S)∗

These maps have the formal dual pair property, namely, that the kernel of the derivatives of each map at a given point are symplectic orthogonals of one another (see Weinstein [1983a]). Sometimes, as in Ortega and Ratiu [2004], this is called the Lie–Weinstein property. Formal proof of the dual pair property. The reduction lemma of Marsden and Weinstein [1974] states that the kernel of the derivative of a momentum map is the symplectic orthogonal of the group orbit. As a consequence, if the group associated to each leg of a potential dual pair of momentum maps acts transitively on the level set of its partner momentum map, then one has a dual pair. In our case, ﬁxing JS at a value, say M, means, according to equation (5.5), that we ﬁx the value of P · dQ. However, as we already noted above, the group Diff (M) acts transitively on the space of Q’s because one can “move the singular surfaces around at will’’ by means of diffeomorphisms of M. The constraint of ﬁxing P · dQ is exactly what

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one needs to transform the P’s properly by means of the cotangent lift (recall that cotangent lift actions are characterized by preserving the canonical one-form). Thus, at least formally, Diff (M) acts transitively on level sets of JS . Similarly, one sees that Diff (S) acts transitively on level sets of JSing since ﬁxing JSing corresponds to ﬁxing the image surface, leaving one only with the parametrization freedom, so that Diff (S) acts transitively on that set.

This is a marvelous framework; it clariﬁes, among other things, the fact that the parameterization of the singular solutions m in terms of Q and P are Clebsch variables in the sense given in Marsden and Weinstein [1983] and that the diffeomorphism group of S corresponds to the gauge group of that Clebsch representation. Also, notice that when we write the singular solutions in Q-P space, we are ﬁnding solutions that are collective and so all the properties of collectivization are valid. See Marsden and Ratiu [1999] for a general discussion and references to the original work of Guillemin and Sternberg on this topic. As explained in Marsden and Weinstein [1983], reduction by the group associated with one leg in a dual pair corresponds to coadjoint orbits on the other leg. Thus the momentum map JS captures the analog of the Kelvin circulation theorem of ﬂuid mechanics, as well as its singular version. It would be interesting to explore in more detail this singular analog of the Kelvin circulation theorem for ﬂuids. The ideal ﬂuid dual pair In Marsden and Weinstein [1983] a dual pair was investigated that captures certain singular situations in ﬂuid mechanics, such as the motion of point vortices in the plane. The general dual pair that generalizes that planar case is the following: Emb(S, P ) @

JP X(P )∗sym

@ JS @ R @ X(S)∗div

Here, S is a volume manifold , that is a manifold with a volume element, while P is a symplectic manifold . The map JP is the momentum map for the left action of the group of symplectic diffeomorphisms on P , while JS is the momentum map for the right action of the group of volume-preserving diffeomorphisms. This is a very beautiful dual pair, but is not the same, of course, as the dual pair we found above. In our case, P = T ∗ M was a cotangent bundle and we identiﬁed Emb(S, P ) with T ∗ Emb(S, M). Also, we always dealt with symplectic diffeomorphisms that were cotangent lifts, while in the above dual pair of Marsden and Weinstein [1983], they are general symplectic diffeomorphisms.

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Relation with ideas of Donaldson Whereas Marsden and Weinstein [1983] focused on ideas related to the dual pair picture, the leg in the above diagram given by the map JS is discussed from various interesting viewpoints in Donaldson [1999]. In particular, that work (apparently being unaware of the above dual pair of Marsden and Weinstein [1983]) makes a number of intriguing comments, including noting that nontrivial topology can cause the momentum map JS to be nonequivariant (so there are links with the Calabi theory). Donaldson also studies reduction and makes links with Kähler geometry, hyper-Kähler geometry, double bracket gradient ﬂows, mean curvature ﬂows, and other intriguing topics. As pointed out by Tony Bloch, there are also interesting links of Donaldson’s paper with the Toda ﬂow and factorization problems. We believe that the pursuit of further connections within the circle of ideas for momentum maps should be a source of inspiration for new research in the context of this paper. For example, this pursuit may lead to new insight into the integrability of the dynamical systems governing these singular solutions. It seems that we are looking at the tip of a rather large and beautiful iceberg.

6 Challenges, future directions, and speculations Numerical issues: Geometric integrators The computations of Martin Staley that illustrated several points in this paper are discussed in detail by Holm and Staley [2003, 2004]. These computations make use of both mimetic differencing and reversibility in a critical way, and this is important for accurate numerical simulations. In other words, integrators that respect the basic geometry underlying the problem obtained accurate singular solutions in numerical simulations. It would be interesting to pursue this aspect further and also incorporate discrete exterior calculus and variational multisymplectic integration methods (see Desbrun et al. [2003] as well as Marsden et al. [1998] and Lew et al. [2003]). Analytical issues: Geodesic incompleteness of H 1 EPDiff The emergence in ﬁnite time of singular solutions from smooth initial data observed numerically in Holm and Staley [2003, 2004] indicates that the diffeomorphism group with respect to the right invariant H 1 metric is geodesically incomplete when the diffeomorphism group has the H s topology s > (n/2) + 1. The degree of its geodesic incompleteness is not known, but we suspect that almost all EPDiff geodesics in H 1 cannot be extended indeﬁnitely. This certainly holds in one spatial dimension, where the discreteness of the CH isospectrum implies that asymptotically in time the CH solution arising from any conﬁned smooth initial velocity data consists only of peakons. It is an important challenge to ﬁnd a context in which one can put the H 1 topology on the diffeomorphism group and reestablish geodesic completeness. The numerics suggests that this might be possible, while known existence theorems, even for the CH equation are not yet capable of showing this—to the best of our knowledge.

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Reversible reconnections of the singular EPDiff solutions EPDiff is a reversible equation, and the collisions of its peakon solutions on the line R1 (or the circle S 1 ) are known to be reversible. The reconnections of the singular EPDiff solutions observed numerically in Holm and Staley [2003, 2004] in periodic domains T2 and T3 are also reversible, and this was used as a test of the numerical method. Reversibility of its reconnections distinguishes the singular solutions of EPDiff from vortex ﬂuid solutions and shocks in ﬂuids, whose reconnections apparently require dissipation, and so are not reversible. The mimetic ﬁnite differencing scheme used for the numerical computation of EPDiff solutions in Holm and Staley [2003, 2004] was indeed found to be reversible for overtaking collisions, but it was found to be only approximately reversible for head-on collisions, which are much more challenging for numerical integration schemes. Applications of EPDiff singular solutions in image processing The singular EPDiff solutions correspond to outlines (or cartoons) of images in applications of geodesic ﬂow for the template, or pattern matching approach. The dynamics of the singular EPDiff solutions described by the momentum map (2.8) introduces the paradigm of momentum exchange in soliton collisions into the mechanics and analysis of image processing by template matching. (See Holm, Trouvé, Ratnanather, and Younes [2004] for more discussions of this new paradigm for image processing.) First, the momentum representation of the image outlines is nonredundant. That is, the momentum has exactly the same dimension as the matched structures; so there is no redundancy of the representation. Second, the reversibility of the collisions among singular solutions and their reconnections under EPDiff ﬂow assures the preservation of the information contained in the image outlines. In addition, the invariance of the manifold of N singular solutions under EPDiff assures that the ﬁdelity of the image is preserved in the sense of approximation theory. That is, an N soliton approximation of the image outlines remains so, throughout the EPDiff ﬂow. A natural approach for numerically simulating EPDiff ﬂows in image processing is to use multisymplectic algorithms. The preservation of the space–time multisymplectic form by these algorithms introduces an initial-value, ﬁnal-value formulation of the numerical solution procedure that is natural for template matching. Rigorous Poisson structures In Vasylkevych and Marsden [2003], the question of the (rigorous) Poisson nature of the time t map of the ﬂow of the Euler equations for an ideal ﬂuid in appropriate Sobolev spaces is explored. Given the smoothness properties in Section 3, it seems reasonable that similar properties should also hold for the EPDiff equations. However, as mentioned earlier, these smoothness properties do not preclude the emergence of singular solutions from smooth initial data in ﬁnite time, because of the possibility for geodesic incompleteness.

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Other groups The general setting of this paper suggests that perhaps one should look for similar measure-valued or singular solutions associated with other problems, including geodesic ﬂows on the group of symplectic diffeomorphisms (relevant for plasma physics, as in Marsden and Weinstein [1982]), Bott–Virasoro central extensions and supersymmetry groups. Scattering It might be interesting to explore the relation of the singular solution momentum map (2.8) to integrability and scattering data. For example, see Vaninsky [2003] for an interesting discussion of the Poisson bracket for the scattering data of CH in 1D. This turns out to be the Atiyah–Hitchin bracket, which is also related to the Toda lattice, and this fascinating observation leads to an inﬁnite-dimensional version of Jacobi elliptic coordinates. Other issues Of course, there are many other issues remaining to explore that are suggested by the above setting, such as convexity of the momentum map, its extension to Riemannian manifolds, and so on. We shall, however, leave these issues for other publications and other researchers. Acknowledgments We are very grateful to Alan Weinstein for his collaboration, help and inspiring discussions over the years. We thank Martin Staley for letting us illustrate some important points using his computations. We also thank Anthony Bloch, Simon Donaldson, Jonathan Munn, Tudor Ratiu, and Richard Thomas for valuable advice and comments.

References M. Alber, R. Camassa, Y. Fedorov, D. Holm, and J. E. Marsden [2001], The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type, Comm. Math. Phys., 221, 197–227. M. S. Alber and J. E. Marsden [1992], On geometric phases for soliton equations, Comm. Math. Phys., 149, 217–240. V. I. Arnold [1966], Sur la géométrie différentielle des groupes de Lie de dimenson inﬁnie et ses applications à l’hydrodynamique des ﬂuids parfaits, Ann. Inst. Fourier Grenoble, 16, 319–361. A. D. Blaom [2001], A Geometric Setting for Hamiltonian Perturbation Theory, Memoirs of the American Mathematical Society, Vol. 727, Americal Mathematical Society, Providence, RI. R. Camassa and D. D. Holm [1993],An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661–1664.

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Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart, Gerstenhaber, and Batalin–Vilkovisky algebras Johannes Huebschmann Université des Sciences et Technologies de Lille UFR de Mathématiques, CNRS-UMR 8524 F-59 655 Villeneuve d’Ascq Cédex France [email protected] Dedicated to Alan Weinstein on the occasion of his 60th birthday. Abstract. Higher homotopy generalizations of Lie–Rinehart algebras, Gerstenhaber, and Batalin–Vilkovisky algebras are explored. These are deﬁned in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer–Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector ﬁelds on the “space of leaves’’ and for the algebra of multivector ﬁelds are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including the Hodge–de Rham spectral sequence.

Introduction In this paper we will explore, in the framework of Lie–Rinehart algebras and suitable higher homotopy generalizations thereof, various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as similar operations involving more than two variables; such operations have recently arisen in algebra, differential geometry, and mathematical physics but are lurking already behind a number of classical developments. Our aim is to somewhat unify these structures by means of the relationship between Lie–Rinehart, Gerstenhaber, and Batalin–Vilkovisky algebras which we ﬁrst observed in Huebschmann [1998a]. This will be, perhaps, a ﬁrst step towards taming the bracket zoo that arose recently in topological ﬁeld theory; see what we wrote in the introduction to Huebschmann [1998a]. The notion of Lie–Rinehart algebra and its generalization are likely to provide a good conceptual

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framework for that purpose. It will also relate new notions like those of Gerstenhaber and Batalin–Vilkovisky algebra, and generalizations thereof, with classical ones like those of connection, curvature, and torsion, as well as with less classical ones like the triple product isolated by Yamaguti [1957–1958] and operations of the kind introduced by Kinyon and Weinstein [2001]; it will connect new developments with old results due to Cartan [1927] and Nomizu [1954] describing the geometry of Lie groups and of reductive homogeneous spaces and, more generally, with more recent results in the geometry of Lie loops; see Kikkawa [1975] and Sabinin and Mikheev [1988]. We will see that the new structures have incarnations of a mathematical nature, e.g., in the theory of foliations. The higher homotopies which are exploited below are of a special kind, where only the ﬁrst of an (in general) inﬁnite family is nonzero. Let R be a commutative ring with 1. A Lie–Rinehart algebra (A, L) consists of a commutative R-algebra A, an R-Lie algebra L, an A-module structure on L, and an action L ⊗R A → A of L on A by derivations. These are required to satisfy suitable compatibility conditions which arise by abstraction from the pair (A, L) = (C ∞ (M), Vect(M)) consisting of the smooth functions C ∞ (M) and smooth vector ﬁelds Vect(M) on a smooth manifold M. A complete deﬁnition will be recalled in Section 1 below. In Huebschmann [1990], [1991a], [1998a], [1998b], [1999a], [1999b], [2000], [2001], [2002] we studied these objects and variants thereof and used them to solve various problems in algebra and geometry. See Huebschmann [2003] for a survey and leisurely introduction. In differential geometry, a special case of a Lie–Rinehart algebra arises from the space of sections of a Lie algebroid. In Huebschmann [1998a], [1998b], and [2000] we have shown that certain Gerstenhaber and Batalin–Vilkovisky algebras admit natural interpretations in terms of Lie–Rinehart algebras. The starting point was the following observation: It is nowadays well understood that a skew-symmetric bracket on a vector space g is a Liebracket (i.e., satisﬁes the Jacobi identity) if and only if the coderivation ∂ on the graded exterior coalgebra [sg] corresponding to the bracket on g has square zero, i.e., is a differential; this coderivation is then the ordinary Lie algebra homology operator. This kind of characterization is not available for a general Lie–Rinehart algebra: Given a commutative algebra A and an A-module L, a Lie–Rinehart structure on (A, L) cannot be characterized in terms of a coderivation on A [sL] with reference to a suitable coalgebra structure on A [sL] (unless the L-action on A is trivial); in fact, in the Lie–Rinehart context, a certain dichotomy between A-modules and chain complexes which are merely deﬁned over R persists thoughout; see, e.g., the Remark 2.5.2 below. On the other hand, Lie–Rinehart algebra structures on (A, L) correspond to Gerstenhaber algebra structures on the exterior A-algebra A [sL]; see, e.g., KosmannSchwarzbach [1995]. In particular, when A is the ground ring and L just an ordinary Lie algebra g, under the obvious identiﬁcation of [sg] and [sg] as graded Rmodules, the (uniquely determined) generator of the Gerstenhaber bracket on [sg] is exactly the Lie algebra homology operator on [sg]. Given a general commutative algebra A and an A-module L, the interpretation of Lie–Rinehart algebra structures on (A, L) in terms of Gerstenhaber algebra structures on A [sL] provides, among other things, a link between Gerstenhaber [1963] and Rinehart [1963] (which seems

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to have been completely missed in the literature). In the present paper, we will extend this link to suitable higher homotopy notions which we refer to by the attribute “quasi’’; we will introduce Lie–Rinehart triples, quasi-Lie–Rinehart algebras, and certain quasi-Gerstenhaber algebras and quasi-Batalin–Vilkovisky algebras, and we will explore the various relationships between these notions. Below we will comment on the relationship with notions of quasi-Gerstenhaber and quasi-Batalin–Vilkovisky algebras already in the literature. When an algebraic structure (e.g., a commutative algebra, Lie algebra, etc.) is “resolved’’ by an object, which we here somewhat vaguely refer to as a “resolution’’ (free, or projective, or variants thereof) having the given structure as its zero-th homology, on the resolution, the algebraic structure is in general deﬁned only up to higher homotopies; likewise, an A∞ structure is deﬁned in terms of a bar construction or variants thereof; see, e.g., Huebschmann and Kadeishvili [1991], Huebschmann and Stasheff [2002], and the references there. Exploiting higher homotopies of this kind, in a series of articles we constructed small free resolutions for certain classes of groups from which we then were able to do explicit calculations in group cohomology which up to now still cannot be done by other methods; see Huebschmann [1989a], [1989b], [1989c], [1991b]. A historical overview related with A∞ -structures may be found in the Addendum to Keller [2001]; see also Huebschmann [1999c] and Huebschmann and Kadeishvili [1991] for more historical comments. In this paper, we will explore a certain higher homotopy related with Lie–Rinehart algebras and variants thereof. A Lie algebra up to higher homotopies (equivalently: L∞ -algebra) on an R-chain complex h may be deﬁned in terms of a coalgebra perturbation of the differential on the graded symmetric coalgebra on the suspension of h; alternatively, it may be deﬁned in terms of a suitable Maurer–Cartan algebra (see below). Since a genuine Lie–Rinehart structure on (A, L) cannot be characterized in terms of a coderivation on A [sL], the ﬁrst alternative breaks down for a general Lie–Rinehart algebra. The higher homotopies we will explore in this paper do not live on an object close to a resolution of the above kind or close to a symmetric coalgebra; they may conveniently be phrased in terms of an object of a rather different nature which, extending terminology introduced by van Est [1989], we refer to as a Maurer–Cartan algebra. A special case thereof arises in the following fashion: Given a ﬁnite dimensional vector space g over a ﬁeld k, skew symmetric brackets on g correspond bijectively to degree −1 derivations of the graded algebra of alternating forms on g (with reference to multiplication of forms), and those brackets which satisfy the Jacobi identity correspond to square zero derivations, i.e., differentials. This observation generalizes to Lie–Rinehart algebras of the kind (A, L) under the assumption that L be a ﬁnitely generated projective A-module; see Theorem 2.2.16 below. For an ordinary Lie algebra g over a ﬁeld k, in van Est [1989], the resulting differential graded algebra Alt(g, k) (which calculates the cohomology of g) has been called Maurer– Cartan algebra. The main point of this paper is that higher homotopy variants of the notion of Maurer–Cartan algebra provide the correct framework to phrase certain higher homotopy versions of Lie–Rinehart, Gerstenhaber, and Batalin–Vilkovisky

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algebras to which we will refer to as quasi-Lie–Rinehart-, quasi-Gerstenhaber, and quasi-Batalin–Vilkovisky algebras. The differential graded algebra of alternating forms on a Lie algebra occurs, at least implicitly, in Cartan [1929] and has a long history of use since then (see Koszul [1985]); once I learned in a talk by van Est that this algebra has been used by E. Cartan in the 1930s to characterize the structure of Lie groups and Lie algebras. For the reader’s convenience, we will explain brieﬂy and somewhat informally a special case of a quasi-Lie–Rinehart algebra at the present stage: Let (M, F) be a foliated manifold, the foliation being written as F, let τF be the tangent bundle of the foliation F, and choose a complement ζ of τF so that the tangent bundle τM of M may be written as τM = τF ⊕ ζ . Let LF ⊆ Vect(M) be the Lie algebra of smooth vector ﬁelds tangent to the foliation F, and let Q be the C ∞ (M)-module (ζ ) of smooth sections of ζ . The Lie bracket in Vect(M) induces a left LF -module structure on Q—the Bott connection— and the space QLF of invariants, that is, of vector ﬁelds on M which are horizontal (with respect to the decomposition τM = τF ⊕ ζ ) and constant on the leaves inherits a Lie bracket. The standard complex A arising from a ﬁne resolution of the sheaf of germs of functions on M which are constant on the leaves acquires a differential graded algebra structure and has H0 (A) equal to the algebra of functions on M which are constant on the leaves, and the Lie algebra QLF of invariants arises as H0 (Q), where Q is the complex coming from a ﬁne resolution of the sheaf VQ of germs of vector ﬁelds on M which are horizontal (with respect to the decomposition (τM ) = LF ⊕ Q) and constant on the leaves. In a sense, QLF is the Lie algebra of vector ﬁelds on the “space of leaves,’’ that is, the space of sections of a certain geometric object which may be seen as a replacement for the in general nonexistant tangent bundle of the “space of leaves.’’ Within our approach, this philosophy is pushed further in the following fashion: The pair (A, Q) acquires what we will call a quasi-Lie–Rinehart structure in an obvious fashion; see (4.12) and (4.15) below for the details. We view A as the algebra of generalized functions and Q as the generalized Lie algebra of vector ﬁelds for the foliation. The pair (H0 (A), H0 (Q)) is necessarily a Lie–Rinehart algebra, and the entire cohomology (H∗ (A), H∗ (Q)) acquires a graded Lie–Rinehart algebra structure. As a side remark, we note that here the resolution of the sheaf VQ is by no means a projective one; indeed, it is a ﬁne resolution of that sheaf, the bracket on Q is not an ordinary Lie(-Rinehart) bracket, in particular, does not satisfy the Jacobi identity, and the entire additional structure is encapsulated in certain homotopies which may conveniently be phrased in terms of a suitable Maurer–Cartan algebra that here arises from the de Rham algebra of M. When the foliation does not come from a ﬁber bundle, the structure of the graded Lie–Rinehart algebra (H∗ (A), H∗ (Q)) will in general be more complicated than in the case when the foliation comes from a ﬁber bundle. Thus the cohomology of a quasi-Lie–Rinehart algebra involves an ordinary Lie–Rinehart algebra in degree zero but in general contains considerably more information. In particular, in the case of a foliation it contains more than just “functions and vector ﬁelds on the space of leaves’’; the additional information partly includes the history of

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the “space of leaves,’’ that is, it includes information as to how this space arises from the foliation, how the leaves sit inside the ambient space, about singularities, etc. In Section 6 we will show that, when the foliation is transversely orientable with a basic transverse volume form ω, a corresponding quasi-Batalin–Vilkovisky algebra isolated in Theorem 6.10 below has an underlying quasi-Gerstenhaber algebra which, in turn, yields a kind of generalized Schouten algebra (generalized algebra of multivector ﬁelds) for the foliation; the cohomology of this quasi-Gerstenhaber algebra may then be viewed as the Schouten algebra for the “space of leaves.’’ See (6.15) below for details. Thus our approach will provide new insight, for example, into the geometry of foliations; see in particular (1.12), (2.10), (4.15), (6.15) below. The formal structure behind foliations which we will phrase in terms of quasi-Lie–Rinehart algebras and its offspring does not seem to have been noticed in the literature before—indeed, it involves, among a number of other things, a suitable grading which seems unfamiliar in the literature on quasi-Gerstenhaber and quasi-Batalin–Vilkovisky algebras (see (6.17) below)—or the formal connections with Yamaguti’s triple product and with Lie loops. A simpliﬁed version of the question we will examine is this: Given a Lie algebra g with a decomposition g = h ⊕ q, where h is a Lie subalgebra, what kind of structure does q then inherit? Variants of this question and possible answers may be found in a number of places in the literature; see, e.g., Cartan [1927], Nomizu [1954], where, in particular, in a global situation, an answer is given for reductive homogeneous spaces. In the framework of Lie–Rinehart algebras, this issue does not seem to have been raised yet, not even for the special case of Lie algebroids. As a by-product, we ﬁnd a certain formal relationship between Yamaguti’s triple product and certain forms ∗∗ which may be found in Koszul [1985]. In particular, the failure of a quasi-Gerstenhaber bracket to satisfy the Jacobi identity is measured by an additional piece of structure which we refer to as an h-Jacobiator; an h-Jacobiator, in turn, is deﬁned in terms of Koszul’s forms 3∗ . Likewise, the quadruple and quintuple products studied in Section 3 below are related with Koszul’s forms, and these, in turn, are related with certain higher order operations which may be found, e.g., in Sabinin and Mikheev [1988]. We do not pursue this here; we hope to eventually come back to it in another article. A Courant algebroid has been shown in Roytenberg and Weinstein [1998] to acquire an L∞ -structure, that is, a Lie algebra structure up to higher homotopies. The present paper paves, perhaps, the way towards ﬁnding a higher homotopy Lie– Rinehart or higher homotopy Lie algebroid structure on a Courant algebroid incorporating the Courant algebroid structure. Graded quasi-Batalin–Vilkovisky algebras have already been explored in Getzler [1995]. Our notions of quasi-Gerstenhaber and quasi-Batalin–Vilkovisky algebra, while closely related, do not coincide with those in Bangoura [2003], Bangoura [2002], Getzler [1995], Kosmann-Schwarzbach [2005], Roytenberg [2002]. In particular, our algebras are bigraded while those in the cited references are ordinary graded algebras; the appropriate totalization (forced, as noted above, by our application of the newly developed algebraic structure to foliations and written in Section 6 below

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as the functor Tot) of our bigraded objects leads to differential graded objects which are not equivalent to those in the quoted references. See Remark 6.17 below for more details on the relationship between the various notions. Also the approaches differ in motivation; the guiding idea behind Getzler [1995] and Kosmann-Schwarzbach [2005] seems to be Drinfeld’s quasi-Hopf algebras. Our motivation, as indicated above, comes from foliations and the search for appropriate algebraic notions encapsulating the inﬁnitesimal structure of the “space of leaves’’ and its history, as well as the search for a corresponding Lie–Rinehart generalization of the operations on a reductive homogenous space isolated by Nomizu and elaborated upon by Yamaguti (mentioned earlier) and taken up again by Kinyon and Weinstein [2001]. Indeed, this paper was prompted by the preprint versions of Kinyon and Weinstein [2001] and Weinstein [2000]. It is a pleasure to dedicate it to Alan Weinstein.

1 Lie–Rinehart triples Let R be a commutative ring with 1, not necessarily a ﬁeld; R could be, for example, the algebra of smooth functions on a smooth manifold; see Huebschmann [1999b]. Furthermore, let A be a commutative R-algebra. An (R, A)-Lie algebra (see Rinehart [1963]) is a Lie algebra L over R which acts on (the left of) A (by derivations) and is also an A-module satisfying the compatibility conditions [α, aβ] = α(a)β + a[α, β], (aα)(b) = a(α(b)), where a, b ∈ A and α, β ∈ L and where the operations of A on L and of L on A are written in terms of elements in the obvious fashion. When the emphasis is on the pair (A, L) with the mutual structure of interaction between A and L, we refer to the pair (A, L) as a Lie–Rinehart algebra. Given an ordinary Lie algebra g over R, the pair (R, g) is a Lie–Rinehart algebra in an obvious fashion. Given a smooth foliated manifold, when A denotes the algebra of smooth functions and H the Lie algebra of vector ﬁelds tangent to the foliation, the pair (A, H ) is a Lie–Rinehart algebra; under such circumstances it may be convenient to take a ground ring R larger than the reals; for example when the foliation comes from a ﬁber bundle, for certain problems (e.g., related with duality), the appropriate ground ring is the algebra of smooth functions on the base manifold; see Huebschmann [1999b]. More details and additional references may be found in our survey paper Huebschmann [2003]. The problem we wish to explore is the following. Question 1.1. Given a Lie–Rinehart algebra (A, L) and an A-module direct sum decomposition L = H ⊕ Q inducing an (R, A)-Lie algebra structure on H , what kind of structure does Q then inherit, and by what additional structure are H and Q related? Question 1.2. Given an (R, A)-Lie algebra structure on H and the (new) structure (which we will isolate below) on Q, what kind of additional structure turns the A-

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module direct sum L = H ⊕ Q into an (R, A)-Lie algebra in such a way that the latter induces the given structure on H and Q? Example 1.3.1. Let g be an ordinary R-Lie algebra with a decomposition g = h ⊕ q, where h is a Lie subalgebra. Recall that the decomposition of g is said to be reductive (see Nomizu [1954]) provided [h, q] ⊆ q. Such a reductive decomposition arises from a reductive homogeneous space; see Cartan [1927], Kikkawa [1975], Nomizu [1954], Sabinin and Mikheev [1988], Yamaguti [1957–1958]. For example, every homogeneous space of a compact Lie group or, more generally, of a reductive Lie group, is reductive. Nomizu [1954] has shown that on such a reductive homogeneous space, the torsion and curvature of the “canonical afﬁne connection of the second kind’’ (afﬁne connection with parallel torsion and curvature) yield a bilinear and a ternary operation which, at the identity, come down to a certain bilinear and ternary operation on the constituent q; Yamaguti [1957–1958] gave an algebraic characterization of pairs of such operations. Example 1.3.2. A quasi-Lie bialgebra (h, q) (see Kosmann-Schwarzbach [1992]) consists of a (real or complex) Lie algebra h and a (real or complex) vector space q with suitable additional structure where q = h∗ , so that g = h ⊕ h∗ is an ordinary Lie algebra; the pair (g, h) is occasionally referred to in the literature as a Manin pair. Quasi-Lie bialgebras arise as classical limits of quasi-Hopf algebras; these, in turn, were introduced by Drinfeld [1990]. Example 1.4.1. Let R be the ﬁeld R of real numbers, let (M, F) be a foliated manifold, let τF be the tangent bundle of the foliation F, and choose a complement ζ of τF so that the tangent bundle τM of M may be written as τM = τF ⊕ ζ . Thus, as a vector bundle, ζ is canonically isomorphic to the normal bundle of the foliation. Let (A, L) be the Lie–Rinehart algebra (C ∞ (M), Vect(M)), let H = LF ⊆ L be the (R, A)-Lie algebra of smooth vector ﬁelds tangent to the foliation F, and let Q be the A-module (ζ ) of smooth sections of ζ . Then L = H ⊕ Q is an A-module direct sum decomposition of the (R, A)-Lie algebra L, and the question arises as to what kind of Lie structure Q carries. This question, in turn, may be subsumed under the more general question to what extent the “space of leaves’’ can be viewed as a smooth manifold. This more general question is not only of academic interest since, for example, in certain physical situations, the true classical state space of a constrained system is the “space of leaves’’ of a foliation which is in general not ﬁbrating, and the Noether theorems are conveniently phrased in the framework of foliations. Example 1.4.2. Let R be the ﬁeld C of complex numbers, M a smooth complex manifold M, A the algebra of smooth complex functions on M, L the (C, A)-Lie algebra of smooth complexiﬁed vector ﬁelds, and let L and L be the spaces of smooth sections of the holomorphic and antiholomorphic tangent bundle of M, respectively. Then L and L are (C, A)-Lie algebras, and (A, L , L ) is a twilled Lie–Rinehart algebra in the sense of Huebschmann [1998b], Huebschmann [2000]. Adjusting the notation to that in Example 1.4.1, let H = L and Q = L . Thus, in this particular case, Q = L is in fact an ordinary (R, A)-Lie algebra, and the additional structure

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relating H and Q is encapsulated in the notion of twilled Lie–Rinehart algebra. The integrability condition for an almost complex structure may be phrased in terms of the twilled Lie–Rinehart axioms; see Huebschmann [1998b], Huebschmann [2000] for details. The situation of Example 1.4.1 is somewhat more general than that of Example 1.4.2 since in Example 1.4.1 the constituent Q carries a structure which is more general than that of an ordinary (R, A)-Lie algebra. Another example for a decomposition of the kind spelled out in Questions 1.1 and 1.2 above arises from combining the situations of Example 1.4.1 and of Example 1.4.2, that is, from a smooth manifold foliated by holomorphic manifolds, and yet another example arises from a holomorphic foliation. Abstracting from these examples, we isolate the notion of a Lie–Rinehart triple. For ease of exposition, we also introduce the weaker concepts of almost preLie–Rinehart triple and pre-Lie–Rinehart triple. Distinguishing among these three notions may appear pedantic but will clarify the statement of Theorem 2.7 below. See also Remark 2.8.4 below. Our usage of the term triple (to be isolated below) is not consistent with that of Manin triple in the literature. A Lie–Rinehart algebra involves a pair consisting of an algebra and a Lie algebra; in this context, it is also common in the literature to refer to this kind of structure as a pair (see Huebschmann [2003] for more details), and this usage of the term pair, in turn, is not consistent with the notion of a Manin pair. Below we will consider triples involving a commutative algebra A and two A-modules H and Q, just as in Example 1.4.1 and Example 1.4.2 above, and (A, H ⊕ Q) will be a Lie–Rinehart algebra if and only if (A, H, Q) is a Lie–Rinehart triple (so far unexplained). These remarks are intended to justify our terminology Lie–Rinehart triple. We now proceed towards the description of Lie–Rinehart triples. Let A be a commutative R-algebra. Consider two A-modules H and Q, together with • • •

skew-symmetric R-bilinear brackets of the kind (1.5.1.H ) and (1.5.1.Q) below, not necessarily Lie brackets; R-bilinear operations of the kind (1.5.2.H ), (1.5.2.Q), (1.5.3), (1.5.4) below; and a skew-symmetric A-bilinear pairing δ of the kind (1.5.5) below: [·, ·]H : H ⊗R H − → H, → Q, [·, ·]Q : Q ⊗R Q − H ⊗R A − → A, x ⊗R a → x(a), Q ⊗R A − → A, ξ ⊗R a → ξ(a), → Q, · : H ⊗R Q − · : Q ⊗R H − → H, δ : Q ⊗A Q − → H.

(1.5.1.H ) (1.5.1.Q) x ∈ H, a ∈ A,

(1.5.2.H )

ξ ∈ Q, a ∈ A,

(1.5.2.Q) (1.5.3) (1.5.4) (1.5.5)

We will say that the data (A, H, Q) constitute an almost pre-Lie–Rinehart triple provided they satisfy (i), (ii), and (iii) below.

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(i) The values of the adjoints H → End R (A) and Q → End R (A) of (1.5.2.H ) and (1.5.2.Q) respectively lie in Der R (A); (ii) (1.5.1.H ), (1.5.2.H ) and the A-module structure on H and, likewise, (1.5.1.Q), (1.5.2.Q) and the A-module structure on Q, satisfy the following Lie–Rinehart axioms (1.5.6.H ), (1.5.7.H ), (1.5.6.Q), (1.5.7.Q): (ax)(b) = a(x(b)), a, b ∈ A, x ∈ H, [x, ay]H = x(a)y + a[x, y]H , a ∈ A, x, y ∈ H, (aξ )(b) = a(ξ(b)), a, b ∈ A, ξ ∈ Q, [ξ, aη]Q = ξ(a)η + a[ξ, η]Q , a ∈ A, ξ, η ∈ Q;

(1.5.6.H ) (1.5.7.H ) (1.5.6.Q) (1.5.7.Q)

(iii) (1.5.3) and (1.5.4) behave like connections, that is, for a ∈ A, x ∈ H, ξ ∈ Q, the identities x · (aξ ) = (x(a))ξ + a(x · ξ ), (ax) · ξ = a(x · ξ ),

(1.5.8) (1.5.9)

ξ · (ax) = (ξ(a))x + a(ξ · x), (aξ ) · x = a(ξ · x),

(1.5.10) (1.5.11)

are required to hold. We will say that an almost pre-Lie–Rinehart triple (A, H, Q) is a pre-Lie–Rinehart triple provided that (i) (A, H ), endowed with the operations (1.5.1.H ) and (1.5.2.H ), is a Lie–Rinehart algebra—equivalently, the bracket (1.5.1.H ) satisﬁes the Jacobi identity—, and that (ii) the operation (1.5.3) turns Q into a left (A, H )-module, that is, the “connection’’ given by this operation is “ﬂat,’’ i.e., satisﬁes the identity [x, y]H · ξ = x · (y · ξ ) − y · (x · ξ ),

x, y ∈ H, ξ ∈ Q.

(1.5.12)

(1.5.13) Thus a pre-Lie–Rinehart triple (A, H, Q) consists of a Lie–Rinehart algebra (A, H ) (the structure of which is given by (1.5.1.H ), (1.5.2.H )) and a left (A, H )module Q (given by the operation (1.5.3) which, in turn, is required to satisfy the axioms (1.5.8) and (1.5.9)) together with the additional structure (1.5.1.Q), (1.5.2.Q), (1.5.4), (1.5.5) subject to the axioms (1.5.6.Q), (1.5.7.Q), (1.5.10), (1.5.11). Given an almost pre-Lie–Rinehart triple (A, H, Q), let L = H ⊕ Q be the Amodule direct sum, and deﬁne an R-bilinear skew-symmetric bracket →L [·, ·] : L ⊗R L −

(1.6.1)

by means of the formula [(x, ξ ), (y, η)] = [x, y]H + [ξ, η]Q + δ(ξ, η) + x · η − η · x + ξ · y − y · ξ (1.6.2) and, furthermore, an operation →A L ⊗R A −

(1.6.3)

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in the obvious way, that is, by means of the association (ξ, x) ⊗R a → ξ(a) + x(a),

x ∈ H, ξ ∈ Q, a ∈ A.

(1.6.4)

By construction, the values of the adjoint of (1.6.3) then lie in Der R (A), that is, this adjoint is then of the form L=H ⊕Q− → Der R (A).

(1.6.5)

An almost pre-Lie–Rinehart triple (A, H, Q) will be said to be a Lie–Rinehart triple if (1.6.1) and (1.6.3) turn (A, L), where L = H ⊕ Q, into a Lie–Rinehart algebra. A Lie–Rinehart triple (A, H, Q), where δ is zero, is a twilled Lie–Rinehart algebra in the sense of Huebschmann [1998b], Huebschmann [2000]. Thus Lie–Rinehart triples generalize twilled Lie–Rinehart algebras. A direct sum decomposition L = H ⊕ Q of an (R, A)-Lie algebra L such that (A, H ) inherits a Lie–Rinehart structure yields a Lie–Rinehart triple (A, H, Q) in an obvious fashion: The brackets (1.5.1.H ) and (1.5.1.Q) result from restriction and projection; the operations (1.5.2.H ) and (1.5.2.Q) are obtained by restriction as well; further, the requisite operations (1.5.3) and (1.5.4) are given by the composites [·,·]|H ⊗R Q

prQ

[·,·]|Q⊗R H

prH

· : H ⊗R Q −−−−−−→ H ⊕ Q −−→ Q and

· : Q ⊗R H −−−−−−→ H ⊕ Q −−→ H,

(1.7.1)

(1.7.2)

where, for M = H ⊗R Q and M = Q ⊗R H , [·, ·]|M denotes the restriction of the Lie bracket to M. The pairing (1.5.5) is the composite [·,·]|Q⊗R Q

prH

δ : Q ⊗A Q −−−−−−→ L = H ⊕ Q −−→ H ;

(1.7.3)

at ﬁrst it is only R-bilinear but is readily seen to be A-bilinear. The formula (1.6.2) is then merely a decomposition of the initially given bracket on L into components according to the direct sum decomposition of L into H and Q, and (1.6.3) is accordingly a decomposition of the L-action on A. Furthermore, given x, y ∈ H and ξ ∈ Q, in L we have the identity [x, y] · ξ − ξ · [x, y] = [[x, y], ξ ] = [x, [y, ξ ]] − [y, [x, ξ ]] = x · (y · ξ ) − (y · ξ ) · x − [x, ξ · y] − y · (x · ξ ) + (x · ξ ) · y − [ξ · x, y] which at once implies (1.5.12). Remark 1.8.1. Thus we see that, in particular, if an almost pre-Lie–Rinehart triple (A, H, Q) is a Lie–Rinehart triple, it is necessarily a pre-Lie–Rinehart triple; see (1.5.13). Remark 1.8.2. In the situation of Example 1.3.2, when g arises from a quasi-Lie bialgebra (so that q = h∗ ), in the literature, the piece of structure δ is often written as an element of 3 h.

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Theorem 1.9. A pre-Lie–Rinehart triple (A, H, Q) is a genuine Lie–Rinehart triple, that is, the bracket [·, ·] (see (1.6.1)) and the operation (1.6.3) turn (A, L), where L = H ⊕ Q, into a Lie–Rinehart algebra, if and only if the brackets [·, ·]H and [·, ·]Q on H and Q, respectively, and the operations (1.5.3), (1.5.4), and (1.5.5), are related by ξ(x(a)) − x(ξ(a)) = (ξ · x)(a) − (x · ξ )(a), x · [ξ, η]Q = [x · ξ, η]Q + [ξ, x · η]Q − (ξ · x) · η + (η · x) · ξ, ξ · [x, y]H = [ξ · x, y]H + [x, ξ · y]H

(1.9.1) (1.9.2)

− (x · ξ ) · y + (y · ξ ) · x,

(1.9.3)

ξ(η(a)) − η(ξ(a)) = [ξ, η]Q (a) + (δ(ξ, η))(a),

(1.9.4)

[ξ, η]Q · x = ξ · (η · x) − η · (ξ · x) − δ(x · ξ, η) − δ(ξ, x · η) + [x, δ(ξ, η)]H , [[ξ, η]Q , ϑ]Q + (δ(ξ, η)) · ϑ = 0,

(ξ,η,ϑ) cyclic

δ([ξ, η]Q , ϑ) =

(ξ,η,ϑ) cyclic

ξ · δ(η, ϑ),

(1.9.5) (1.9.6) (1.9.7)

(ξ,η,ϑ) cyclic

where a ∈ A, x, y ∈ H , ξ, η, ϑ ∈ Q. Recall that given a commutative algebra A and Lie–Rinehart algebras (A, L ), (A, L) and (A, L ), where L is an ordinary A-Lie algebra, an extension of Lie– Rinehart algebras 0− → L − →L− → L − →0 is an extension of A-modules which is also an extension of ordinary Lie algebras so that the projection from L to L is a morphism of Lie–Rinehart algebras; see Huebschmann [1999a]. Theorem 1.9 entails at once the following. Corollary 1.9.8. Given a Lie–Rinehart triple (A, H, Q), the left (A, H )-module structures (1.5.2.H ) on A and (1.5.3) on Q are trivial if and only if (A, Q) is a Lie– Rinehart algebra in such a way that the projection from E = H ⊕ Q to Q ﬁts into an extension 0− →H − →E− →Q− →0 of Lie–Rinehart algebras. Thus Lie–Rinehart triples (A, H, Q) having trivial left (A, H )-module structures on A and Q and extensions of Lie–Rinehart algebras of the kind (A, L) together with an A-module section of the projection map are equivalent notions. Proof of Theorem 1.9. The bracket (1.6.1) is plainly skew-symmetric. Hence the proof comes down to relating the Jacobi identity in L and the Lie–Rinehart compatibility properties with (1.9.1)–(1.9.7).

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Thus, suppose that the bracket [·, ·] on L = H ⊕ Q given by (1.6.1) and the operation L ⊗R A → A given by (1.6.3) turn (A, L) into a Lie–Rinehart algebra. Given ξ ∈ Q and x ∈ H , we have [ξ, x] = ξ · x − x · ξ ; since L acts on A by derivations, for a ∈ A, we conclude that ξ(x(a)) − x(ξ(a)) = [ξ, x](a) = (ξ · x)(a) − (x · ξ )(a), that is, (1.9.1) holds. Likewise, given ξ, η ∈ Q, [ξ, η] = [ξ, η]Q + δ(ξ, η) ∈ L, whence, for a ∈ A, ξ(η(a)) − η(ξ(a)) = [ξ, η](a) = [ξ, η]Q (a) + (δ(ξ, η))(a), that is, (1.9.4) holds. Next, since L is a Lie algebra, its bracket satisﬁes the Jacobi identity. Hence, given x ∈ H and ξ, η ∈ Q, x · [ξ, η]Q − [ξ, η]Q · x = [x, [ξ, η]Q ] = [x, [ξ, η]] − [x, δ(ξ, η)] = [[x, ξ ], η] + [ξ, [x, η]] − [x, δ(ξ, η)] = [x · ξ − ξ · x, η] + [ξ, x · η − η · x] − [x, δ(ξ, η)]H = [x · ξ, η] − [ξ · x, η] + [ξ, x · η] − [ξ, η · x] − [x, δ(ξ, η)]H = [x · ξ, η] + [ξ, x · η] − (ξ · x) · η + η · (ξ · x) + (η · x) · ξ − ξ · (η · x) − [x, δ(ξ, η)]H = [x · ξ, η]Q + δ(x · ξ, η) + [ξ, x · η]Q + δ(ξ, x · η) − (ξ · x) · η + η · (ξ · x) + (η · x) · ξ − ξ · (η · x) − [x, δ(ξ, η)]H , whence, comparing components in H and Q, we conclude that x · [ξ, η]Q = [x · ξ, η]Q + [ξ, x · η]Q − (ξ · x) · η + (η · x) · ξ, [ξ, η]Q · x = ξ · (η · x) − η · (ξ · x) − δ(x · ξ, η) − δ(ξ, x · η) + [x, δ(ξ, η)]H , that is, (1.9.2) and (1.9.5) hold. Likewise, given ξ ∈ Q and x, y ∈ H , ξ · [x, y]H − [x, y]H · ξ = [ξ, [x, y]H ] = [[ξ, x], y] + [x, [ξ, y]] = [ξ · x − x · ξ, y] + [x, ξ · y − y · ξ ] = [ξ · x, y] − [x · ξ, y] + [x, ξ · y] − [x, y · ξ ] = [ξ · x, y] + [x, ξ · y] − (x · ξ ) · y + y · (x · ξ ) + (y · ξ ) · x − x · (y · ξ ) = [ξ · x, y]H + [x, ξ · y]H − (x · ξ ) · y + y · (x · ξ ) + (y · ξ ) · x − x · (y · ξ ),

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whence, comparing components in Q and H , we conclude that ξ · [x, y]H = [ξ · x, y]H + [x, ξ · y]H − (x · ξ ) · y + (y · ξ ) · x [x, y]H · ξ = x · (y · ξ ) − y · (x · ξ ), that is, (1.9.3) and (1.5.12) hold; notice that (1.5.12) holds already by assumption. Next, given ξ, η, ϑ ∈ Q, [[ξ, η], ϑ] = [[ξ, η]H , ϑ] + [[ξ, η]Q , ϑ] = [δ(ξ, η), ϑ] + [[ξ, η]Q , ϑ] = [δ(ξ, η), ϑ]H + [δ(ξ, η), ϑ]Q + [[ξ, η]Q , ϑ]H + [[ξ, η]Q , ϑ]Q = (δ(ξ, η)) · ϑ − ϑ · δ(ξ, η) + δ([ξ, η]Q , ϑ) + [[ξ, η]Q , ϑ]Q . Hence [[ξ, η], ϑ] + [[η, ϑ], ξ ] + [[ϑ, ξ ], η] = (δ(ξ, η)) · ϑ + (δ(η, ϑ)) · ξ + (δ(ϑ, ξ )) · η + [[ξ, η]Q , ϑ]Q + [[η, ϑ]Q , ξ ]Q + [[ϑ, ξ ]Q , η]Q − ξ · δ(η, ϑ) − η · δ(ϑ, ξ ) − ϑ · δ(ξ, η) + δ([ξ, η]Q , ϑ) + δ([η, ϑ]Q , ξ ) + δ([ϑ, ξ ]Q , η). Thus the Jacobi identity implies that [[ξ, η]Q , ϑ]Q + [[η, ϑ]Q , ξ ]Q + [[ϑ, ξ ]Q , η]Q +(δ(ξ, η)) · ϑ + (δ(η, ϑ)) · ξ + (δ(ϑ, ξ )) · η = 0 δ([ξ, η]Q , ϑ) + δ([η, ϑ]Q , ξ ) + δ([ϑ, ξ ]Q , η) −ξ · δ(η, ϑ) − η · δ(ϑ, ξ ) − ϑ · δ(ξ, η) = 0, that is, (1.9.6) and (1.9.7) are satisﬁed. Conversely, suppose that the brackets [·, ·]H and [·, ·]Q on H and Q, respectively, and the operations (1.5.3), (1.5.4), and (1.5.5), are related by (1.9.1)–(1.9.7). We can then read the above calculations backwards and conclude that the bracket (1.6.1) on L satisﬁes the Jacobi identity and the operation (1.6.3) yields a Lie algebra action of L on A by derivations. The remaining Lie–Rinehart algebra axioms hold by assumption. Thus (A, L) is a Lie–Rinehart algebra.

Remark 1.10. Under the circumstances of Example 1.3, the requirements (1.5.6.Q), (1.5.6.H ), (1.5.7.Q), (1.5.7.H ), (1.5.8)–(1.5.11) are vacuous, and so are (1.9.1) and (1.9.4) as well. Given an (R, A) Lie algebra L and an (R, A) Lie subalgebra H , the invariants AH ⊆ A constitute a subalgebra of A; we will then denote the normalizer of H in L in the sense of Lie algebras by LH , that is, LH consists of all α ∈ L having the property that [α, β] ∈ H whenever β ∈ H .

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Corollary 1.11. Given a Lie–Rinehart triple (A, H, Q), the corresponding (R, A)Lie algebra being written as L = H ⊕ Q, the intersection Q ∩ LH coincides with the invariants QH under the H -action on Q (given by the corresponding operation (1.5.3)), the pair (AH , QH ) acquires a Lie–Rinehart algebra structure, and the projection from LH to QH ﬁts into an extension 0− →H − → LH − → QH − →0

(1.11.1)

of (R, AH )-Lie algebras. Furthermore, the restriction of δ to QH is a cocycle for this extension, that is, it yields the curvature of the connection for the extension determined by the AH -module direct sum decomposition LH = H ⊕ QH . Notice that H is here viewed as an ordinary AH -Lie algebra, the H -action on AH being trivial by construction. Proof . Indeed, given α ∈ Q and β ∈ H , [α, β] = α · β − β · α ∈ L, whence [α, β] ∈ H for every β ∈ H if and only if β · α = 0 ∈ Q for every β ∈ H , that is, if and only if α is invariant under the H -action on Q. The rest of the claim is an immediate consequence of Theorem 1.9.

Illustration 1.12. Under the circumstances of Example 1.4.1, Corollary 1.11 obtains, with H = LF . Now AH = ALF ⊆ A is the algebra of smooth functions which are constant on the leaves, that is, the algebra of functions on the “space of leaves,’’ and LH consists of the vector ﬁelds which “project’’ to the “space of leaves.’’ Indeed, given a function f which is constant on the leaves and vector ﬁelds X ∈ LH and Y ∈ LF , necessarily Y (Xf ) = [Y, X]f + X(Yf ) = 0, whence Xf is constant on the leaves as well. Thus we may view QH as the Lie algebra of vector ﬁelds on the “space of leaves,’’ that is, as the space of sections of a certain geometric object which serves as a replacement for the in general nonexistant tangent bundle of the “space of leaves.’’ Remark 1.13. In analogy with the deformation theory of complex manifolds, given a Lie–Rinehart triple (A, H, Q), we may view H and Q as corresponding to the antiholomorphic and holomorphic tangent bundle, respectively, and accordingly study deformations of the Lie–Rinehart triple via morphisms ϑ : H → Q and spell out the resulting inﬁnitesimal obstructions. This will include a theory of deformations of foliations. Details will be given elsewhere.

2 Lie–Rinehart triples and Maurer–Cartan algebras In this section we will explore the relationship between Lie–Rinehart triples and suitably deﬁned Maurer–Cartan algebras. In particular, we will show that, under an additional assumption, the two notions are equivalent; see Theorem 2.8.3 below for details. As an application we will explain how the spectral sequence of a foliation and the Hodge–de Rham spectral sequence arise as special cases of a single conceptually simple construction. More applications will be given in subsequent sections.

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2.1. Maurer–Cartan algebras Given an A-module L and an R-derivation d of degree −1 on the graded A-algebra Alt A (L, A), we will refer to (Alt A (L, A), d) as a Maurer–Cartan algebra (over L) provided d has square zero, i.e., is a differential. Recall that a multicomplex (over R) is a bigraded R-module {M p,q }p,q together with an operator dj : M p,q → M p+j,q−j +1 for every j ≥ 0 such that the sum d = d0 + d1 + . . . is a differential, i.e., dd = 0; under such circumstances, an inﬁnite sequence of operators of the kind (d2 , d3 , . . . ) is a system of higher homotopies. We will refer to a multicomplex (M; d0 , d1 , d2 , . . . ) whose underlying bigraded object M is endowed with a bigraded algebra structure such that the operators dj are derivations with respect to this algebra structure as a multialgebra and, when it is necessary to spell out the ground ring explicitly, we will refer to a multi-R-algebra. The idea of multicomplex already occurs in Heller [1954], Liulevicius [1963], Wall [1961] and was exploited in various other places in the literature including Huebschmann [1989c], Huebschmann [1991b]. The terminology multicomplex goes back at least to unpublished mimeographed notes of Liulevicius from the 1960s. Given A-modules H and Q, consider the bigraded A-algebra (Alt A (Q, Alt A (H, A)); we will refer to a multi-R-algebra structure (beware: not multi-A-algebra structure) on this bigraded A-algebra having at most d0 , d1 , d2 nonzero as a Maurer–Cartan algebra structure. The resulting multi-R-algebra will then be written as (Alt A (Q, Alt A (H, A)); d0 , d1 , d2 )

(2.1.1)

and referred to as a (multi-)Maurer–Cartan algebra (over (Q, H )). Usually we will discard “multi’’ and more simply refer to a Maurer–Cartan algebra. We note that, for degree reasons, when (2.1.1) is a Maurer–Cartan algebra, the operator d2 is necessarily an A-derivation (since d2 (a) = 0 for every a ∈ A ∼ = Alt 0A (Q, Alt 0A (H, A))). Remark 2.1.2. In this deﬁnition, we could allow for nonzero derivations of the kind dj for j ≥ 3 as well. This would lead to a more general notion of multi-Maurer– Cartan algebra not studied here. The presence of a nonzero operator at most of the kind d2 is an instance of a higher homotopy of a special kind which sufﬁces to explain the “quasi’’ structures explored later in the paper. Remark 2.1.3. Given a (multi-)Maurer–Cartan algebra of the kind (2.1.1), the sum d = d0 + d1 + d2 turns Alt A (Q ⊕ H, A) into a Maurer–Cartan algebra. However, not every Maurer–Cartan structure on Alt A (Q ⊕ H, A) arises in this fashion, that is, a multi-Maurer–Cartan algebra structure captures additional structure of interaction between A, Q, and H , indeed, it captures essentially a Lie–Rinehart triple structure. The purpose of the present section is to make this precise. For later reference, we spell out the following, the proof of which is immediate.

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Proposition 2.1.4. Given the three derivations d0 , d1 , d2 , (Alt ∗A (Q, Alt ∗A (H, A)), d0 , d1 , d2 ) is a (multi-)Maurer–Cartan algebra if and only if the following identities are satisﬁed. d0 d0 = 0

(2.1.4.1)

d0 d1 + d1 d0 = 0 d0 d2 + d1 d1 + d2 d0 = 0 d1 d2 + d2 d1 = 0

(2.1.4.2)

d2 d2 = 0.

(2.1.4.5)

(2.1.4.3) (2.1.4.4)

2.2. Lie–Rinehart and Maurer–Cartan algebras Let A be a commutative R-algebra and L an A-module, together with a skewsymmetric R-bilinear bracket [·, ·]L : L ⊗R L − →L

(2.2.1)

and an operation L ⊗R A − → A,

x ⊗R a → x(a),

x ∈ H, a ∈ A

(2.2.2)

such that the values of the adjoint L − → End R (A) lie in Der R (A) and that (2.2.1), (2.2.2) and the A-module structure on L satisfy the Lie–Rinehart axioms (2.2.3) and (2.2.4) below: (ax)(b) = a(x(b)), a, b ∈ A, x ∈ L, [x, ay]L = x(a)y + a[x, y]L , a ∈ A, x, y ∈ L.

(2.2.3) (2.2.4)

Let M be a graded A-module, together with an operation L ⊗R M − → M,

x ⊗ m → x(m),

x ∈ L, m ∈ m

(2.2.5)

subject to the following requirement: For α ∈ L, a ∈ A, m ∈ M, (aα)(m) = a(α(m)), α(am) = aα(m) + α(a)m.

(2.2.6) (2.2.7)

We refer to an operation of the kind (2.2.5) as a generalized L-connection on M. Under these circumstances, the ordinary Cartan–Chevalley–Eilenberg (CCE) operator d is deﬁned, at ﬁrst on the bigraded object AltR (L, M) of M-valued R-multilinear alternating forms on L. Indeed, given an R-multilinear alternating function f on L of n − 1 variables which is homogeneous, i.e., the values of f lie in a homogeneous constituent of M, the Cartan–Chevalley–Eilenberg (CCE) formula yields

Homotopies and Maurer–Cartan algebras n

(−1)|f |+1 (df )(α1 , . . . , αn ) = +

253

(−1)(i−1) αi (f (α1 , . . . α=i . . . , αn ))

i=1

(−1)(j +k) f ([αj , αk ], α1 , . . . α=j . . . α=k . . . , αn ),

1≤j
(2.2.8) where α1 , . . . , αn ∈ L and where as usual “=’’ indicates omission of the corresponding term. We note that when the values of the homogeneous alternating function f on L of n − 1 variables lie in Mq , |f | = q − n + 1. Here and below our convention is that given graded objects N and M, a homogeneous morphism h : Np → Mq has degree |h| = q − p. This is the standard grading on the Hom-functor for graded objects. The requirements (2.2.3), (2.2.4), (2.2.6), and (2.2.7) entail that the operator d on Alt R (L, M) passes to an R-linear operator on the (bi)graded A-submodule Alt A (L, M) of A-multilinear functions, written here and henceforth as d : Alt A (L, M) − → Alt A (L, M)

(2.2.9)

as well. The sign (−1)|f |+1 in (2.2.8) is the appropriate one according to the customary Eilenberg–Koszul convention in differential homological algebra since (2.2.9) involves graded objects. For M = A, the operator d is plainly a derivation on Alt A (L, A). Let M1 and M2 be graded A-modules endowed with generalized L-connections of the kind (2.2.5), and let ·, · : M1 ⊗A M2 − →M be an A-module pairing which is compatible with the generalized L-connections in the sense that x(m1 , m2 ) = (x(m1 ), m2 ) + (m1 , x(m2 )),

x ∈ L, m1 ∈ M1 , m2 ∈ M2 .

This pairing induces a (bi)graded pairing Alt A (L, M1 ) ⊗R Alt A (L, M2 ) − → Alt A (L, M)

(2.2.10)

which is compatible with the generalized CCE operators. An A-module M will be said to have property P provided that for x ∈ M, φ(x) = 0 for every φ : M → A implies that x is zero. For example, a projective A-module has property P, or a reﬂexive A-module has this property as well or, more generally, any A-module M such that the canonical map from M into its double A-dual is injective. On the other hand, for example, for a smooth manifold X, the C ∞ (X)module D of formal (= Kähler) differentials does not have property P: On the real line, with coordinate x, consider the functions f (x) = sin x and g(x) = cos x. The formal differential df − gdx is nonzero in D; however, the C ∞ (X)-linear maps from D to C ∞ (X) are the smooth vector ﬁelds, whence every such C ∞ (X)-linear map annihilates the formal differential df − gdx.

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Lemma 2.2.11. When L has the property P, the pair (A, L), endowed with the bracket [·, ·]L (see (2.2.1)) and operation (2.2.2) is a Lie–Rinehart algebra, that is, the bracket [x, y]L satisﬁes the Jacobi identity and the adjoint of (2.2.2) is a morphism of R-Lie algebras, if and only if (Alt A (L, A), d) is a Maurer–Cartan algebra. Proof . A familiar calculation shows that d is a differential if and only if the bracket [x, y]L satisﬁes the Jacobi identity and if the adjoint of (2.2.2) is a morphism of R-Lie algebras. See also 2.8.5(i) below.

Example 2.2.12. The Lie algebra L of derivations of a polynomial algebra A in inﬁnitely many indeterminates (over a ﬁeld) has property P as an A-module but is not a projective A-module. To include this kind of example and others, it is necessary to build up the theory for modules having property P rather than just projective ones or even ﬁnitely generated projective modules. Now let (A, L) be an (ungraded) Lie–Rinehart algebra, and let (Alt A (L, A), d) be the corresponding Maurer–Cartan algebra; notice that the operator d is not Alinear unless L acts trivially on A. For reasons explained in Huebschmann [2003] we will refer to this operator as a Lie–Rinehart differential. We will say that the graded A-module M, endowed with the operation (2.2.5), is a graded (left) (A, L)module provided this operation is an ordinary Lie algebra action on M. When M is concentrated in degree zero, we simply refer to M as a (left) (A, L)-module. In particular, with the obvious L-module structure, the algebra A itself is a (left) (A, L)module. The proof of the following is straightforward and left to the reader. Lemma 2.2.13. When (A, L) is a Lie–Rinehart algebra and when M has the property P, the operation (2.2.5) turns M into a left (A, L)-module if and only if the operator d on Alt A (L, M) turns (Alt A (L, M), d) into a differential graded (Alt A (L, A), d)module via (2.2.10) (with M1 = A and M2 = M). Given a graded (A, L)-module M, we will refer to the resulting (co)chain complex (Alt A (L, M), d)

(2.2.14)

as the Rinehart complex of M-valued forms on L; often we write this complex more simply in the form Alt A (L, M). It inherits a differential graded Alt A (L, A)-module structure via (2.2.10). We now spell out the passage from Maurer–Cartan algebras to Lie–Rinehart algebras. Lemma 2.2.15. Let L be a ﬁnitely generated projective A-module. Then an Rderivation d on the graded A-algebra Alt A (L, A) determines a skew-symmetric R-bilinear bracket [·, ·]L on L of the kind (2.2.1) and an operation L ⊗ A → A of the kind (2.2.2) such that the identities (2.2.3) and (2.2.4) are satisﬁed and the corresponding CCE operator (2.2.8) (for M = A) coincides with d. Furthermore, Alt A (L, A) is then a Maurer–Cartan algebra if and only if (A, L) is a Lie–Rinehart algebra.

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Proof . The operator q

q+1

d : Alt A (L, A) − → Alt A (L, A)

(q ≥ 0)

induces, for q = 0, an operation L ⊗R A − → A of the kind (2.2.2) and, for q = 1, a skew-symmetric R-bilinear bracket [·, ·]L on L of the kind (2.2.1). More precisely: Given x ∈ L and a ∈ A, let x(a) = −(d(a))(x). This yields an operation of the kind (2.2.2). Given x, y ∈ L, using the hypothesis that L is a ﬁnitely generated projective A-module, identify x and y with their images in the double A-dual L∗∗ and deﬁne the value [x, y]L by [x, y]L (α) = x(α(y)) − y(α(x)) − dα(x, y), where α ∈ L∗ = HomA (L, A). This yields a bracket of the kind (2.2.1), that is, an R-bilinear (beware: not A-bilinear) skew-symmetric bracket on L. Notice that, at this stage, the operation of the kind (2.2.2) is already deﬁned, hence the deﬁnition of the bracket makes sense. Since, by assumption, d is a derivation on Alt A (L, A), the identities (2.2.3) and (2.2.4) are satisﬁed. By construction, the resulting CCE operator coincides with d in degree 0 and in degree −1, hence the two operators coincide. Since a ﬁnitely generated projective A-module has property P, Lemma 2.2.11 completes the proof.

Combining Lemmas 2.2.13 and 2.2.15, we arrive at the following. Theorem 2.2.16. Given a ﬁnitely generated projective A-module L, Lie–Rinehart algebra structures on (A, L) and Maurer–Cartan algebra structures on Alt A (L, A) are equivalent notions. 2.3. Connections Let (A, L) be a Lie–Rinehart algebra. Given a graded A-module M, a degree zero operation L ⊗R M → M, not necessarily a graded left L-module structure but still satisfying (2.2.6) and (2.2.7), is referred to as an (A, L)-connection (see Huebschmann [1990], Huebschmann [1998a]) or, somewhat more precisely, as a graded left (A, L)-connection; in this language, a (graded) (A, L)-module structure is a (graded) ﬂat (A, L)-connection. Given a graded A-module M, together with a graded (A, L)connection, we extend the deﬁnition of the Lie–Rinehart operator to an operator d : Alt A (L, M) − → Alt A (L, M)

(2.3.1)

by means of the formula (2.2.8). The resulting operator d is well deﬁned; it is a differential if and only if the (A, L)-connection on M is ﬂat, i.e., an ordinary (A, L)module structure.

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2.4. From Lie–Rinehart triples to Maurer–Cartan algebras Let (A, H, Q) be an almost pre-Lie–Rinehart triple. Consider the bigraded A-module ∗,∗ Alt A (Q ⊕ H, A) ∼ = Alt ∗A (Q, Alt ∗A (H, A)).

(2.4.1)

Henceforth we spell out a particular homogeneous constituent of bidegree (p, q) (according to the conventions used below, such a homogenous constituent will be of bidegree (−p, −q) but for the moment this usage of negative degrees is of no account) in the form p q Alt A (Q, Alt A (H, A)). (2.4.2) The operations (1.5.3) and (1.5.4) induce degree zero operations H ⊗R Alt ∗A (Q, A) − → Alt ∗A (Q, A) Q ⊗R Alt ∗A (H, A) − → Alt ∗A (H, A)

(2.4.3) (2.4.4)

on Alt ∗A (Q, A) and Alt ∗A (H, A), respectively, when (1.5.3) and (1.5.4) are treated like connections. By evaluation of the expression given on the right-hand side of (2.2.8), with (1.5.1.H ) and (1.5.1.Q) instead of (2.2.1), and with (2.5.1) and (2.5.2) instead of (2.2.5), these operations, in turn, induce two operators p

p

q

q+1

d0 : Alt A (Q, Alt A (H, A)) − → Alt A (Q, Alt A (H, A)) d1 :

p q Alt A (Q, Alt A (H, A))

− →

p+1 q Alt A (Q, Alt A (H, A)).

(2.4.5) (2.4.6)

A little thought reveals that in view of (1.5.6.H ), (1.5.6.Q), (1.5.7.H ), and (1.5.7.Q), (1.5.8)–(1.5.11), these operators, which are at ﬁrst deﬁned only on the R-multilinear alternating functions, in fact pass to operators on A-multilinear alternating functions. Furthermore, the skew-symmetric A-bilinear pairing δ (see (1.5.5)) induces an operator p q p+2 q−1 d2 : Alt A (Q, Alt A (H, A)) − → Alt A (Q, Alt A (H, A)). (2.4.7) Hence, when (A, H, Q) is a Lie–Rinehart triple, that is, when (1.6.1) and (1.6.3) turn (A, H ⊕ Q) into a Lie–Rinehart algebra, (Alt A (Q, Alt A (H, A)); d0 , d1 , d2 ) is a (multi-)Maurer–Cartan algebra. 2.5. Explicit description of the operators d0 , d1 , d2 Let f be an alternating A-multilinear function on Q of p variables with values in q Alt A (H, A), so that |f | = −q − p and (−1)|f |+1 = (−1)p+q+1 . Let ξ1 , . . . , ξp+2 ∈ Q and x1 , . . . , xq+1 ∈ H . The operator d0 . (−1)p+q+1 (d0 f )(ξ1 , . . . , ξp ) (x1 , . . . , xq+1 ) =

q+1 j =1

(−1)p+j −1 xj

f (ξ1 , . . . , ξp ) (x1 , . . . x=j . . . , xq+1 )

Homotopies and Maurer–Cartan algebras

+

257

(−1)p+j +k f (ξ1 , . . . , ξp ) ([xj , xk ]H , x1 , . . . x=j . . . x=k . . . , xq+1 )

1≤j
+

q+1 p

(−1)j +k+p+1 f (xk · ξj , ξ1 , . . . ξ=j . . . , ξp ) (x1 , . . . x=k . . . , xq+1 ).

j =1 k=1

(2.5.1) The last term involving the double summation necessarily appears since, for 1 ≤ j ≤ p and 1 ≤ k ≤ q + 1, the bracket [xk , ξj ] in Q ⊕ H (see (1.6.2)) is given by [xk , ξj ] = xk · ξj − ξj · xk . Remark 2.5.2. A crucial observation is this: The operator d0 may be written as the sum d0 = dH + dQ of certain operators dH and dQ deﬁned on Alt R (Q, Alt R (H, A)) by (−1)p+q+1 (dH f )(ξ1 , . . . , ξp ) (x1 , . . . , xq+1 ) =

q+1 j =1

+

(−1)p+j −1 xj

f (ξ1 , . . . , ξp ) (x1 , . . . x=j . . . , xq+1 )

(−1)j +k f (ξ1 , . . . , ξp ) ([xj , xk ]H , x1 , . . . x=j . . . x=k . . . , xq+1 ),

1≤j
(−1)p+q+1 (dQ f )(ξ1 , . . . , ξp ) (x1 , . . . , xq+1 ) (−1)j +k+p+1 f (xk · ξj , ξ1 , . . . ξ=j . . . , ξp ) (x1 , . . . x=k . . . , xq+1 ). = 1≤j ≤p,1≤k≤q+1

However, even when (A, H, Q) is a (pre-)Lie–Rinehart triple, the individual operators dH and dQ are well deﬁned merely on Alt R (Q, Alt R (H, A)); only their sum is well deﬁned on Alt A (Q, Alt A (H, A)). The operator d1 . (−1)p+q+1 (d1 f )(ξ1 , . . . , ξp+1 ) (x1 , . . . , xq ) =

p+1 j =1

+

(−1)j −1 ξj

f (ξ1 , . . . ξ=j . . . , ξp+1 ) (x1 , . . . , xq )

(−1)j +k f ([ξj , ξk ]Q , ξ1 . . . ξ=j . . . ξ=k . . . , ξp+1 ) (x1 , . . . , xq )

1≤j
+

q p+1

(−1)j +k+1 f (ξ1 , . . . ξ=j . . . , ξp+1 ) (ξj · xk , x1 , . . . x=k . . . , xq ).

j =1 k=1

(2.5.3)

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The last term involving the double summation necessarily appears in view of (1.6.4). With the generalized operation of Lie-derivative (ξ, α) −→ ξ(α),

q

ξ ∈ Q, α ∈ Alt A (H, A)

(q ≥ 0)

which, for x1 , . . . , xq ∈ H , is given by (ξ(α))(x1 , . . . , xq ) = ξ(α(x1 , . . . , xq )) −

q

α(x1 , . . . , xk−1 , ξ · xk , xk+1 , . . . , xq ),

k=1

the identity (2.5.3) may be written as (−1)p+q+1 (d1 f )(ξ1 , . . . , ξp+1 ) = (−1)|f |+1 (d1 f )(ξ1 , . . . , ξp+1 ) =

p+1 j =1

+

(−1)j −1 ξj f (ξ1 , . . . ξ=j . . . , ξp+1 )

(2.5.3 )

(−1)j +k f ([ξj , ξk ]Q , ξ1 . . . ξ=j . . . ξ=k . . . , ξp+1 ).

1≤j
The operator d2 . (−1)p+q+1 (d2 f )(ξ1 , . . . , ξp+2 ) (x1 , . . . , xq−1 ) = (−1)j +k+p f (ξ1 , . . . ξ=j . . . ξ=k . . . , ξp+2 ) (δ(ξj , ξk ), x1 , . . . , xq−1 ). 1≤j
(2.5.4) Remark 2.5.5. The operator d2 does not involve the pieces of structure (1.5.1.H ), (1.5.1.Q), (1.5.2.H ), (1.5.2.Q), (1.5.3), (1.5.4). Hence, for an arbitrary A-module M, the formula (2.5.4) given above yields an operator p

q

p+2

q−1

d2 : Alt A (Q, Alt A (H, M)) − → Alt A (Q, Alt A (H, M)),

(2.4.7 )

for p ≥ 0 and q ≥ 1. We will use this observation in (5.8.7) and (5.8.8) below. Remark 2.6. Given an almost pre-Lie–Rinehart triple (A, H, Q), the vanishing of d2 d2 is automatic for the following reason: View H and Q as abelian A-Lie algebras and H as being endowed with the trivial Q-module structure. Since δ is a skewsymmetric A-bilinear pairing, we may use it to endow the A-module direct sum L = H ⊕ Q with a nilpotent A-Lie algebra structure (of class two) by setting [(x, ξ ), (y, η)] = (δ(ξ, η), 0),

ξ, η ∈ Q, x, y ∈ H.

We write Lnil for this nilpotent A-Lie algebra. The ordinary CCE complex for calculating the Lie algebra cohomology H∗ (Lnil , A) (with trivial Lnil -action on A) is just (Alt A (L, A), d2 ). Thus the vanishing of d2 d2 is automatic.

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Theorem 2.7. An almost pre-Lie–Rinehart triple (A, H, Q) such that H and Q have property P is a Lie–Rinehart triple, that is, (1.6.1) and (1.6.3) then turn (A, H ⊕ Q) into a Lie–Rinehart algebra, if and only if (Alt A (Q, Alt A (H, A)); d0 , d1 , d2 ) is a (multi-)Maurer–Cartan algebra. Proof . The direct A-module sum L = Q ⊕ H has the property P. The sum d = d0 + d1 + d2 is an R-derivation on Alt A (L, A). Hence the claim is an immediate consequence of Lemma 2.2.11.

2.8. From Maurer–Cartan algebras to Lie–Rinehart triples Let H and Q be ﬁnitely generated projective A-modules, and let d0 , d1 , d2 be homogeneous R-derivations of the bigraded A-algebra Alt A (Q, Alt A (H, A)) of the kind p

q

p+j

q−j +1

dj : Alt A (Q, Alt A (H, A)) − → Alt A (Q, Alt A

(H, A)).

Proposition 2.8.1. The operators d0 , d1 , d2 induce an almost pre-Lie–Rinehart triple structure on (A, H, Q). Proof . Write L = Q ⊕ H . The sum d = d0 + d1 + d2 is a derivation on Alt A (L, A). By Lemma 2.2.15, d induces a bracket [·, ·]L on L (of the kind (2.2.1)) and an operation L ⊗R A − → A of the kind (2.2.2). Taking homogeneous components with reference to the direct sum decomposition L = Q ⊕ H , we obtain an almost pre-Lie– Rinehart triple structure of the kind (1.5.1.H ), (1.5.2.H ), (1.5.1.Q), (1.5.2.Q), (1.5.3), (1.5.4), (1.5.5) on (A, H, Q). The three almost pre-Lie–Rinehart triple axioms are implied by the fact that the operators d0 , d1 , d2 are derivations of the bigraded algebra Alt A (Q, Alt A (H, A)).

Theorem 2.8.2. The triple (A, H, Q), endowed with the induced operations of the kind (1.5.1.H ), (1.5.2.H ), (1.5.1.Q), (1.5.2.Q), (1.5.3), (1.5.4), (1.5.5) given in (2.8.1) above, is a pre-Lie–Rinehart triple if and only if d0 is a differential; (A, H, Q) is a Lie–Rinehart triple if and only if (Alt A (Q, Alt A (H, A)); d0 , d1 , d2 ) is a Maurer– Cartan algebra. Proof . This is a consequence of Lemmas 2.2.13 and 2.2.15.

Combining Theorem 2.7 and Theorem 2.8.2, we arrive at the following. Theorem 2.8.3. Given ﬁnitely generated projective A-modules H and Q, Lie– Rinehart triple structures on (A, H, Q) and (multi-)Maurer–Cartan algebra structures on Alt A (Q, Alt A (H, A)) are equivalent notions. Remark 2.8.4. Concerning the hypotheses and hence the range of applications (see, e.g., Example 2.2.12 above), Theorem 2.7 is somewhat more general than Theorem 2.8.3. This justiﬁes, hopefully, the terminology “almost-’’ and “pre-Lie–Rinehart triple,’’ admittedly a bit cumbersome. In fact, it would be interesting and important to establish the statement of Theorem 2.8.3 for A-modules more general than ﬁnitely generated and projective.

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2.8.5. Direct veriﬁcation of the Lie–Rinehart triple structure Let (A, H, Q) be an almost pre-Lie–Rinehart triple such that H and Q have property P, and suppose that (Alt A (Q, Alt A (H, A)); d0 , d1 , d2 ) is a (multi-)Maurer–Cartan algebra. It is then instructive to deduce directly that (A, H, Q) is a Lie–Rinehart triple. (i) Consider the operator j +2

j

d0 d0 : Alt A (H, A) − → Alt A (H, A) j

j

for j = 0 and j = 1. Notice that Alt A (H, A) equals Alt A (H, Alt 0A (Q, A)) and that j +2 j +2 Alt A (H, A) equals Alt A (H, Alt 0A (Q, A)). For j = 1, given x, y, z ∈ H and j φ ∈ HomA (H, A) = Alt A (H, A), we ﬁnd that (d0 d0 φ)(x, y, z) = φ([[x, y]H , z]H + [[y, z]H , x]H + [[z, x]H , y]H ). Since H has property P, we conclude that the bracket on H satisﬁes the Jacobi identity, that is, H is an R-Lie algebra. Likewise, for j = 0, given x, y ∈ H and a ∈ A, we ﬁnd that (d0 d0 a)(x, y) = x(y(a)) − y(x(a)) − [x, y](a). Consequently the adjoint H → Der R (A) of (1.5.2.H ) is a morphism of R-Lie algebras. In view of (1.5.6.H ) and (1.5.7.H ), we conclude that (1.5.1.H ) and (1.5.2.H ) turn (A, H ) into a Lie–Rinehart algebra. (ii) Next, consider the operator d0 d0 : Alt 1A (Q, Alt 0A (H, A)) − → Alt 1A (Q, Alt 2A (H, A)). We note that Alt 1A (Q, Alt 0A (H, A)) = Alt 1A (Q, A) = HomA (Q, A). Let ξ ∈ Q, x, y ∈ H , and φ ∈ HomA (H, A). A straightforward calculation gives ((d0 d0 φ)(ξ ))(x, y) = φ(y · (x · ξ ) − x · (y · ξ ) + [x, y]H · ξ ). Since H is assumed to have property P, we conclude that, for every ξ ∈ Q, x, y ∈ H , [x, y]H · ξ = x · (y · ξ ) − y · (x · ξ ), that is, (1.5.3) is a left (A, H )-module structure on Q. (iii) Pursuing the same kind of reasoning, consider the operator d0 d1 + d1 d0 : A = Alt 0A (Q, Alt 0A (H, A)) − → Alt 1A (Q, Alt 1A (H, A)). Let a ∈ A, ξ ∈ Q, x ∈ H . Again a calculation shows that ((d0 d1 + d1 d0 )a)(ξ )(x) = x(ξ(a)) − ξ(x(a)) − ((x · ξ )(a) − (ξ · x)(a)),

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whence the vanishing of d0 d1 + d1 d0 in bidegree (0, 0) entails the compatibility property (1.9.1). Likewise, consider the operator d0 d1 + d1 d0 : HomA (H, A) = Alt 0A (Q, Alt 1A (H, A)) − → Alt 1A (Q, Alt 2A (H, A)). Again a calculation shows that, for ξ ∈ Q, x, y ∈ H , φ ∈ HomA (H, A), ((d0 d1 + d1 d0 )φ)(ξ )(x, y) = φ (ξ · [x, y]H − ([ξ · x, y]H + [x, ξ · y]H − (x · ξ ) · y + (y · ξ ) · x)) , whence the vanishing of d0 d1 + d1 d0 in bidegree (0, 1) entails the compatibility property (1.9.3). Likewise, the vanishing of the operator d0 d1 + d1 d0 in bidegree (1, 0), that is, of d0 d1 + d1 d0 : Alt 1A (Q, Alt 0A (H, A)) − → Alt 2A (Q, Alt 1A (H, A)), entails the compatibility property (1.9.2). In the same vein, we have the following: (iv) The vanishing of the operator → Alt 2A (Q, Alt 0A (H, A)) d1 d1 + d2 d0 = d0 d2 + d1 d1 + d2 d0 : Alt 0A (Q, Alt 0A (H, A)) − entails the compatibility property (1.9.4). (v) The vanishing of the operator d1 d1 +d2 d0 = d0 d2 +d1 d1 +d2 d0 : Alt 1A (Q, Alt 0A (H, A)) − → Alt 3A (Q, Alt 0A (H, A)), together with (1.9.4), entails the compatibility property (1.9.6), the “generalized Jacobi identity for the bracket [·, ·]Q .’’ For intelligibility and later reference (see (4.10) and (6.11) below), we sketch the argument: Let α ∈ Alt 1A (Q, Alt 0A (H, A)) and ξ, η, ϑ ∈ Q. A straightforward calculation yields α([[ξ, η]Q , ϑ]Q (d1 d1 α)(ξ, η, ϑ) = − (ξ,η,ϑ) cyclic

−

(ξ(η(α(ϑ))) − η(ξ(α(ϑ))) − [ξ, η]H (α(ϑ))) .

(ξ,η,ϑ) cyclic

Using (1.9.4), we substitute (δ(ξ, η))(α(ϑ)) for ξ(η(α(ϑ))) − η(ξ(α(ϑ))) − [ξ, η]H (α(ϑ)) and obtain (d1 d1 α)(ξ, η, ϑ) = −

α([[ξ, η]Q , ϑ]Q −

(ξ,η,ϑ) cyclic

Likewise, a calculation gives

(δ(ξ, η))(α(ϑ)).

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(d2 d0 α)(ξ, η, ϑ) =

(δ(ξ, η))(α(ϑ)) −

(ξ,η,ϑ) cyclic

α(δ(ξ, η) · ϑ),

(ξ,η,ϑ) cyclic

whence the vanishing of the operator d1 d1 + d2 d0 on Alt 1A (Q, Alt 0A (H, A)) implies α [[ξ, η]Q , ϑ]Q + (δ(ξ, η)) · ϑ = 0. (ξ,η,ϑ) cyclic

For later reference, we note that (d2 d0 α(ϑ))(ξ, η) = (δ(ξ, η))(α(ϑ)), whence

α([[ξ, η]Q , ϑ]Q ) = (d2 d0 α)(ξ, η, ϑ)

(ξ,η,ϑ) cyclic

+

(2.8.6) (d2 d0 α(ϑ))(ξ, η).

(ξ,η,ϑ) cyclic

(vi) The vanishing of the operator d0 d2 + d1 d1 + d2 d0 : Alt 0A (Q, Alt 1A (H, A)) − → Alt 2A (Q, Alt 1A (H, A)) entails the compatibility property (1.9.5), the “generalized Q-module structure on H .’’ (vii) The vanishing of the operator d1 d2 + d2 d1 : Alt 0A (Q, Alt 1A (H, A)) − → Alt 3A (Q, Alt 0A (H, A)) entails the compatibility property (1.9.7). Indeed, given ξ, η, ϑ ∈ Q and α : H → A, ((d1 d2 + d2 d1 )α)(ξ, η, ϑ) = α δ([ξ, η]Q , ϑ) − ξ · δ(η, ϑ) . (ξ,η,ϑ)

cyclic

2.9. The spectral sequence Let (A, H, Q) be a Lie–Rinehart triple. The ﬁltration of Alt A (Q, Alt A (H, A)) by Q-degree leads to a spectral sequence (Er∗,∗ , dr )

(2.9.1)

(E0 , d0 ) = (Alt A (Q, Alt A (H, A)), d0 ),

(2.9.2)

having p,q whence E1

p amounts to the Lie–Rinehart cohomology Hq (H, Alt A (Q, A)) of H p

with values in the left (A, H )-module Alt A (Q, A). There is a slight conﬂict of notation here but it will always be clear from the context whether dj (j ≥ 0) refers to the differentials of a spectral sequence or to a system of multicomplex operators. The spectral sequence (2.9.1) is an invariant of the Lie–Rinehart triple structure. In particular, E10,0 = AH

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and E11,0 = Hom(Q, A)H , and H∗ (H, A) inherits an (AH , QH )-module structure, with reference to the Lie–Rinehart structure on (AH , QH ); see Corollary 1.11. Thus the Rinehart complex (Alt AH (QH , H∗ (H, A)), d) is deﬁned. Illustration 2.10. The spectral sequence (2.9.1) includes as special cases that of a foliation and the Hodge–de Rham spectral sequence. This provides a conceptually simple approach to these spectral sequences and subsumes them under a single more general construction. We will now make this precise. (i) Consider a foliated manifold M, the foliation being written as F. Recall that a p-form ω on M is called horizontal (with reference to the foliation F) provided ω(X1 , . . . , Xp ) = 0 if some Xj is vertical, i.e., tangent to the foliation, or, equivalently, iX ω = 0 whenever X is vertical; a horizontal p-form ω is said to be basic provided it is constant on the leaves (i.e., λX ω = 0 whenever X is vertical). The sheaf of germs of basic p-forms is in general not ﬁne and hence gives rise to in general nontrivial cohomology in nonzero degrees; see Reinhart [1959]. Thus, under the circumstances of the Example 1.4.1, and those of (1.12) as well, so that (A, H ) is the Lie–Rinehart algebra (C ∞ (M), LF ) arising from a foliation F of a smooth manifold M, for every p ≥ 0, the Rinehart complex (Alt ∗A (H, Alt p (Q, A)), d) for the Lie–Rinehart algebra (A, H ) = (C ∞ (M), LF ) with coefﬁcients in Alt p (Q, A) which computes the cohomology H∗ (LF , Alt p (Q, A)), is the standard complex arising from a ﬁne resolution of the sheaf of germs of basic p-forms on M. Thus the p,∗ cohomology E1 − H∗ (LF , Alt p (Q, A)) is the cohomology of M with values in the sheaf of germs of basic p-forms on M. The corresponding spectral sequence (2.9.1) comes down to the ordinary spectral sequence of a foliation, already studied in the literature (see Reinhart [1959], Sarkharia [1974], Sarkharia [1984]); this spectral sep,0 quence is an invariant of the foliation. The cohomology E2 is sometimes called “basic cohomology,’’ since it may be viewed as the cohomology of the “space of leaves.’’ (ii) Suppose that the foliation F arises from a ﬁber bundle with ﬁber F , and write ξ : P → B for an associated principal bundle, the structure group being written as G. In this case, the spectral sequence (2.9.1) comes down to that of the ﬁbration. Furthermore, as a C ∞ (B)-module, the cohomology H∗ (LF , A) is the space of sections of the induced graded vector bundle ζ ∗ : P ×G H∗ (F, R) → B. This vector bundle is ﬂat and therefore inherits a left (C ∞ (B), Vect(B))-module structure, and (E1∗,∗ , d1 ) coincides with the Rinehart complex (Alt ∗C ∞ (B) (Vect(B), (ζ ∗ )), d) which, in turn, is just the de Rham complex of B with values in the ﬂat vector bundle ζ ∗ and thus computes the cohomology E2∗,∗ = H∗ (B, ζ ∗ ); equivalently, the ﬂat connection on ζ ∗ turns H∗ (F, R) into a local system on B, and the de Rham complex of B with values in the ﬂat vector bundle ζ ∗ computes the cohomology E2∗,∗ = H∗ (B, H∗ (F, R)) of B with coefﬁcients in this local system. (iii) Returning to (i) above, suppose in particular that the foliation is transversely complete; see Almeida and Molino [1985]. Then the closures of the leaves constitute a smooth ﬁber bundle M → W , the algebra AH is isomorphic to that of smooth functions on W in an obvious fashion, and the obvious map from QH to Vect(W ) ∼ = Der(AH ) which is part of the Lie–Rinehart structure of (AH , QH ) is surjective (see

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Molino [1988]), and hence ﬁts into an extension of (R, AH )-Lie algebras of the kind → QH − → Vect(W ) − → 0. 0− → L −

(2.10.1)

Here L is the space of sections of a Lie algebra bundle on W and, following Molino [1988], we refer to the underlying extension of Lie algebroids on W as the Atiyah sequence of the (transversely complete) foliation F. Thus we see that the interpretation of QH as the space of vector ﬁelds on the “space of leaves’’ requires, perhaps, some care, since L will then consist of the “vector ﬁelds on the “space of leaves’’ which act trivially on every function.’’ To get a concrete example, let M = SU(2) × SU(2), and let F be the foliation deﬁned by a dense one-parameter subgroup in a maximal torus S 1 × S 1 in SU(2) × SU(2). Then the space W is S 2 × S 2 , and L is the space of sections of a real line bundle on S 2 × S 2 , necessarily trivial. One easily chooses a vector bundle ζ on SU(2) × SU(2) which is complementary to τF , and the Lie–Rinehart triple structure is deﬁned on (C ∞ (M), LF , (ζ )). In particular, the operation δ is nonzero. We note that the Chern–Weil construction in Huebschmann [1999a] yields a characteristic 2 class in HdeRham (S 2 × S 2 , R) for the extension (2.10.1), and this class may be viewed as an irrational Chern class Huebschmann [1999a] (Section 4). The nontriviality of this class entails that the differential d2 of the spectral sequence (2.9.1) be nontrivial. We also note that in view of a result of Almeida and Molino [1985], the transitive Lie algebroid corresponding to (2.10.1) does not integrate to a principal bundle; in fact, the integrability obstruction developed in Mackenzie [1987] is nonzero. (iv) Under the circumstances of the Example 1.4.2, the cohomology H∗ (H, Alt ∗ (Q, A)) is the Hodge cohomology of the smooth complex manifold M, i.e., amounts to the cohomology of M with values in the sheaf of germs of holomorphic p-forms, and the spectral sequence (2.9.1) is the Hodge–de Rham spectral sequence, sometimes referred to as the Frölicher spectral sequence in the literature. (v) Under the circumstances of Corollary 1.9.8, so that (A, Q, H ) is a Lie– Rinehart triple with trivial (A, H )-module structures on A and Q, the spectral sequence (2.9.1) is the ordinary spectral sequence for the corresponding extension of Lie–Rinehart algebras. If, furthermore, A is the ground ring so that Q and H are ordinary Lie algebras, this comes down to the Hochschild–Serre spectral sequence of the Lie algebra extension.

3 The additional structure on Q Let (A, H, Q) be a Lie–Rinehart triple. Theorem 1.9 gives a possible answer to Question 1.2 as well as to Question 1.1. What is missing is an intrinsic description of the structure induced on the constituent (A, Q) which, in turn, should then in particular encapsulate the Lie–Rinehart triple structure on (A, H, Q).We now proceed towards ﬁnding such an intrinsic description. To this end, we will introduce, on the constituent

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Q, certain operations similar to those introduced by Nomizu [1954] on the constituent q of a reductive decomposition g = h ⊕ q of a Lie algebra; the operations isolated by Nomizu come from the curvature and torsion of an afﬁne connection of the second kind. We note that the naive generalization to Lie–Rinehart algebras of the notion of reductive decomposition of a Lie algebra is not consistent with the Lie–Rinehart axioms. Given a Lie–Rinehart algebra L and an A-module decomposition L = H ⊕Q, where (A, H ) inherits a Lie–Rinehart structure, since for x ∈ H , ξ ∈ Q, and a ∈ A, necessarily [x, aξ ] = a[x, ξ ] − ξ(a)x, the deﬁning property [H, Q] ⊂ Q of a reductive decomposition cannot be satisﬁed unless the constituent Q acts trivially on A. Let (A, H, Q) be an almost pre-Lie–Rinehart triple. We will now deﬁne triple-, quadruple-, and quintuple products of the kind {·, ·; ·} : Q ⊗R Q ⊗R A − →A {·; ·, ·; ·} : Q ⊗R Q ⊗R Q ⊗R A − →A {·; ·; ·, ·; ·} : Q ⊗R Q ⊗R Q ⊗R Q ⊗R A − →A

(3.1)

{·, ·; ·} : Q ⊗R Q ⊗R Q − →Q {·; ·, ·; ·} : Q ⊗R Q ⊗R Q ⊗R Q − →Q {·; ·; ·, ·; ·} : Q ⊗R Q ⊗R Q ⊗R Q ⊗R Q − → Q.

(3.4)

(3.2) (3.3) (3.5) (3.6)

To this end, pick α, β, γ , ξ, η, ϑ, κ ∈ Q and a ∈ A. For 1 ≤ j ≤ 6, we will spell out an explicit description of each of the operations (3.j ) and label it as (3.j ), as follows: {ξ, η; a} = (δ(ξ, η))(a)

(3.1 )

{α; ξ, η; a} = (α · δ(ξ, η))(a)

(3.2 )

{α; β; ξ, η; a} = (α · (β · δ(ξ, η))(a)

(3.3 )

{ξ, η; ϑ} = (δ(ξ, η)) · ϑ {α; ξ, η; κ} = (α · δ(ξ, η)) · κ {α; β; ξ, η; γ } = (α · (β · δ(ξ, η)) · γ .

(3.4 ) (3.5 ) (3.6 )

Proposition 3.7. Suppose that (A, H, Q) is a pre-Lie–Rinehart triple. (i) The operations {ξ, η; ·} : A → A, {α; ξ, η; ·} : A → A, {α; β; ξ, η; ·} : A → A are derivations. (ii) The operations {ξ, η; ·} : A → A, {α; ξ, η; ·} : A → A, {α; β; ξ, η; ·} : A → A are skew in the variables ξ and η. (iii) The operations {ξ, η; ·} (on A as well as on Q) are A-linear in the variables ξ and η, and the operations (on A as well as on Q) {α; ξ, η; ·} and {α; β; ξ, η; ·} are A-linear in the variable α. (iv) The triple, quadruple, and quintuple products {ξ, η; ϑ}, {α; ξ, η; κ}, and {α; β; ξ, η; γ } are skew in the variables ξ and η.

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(v) Furthermore, these operations are related by the following identities: {ξ, η; aϑ} = a{ξ, η; ϑ} + {ξ, η; a}ϑ {α; aξ, η; b} = {α; ξ, aη; b} = a{α; ξ, η; b} + α(a){ξ, η; b} {α; aξ, η; κ} = {α; ξ, aη; κ} = a{α; ξ, η; κ} + α(a){ξ, η; κ} {α; ξ, η; aκ} = a{α; ξ, η; κ} + {α; ξ, η; a}κ {α; aβ; ξ, η; b} = a{α; β; ξ, η; b} + α(a){β; ξ, η; b} {α; β; aξ, η; b} = {α; β; ξ, aη; b} = a{α; β; ξ, η; b} + α(β(a)){ξ, η; b} + β(a){α; ξ, η; b} + α(a){β; ξ, η; b} {α; aβ; ξ, η; γ } = a{α; β; ξ, η; γ } + α(a){β; ξ, η; γ } {α; β; aξ, η; γ } = {α; β; ξ, aη; γ } = a{α; β; ξ, η; γ } + α(β(a)){ξ, η; γ } + β(a){α; ξ, η; γ } + α(a){β; ξ, η; γ } {α; β; ξ, η; aγ } = a{α; β; ξ, η; γ } + {α; β; ξ, η; a}γ . Proof . These assertions are immediate consequences of the pre-Lie–Rinehart triple properties of (A, H, Q).

Proposition 3.8. Suppose that (A, H, Q) is a pre-Lie–Rinehart triple, and let a ∈ A and α, β, γ , ζ, ξ, η, ϑ, κ ∈ Q. With the notation x = δ(α, β) and y = δ(γ , ζ ), the compatibility properties (1.9.1)–(1.9.7) take the following form. ξ {α, β; a} − {α, β; ξ(a)} = {ξ ; α, β; a} − {α, β; ξ }(a), {α, β; [ξ, η]Q } = [{α, β; ξ }, η]Q + [ξ, {α, β; η}]Q − {ξ ; α, β; η} + {η; α, β; ξ },

(3.8.1) (3.8.2)

(ξ · [δ(α, β), δ(γ , ζ )]H ) · κ = {ξ ; α, β; {γ , ζ ; κ}} − {γ , ζ, {ξ ; α, β; κ}} + {α, β; {ξ ; γ , ζ ; κ}} − {ξ ; γ , ζ ; {α, β; κ}} − {{α, β; ξ }; γ , ζ ; κ} + {{γ , ζ ; ξ }; α, β; κ}, (3.8.3) ξ(η(a)) − η(ξ(a)) = [ξ, η]Q (a) + {ξ, η; a},

(3.8.4)

{[ξ, η]Q ; α, β; γ } = {ξ ; η; α, β; γ } − {η; ξ ; α, β; γ } − {{α, β; ξ }, η; γ } − {ξ, {α, β; η}; γ } + {α, β; {ξ, η; γ }} − {ξ, η; {α, β; γ }} [[ξ, η]Q , ϑ]Q + {ξ, η; ϑ} = 0,

(ξ,η,ϑ) cyclic

(ξ,η,ϑ) cyclic

{[ξ, η]Q , ϑ; κ} =

{ξ ; η, ϑ; κ}.

(ξ,η,ϑ) cyclic

Furthermore, the compatibility property (1.5.12) takes the form

(3.8.5) (3.8.6) (3.8.7)

Homotopies and Maurer–Cartan algebras

[δ(α, β), δ(ξ, η)]H · ξ = {α, β; {ξ, η; ξ }} − {ξ, η; {α, β; ξ }}.

267

(3.8.8)

Proof . This is an immediate consequence of Theorem 1.9. We leave the details to the reader.

We note that (3.8.5) is equivalent to {α, β; {ξ, η; γ }} − {ξ, η; {α, β; γ }} = {{α, β; ξ }, η; γ } + {ξ, {α, β; η}; γ } + {[ξ, η]Q ; α, β; γ }

(3.8.5 )

− {ξ ; η; α, β; γ } + {η; ξ ; α, β; γ }. Somewhat more explicitly, (3.8.7) reads {[ξ, η]Q , ϑ, κ} + {[η, ϑ]Q , ξ, κ} + {[ϑ, ξ ]Q , η, κ} = (ξ · δ(η, ϑ)) · κ + (η · δ(ϑ, ξ )) · κ + (ϑ · δ(ξ, η)) · κ. Moreover, with the notation x = δ(ξ, η), (3.8.2) comes down to x · [ϑ, κ]Q = [x · ϑ, κ]Q + [ϑ, x · κ]Q − (ϑ · x) · κ + (κ · x) · ϑ, which is just (1.9.2), and (3.8.5) reads [x, δ(α, β)]H = δ(x · α, β) + δ(α, x · β) + [α, β]Q · x − α · (β · x) + β · (α · x), which is (1.9.5). Remark 3.9. The description of the structure on (A, Q) given in Propositions 3.7 and 3.8 is nearly intrinsic: Only the left-hand side (ξ · [δ(α, β), δ(γ , ζ )]H )(κ) of the equation (3.8.3) and the left-hand side [δ(α, β), δ(ξ, η)]H · ξ of (3.8.8) involve the Lie–Rinehart bracket [·, ·]H on H explicitly, and this bracket is not covered by the structure on (A, Q). The Lie–Rinehart structure of (A, H ) encapsulates a whole bunch of additional compatibility conditions which the triple-, quadruple-, quintuple products necessarily satisfy. 3.10. Reconstruction of the Lie–Rinehart triple structure Starting from (A, Q), endowed with the pieces of structure (1.5.1.Q) and (1.5.2.Q) which are supposed to satisfy (1.5.6.Q) and (1.5.7.Q) and, furthermore, with the triple-, quadruple-, quintuple products (3.1)–(3.6), to reconstruct an (R, A)-Lie algebra complement H such that E = H ⊕ Q inherits an (R, A)-Lie algebra structure which, in turn, then determines the given structure on (A, Q), we might proceed as follows, where we pursue a reasoning similar to that in the proof of Theorem 18.1 in Nomizu [1954] and that of Theorem 7.1 in Kikkawa [1975]: Suppose that those compatibility properties spelled out in (3.7) and (3.8) which are merely phrased in

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terms of Q and, in particular, do not involve the bracket [·, ·]H on H explicitly, are satisﬁed. Given ξ, η ∈ Q, deﬁne δ(ξ, η) ∈ End R (Q) by δ(ξ, η)(ϑ) = {ξ, η; ϑ} and let H ⊆ End R (Q) be the A-linear span of the δ(ξ, η)s in End R (Q) (where ξ, η ∈ Q); notice that, by assumption, Q comes with an A-module structure, whence it makes sense to take the A-linear span of the δ(ξ, η)s in End R (Q) (ξ, η ∈ Q). The restriction of the evaluation pairing End R (Q) ⊗R Q → Q to H yields the pairing (1.5.3), to be written as the association (δ(ξ, η), ϑ) −→ δ(ξ, η) · ϑ,

ξ, η, ϑ ∈ Q,

and the requisite bilinear pairing (1.5.5) is just δ, viewed as a function from Q ⊗A Q to H . Since the triple product (3.4) is A-bilinear, the pairing (1.5.3) will then satisfy (1.5.9), and δ is well deﬁned on Q ⊗A Q. Next, deﬁne a pairing H ⊗R A − → A,

(x, a) → x(a),

by means of δ(ξ, η)(a) = {ξ, η; a},

ξ, η ∈ Q, a ∈ A.

This yields the requisite pairing (1.5.2.H ). Since the triple product (3.1) is A-bilinear, (1.5.6.H ) will hold. Thereafter, deﬁne a pairing · : Q ⊗R H − →H by setting (α · δ(ξ, η))(κ) = {α; ξ, η; κ},

α, ξ, η, κ ∈ Q.

This yields the requisite pairing (1.5.4). Since the quadruple product is A-linear in α, (1.5.11) will hold. The compatibility properties in (3.7) and (3.8) imply that the pairings (1.5.3) and (1.5.4) will satisfy (1.5.8) and (1.5.10). To complete the construction, we must require that the ordinary commutator bracket on End R (Q) descend to a bracket [·, ·]H on H in such a way that (A, H ), with this bracket and the pairing (1.5.2.H ) (which we reconstructed from the triple product (3.4)), be a Lie–Rinehart algebra in such a way that (3.8.3) and (3.8.8) are satisﬁed . The remaining compatibility properties in order for (A, H, Q) to be a Lie–Rinehart triple will then be implied by the structure isolated in (3.7) and (3.8).

4 Quasi-Lie–Rinehart algebras Let (A, H, Q) be a Lie–Rinehart triple. Thus (A, H ) is a Lie–Rinehart algebra, whence the Rinehart complex A = (Alt A (H, A), d) inherits a differential graded R-algebra structure and Q is, in particular, an (A, H )-module, whence the Rinehart complex Q = (Alt A (H, Q), d) is a differential graded A-module in an obvious fashion. For the special case where (A, H, Q) is a twilled Lie–Rinehart algebra (i.e.,

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269

the operation δ : Q ⊗A Q → H (see (1.5.5)) is zero), we have shown in Huebschmann [1998b] (3.2) (see also Huebschmann [2000]) that the pair (A, Q) acquires a differential graded Lie–Rinehart structure and that the twilled Lie–Rinehart algebra compatibility conditions can be characterized in terms of this differential graded Lie– Rinehart structure. We will now show that for a general Lie–Rinehart triple (A, H, Q) (i.e., with in general nonzero δ), the pair (A, Q) inherits a higher homotopy version of a differential graded Lie–Rinehart algebra structure; abstracting from the structure which thus emerges, we isolate the notion of a quasi-Lie–Rinehart algebra. This structure provides a complete solution of the problem of describing the structure on the constituent of a Lie–Rinehart triple written as Q and hence yields a complete answer to Question 1.1. We begin by describing the requisite pieces of structure, independently of any given (pre-)Lie–Rinehart triple, in the following fashion: Let A be a graded commutative algebra concentrated in nonnegative degrees (Aq = 0 for q < 0), at this stage not a differential graded commutative algebra, and let Q be a graded (left) A-module which we suppose to be an induced graded A-module of the kind Q = A ⊗A Q, where A = A0 and where Q is concentrated in degree zero; the notation (A, Q) will refer to this kind of structure throughout, perhaps endowed with additional structure. A homogeneous A-multilinear function φ on Q in variables with values in a graded A-module M is said to be A-graded multilinear if, for every α1 , . . . , α ∈ A and every ξ1 , . . . , ξ ∈ Q, φ(ξ1 , . . . , ξj −1 , αj ξj , ξj +1 , . . . , ξ ) = (−1)(|φ|+|ξ1 |+···+|ξj −1 |)|αj | αj φ(ξ1 , . . . , ξj −1 , ξj , ξj +1 , . . . , ξ ); it is called graded alternating if, for every ξ1 , . . . , ξ ∈ Q, φ(ξ1 , . . . , ξj , ξj +1 , . . . , ξ ) = −(−1)|ξj ||ξj +1 | φ(ξ1 , . . . , ξj +1 , ξj , . . . , ξ ). A pairing is graded skew-symmetric provided it is graded alternating as a graded bilinear function. With these preparations out of the way suppose that, in addition, (A, Q) carries •

a graded skew-symmetric R-bilinear pairing of degree zero [·, ·]Q : Q ⊗R Q − → Q,

•

an R-bilinear pairing of degree zero Q ⊗R A − → A,

•

(4.1)

(ξ, α) → ξ(α),

(4.2)

an A-trilinear operation of degree −1 ·, ·; ·Q : Q ⊗A Q ⊗A A − →A

(4.3.Q)

which is graded skew-symmetric in the ﬁrst two variables (i.e., in the Q-variables).

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We will say that the pair (A, Q) constitutes a pre-quasi-Lie–Rinehart algebra provided it satisﬁes (i) and (ii) below. (i) The values of the adjoints Q → End R (A) and Q ⊗A Q → End R (A) of (4.2) and (4.3.Q) respectively, lie in Der R (A) so that, in particular, given ξ, η ∈ Q and homogeneous α, β ∈ A, ξ, η; βαQ = ξ, η; βQ α + (−1)|β| βξ, η; αQ ; (ii) the bracket (4.1), the operation (4.2), and the graded A-module structure on Q satisfy the following graded Lie–Rinehart axioms (4.4) and (4.5): (aξ )(b) = a(ξ(b)), a, b ∈ A, ξ ∈ Q, [ξ, aη]Q = ξ(a)η + a[ξ, η]Q , a ∈ A, ξ, η ∈ Q.

(4.4) (4.5)

The graded Lie–Rinehart algebra axioms (4.4) and (4.5) imply that (4.1) and (4.2) are determined by their restrictions [·, ·]Q : Q ⊗R Q − →Q → A, Q ⊗R A −

(ξ, α) → ξ(α).

(4.1.Q) (4.2.Q)

Here the values of (4.1.Q) necessarily lie in Q since [·, ·]Q is supposed to be of degree zero; in particular, (4.1.Q) is skew-symmetric in the usual sense. We note that, when A is concentrated in degree zero, the operation (4.3.Q) is necessarily zero. Given a pre-quasi-Lie–Rinehart algebra (A, Q), consider the bigraded algebra Alt A (Q, A) ∼ = Alt A (Q, A),

(4.6)

of A-valued A-multilinear alternating functions on Q and deﬁne the operators p

p+1

d1 : Alt A (Q, Aq ) − → Alt A (Q, Aq ) and

p

p+2

d2 : Alt A (Q, Aq ) − → Alt A (Q, Aq−1 )

(p, q ≥ 0) (p, q ≥ 0)

(4.7.1) (4.8.1)

by |f |+1

(−1)

+

(d1 f )(ξ1 , . . . , ξp+1 ) =

p+1

(−1)j −1 ξj (f (ξ1 , . . . ξ=j . . . , ξp+1 ))

j =1

(−1)

j +k

(4.7.2)

f ([ξj , ξk ]Q , ξ1 , . . . ξ=j . . . ξ=k . . . , ξp+1 )

1≤j
(the graded CCE formula), (−1)|f |+1 (d2 f )(ξ1 , . . . , ξp+2 ) = (−1)p (−1)j +k ξj , ξk ; f (ξ1 , . . . ξ=j . . . ξ=k . . . , ξp+2 )Q , 1≤j
(4.8.2)

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where ξ1 , . . . , ξp+2 ∈ Q. The graded Lie–Rinehart axioms (4.4) and (4.5) imply that the operator d1 is well deﬁned on Alt A (Q, A) as an R-linear (beware, not A-linear) operator. The usual argument shows that d1 is a derivation on the bigraded A-algebra Alt A (Q, A). Since the operation ·, ·; ·Q (see (4.3.Q)) is A-trilinear, the operator d2 is well deﬁned on A-valued A-multilinear functions on Q. Since (4.3.Q) is skewsymmetric in the ﬁrst two variables, the operator d2 automatically has square zero, i.e., is a differential. Lemma 4.8.3. The operator d2 is an A-linear derivation on the bigraded A-algebra Alt A (Q, A). Proof . Since, as a graded A-module, Q is an induced graded A-module, the bigraded algebra Alt A (Q, A) may be written as the bigraded tensor product Alt A (Q, A) ∼ = Alt A (Q, A) ⊗ A, and it sufﬁces to consider forms which may be written as βα, where β ∈ Alt A (Q, A) and α ∈ A; the formula (4.8.2) yields d2 (βα) = (−1)|β| βd2 (α)

d2 (β) = 0,

and, since for ξ, η ∈ Q, the operation ξ, η; ·Q is a derivation of A, we conclude that the operator d2 is an R-linear derivation on Alt A (Q, A). Furthermore, since for a ∈ A = A0 , for degree reasons, d2 (a) is necessarily zero the operator d2 is plainly well deﬁned on A-valued A-multilinear functions on Q and in fact an Alinear derivation on Alt A (Q, A) as asserted.

Remark 4.8.4. On the formal level, the notion of a quasi-Lie–Rinehart algebra isolated above is somewhat unsatisfactory since the deﬁnition involves the structure of Q as an induced A-module. The operator d1 may be written as an operator on the bigraded Amodule Alt A (Q, A) of A-graded multilinear alternating forms on Q directly in terms of the operations (4.1) and (4.2), that is, in terms of the arguments of these operations, without explicit reference to the induced A-module structure. Indeed, given an ntuple η = (η1 , . . . , ηn ) of homogeneous elements of Q, write |η| = |η1 | + · · · + |ηn | and |η|(j ) = |η1 | + · · · + |ηj |, for 1 ≤ j ≤ n, and deﬁne the operator p

p+1

d(·,·) : Alt R (Q, Aq ) − → Alt R (Q, Aq ) by means of the formula (−1)|f |+1+|η| (d(·,·) (f ))(η1 , . . . , ηp+1 ) =

p+1

(j −1) +|f |)|η | j

(−1)j −1+(|η|

ηj f (η1 , . . . η=j . . . , ηp+1 )

j =1

and the operator p

p+1

d[·,·] : Alt R (Q, Aq ) → Alt R (Q, Aq ),

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by means of the formula (−1)|f |+1+|η| (d[·,·] (f ))(η1 , . . . , ηp+1 ) = ε(j, k, η)f ([ηj , ηk ], η1 , . . . = ηj . . . η=k . . . , ηp+1 ), 1≤j
where η1 , . . . , ηp+1 are homogeneous elements of Q and (j −1) |η |+(|η|(k−1) −|η |)|η | j j k

ε(j, k, η) = (−1)j +k+|η|

.

Then the sum d(·,·) + d[·,·] descends to an operator on Alt A (Q, A) which, in turn, coincides with d1 . In this fashion, d1 appears as being given by the CCE formula (2.2.8) with respect to (4.1) and (4.2). We were so far unable to give a similar description of the operator d2 , although, in terms of a suitable extension of (4.3.Q) to an operation of the kind Q ⊗A Q ⊗A A → A. Deﬁnition 4.9. Let (A, Q) be a pre-quasi-Lie–Rinehart algebra so that, in particular, A is a differential graded commutative algebra and Q a differential graded A-module. Consider the bigraded A-algebra Alt A (Q, A) ∼ = Alt A (Q, A) ⊆ Mult R (Q, A) (see (4.6) above), where Mult R (Q, A) refers to the bigraded algebra of A-valued R-multilinear forms on Q. The differentials on Q and A (both written as d, with an abuse of notation,) induce a differential D on Mult R (Q, A) in the usual way, that is, given an R-multilinear A-valued form f on Q, Df = df + (−1)|f |+1 f d, where, with a further abuse of notation, the “d’’ in the constituent f d signiﬁes the induced operator on any of the tensor powers Q⊗R ( ≥ 1). We will say that the pre-quasi-Lie–Rinehart algebra (A, Q) is a quasi-Lie–Rinehart algebra provided it satisﬁes the requirements (4.9.1)–(4.9.6) below where d1 and d2 are the operators (4.7.1) and (4.8.1), respectively. (4.9.1) The differential D descends to an operator on Alt A (Q, A), necessarily a differential, which we then write as d0 . (4.9.2) The differential on Q is a derivation for the bracket (4.1). (4.9.3) The pairing (4.2) is compatible with the differentials on A and Q. (4.9.4) For every ξ, η ∈ Q and α ∈ A, ξ(η(α)) − η(ξ(α)) − [ξ, η]Q (α) = ((d0 d2 + d2 d0 )(α)) (ξ, η). (4.9.5) For every ξ, η, ϑ ∈ Q and α ∈ Alt 1A (Q, A0 ) = HomA (Q, A),

Homotopies and Maurer–Cartan algebras

α([[ξ, η]Q , ϑ]Q ) = (d2 d0 α)(ξ, η, ϑ) +

(ξ,η,ϑ) cyclic

273

(d2 d0 α(ϑ))(ξ, η).

(ξ,η,ϑ) cyclic

(4.9.6) The operators d1 and d2 satisfy the commutation relation d1 d2 + d2 d1 = 0. In (4.9.6), it sufﬁces to require the vanishing of the operator d1 d2 + d2 d1 on Alt 0A (Q, A1 ). We leave it to the reader to spell out a description of this requirement directly in terms of the structure (4.1)–(4.3); this description would be less concise than the requirement given as (4.9.6). Theorem 4.10. Let (A, Q) be a pre-quasi-Lie–Rinehart algebra, consider the bigraded A-algebra Alt A (Q, A) ∼ = Alt A (Q, A) ⊆ Mult R (Q, A), suppose that the operator D on Mult R (Q, A) descends to an operator d0 on Alt A (Q, A), and let d1 and d2 be the operators on Alt A (Q, A) given by (4.7.1) and (4.8.1), respectively. Then (A, Q) is a quasi-Lie–Rinehart algebra if and only if (Alt A (Q, A), d0 , d1 , d2 ) is a multialgebra. Proof . (i) The identity 0 = d0 d1 + d1 d0 on Alt 1A (Q, A0 ) is equivalent to (4.9.2), that is, to the differential on Q being a derivation for the bracket [·, ·]Q ; see (4.1). See also (2.8.5(iii)). (ii) The identity 0 = d0 d1 + d1 d0 on Alt 0A (Q, A∗ ) is equivalent to (4.9.3), that is, to the differentials on A and Q being compatible with the pairing (4.2). (iii) The identity 0 = d0 d2 + d1 d1 + d2 d0 on Alt 0A (Q, A0 ) is equivalent to the special case of (4.9.4) where α ∈ A = A0 . See (2.8.5(iv)). (iv) Once (4.9.4) holds, the identity 0 = d0 d2 + d1 d1 + d2 d0 on Alt 1A (Q, A0 ) is equivalent to (4.9.5). See (2.8.5(v)). (v) The identity 0 = d0 d2 + d1 d1 + d2 d0 on Alt 0A (Q, A1 ) is equivalent to the special case of (4.9.4) where α ∈ A1 . See (2.8.5(vi)).

Under the circumstances of Theorem 4.10, we will refer to the multialgebra (Alt A (Q, A), d0 , d1 , d2 ) as the Maurer–Cartan algebra for the quasi-Lie–Rinehart algebra structure on (A, Q). 4.11. Relationhip with almost pre-Lie–Rinehart triples Our goal is to show how a Lie–Rinehart triple determines a quasi-Lie–Rinehart algebra. Here we explain the ﬁrst step, that is, how a structure of the kind (4.1.Q)– (4.3.Q) that underlies a pre-quasi-Lie–Rinehart algebra arises: Let (A, Q, H ) be an

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almost pre-Lie–Rinehart triple, and let A = Alt A (H, A) and Q = Alt A (H, Q). Then A = Alt A (H, A) is a graded commutative algebra (beware, not necessarily a differential graded commutative algebra) and Q = Alt A (H, Q) is a graded A-module (not necessarily a differential graded module). The pairings (1.5.2.Q) and (1.5.4) induce a pairing Q ⊗R A → A of the kind (4.2.Q) by means of the association ξ ⊗ α → ξ(α),

ξ ∈ Q, α ∈ A = Alt A (H, A),

(4.11.1)

where (ξ(α))(x1 , . . . , xn ) = ξ(α(x1 , . . . , xn )) −

n

α(x1 , . . . , ξ · xj , . . . , xn ).

(4.11.2)

j =1

The corresponding induced pairing of the kind (4.2) has the form Q ⊗R A − → A,

(ξ, α) → ξ(α), ξ ∈ Q, α ∈ A.

(4.11.3)

Furthermore, the bracket [·, ·]Q is exactly of the kind (4.1.Q). It extends to a graded skew-symmetric bracket [·, ·]Q : Q ⊗R Q − →Q (4.11.4) of the kind (4.1). To get an explicit formula for this bracket we suppose, for simplicity, that the canonical map from A⊗A Q to Q = Alt A (H, Q) is an isomorphism of graded A-modules so that Q is indeed an induced graded A-module of the kind considered above. This will be the case, for example, when H is ﬁnitely generated and projective as an A-module or when Q is projective as an A-module. Under these circumstances, given homogeneous elements α, β ∈ A and ξ, η ∈ Q, the value [α ⊗ ξ, β ⊗ η]Q of the bracket (4.11.4) is given by [α ⊗ ξ, β ⊗ η]Q = (αξ(β)) ⊗ η − (βη(α)) ⊗ ξ + (αβ) ⊗ [ξ, η]Q .

(4.11.5)

Furthermore, setting ξ, η; αQ = iδ(ξ,η) α,

α ∈ A, ξ, η ∈ Q,

(4.11.6)

where, for x ∈ H , ix refers to the operation of contraction, that is, for x1 , . . . , xq−1 ∈ H , ξ, η; αQ (x1 , . . . , xq−1 ) = α(δ(ξ, η), x1 , . . . , xq−1 ),

(4.11.7)

we obtain a pairing of the kind (4.3.Q). Thus, summing up, we conclude that, on (A, Q), the operations (4.11.1), (4.11.4), and (4.11.6) which, in turn, come from the almost pre-Lie–Rinehart triple structure on (A, Q, H ) determine a structure of the kind (4.1.Q)–(4.3.Q) which underlies that of a pre-quasi-Lie–Rinehart algebra. Indeed, the structure on (A, Q) given by (4.11.1), (4.11.4), and (4.11.6) is essentially a rewrite of the almost pre-Lie–Rinehart triple structure on (A, Q, H ); the two structures are equivalent when H is ﬁnitely generated projective as an A-module and when Q has property P. At this stage we do not make any claim as to whether or not the structure given by (4.11.1), (4.11.4), and (4.11.6) turns (A, Q) into a pre-quasi-Lie– Rinehart algebra.

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275

4.12. Lie–Rinehart triples and quasi-Lie–Rinehart algebras Suppose now that (A, Q, H ) is a pre-Lie–Rinehart triple; with reference to the Lie– Rinehart structure on (A, H ) and the left (A, H )-module structure on Q, the Lie– Rinehart differentials then turn A = Alt A (H, A) into a differential graded commutative algebra and Q = Alt A (H, Q) into a differential graded (left) A-module; see (1.5.13). Furthermore, the bigraded algebra Alt A (Q, A) of alternating A-multilinear A-valued forms on Q may be rewritten in the form Alt A (H, Alt A (Q, A)); equivalently, the algebra Alt A (Q, A) may be viewed as the bigraded algebra Alt A (Q, A) of alternating A-multilinear A-valued forms on Q. We write the resulting operator d0 (see (2.4.5) and (2.5.1)) as p

p

d0 : Alt A (Q, Aq ) − → Alt A (Q, Aq+1 )

(p, q ≥ 0).

(4.12.1)

Consider the operators d1 and d2 on Alt A (Q, A) given as (4.7.1) and (4.8.1) above, respectively. These operators now come down to the operators (2.4.6) and (2.4.7), respectively. By Theorem 2.7, when (A, Q, H ) is a genuine Lie–Rinehart triple, (Alt A (Q, A), d0 , d1 , d2 ) = (Alt A (Q, Alt A (H, A)), d0 , d1 , d2 )

(4.12.2)

is a Maurer–Cartan algebra, that is, d = d0 + d1 + d2 turns Alt A (Q, A) into a differential graded algebra. Furthermore, still by Theorem 2.7, under the assumption that H and Q both have property P, the converse holds, i.e., when (4.12.2) is a Maurer– Cartan algebra, (A, Q, H ) is a genuine Lie–Rinehart triple. In view of Theorem 4.10 we conclude the following. Theorem 4.13. Let (A, H, Q) be a pre-Lie–Rinehart triple and suppose that both H and Q have property P (e.g., H and Q are both projective as A-modules). Then (A, H, Q) is a genuine Lie–Rinehart triple if and only if (A, Q) = (Alt A (H, A), Alt A (H, Q)), endowed with the pairing (4.11.1), the bracket [·, ·]Q (see (4.11.4)), and the operation ξ, η; αQ (see (4.11.6)), is a quasi-Lie–Rinehart algebra. The proof of the following is straightforward and left to the reader. Proposition 4.14. The homology (H∗ (A), H∗ (Q)) of a quasi-Lie–Rinehart algebra (A, Q) inherits a graded Lie–Rinehart algebra structure. Given a Lie–Rinehart triple (A, H, Q), the graded Lie–Rinehart algebra (H∗ (A), H∗ (Q)) of the corresponding quasi-Lie–Rinehart algebra (A, Q) = (Alt A (H, A), Alt A (H, Q)) contains more information than the Lie–Rinehart algebra

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(AH , QH ) = (H0 (A), H0 (Q)) spelled out in Corollary 1.11. Illustration 4.15. Let (M, F) be a foliated manifold, maintain the notation established earlier in (1.4.1), (1.12), and (2.11), let (A, H, Q) = (C ∞ (M), LF , Q), the corresponding Lie–Rinehart triple, and consider the resulting quasi-Lie–Rinehart algebra (A, Q) = (Alt A (H, A), Alt A (H, Q)). We may view A as the algebra of generalized functions and Q as the generalized Lie algebra of vector ﬁelds for the foliation. Thus A is the standard complex arising from a ﬁne resolution of the sheaf of germs of functions on M which are constant on the leaves. Likewise, the constituent QH of the Lie–Rinehart algebra (AH , QH ) (discussed earlier) (see (1.12) and (2.10) (iii)) amounts to the space of global sections of the sheaf VQ of germs of vector ﬁelds on M which are horizontal (with respect to the decomposition (τM ) = LF ⊕ Q) and constant on the leaves, and Q is the standard complex arising from a ﬁne resolution of this sheaf. Thus H∗ (A) is the cohomology of M with values in the sheaf of germs of functions which are constant on the leaves, and H∗ (Q) is the cohomology of M with values in the sheaf VQ . Under the circumstances of (2.10(ii)), so that the foliation F comes from a ﬁber bundle and the space of leaves coincides with the base B of the corresponding ﬁbration, from the graded commutative R-algebra structure of H∗ (F, R), the space (ζ ∗ ) of sections of the induced graded vector bundle ζ ∗ : P ×G H∗ (F, R) → B inherits a graded C ∞ (B)-algebra structure and, as a graded C ∞ (B)-algebra, H∗ (A) coincides with the graded commutative algebra (ζ ∗ ) of sections of ζ ∗ ; in particular, H0 (A) = C ∞ (B). Furthermore, H0 (Q) is the (R, C ∞ (B))-Lie algebra Vect(B) of smooth vector ﬁelds on the base B and, as a graded (R, H∗ (A))-Lie algebra, H∗ (Q) is the graded crossed product H∗ (Q) = H∗ (A) ⊗C ∞ (B) Vect(B)

(4.15.1)

(see Huebschmann [1998b] for the notion of graded crossed product Lie–Rinehart algebra). Under the circumstances of (2.10(i)), when the foliation does not come from a ﬁber bundle, the structure of the graded Lie–Rinehart algebra (H∗ (A), H∗ (Q)) will in general be more complicated than that for the case when the foliation comes from a ﬁber bundle. The signiﬁcance of this more complicated structure has been commented on already in the introduction. Remark 4.16. We are indebted to P. Michor for having pointed out to us a possible relationship between the above notion of quasi-Lie–Rinehart bracket and the familiar Frölicher–Nijenhuis bracket; see Frölicher and Nijenhuis [1956] and Nijenhuis

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277

[1955] for the latter. Given a smooth manifold M, the Frölicher–Nijenhuis bracket is deﬁned on the graded vector space of forms on M with values in the tangent bundle τM of M and endows this graded vector space with a graded Lie algebra structure which in degree zero amounts to the ordinary Lie bracket of vector ﬁelds on M. Given a Lie–Rinehart algebra (A, L), an obvious generalization of the Frölicher–Nijenhuis bracket endows the graded A-module Alt A (L, L) with a graded R-Lie algebra structure. Given a Lie–Rinehart triple (A, H, Q), with correponding Lie–Rinehart algebra (A, L), where L = H ⊕Q, the induced quasi-Lie–Rinehart bracket (4.11.4) is deﬁned on Alt A (H, Q), and the obvious question arises as to how this quasi-Lie–Rinehart bracket is related with the Frölicher–Nijenhuis bracket on Alt A (L, L).

5 Quasi-Gerstenhaber algebras The notion of a Gerstenhaber algebra has recently been isolated in the literature but implicitly already occurs in Gerstenhaber [1963]; see Huebschmann [1998a] for details and more references. In this section we will introduce the notion of quasiGerstenhaber algebra which generalizes that of strict differential bigraded Gerstenhaber algebra isolated in Huebschmann [1998b] and Huebschmann [2000] (where the attribute “strict’’ refers to the requirement that the differential be a derivation for the Gerstenhaber bracket). The generalization consists in admitting a bracket which does not necessarily satisfy the graded Jacobi identity and incorporating an additional piece of structure which measures the deviation from the graded Jacobi identity. For intelligibility, we recall the notion of graded Lie algebra, tailored to our purposes. As before, R denotes a commutative ring with 1. A graded R-module g, endowed with a graded skew-symmetric degree zero bracket [·, ·] : g ⊗ g → g, is called a graded Lie algebra provided the bracket satisﬁes the graded Jacobi identity (−1)|a||c| [a, [b, c]] = 0, (a,b,c) cyclic

for every triple (a, b, c) of homogeneous elements of g. Given a graded commutative algebra A, an (ordered) m-tuple a = (a1 , . . . , am ) of homogeneous elements thereof, and a permutation σ of m objects, we denote by ε(a, σ ) the sign deﬁned by a1 · · · · · am = ε(a, σ )aσ 1 · · · · · aσ m according to the Eilenberg–Koszul convention. We will consider bigraded R-algebras. Such a bigraded algebra is said to be bigraded commutative provided it is commutative in the bigraded sense, that is, graded commutative with respect to the total degree. Given such a bigraded commutative algebra G, for bookkeeping purposes, we will write its homogeneous components in q the form Gp , the superscript being viewed as a cohomology degree and the subscript q as a homology degree; the total degree |α| of an element α of Gp is, then, |α| = p −q.

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We will explore differential operators in the bigraded context. We recall the requisite notions from Koszul [1985] (Section 1); see also Akman [1997]. Let G be a bigraded commutative R-algebra with 1, and let r ≥ 1. A (homogeneous) differential operator on G of order ≤ r is a homogeneous R-endomorphism D of G such that a certain G-valued (r + 1)-form r+1 D on G (the deﬁniton of which for general r we do not reproduce here) vanishes. For our purposes, it sufﬁces to recall explicit descriptions of these forms in low degrees. Thus, given the homogeneous R-endomorphism D of G, for homogeneous ξ, η, ϑ, 1D (ξ ) = D(ξ ) − D(1)ξ 2D (ξ, η) = D(ξ η) − D(ξ )η − (−1)|ξ ||η| D(η)ξ + D(1)ξ η 3D (ξ, η, ϑ) = D(ξ ηϑ) − D(ξ η)ϑ − (−1)|ξ |(|η|+|ϑ|) D(ηϑ)ξ − (−1)|ϑ|(|ξ |+|η|) D(ϑξ )η + D(ξ )ηϑ + (−1)|ξ |(|η|+|ϑ|) D(η)ϑξ + (−1)|ϑ|(|ξ |+|η|) D(ϑ)ξ η − D(1)ξ ηϑ. In the literature, a (homogeneous) differential operator D of order ≤ r with D(1) = 0 is also referred to as a (homogeneous) derivation of order ≤ r. In particular, a homogeneous derivation d of (total) degree 1 and order 1 is precisely a differential turning G into a differential graded R-algebra. With these preparations out of the way, consider a bigraded commutative Rq algebra G with 1, with Gp zero when q < 0 or p < 0, together with •

a homogeneous bracket [·, ·] : G ⊗R G → G of bidegree (0, −1), where “bidegree (0, −1)’’ means that, in given bidegrees (q1 , p1 ) and (q2 , p2 ), the bracket takes the form q q q +q [·, ·] : Gp11 ⊗ Gp22 − → Gp11 +p22 −1 ;

•

a differential d : G∗∗ → G∗∗+1 of bidegree (1, 0) which endows G (with respect to the total degree) with a differential graded R-algebra structure; and a homogeneous differential operator # : G → G of order ≤ 3 with #(1) = 0 which is G00 -linear and of bidegree (−1, −2), i.e., in bidegree (q, p), # may be described as q q−1 # : Gp − → Gp−2 (q ≥ 1, p ≥ 2). (5.1)

•

In particular, # is zero on G∗0 , G0∗ , G1∗ . Notice that d and # both lower total degree by 1, that is, are homogeneous operators on G of degree −1. We will refer to the bracket [·, ·] as a quasi-Gerstenhaber bracket and to # as an h-Jacobiator for the bracket [·, ·] provided [·, ·] and # satisfy (5.i)–(5.vi) below: (5.i)

The bracket [·, ·] is graded skew-symmetric when the total degree of G is regraded down by one, that is, for homogeneous α, β ∈ G, [α, β] = −(−1)(|α|−1)(|β|−1) [β, α].

(5.2)

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279

(5.ii) For each homogeneous element α of G of bidegree (q, p), the operation [α, ·] is a derivation of G of bidegree (q − 1, p) for the multiplicative structure on G; that is, [α, ·] may be described as ∗+q−1

[α, ·] : G∗∗ − → G∗+p and, for homogeneous β, γ ∈ G,

[α, βγ ] = [α, β]γ + (−1)(|α|−1)|β| β[α, γ ].

(5.3)

(5.iii) The differential d behaves as a derivation for the bracket [·, ·], that is, for homogeneous x, y ∈ G, d[x, y] = [dx, y] − (−1)|x| [x, dy].

(5.4)

(5.iv) Given homogeneous elements ξ, η, ϑ of G, (−1)(|ξ |−1)(|ϑ|−1) [ξ, [η, ϑ]] = (−1)(|ξ |+|η|+|ϑ|) 3d#+#d (ξ, η, ϑ). (ξ,η,ϑ) cyclic

(5.5) (5.v) The differential operator # has square zero. (5.vi) The bracket [·, ·] and # are related by the following requirement: For every ordered quadruple a = (a1 , a2 , a3 , a4 ) of homogeneous elements of G, ε(σ )ε(a, σ )[3# (aσ 1 , aσ 2 , aσ 3 ), aσ 4 ] σ

=

ε(τ )ε(a, τ )3# ([aτ 1 , aτ 2 ], aτ 3 , aτ 4 ),

(5.6)

τ

where σ runs through (3,1)-shufﬂes and τ through (2,2)-shufﬂes and where ε(σ ) and ε(τ ) are the signs of the permutations σ and τ . The data (G; d, [·, ·], #) will then be referred to as a quasi-Gerstenhaber algebra. Notice that (5.3) implies that [α, 1] = 0 for every homogeneous element α of G. We note that, given an L∞ -algebra h with only two-variable and three-variable brackets [·, ·] and [·, ·, ·], respectively (and no nonzero higher order bracket operation), the compatibility condition which relates [·, ·] and [·, ·, ·] is exactly an identity of the kind (5.6) when [·, ·, ·] is substituted for 3# . A quasi-Gerstenhaber algebra having # zero is just an ordinary strict differential bigraded Gerstenhaber algebra. Indeed, in the general (quasi-) case, in view of the requirement (5.iv), the operation # measures the failure of the quasi-Gerstenhaber bracket [·, ·] to satisfy the graded Jacobi identity in a coherent fashion. A strict differential bigraded Gerstenhaber algebra having zero differential is called a bigraded Gerstenhaber algebra (see Huebschmann [1998b] and Huebschmann [2000]). Given a quasi-Gerstenhaber algebra (G; d, [·, ·], #), we denote its d-homology by H∗∗ (G)d . The following is straightforward.

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Proposition 5.7. Given a quasi-Gerstenhaber algebra (G; d, [·, ·], #), the quasiGerstenhaber bracket [·, ·] induces a bracket q

q

q +q

[·, ·] : Hp11 (G)d ⊗ Hp22 (G)d − → Hp11 +p22 −1 (G)d

(5.7.1)

on the d-homology H∗∗ (G)d which turns H∗∗ (G)d into an ordinary bigraded Gerstenhaber algebra. 5.8. Relationship with Lie–Rinehart triples We will now explain how quasi-Gerstenhaber algebras arise from Lie–Rinehart triples. To this end, we recall that given an ordinary Lie–Rinehart algebra (A, L), the Lie bracket on L and the L-action on A determine a Gerstenhaber bracket on the exterior A-algebra A L on L; for α1 , . . . , αn ∈ L, the bracket [u, v] in A L of u = α1 ∧ · · · ∧ α and v = α+1 ∧ · · · ∧ αn is given by the expression (−1)j +k [αj , αk ] ∧ α1 ∧ . . . α=j . . . α=k · · · ∧ αn , (5.8.1) [u, v] = (−1) 1≤j ≤
where = |u| is the degree of u; see Huebschmann [1998a] (1.1). In fact, given the R-algebra A and the A-module L, a bracket of the kind (5.8.1) yields a bijective correspondence between Lie–Rinehart structures on (A, L) and Gerstenhaber algebra structures on A L. Our goal, which will be achieved in the next section, is now to extend this observation to a relationship between Lie–Rinehart triples, quasi-Lie– Rinehart algebras, and quasi-Gerstenhaber algebras. Thus, let (A, Q, H ) be a pre-Lie–Rinehart triple. Consider the graded exterior Aq q p algebra A Q, and let G = Alt A (H, A Q), with the bigrading Gp = Alt A (H, A Q) (p, q ≥ 0). Suppose for the moment that (A, Q, H ) is merely an almost pre-Lie– Rinehart triple. Recall that the almost pre-Lie–Rinehart triple structure induces operations of the kind (4.11.3), (4.11.4), and (4.11.6) on the pair (A, Q) = (Alt A (H, A), Alt A (H, Q)) but, at the present stage, this pair is not necessarily a quasi-Lie–Rinehart algebra. Consider the bigraded algebra Alt A (H, A Q); at times we will view it as the exterior A-algebra on Q, and we will accordingly write A Q = Alt A (H, A Q).

(5.8.2)

The graded skew-symmetric bracket (4.11.4) on Q (= Alt A (H, Q)) extends to a (bigraded) bracket [·, ·] : A Q ⊗R A Q − → A Q (5.8.3) on A Q = Alt A (H, A Q). Indeed, with reference to the graded bracket [·, ·] on Q spelled out as (4.11.4) (and written there as [·, ·]Q ) and the pairing (4.11.1), the bigraded bracket (5.8.3) on A Q = Alt A (H, A Q) is determined by the formulas

Homotopies and Maurer–Cartan algebras

[αβ, γ ] = α[β, γ ] + (−1)|α||β| β[α, γ ], [ξ, a] = ξ(a), [α, β] = −(−1)

(|α|−1)(|β|−1)

281

(5.8.4)

[β, α],

where α, β, γ are homogeneous elements of A Q = Alt A (H, A Q), and where ξ ∈ Q and a ∈ A. We now construct an operation # of the kind (5.1) from the operation ·, ·; ·Q , that is, one which formally looks like an h-Jacobiator for (5.8.3). To this end we suppose that as an A-module, at least one of H or Q is ﬁnitely generated and projective; then the canonical A-linear morphism from Alt A (H, A) ⊗ A Q to Alt A (H, A Q) is an isomorphism of bigraded A-algebras. Let ξ1 , . . . , ξp ∈ Q. Now, given a homogeneous element β of Alt A (H, A), with reference to the operation ·, ·; ·Q induced by δ (see (4.11.6)), let #(βξ1 ∧ · · · ∧ ξp ) = (−1)j +k ξj , ξk ; βQ ξ1 ∧ . . . = ξj . . . = ξk · · · ∧ ξp ; 1≤j
(5.8.5) we will write #δ rather than just # whenever appropriate. As an operator on the graded A-algebra Alt A (H, A Q), # may be written as a ﬁnite sum of operators which are three consecutive contractions each; since an operator which consists of three consecutive contractions is a differential operator of order ≤ 3, the operator # is a differential operator of order ≤ 3. Furthermore, since for ξ, η ∈ Q, the operation ξ, η; ·Q is a derivation of the graded A-algebra Alt A (H, A), given homogeneous elements β1 and β2 of Alt A (H, A), #(β1 β2 ξ1 ∧ · · · ∧ ξp ) = (−1)|β1 | β1 #(β2 ξ1 ∧ · · · ∧ ξp ) + (−1)(|β1 |+1)|β2 | β2 #(β1 ξ1 ∧ · · · ∧ ξp ).

(5.8.6)

A somewhat more intrinsic description of # results from the observation that the operation # : Alt 1A (H, 2A Q) = HomA (H, 2A Q) − →A∼ = Alt 0A (H, 0A Q) is simply given by the assignment to χ : H → 2A Q of the trace of the A-module endomorphism δ◦χ of H when H is ﬁnitely generated and projective as an A-module, and of the trace of the A-module endomorphism χ ◦ δ of 2A Q when Q is ﬁnitely generated and projective as an A-module. We now give another description of # (see (5.8.11) below) under an additional hypothesis: Suppose that, as an A-module, Q is ﬁnitely generated and projective of constant rank n. Then the canonical A-module isomorphism n φ : ∗A Q → Alt n−∗ A (Q, A Q)

extends to an isomorphism n φ : Alt ∗A (H, ∗A Q) − → Alt ∗A (H, Alt n−∗ A (Q, A Q))

(5.8.7)

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of graded A-modules. In this fashion, Alt ∗A (H, ∗A Q) acquires a bigraded Alt ∗A (H, Alt ∗A (Q, A))-module structure, induced from the graded A-module nA Q. Further, the skew-symmetric A-bilinear pairing (1.5.5) induces an operator n → Alt ∗−1 d2 : Alt ∗A (H, Alt n−∗ A (Q, A Q)) − A (H, Alt A

n−(∗−2)

(Q, nA Q)).

(5.8.8)

This is just the operator (2.4.7 ), suitably rewritten, with M = nA Q, where the degree of the latter A-module forces the correct sign: The A-module nA Q is concentrated n−p in degree n, and a form in AltA (Q, nA Q) has degree p. In bidegree (q, p), given q

n−p

ψ ∈ Alt A (H, Alt A (Q, nA Q)), the value

q−1

n−p+2

d2 (ψ) ∈ Alt A (H, Alt A

(Q, nA Q))

of the operator (5.8.8) is given by the formula (−1)|ψ|+1 (d2 ψ)(x1 , . . . , xq−1 ) (ξp−1 , . . . , ξn ) = (−1)j +k ψ(δ(ξj , ξk ), x1 , . . . , xq−1 ) (ξp−1 , . . . = ξj . . . = ξk . . . , ξn ), p−1≤j
where x1 , . . . , xq−1 ∈ H and ξp−1 , . . . , ξn ∈ Q and, with |ψ| = q + p (the correct degree would be |ψ| = p − q but modulo 2 this makes no difference), this simpliﬁes to (−1)p (d2 ψ)(x1 , . . . , xq−1 ) (ξp−1 , . . . , ξn ) = (−1)j +k ψ(x1 , . . . , xq−1 , δ(ξj , ξk )) (ξp−1 , . . . = ξj . . . = ξk . . . , ξn ); p−1≤j
(5.8.9) see (2.5.4). Lemma 5.8.10. The operator # makes the diagram ∗−2 Alt ∗−1 A (H, A Q) ⏐ ⏐φ ?

#

Alt ∗A (H, ∗A Q) ⏐ ⏐ φ?

−−−−→

n Alt ∗A (H, Alt n−∗ −−−→ Alt ∗−1 A (Q, A Q)) − A (H, Alt A

n−(∗−2)

d2

(Q, nA Q))

commutative. Thus, under the isomorphism (5.8.7), the operator # is induced by the operator d2 (on the right-hand side of (5.8.7)). Proof . In a given bidegree (q, p), under the isomorphism (5.8.7), an element α ∈ q p Alt A (H, A Q) goes to q

n−p

φα ∈ Alt A (H, Alt A (Q, nA Q))

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283

determined by the identity φα (x1 , . . . , xq ) (ξp+1 , . . . , ξn ) = α(x1 , . . . , xq ) ∧ ξp+1 ∧ · · · ∧ ξn , for arbitrary x1 , . . . , xq ∈ H and ξp+1 , . . . , ξn ∈ Q. Under the isomorphism (5.8.7), the operator d2 (on the right-hand side of (5.8.7)) induces an operator q

p

q−1

p−2

' = 'δ : Alt A (H, A Q) − → Alt A (H, A Q)

(5.8.11)

of the kind (5.1) on the left-hand side of (5.8.7); by construction, for x1 , . . . , xq−1 ∈ H and ξp−1 , . . . , ξn ∈ Q, in view of (5.8.9), (−1)p ('α)(x1 , . . . , xq−1 ) ∧ ξp−1 ∧ · · · ∧ ξn = (−1)j +k α(x1 , . . . , xq−1 , δ(ξj , ξk )) ∧ ξp−1 ∧ . . . = ξj . . . = ξk · · · ∧ ξn . p−1≤j
Let β ∈ Alt A (H, A), η1 , . . . , ηp ∈ Q, and α = (η1 ∧ · · · ∧ ηp )β; then α(x1 , . . . , xq−1 , δ(ξj , ξk )) = (η1 ∧ · · · ∧ ηp )β(x1 , . . . , xq−1 , δ(ξj , ξk )) = β(x1 , . . . , xq−1 , δ(ξj , ξk ))η1 ∧ · · · ∧ ηp = (−1)q−1 β(δ(ξj , ξk ), x1 , . . . , xq−1 )η1 ∧ · · · ∧ ηp = (−1)q−1 ξj , ξk ; βQ (x1 , . . . , xq−1 )η1 ∧ · · · ∧ ηp = (−1)q−1+p(q−1) ξj , ξk ; βQ η1 ∧ · · · ∧ ηp (x1 , . . . , xq−1 ), for p − 1 ≤ j < k ≤ n, whence (−1)p+q−1 ('α)(x1 , . . . , xq−1 ) ∧ ξp−1 ∧ · · · ∧ ξn = (−1)j +k ξj , ξk ; βQ (x1 , . . . , xq−1 )η1 ∧ . . . = ξj . . . = ξk · · · ∧ ξn , p−1≤j
where, for p − 1 ≤ j < k ≤ n, η1 ∧ . . . = ξj . . . = ξk · · · ∧ ξn = η1 ∧ · · · ∧ ηp ∧ ξp−1 ∧ . . . = ξj . . . = ξk · · · ∧ ξn . Let (η1 , . . . , ηp ) = (ξ1 , . . . , ξp ). With j = p − 1 and k = p, the above expression yields (−1)p+q−1 ('α)(x1 , . . . , xq−1 ) ∧ ξp−1 ∧ · · · ∧ ξn = −ξp−1 , ξp ; βQ (x1 , . . . , xq−1 )ξ1 ∧ · · · ∧ ξp ∧ ξp+1 ∧ · · · ∧ ξn or, equivalently, since |x1 | + · · · + |xq−1 | = q − 1 and |ξ1 ∧ · · · ∧ ξp | = p, (−1)pq ('α)(x1 , . . . , xq−1 ) ∧ ξp−1 ∧ · · · ∧ ξn = − ξp−1 , ξp ; βQ η1 ∧ · · · ∧ ηp (x1 , . . . , xq−1 ) ∧ ξp+1 ∧ · · · ∧ ξn , = − ξp−1 , ξp ; βQ ξ1 ∧ · · · ∧ ξp−2 (x1 , . . . , xq−1 ) ∧ ξp−1 ∧ · · · ∧ ξn .

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Hence (−1)pq ('α) = −ξp−1 , ξp ; βQ ξ1 ∧ · · · ∧ ξp−2 ± . . . or, equivalently, '(βξ1 ∧ · · · ∧ ξp ) = −ξp−1 , ξp ; βQ ξ1 ∧ · · · ∧ ξp−2 ± . . . , where . . . stands for terms involving ξ1 ∧. . . = ξj . . . = ξk · · ·∧ξp with (j, k) = (p−1, p). Consequently, '(βξ1 ∧ · · · ∧ ξp ) = (−1)j +k ξj , ξk ; βQ ξ1 ∧ . . . = ξj . . . = ξk · · · ∧ ξp . 1≤j
However, this is exactly the deﬁnition (5.8.5) of #.

In view of Remark 2.6, the operator # thus calculates essentially the Lie algebra cohomology H∗ (Lnil , nA Q) of the (nilpotent) A-Lie algebra Lnil (= H ⊕ Q as an A-module) with values in the A-module nA Q, viewed as a trivial Lnil -module. In particular, # is A-linear. Suppose ﬁnally that (A, Q, H ) is a genuine Lie–Rinehart triple, not just an almost pre-Lie–Rinehart triple. By Proposition 4.13, (A, Q) then acquires a quasi-Lie– Rinehart structure. Our ultimate goal is now to prove that, likewise, A Q endowed with the bigraded bracket (5.8.3) and the operation # (see (5.8.5)), which formally looks like an h-Jacobiator, acquires a quasi-Gerstenhaber structure. The veriﬁcation of the requirements (5.2)–(5.4) does not present any difﬁculty at this stage, and the vanishing of ## is immediate. However we were so far unable to establish (5.5) and (5.6) without an additional piece of structure, that of a generator of a (quasiGerstenhaber) bracket. The next section is devoted to the notion of generator and the consequences it entails. A precise statement is given as Corollary 6.10.4 below.

6 Quasi-Batalin–Vilkovisky algebras and quasi-Gerstenhaber algebras Let G = G∗∗ be a bigraded commutative R-algebra, endowed with a bigraded bracket [·, ·] : G ⊗R G − →G of bidegree (0, −1) which is graded skew-symmetric when the total degree is regraded down by 1. Extending terminology due to Koszul (see the deﬁnition of [·, ·]D on p. 260 of Koszul [1985]), we will say that an R-linear operator on G of bidegree (0, −1) generates the bracket [·, ·] provided, for every homogeneous a, b ∈ G,

[a, b] = (−1)|a| (6.1) (ab) − ( a)b − (−1)|a| a( b) ; we then refer to the operator as a generator. We remind the reader that the righthand side of (6.1) equals (−1)|a| 2 (a, b). In particular, let (G; d, [·, ·], #) be a

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285

quasi-Gerstenhaber algebra over R. In view of the identity (1.4) on p. 260 of Koszul [1985], a generator is then necessarily a differential operator on G of order ≤ 2. Indeed, given a differential operator D, this identity reads as 3D (a, b, c) = 2D (a, bc) − 2D (a, b)c − (−1)|b||c| 2D (a, c)b. Hence, when a differential operator generates a quasi-Gerstenhaber bracket [·, ·],

3 (a, b, c) = (−1)|a| [a, bc] − [a, b]c − (−1)|b||c| [a, c]b . However, by virtue of (5.3), the right-hand side of this identity is zero, whence is necessarily of order ≤ 2. A generator of a quasi-Gerstenhaber bracket satisﬁes (1)a = 0 for every a ∈ G since, with respect to the multiplication map on G, the quasi-Gerstenhaber bracket behaves as a derivation of the appropriate degree in each variable of the bracket; see (5.3). We will say that a generator is strict provided (1) = 0 and d d# +

+ d = 0, + #d = 0;

# +#

(6.2) (6.3)

= 0;

(6.4)

a strict generator will henceforth often be written as ∂. Let G be a bigraded commutative algebra, with differential operators d : G∗∗ → G∗∗+1 ,

∗ : G∗∗ → G∗−1 ,

∗−1 # : G∗∗ → G∗−2 ,

having orders, respectively, ≤ 1, ≤ 2, ≤ 3, and having the properties d(1) = 0, (1) = 0, #(1) = 0. We will say that (G; d, , #) is a quasi-Batalin–Vilkovisky algebra provided dd = 0 and the operators d, , # satisfy the identities (6.2)–(6.4) as well as the identity ## = 0. (6.5) Thus (G; d, , #) is a quasi-Batalin–Vilkovisky algebra if and only if, on the totalization, the operator D = d + + # has square zero. 6.6. From quasi-Batalin–Vilkovisky algebras to quasi-Gerstenhaber algebras q

Theorem 6.6.1. Given a quasi-Batalin–Vilkovisky algebra (G; d, ∂, #) with Gp = 0 for q < 0 and p < 0, let [·, ·] be the bracket on G generated by ∂. Then (G; d, [·, ·], #) is a quasi-Gerstenhaber algebra provided, as a bigraded G00 -algebra, G is generated by its homogeneous constituents G01 and G10 . To prepare for the proof, we need the following. Lemma 6.6.2. Let G = {G∗∗ } be a bigraded commutative R-algebra with Gp = 0 for ∗−1 ∗ and # : G∗∗ → G∗−2 be differential operators q < 0 and p < 0, let : G∗∗ → G∗−1 ∗ ∗ be the bracket (6.1) of orders ≤ 2 and ≤ 3, respectively, and let [·, ·] : G∗ → G∗−1 q

286

J. Huebschmann

generated by . Let A = G00 , and suppose that, as a bigraded A-algebra, G is generated by its homogeneous constituents G01 and G10 and that # is A-linear. Then # +#

=0

(6.6.3)

if and only if, for every ordered quadruple a = (a1 , a2 , a3 , a4 ) of homogeneous elements of G, ε(σ )ε(a, σ )[3# (aσ 1 , aσ 2 , aσ 3 ), aσ 4 ] σ

=

ε(τ )ε(a, τ )3# ([aτ 1 , aτ 2 ], aτ 3 , aτ 4 ),

(6.6.4)

τ

where σ runs through (3,1)-shufﬂes and τ through (2,2)-shufﬂes and where ε(σ ) and ε(τ ) are the signs of the permutations σ and τ . We note the identity (6.6.4) is formally the same as (5.6), but the circumstances are now more general. We also note that the hypotheses of the lemma imply that (1) = 0 and #(1) = 0. Remark 6.6.5. For a graded commutative (not bigraded) algebra, multiplicatively generated by its homogeneous degree 1 constituent and endowed with a suitable Batalin–Vilkovisky structure, formally the same identity as (5.6) has been derived in Theorem 3.2 of Bangoura [2002]. Our totalization Tot yields a notion of Batalin– Vilkovisky algebra that is not equivalent to that explored in Bangoura [2002]; see Remark 6.17 below for details. The distinction between the ground ring R and the R-algebra A, crucial for our approach (involving in particular Lie–Rinehart algebras and variants thereof), complicates the situation further. We therefore give a complete proof of the lemma. Proof of Lemma 6.6.2. We start by exploring the operator # + # : G31 − → G00 = A. Let α ∈ G01 and ξ1 , ξ2 , ξ3 ∈ G10 ; then αξ1 ξ2 ξ3 ∈ G31 . Since # is of order ≤ 3, 4# (α, ξ1 , ξ2 , ξ3 ) = 0 and, for degree reasons, this identity boils down to #(αξ1 ξ2 ξ3 ) = #(αξ1 ξ2 )ξ3 + #(αξ2 ξ3 )ξ1 + #(αξ3 ξ1 )ξ2 . In view of the deﬁnition (6.1) of the bracket [·, ·], [#(αξ1 ξ2 ), ξ3 ] =

(#(αξ1 ξ2 )ξ3 ) − #(αξ1 ξ2 ) (ξ3 ),

whence #(αξ1 ξ2 ξ3 ) = [#(αξ1 ξ2 ), ξ3 ] + [#(αξ2 ξ3 ), ξ1 ] + [#(αξ3 ξ1 ), ξ2 ] + #(αξ1 ξ2 ) (ξ3 ) + #(αξ2 ξ3 ) (ξ1 ) + #(αξ3 ξ1 ) (ξ2 ).

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287

On the other hand, (ξ1 ξ2 ξ3 ) =

(ξ1 )ξ2 ξ3 +

(ξ2 )ξ3 ξ1 +

(ξ3 )ξ1 ξ2

− [ξ1 , ξ2 ]ξ3 − [ξ2 , ξ3 ]ξ1 − [ξ3 , ξ1 ]ξ2 [α, ξ1 ξ2 ξ3 ] = − ( (αξ1 ξ2 ξ3 ) + α (ξ1 ξ2 ξ3 )) . Hence # (αξ1 ξ2 ξ3 ) = −# ([α, ξ1 ξ2 ξ3 ] + α (ξ1 ξ2 ξ3 )) = −#[α, ξ1 ξ2 ξ3 ] − #(α (ξ1 )ξ2 ξ3 ) − #(α (ξ2 )ξ3 ξ1 ) − #(α (ξ3 )ξ1 ξ2 ) + #(α[ξ1 , ξ2 ]ξ3 ) + #(α[ξ2 , ξ3 ]ξ1 ) + #(α[ξ3 , ξ1 ]ξ2 ), that is, # (αξ1 ξ2 ξ3 ) = −#[α, ξ1 ξ2 ξ3 ] − #(αξ2 ξ3 ) (ξ1 ) − #(αξ3 ξ1 ) (ξ2 ) − #(αξ1 ξ2 ) (ξ3 ) + #(α[ξ1 , ξ2 ]ξ3 ) + #(α[ξ2 , ξ3 ]ξ1 ) + #(α[ξ3 , ξ1 ]ξ2 ) since # is A-linear. Exploiting the identity #[α, ξ1 ξ2 ξ3 ] = #[α, ξ1 ]ξ2 ξ3 + #[α, ξ2 ]ξ3 ξ1 + #[α, ξ3 ]ξ1 ξ2 , we conclude that ( # + # )(αξ1 ξ2 ξ3 ) = [#(αξ1 ξ2 ), ξ3 ] + [#(αξ2 ξ3 ), ξ1 ] + [#(αξ3 ξ1 ), ξ2 ] + #(α[ξ1 , ξ2 ]ξ3 ) + #(α[ξ2 , ξ3 ]ξ1 ) + #(α[ξ3 , ξ1 ]ξ2 ) − #([α, ξ1 ]ξ2 ξ3 ) − #([α, ξ2 ]ξ3 ξ1 ) − #([α, ξ3 ]ξ1 ξ2 ). Thus the graded commutator α ∈ G01 and ξ1 , ξ2 , ξ3 ∈ G10 ,

#+#

vanishes on G31 if and only if, for every

[#(αξ1 ξ2 ), ξ3 ] + [#(αξ2 ξ3 ), ξ1 ] + [#(αξ3 ξ1 ), ξ2 ] = #([α, ξ1 ]ξ2 ξ3 ) + #([α, ξ2 ]ξ3 ξ1 ) + #([α, ξ3 ]ξ1 ξ2 ) − #(α[ξ1 , ξ2 ]ξ3 ) − #(α[ξ2 , ξ3 ]ξ1 ) − #(α[ξ3 , ξ1 ]ξ2 ); since #(ξ1 ξ2 ξ3 ) = 0, the latter identity is equivalent to [#(αξ1 ξ2 ), ξ3 ] − [#(ξ1 ξ2 ξ3 ), α] + [#(ξ2 ξ3 α), ξ1 ] − [#(ξ3 αξ1 ), ξ2 ] = #([α, ξ1 ]ξ2 ξ3 ) + #([α, ξ2 ]ξ3 ξ1 ) + #([α, ξ3 ]ξ1 ξ2 ) + #([ξ1 , ξ2 ]αξ3 ) + #([ξ2 , ξ3 ]αξ1 ) + #([ξ3 , ξ1 ]αξ2 ). With the more neutral notation (a1 , a2 , a3 , a4 ) = (α, ξ1 , ξ2 , ξ3 ) since, for degree reasons, 3# (a1 , a2 , a3 ) = −#(a1 a2 a3 ),

3# ([a1 , a2 ], a3 , a4 ) = −#([a1 , a2 ]a3 a4 )

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etc., the identity takes the form [3# (a1 , a2 , a3 ), a4 ] − [3# (a2 , a3 , a4 ), a1 ] + [3# (a3 , a4 , a1 ), a2 ] − [3# (a4 , a1 , a2 ), a3 ] = 3# ([a1 , a2 ], a3 , a4 ) − 3# ([a1 , a3 ], a2 , a4 ) + 3# ([a1 , a4 ], a2 , a3 ) + 3# ([a2 , a3 ], a1 , a4 ) − 3# ([a2 , a4 ], a1 , a3 ) + 3# ([a3 , a4 ], a1 , a2 ). This is the identity (6.6.4) for the special case where the elements a1 , a2 , a3 , a4 are from the union G01 ∪ G10 . The operator being of order ≤ 2 means precisely that the bracket [·, ·] = ±2 (generated by it) behaves as a derivation in each argument and, accordingly, the operator # being of order ≤ 3 means that the operation 3# is a derivation in each of its three arguments. The equivalence between the identities (6.6.3) and (5.7.1) for arbitrary arguments is now etablished by induction on the degrees of the arguments.

Proof of Theorem 6.6.1. The quasi-Gerstenhaber bracket [·, ·] on G is that generated by = ∂ via (6.1). This bracket is plainly graded skew-symmetric in the correct sense, and the reasoning in Section 1 of Koszul [1985] shows that this bracket satisﬁes the identities (5.3)–(5.5). In particular, the identity (5.5) is a consequence of the identity (6.3): This identity may be rewritten as d# + #d = −

.

(6.3 )

Hence, given homogeneous elements ξ, η, ϑ of G, the identity (5.5) takes the form (−1)(|ξ |−1)(|ϑ|−1) [ξ, [η, ϑ]] = −(−1)(|ξ |+|η|+|ϑ|) 3 2 (ξ, η, ϑ). (5.5 ) (ξ,η,ϑ) cyclic

This is exactly the identity in line −5 on p. 260 of Koszul [1985], which measures the failure of the bracket [·, ·] to satisfy the graded Jacobi identity in terms of the square 2 of the generating operator . The identity (5.6) holds by virtue of Lemma 6.6.2.

An observation due to Koszul [1985] (p. 261) extends to the present case in the following fashion: For any quasi-Batalin–Vilkovisky algebra (G; d, ∂, #), the operator ∂ (which is strict by assumption) behaves as a derivation for the quasiGerstenhaber bracket [·, ·], up to a suitable correction term which we now determine: The identity in line 6 on p. 261 of Koszul [1985] implies that for homogeneous a, b ∈ G, ∂[a, b] − ([∂a, b] − (−1)|a| [a, ∂b]) = (−1)|a| 2∂ 2 (a, b). Since, by virtue of (6.3), ∂∂ + d# + #d = 0, we conclude that ∂[a, b] − ([∂a, b] − (−1)|a| [a, ∂b]) = (−1)|a|−1 2d#+#d (a, b). The correction term 2d#+#d (a, b) is plainly an instance of the occurrence of a homotopy. We also note that, in view of (6.1), a generator, even if strict, behaves as

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a derivation for the multiplication of G only if the quasi-Gerstenhaber bracket [·, ·] is zero. A quasi-Batalin–Vilkovisky algebra having # zero is just an ordinary differential bigraded Batalin–Vilkovisky algebra, and a differential bigraded Batalin–Vilkovisky algebra having zero differential is called a bigraded Batalin–Vilkovisky algebra (see Huebschmann [1998b] and Huebschmann [2000]). Maintaining notation introduced in the previous section, given a quasi-Batalin–Vilkovisky algebra (G; d, ∂, #), we denote its d-homology by H∗∗ (G)d ; Proposition 5.7 above says that H∗∗ (G)d inherits a bigraded Gerstenhaber bracket. Plainly, under the present circumstances this homology inherits more structure; indeed, the proof of the following is straightforward and left to the reader. Proposition 6.7. Given a quasi-Batalin–Vilkovisky algebra (G; d, ∂, #), the strict operator ∂ induces a generator ∗ → H∗−1 (G)d ∂ : H∗∗ (G)d −

(6.7.1)

for the bigraded Gerstenhaber bracket on its d-homology H∗∗ (G)d and hence turns the latter into a bigraded Batalin–Vilkovisky algebra. A quasi-Batalin–Vilkovisky algebra has an invariant which is ﬁner than just ordinary homology. Let (G; d, ∂, #) be a quasi-Batalin–Vilkovisky algebra, and consider the following Tot G of G given by q (Tot G)n = Gp = G0n ⊕ G1n+1 ⊕ · · · ⊕ Gkn+k ⊕ . . . (6.7.2) q−p=n

This totalization is forced by the isomorphism (6.8) and by Theorem 6.10 below. In a given bidegree (q, p), the operators d, ∂, # may be described as q

q+1

d : Gp → Gp

,

q

q

∂ : Gp → Gp−1 ,

q

q−1

# : Gp → Gp−2 ,

(6.7.3)

and the deﬁning properties (6.2)–(6.5) say that the sum D =d +∂ +#

(6.7.4)

is a square zero operator on Tot G, i.e., a differential. Consider the ascending ﬁltration {Fr }r≥0 of Tot G given by q Gp = G0n ⊕ G1n+1 ⊕ · · · ⊕ Grn+r . (6.7.5) Fr (Tot G)n = q−p=n,p≤r

This ﬁltration gives rise to a spectral sequence (E∗∗ (r), d(r)),

q

q−r+1

d(r) : Ep (r) − → Ep−r

(r)

(6.7.6)

having (E(0), d(0)) = (G, d),

(6.7.7)

290

whence

J. Huebschmann

(E(1), d(1)) = (H∗∗ (G)d , ∂),

(6.7.8)

which is the bigraded homology Batalin–Vilkovisky algebra spelled out in Proposition 6.7 above. This spectral sequence is an invariant for the quasi-Batalin–Vilkovisky algebra G which is ﬁner than just the bigraded homology Batalin–Vilkovisky algebra (H∗∗ (G)d , ∂). We will now take up and extend the discussion in (5.8) and describe how quasiGerstenhaber and quasi-Batalin–Vilkovisky algebras arise from Lie–Rinehart triples. To this end, let (A, H, Q) be a pre-Lie–Rinehart triple and suppose that as an Amodule, Q is ﬁnitely generated and projective, of constant rank n. Consider the graded q q p exterior A-algebra A Q, and let G = Alt A (H, A Q), with Gp = Alt A (H, A Q); this is a bigraded commutative A-algebra. The Lie–Rinehart differential d, with respect to the canonical graded (A, H )-module structure on A Q, turns G into a differential graded R-algebra. Our aim is to determine when (A, Q, H ) is a genuine Lie–Rinehart triple in terms of conditions on G. The graded A-module Alt ∗A (Q, nA Q) acquires a canonical graded (A, H )module structure. Further, since (A, H, Q) is a pre-Lie–Rinehart triple (not just an almost pre-Lie–Rinehart triple), the canonical bigraded A-module isomorphism (5.8.7) is now an isomorphism n φ : (Alt ∗A (H, ∗A Q), d) − → (Alt ∗A (H, Alt n−∗ A (Q, A Q)), d)

(6.8)

of Rinehart complexes, with reference to the graded (A, H )-module structures on n n ∗A Q and Alt n−∗ A (Q, A Q). We will say that (A, H, Q) is weakly orientable if A Q n is a free A-module, that is, if there is an A-module isomorphism ω : A Q → A, and ω will then be referred to as a weak orientation form. Under the circumstances of Example 1.4.1, this notion of weak orientability means that the foliation F is transversely orientable, with transverse volume form ω. For a general pre-Lie–Rinehart triple (A, H, Q), we will say that a weak orientation form ω is invariant provided it is invariant under the H -action; we will then refer to ω as an orientation form, and we will say that (A, H, Q) is orientable. In the situation of Example 1.4.1, with a grain of salt, an orientation form in this sense amounts to an orientation for the “space of leaves,’’ that is, with reference to the spectral sequence (2.9.1), the class in the top basic cohomology group E2n,0 (see 2.10(i)) of such a form is nonzero and generates this cohomology group. Likewise, in the situation of Example 1.4.2, an orientation form is a holomorphic volume form, and the requirement that an (invariant) orientation form exist is precisely the Calabi–Yau condition. Let (A, H, Q) be a general orientable pre-Lie–Rinehart triple, and let ω be an invariant orientation form. Then ω induces an isomorphism Alt ∗A (Q, nA Q) → Alt ∗A (Q, A) of graded (A, H )-modules and hence an isomorphism φ ω : (Alt ∗A (H, ∗A Q), d) − → (Alt ∗A (H, Alt n−∗ A (Q, A)), d0 )

(6.9)

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291

of Rinehart complexes. Here, on the right-hand side of (6.9), the operator d0 is that given earlier as (2.4.5), with the orders of H and Q interchanged. On the right-hand side of (6.9), we have as well the operator d1 given as (2.4.6) and the operator d2 given as (2.4.7) (the order of H and Q being interchanged); see also (5.8.8). The operator d1 induces an operator ω:

Alt ∗A (H, ∗A Q) − → Alt ∗A (H, ∗−1 A Q)

(6.10.1)

on the left-hand side of (6.9) by means of the the relationship φ ω ω (α) = (−1)n+1 d1 (φαω ),

α ∈ ∗A Q.

By Lemma 5.8.10, the operator d2 on the right-hand side of (6.9) corresponds to the operator #δ on the left-hand side of (6.9) given as (5.8.5) above. Notice that ω is an R-linear operator on G∗∗ = Alt ∗A (H, ∗A Q) of bidegree (0, −1) which looks like a generator for the corresponding bracket (5.8.3). We will now describe the circumstances where ω is a generator. Theorem 6.10. Let (A, H, Q) be an orientable pre-Lie–Rinehart triple, with invariant orientation form ω. If (A, H, Q) is a genuine Lie–Rinehart triple, then (Alt ∗A (H, ∗A Q); d, ω , #δ ) is a quasi-Batalin–Vilkovisky algebra, and ω is a strict generator for the bracket [·, ·] given by (5.8.3). Conversely, under the additional hypothesis that H satisfy the property P, if (Alt ∗A (H, ∗A Q), d, ω , #δ ) is a quasi-Batalin–Vilkovisky algebra, then (A, H, Q) is a genuine Lie–Rinehart triple. Proof . We note ﬁrst that, when (A, d0 , d1 , d2 ) is a multialgebra, so is (A, d0 , −d1 , d2 ). Furthermore, when (Alt ∗A (H, Alt ∗A (Q, A)), d0 , −d1 , d2 ) is a multialgebra, (Alt ∗A (H, Alt ∗A (Q, nA Q)), d0 , −d1 , d2 ) is a multicomplex, the operators dj (0 ≤ j ≤ 2) (where the notation dj is abused somewhat) being the induced ones, with the correct sign, that is, ω∗ (dj (·)) = (−1)n dj ω∗ ((·)), where ω∗ is the induced bigraded morphism of degree n. Hence the equivalence between the Lie–Rinehart triple and quasi-Batalin–Vilkovisky properties is straightforward in view of Theorem 2.7 and Theorem 6.6.1. In particular, the identities (2.1.4.2)–(2.1.4.5) correspond to the identities (6.2)–(6.5) which characterize (Alt ∗A (H, ∗A Q), d, ω , #δ ) being a quasi-Batalin–Vilkovisky algebra. It remains to show that when (A, H, Q) is a genuine Lie–Rinehart triple, the operator ω (given by (6.10.1)) is indeed a strict generator for the bigraded bracket (5.8.3) to which the rest of the proof is devoted. 6.10.2. Veriﬁcation of the generating property We note ﬁrst that in view of the derivation properties of a quasi-Gerstenhaber bracket, it sufﬁces to establish the generating property (6.1) on A Q, viewed as the bidegree

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(0, ∗)-constituent of Alt A (H, A Q) = A Q. To make the operator ω somewhat more explicit, we note that the pairing (1.5.2.Q) and the choice of ω determine a generalized Q-connection → nA Q ∇ : Q ⊗ nA Q − on nA Q determined by requiring that the diagram ∇

Q ⊗R nA Q −−−−→ nA Q ⏐ ⏐ ⏐ ⏐ω Id⊗ω? ? Q ⊗R A

−−−−−→ (1.5.2.Q)

A

be commutative, and the multialgebra compatibility property d0 d1 + d1 d0 = 0 (see (2.1.4.2)) is equivalent to this generalized Q-connection being compatible with the H module structures. In the situation of Example 1.4.2, such a generalized Q-connection on nA Q amounts to a ﬂat holomorphic connection on the highest exterior power of the holomorphic tangent bundle. In the present general case, the operator d1 on Alt A (H, Alt A (Q, A)) (given by (2.5.3) then corresponds to an operator → Alt A (H, Alt A (Q, nA Q)) d ∇ : Alt A (H, Alt A (Q, nA Q)) − p

p+1

(p ≥ 0)

determined by the commutativity of the diagram d∇

p+1

−−−− → n

Alt A (H, Alt A (Q, A))

p

Alt A (H, Alt A (Q, nA Q)) −−−−→ Alt A (H, Alt A (Q, nA Q)) ⏐ ⏐ ⏐ ⏐ ? ? p

Alt A (H, Alt A (Q, A))

(−1) d1

p+1

whose vertical arrows are induced by ω. Consider, then, the operator D determined by the requirement that the diagram φ

p

Alt A (H, A Q) −−−−→ ⏐ ⏐ D?

n−p

Alt A (H, Alt A (Q, nA Q)) ⏐ ⏐ ∇ ?−d

p−1

n−(p−1)

Alt A (H, A Q) −−−−→ Alt A (H, Alt A φ

(Q, nA Q))

be commutative. This operator coincides with the operator ω but we prefer to use a neutral notation. In view of the derivation properties of a quasi-Gerstenhaber bracket, to establish the generating property, it will sufﬁce to study the restriction p

p−1

→ A Q D : A Q − of this operator.

(1 ≤ p ≤ n)

Homotopies and Maurer–Cartan algebras p

293

n−p

Given α ∈ A Q, we will write φα ∈ Alt A (Q, nA Q) for the image under φ so that, for ξp+1 , . . . , ξn , φα (ξp+1 , . . . , ξn ) = α ∧ ξp+1 ∧ · · · ∧ ξn . Let α1 and α2 be homogeneous elements of ∗A Q. We will now establish the generating property (−1)|α1 | [α1 , α2 ] = D(α1 α2 ) − (Dα1 )α2 − (−1)|α1 | α1 (Dα2 ). n+1−|α1 |−|α2 |

Let β ∈ A

(6.10.3)

Q; it will sufﬁce to study the expression

(D(α1 α2 )) ∧ β − ((Dα1 )α2 ) ∧ β − (−1)|α1 | (α1 (Dα2 )) ∧ β − (−1)|α1 | [α1 , α2 ] ∧ β (which yields an element of of nA Q) or, equivalently, the expression φD(α1 α2 ) (β) − φ(Dα1 )α2 (β) − (−1)|α1 | φα1 Dα2 (β) − (−1)|α1 | φ[α1 ,α2 ] (β) ∈ nA Q. To this end, we note ﬁrst that − φD(α1 α2 ) (β) = (d ∇ φα1 α2 )(β) −φ(Dα1 )α2 (β) = −(Dα1 ) ∧ α2 ∧ β = (d ∇ φDα1 )(α2 ∧ β) −φα1 (Dα2 ) (β) = −α1 ∧ (Dα2 ) ∧ β = −(−1)|α1 |(|α2 |−1) (Dα2 ) ∧ α1 ∧ β = (−1)|α1 |(|α2 |−1) (d ∇ φDα2 )(α1 ∧ β). Let ϑ1 , ϑ2 ∈ Q and ξ2 , . . . , ξn ∈ Q. Letting ξ1 = ϑ2 we obtain (d ∇ φϑ1 )(ϑ2 , ξ2 , . . . , ξn ) = (d ∇ φϑ1 )(ξ1 , ξ2 , . . . , ξn ) = (−1)j −1 ∇ξj (ϑ1 ∧ ξ1 ∧ . . . = ξj · · · ∧ ξn ) 1≤j ≤n

+

(−1)j +k ϑ1 ∧ [ξj , ξk ] ∧ ξ1 ∧ . . . = ξj . . . = ξk · · · ∧ ξn

1≤j
and a straightforward calculation gives (d ∇ φϑ1 )(ξ1 , ξ2 , . . . , ξn ) = ∇ϑ2 (ϑ1 ∧ ξ2 ∧ · · · ∧ ξn ) + (−1)j −1 ∇ξj (ϑ1 ∧ ϑ2 ∧ ξ2 ∧ . . . = ξj · · · ∧ ξn ) 2≤j ≤n

+

(−1)1+k ϑ1 ∧ [ϑ2 , ξk ] ∧ ξ2 ∧ . . . = ξk · · · ∧ ξn

1
+

(−1)j +k ϑ1 ∧ [ξj , ξk ] ∧ ϑ2 ∧ ξ2 ∧ . . . = ξj . . . = ξk · · · ∧ ξn .

2≤j
Likewise, letting ξ1 = ϑ1 , we obtain

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J. Huebschmann

(d ∇ φϑ2 )(ϑ1 , ξ2 , . . . , ξn ) = (d ∇ φϑ2 )(ξ1 , ξ2 , . . . , ξn ) = (−1)j −1 ∇ξj (ϑ2 ∧ ξ1 ∧ . . . = ξj · · · ∧ ξn ) 1≤j ≤n

+

(−1)j +k ϑ2 ∧ [ξj , ξk ] ∧ ξ1 ∧ . . . = ξj . . . = ξk · · · ∧ ξn

1≤j
and again a calculation yields (d ∇ φϑ2 )(ξ1 , ξ2 , . . . , ξn ) = ∇ϑ1 (ϑ2 ∧ ξ2 ∧ · · · ∧ ξn ) + (−1)j −1 ∇ξj (ϑ2 ∧ ϑ1 ∧ ξ2 ∧ . . . = ξj · · · ∧ ξn ) 2≤j ≤n

+

(−1)1+k ϑ2 ∧ [ϑ1 , ξk ] ∧ ξ2 ∧ . . . = ξk · · · ∧ ξn

1

+

(−1)j +k ϑ2 ∧ [ξj , ξk ] ∧ ϑ1 ∧ ξ2 ∧ . . . = ξj . . . = ξk · · · ∧ ξn .

2≤j
(−1)3 (d ∇ φα )(ξ2 , . . . , ξn ) =− (−1)j −1 ∇ξj (ϑ1 ∧ ϑ2 ∧ ξ2 ∧ . . . = ξj · · · ∧ ξn ) 2≤j ≤n

+

(−1)j +k ϑ1 ∧ ϑ2 ∧ [ξj , ξk ] ∧ ξ2 ∧ . . . = ξj . . . = ξk · · · ∧ ξn ,

2≤j
that is, (d ∇ φα )(ξ2 , . . . , ξn ) = (−1)j −1 ∇ξj (ϑ1 ∧ ϑ2 ∧ ξ2 ∧ . . . = ξj · · · ∧ ξn ) 2≤j ≤n

−

(−1)j +k ϑ1 ∧ ϑ2 ∧ [ξj , ξk ] ∧ ξ2 ∧ . . . = ξj . . . = ξk · · · ∧ ξn .

2≤j
We now take (ϑ1 , ϑ2 ) = (ξ1 , ξ2 ). Then (d ∇ φα )(ξ2 , . . . , ξn ) (−1)j −1 ∇ξj (ϑ1 ∧ ϑ2 ∧ ξ2 ∧ . . . = ξj · · · ∧ ξn ) = 2≤j ≤n

−

(−1)j +k ϑ1 ∧ ϑ2 ∧ [ξj , ξk ] ∧ ξ2 ∧ . . . = ξj . . . = ξk · · · ∧ ξn

2≤j
= −∇ξ2 (ξ1 ∧ ξ2 ∧ ξ3 ∧ · · · ∧ ξn )

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+

295

(−1)k+1 ξ1 ∧ ξ2 ∧ [ξ2 , ξk ] ∧ ξ3 ∧ . . . = ξk · · · ∧ ξn

2
and

(d ∇ φϑ1 )(ϑ2 , ξ2 , . . . , ξn ) = 0,

whereas, by a calculation the details of which are not given here, (d ∇ φϑ2 )(ϑ1 , ξ2 , . . . , ξn ) = ∇ξ2 (ξ1 ∧ ξ2 ∧ ξ3 ∧ · · · ∧ ξn ) + [ξ1 , ξ2 ] ∧ ξ2 ∧ ξ3 ∧ · · · ∧ ξn + (−1)k ξ1 ∧ ξ2 ∧ [ξ2 , ξk ] ∧ ξ3 ∧ . . . = ξk · · · ∧ ξn . 2
Consequently, (d ∇ φϑ1 ∧ϑ2 )(ξ2 , . . . , ξn ) − (d ∇ φϑ1 )(ϑ2 , ξ2 , . . . , ξn ) + (d ∇ φϑ2 )(ϑ1 , ξ2 , . . . , ξn ) = (−1)k+1 ξ1 ∧ ξ2 ∧ [ξ2 , ξk ] ∧ ξ3 ∧ . . . = ξk · · · ∧ ξn 2
+ [ξ1 , ξ2 ] ∧ ξ2 ∧ ξ3 ∧ · · · ∧ ξn + (−1)k ξ1 ∧ ξ2 ∧ [ξ2 , ξk ] ∧ ξ3 ∧ . . . = ξk · · · ∧ ξn 2
= [ξ1 , ξ2 ] ∧ ξ2 ∧ ξ3 · · · ∧ ξn , that is, with β = ξ2 ∧ ξ3 · · · ∧ ξn , (D(ϑ1 ∧ ϑ2 )) ∧ β − ((Dϑ1 )ϑ2 ) ∧ β − (ϑ1 (Dϑ2 )) ∧ β = −[ϑ1 , ϑ2 ] ∧ β ∈ nA Q. This etablishes the generating property (6.10.3) for α1 and α2 homogeneous of degree 1 since, as an A-module, Q is ﬁnitely generated and projective of constant rank n. Since, as an A-algebra, A Q is generated by its elements of degree 1, a straightforward induction completes the proof of Theorem 6.10.

For the special case where δ and hence #δ is zero, the statement of the theorem is a consequence of Theorem 5.4.4 in Huebschmann [1998b]. Corollary 6.10.4. Let (A, H, Q) be an orientable Lie–Rinehart triple, and let G = Alt A (H, A Q) be endowed with the Lie–Rinehart differential d, the bigraded bracket (5.8.3), and Jacobiator (5.8.5). Then (G, d, [·, ·], #) is a quasi-Gerstenhaber algebra. Indeed, the identity (5.6) then corresponds to (1.9.7); see also (4.9.6) and (6.11)(vii) below. Remark 6.11. It is instructive to spell out the relationship between the quasi-Batalin– Vilkovisky compatibility conditions (6.2)–(6.4) and the Lie–Rinehart triple axioms (1.9.1)–(1.9.7); see (2.8.5) above. As before, write G = Alt A (H, A Q) and recall that n is the rank of Q as a projective A-module.

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1 (i) The vanishing of d + d : Gn0 → Gn−1 (a special case of (6.2)) corresponds to (1.9.1). 2 (ii) The vanishing of the operator d + d : Gn1 → Gn−1 (a special case of (6.2), too) corresponds to (1.9.2). 0 1 (iii) The vanishing of d + d : Gn−1 → Gn−2 (still a special case of (6.2)) corresponds to (1.9.3). 0 → Gn−2 (a special case of (iv) The vanishing of + #d = d# + + #d : Gn0 − (6.3)) corresponds to (1.9.4). 0 0 (v) The vanishing of + #d = d# + + #d : Gn−1 − → Gn−3 (a special case of (6.3), too) corresponds to (1.9.5). 1 (vi) The vanishing of d# + + #d : Gn1 − → Gn−2 (still a special case of (6.3)) corresponds to (1.9.6). 0 (a special case of (6.4)) corresponds (vii) The vanishing of # + # : Gn1 → Gn−3 to (1.9.7). See also (4.9.6) above.

When (A, H, Q) is an orientable Lie–Rinehart triple, with orientation form ω, pursuing the philosophy developed in Section 7 of Huebschmann [1998b] (see, in particular, (7.14)), we may view (Alt ∗A (H, ∗A Q), d, ω , #δ ) as an object the category of A-modules calculating the “quasi-Lie–Rinehart homology H∗∗ (Q, A ) of the quasi-Lie–Rinehart algebra (A, Q), with values in the right (A, Q)-module A ,’’ the right (A, Q)-module structure being induced by . The isomorphism (Alt ∗A (H, ∗A Q); d,

ω , #δ )

− → (Alt ∗A (H, Alt n−∗ A (Q, A)); d0 , −d1 , d2 )

(6.12)

is then a kind of “duality isomorphism’’ of chain complexes inducing a “duality isomorphism’’ which, in bidegree (q, p), is of the kind q → Hq,n−p (Q, A) ∼ Hp (Q, A ) − = Hq,n−p (L, A),

(6.13)

where L = H ⊕ Q is the (R, A)-Lie algebra which corresponds to the given Lie–Rinehart triple (A, H, Q). Proposition 7.14 in Huebschmann [1998b] makes this precise for the special case where (A, Q, H ) is a twilled Lie–Rinehart algebra. In our case, pushing further, consider the ﬁltrations of Alt∗A (H, ∗A Q) and n Alt ∗A (H, Alt n−∗ A (Q, A Q)) by Q-degree. In view of what was said above, the corresponding spectral sequence (6.7.6), which we now write in the form (= E∗∗ (r), d(r)), has whence

d(r) : = Ep (r) − →= Ep−r q

q−r+1

(r),

(6.14.1)

(= E(0), d(0)) = (Alt ∗A (H, ∗A Q), d),

(6.14.2)

(= E(1), d(1)) = (H∗∗ (G)d , ∂);

(6.14.3)

this is the bigraded homology Batalin–Vilkovisky algebra noted in Proposition 6.7 above, for the quasi-Batalin–Vilkovisky algebra G∗∗ = (Alt ∗A (H, ∗A Q); d,

ω , #δ ).

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297

The isomorphism (6.8) is compatible with these ﬁltrations. Hence it identiﬁes the corresponding spectral sequence (2.9.1) with (6.14.1). Illustration 6.15. Return to the situation of (1.4.1), and maintain the notation etablished there as well as in (2.10); see also (4.15). Thus (M, F) is a foliated manifold, (A, H, Q) = (C ∞ (M), LF , Q) is the corresponding Lie–Rinehart triple, and (A, Q) = Alt A (H, A), Alt A (H, Q) is the corresponding quasi-Lie–Rinehart algebra. We now push further the interpretation, already stated in (4.15) above, of A as the algebra of generalized functions and of Q as the generalized Lie algebra of vector ﬁelds for the foliation. This interpretation relies crucially on the totalization spelled out as (6.7.2) above; with the more familiar totalization Tot G given by q (Tot G)n = Gp , p+q=n

such an interpretation is not visible. Thus, consider the bigraded algebra G∗∗ = Alt ∗A (H, ∗A Q) = A Q, where as before A = Alt A (H, A) and Q = Alt ∗A (H, Q). Suppose that the foliation is transversely orientable with a basic transverse volume form ω, and consider the resulting quasi-Batalin–Vilkovisky algebra (Alt ∗A (H, ∗A Q); d, ω , #δ ); see Theorem 6.10. In particular, G∗∗ is then a quasi-Gerstenhaber algebra. This quasi-Gerstenhaber algebra yields a kind of generalized Schouten algebra (algebra of multivector ﬁelds) for the foliation; the cohomology H∗0 (G) may be viewed as the Schouten algebra for the “space of leaves.’’ However the entire cohomology contains more information about the foliation than just H∗0 (G). Under the circumstances of (2.10(ii)), where the foliation comes from a ﬁber bundle (see also (4.15)), let B denote the “space of leaves’’ or, equivalently, the base of the corresponding bundle; an orientation ω in our sense is now essentially equivalent to a volume form ωB for the base B. Let LB = Vect(B). The volume form ωB induces an exact generator ∂ωB for the ordinary Gerstenhaber algebra G∗ = C ∞ (B) LB , and the corresponding bigraded homology Batalin–Vilkovisky algebra (H∗∗ (Alt A (H, ∗A Q))d , ∂ω ) coming into play in Theorem 6.10 may then be written as the bigraded crossed product (H∗∗ (Alt A (H, ∗A Q))d , ∂ω ) = H∗ (A) ⊗C ∞ (B) (G∗ , ∂ωB )

(6.15.1)

of H∗ (A) with the ordinary Batalin–Vilkovisky algebra (G∗ , ∂) = (∗C ∞ (B) LB , ∂ωB ) (see Huebschmann [1998b] for the notion of bigraded crossed product Batalin– Vilkovisky algebra); here A = (Alt A (H, A), d) which (see (2.10(ii))) computes the cohomology of M with values in the sheaf of germs of functions which are constant on the leaves, i.e., ﬁbers.

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Under the circumstances of (2.10(i)), when the foliation does not come from a ﬁber bundle, the structure of the bigraded homology Batalin–Vilkovisky algebra H∗∗ (Alt A (H, ∗A Q))d may be more intricate. Illustration 6.16. For a (ﬁnite-dimensional) quasi-Lie bialgebra (h, h∗ ) , with Manin pair (g, h), where g = h ⊕ h∗ , see Kosmann-Schwarzbach [1992] for details, the resulting quasi-Batalin–Vilkovisky algebra has the form Alt(h, h∗ ) ∼ = (h∗ ⊕ h∗ ). = h∗ ⊗ h∗ ∼ Remark 6.17. Given the bigraded commutative algebra G∗∗ , consider the totalization Tot G spelled out above. Suppose there be given operators d, ∂, # which endow G with a quasi-Batalin–Vilkovisky algebra structure in our sense. These operators induce operators d : (Tot G)∗ − → (Tot G)∗+1 , ∂ : (Tot G)∗ − → (Tot G)∗−1 , → (Tot G)∗−3 , # : (Tot G)∗ − such that L = d∂ + ∂d = 0,

d# + ∂∂ + #d = 0,

∂# + #∂ = 0,

## = 0,

whence, endowed with these operators, Tot G is precisely a quasi-Batalin–Vilkovisky algebra in the sense of Getzler [1995] with zero Laplacian L. This notion of quasi-Batalin–Vilkovisky algebra extends that of differential GBV-algebra in Manin [1999] (III.9.5) (which corresponds to the structure under discussion with # = 0, with reference to the totalization Tot G,) and is a special case of a more general notion of generalized BV-algebra explored in Kravchenko [2000]. In Bangoura [2002] (Definition 3.2), a corresponding notion of quasi-Gerstenhaber algebra has been isolated. When (G, d, [·, ·], #) is a quasi-Gerstenhaber algebra in our sense, the operations d, [·, ·], # induce as well corresponding pieces of structure d, [·, ·], # on Tot G and, in view of Lemma 2.2 in Bangoura [2002], the requirement (5.5) above (which makes precise how under our circumstances the h-Jacobiator # controls the failure of the strict Jacobi identity) entails the requirement (3.7) in Bangoura [2002] which, in turn, describes the failure of the strict Jacobi identity under the circumstances of Bangoura [2002]. Moreover, our requirements (5.i)–(5.iii) in Section 5 above now amount to the corresponding requirements (3.6)–(3.8) in Bangoura [2002]. Likewise, the requirement (5.6) corresponds to the requirement (3.9) in Bangoura [2002]. These observations make precise the relationship between our notions of quasi-Gerstenhaber and of quasi-Batalin–Vilkovisky algebra and that of quasi-Gerstenhaber algebra in Bangoura [2002] and those of quasi-Batalin–Vilkovisky algebra (with zero Laplacian) explored in Bangoura [2002] and Getzler [1995]. However the notion of Laplacian does not seem to have a meaning for the totalization Tot which we use in this paper, in particular, does not have an interpretation (at least not an obvious one) in terms of foliations.

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Acknowledgments Throughout this work, I have been stimulated by M. Kinyon via some email correspondence at an early stage of the project as well as by M. Bangoura, P. Michor, D. Roytenberg, and Y. Kosmann-Schwarzbach. I am indebted to J. Stasheff and to the referees for a number of comments on a draft of the manuscript which helped improve the exposition. This work was partly carried out and presented during two stays at the Erwin Schrödinger Institute at Vienna. I wish to express my gratitude for hospitality and support.

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Localization theorems by symplectic cuts∗ Lisa Jeffrey1 and Mikhail Kogan2 1 Department of Mathematics

University of Toronto Toronto, ON M5S 3G3 Canada [email protected] 2 Department of Mathematics Institute for Advanced Study Princeton, NJ 08540 USA [email protected] This paper is dedicated to Alan Weinstein on the occasion of his 60th birthday. Abstract. Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a regular value of the moment map, and M0 the symplectic reduction at zero. Denote by κ0 the Kirwan map HT∗ (M) → H ∗ (M0 ). For an equivariant cohomology class η ∈ HT∗ (M) we present new localization formulas which express M0 κ0 (η) as sums of certain integrals over

the connected components of the ﬁxed point set M T . To produce such a formula we apply a residue operation to the Atiyah–Bott–Berline–Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T ) we give a new proof of the Jeffrey–Kirwan localization formula [JK1].

1 Introduction Assume we are given a compact symplectic manifold M with the Hamiltonian action of a torus T . There are two kinds of localization theorems which express the integral over M of an equivariant cohomology class η ∈ HT∗ (M) and the integral over the reduced space M0 of the Kirwan map κ0 (η) as sums of certain terms which involve integration over the connected components of the ﬁxed point set M T . In particular, ∗ The ﬁrst author was supported by a grant from NSERC. The ﬁrst author’s work was be-

gun during a visit to Harvard University during the spring of 2003, whose support during this period is acknowledged. The second author was supported by the National Science Foundation under agreement DMS-0111298.

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these localization theorems say that both M η and M0 κ0 (η) depend only on the restriction of η to M T . More speciﬁcally, the localization theorem of Atiyah–Bott [AB] and Berline– Vergne [BV] (or just the ABBV localization theorem) expresses the integral over M (that is, the pushforward in equivariant cohomology with respect to the map M → pt) of a class η ∈ HT∗ (M, C) as a sum of integrals over the connected components F of the ﬁxed point set M T of the restriction of η to F divided by the equivariant Euler class of the normal bundle of F : ι∗ (η) F , η= e(ν(F )) M F F

where ιF is the natural inclusion F → M and e(ν(F )) is the equivariant Euler class of the normal bundle ν(F ). To treat the other kind of localization theorems, let zero be a regular value of the moment map µ : M → t, (M0 , ω0 ) the symplectic reduction at zero and κ0 : HT∗ (M) → H ∗ (M0 ) the Kirwan map. Then the Jeffrey–Kirwan [JK1] and Guillemin– Kalkman [GK] localization theorems express the integral over M0 of classes κ0 (η)eω0 and κ0 (η) respectively as sums over some connected components of M T of certain terms similar to those appearing in the ABBV localization theorem. We postpone the precise statement of the Guillemin–Kalkman theorem until Section 4. However the Jeffrey–Kirwan localization theorem applied to the case of abelian group actions states the following under the above assumptions. Theorem A ([JK1]). For η ∈ HT∗ (M), we have ' * ∗ ιF (η(X)eω ) ω0 i(µ(F ))(X) κ0 (η)e = c · Res [dX] , e M0 F e(ν(F ))(X)

(1.1)

F

where c is a nonzero constant, X is a variable in t ⊗ C so that a T -equivariant cohomology class can be evaluated at X, and Res is a multidimensional residue with respect to the cone ⊂ t deﬁned in Section 3. The similarity of the ABBV localization theorem to the Jeffrey–Kirwan localization theorems is transparent, and is of course not a coincidence. In the case of Hamiltonian circle actions, Lerman [Le] showed that it is possible to deduce the Jeffrey–Kirwan and Guillemin–Kalkman theorems from ABBV localization using the techniques of symplectic cutting. The idea, which was also present, though not explicitly stated, in [GK], is to use symplectic cutting to produce a Hamiltonian space whose connected components of the T -ﬁxed set are either the reduced space M0 or some connected components of the set M T . Then the ABBV localization theorem on this symplectic cut yields a formula which relates integration over some of the connected components of M T and over the reduced space M0 . To arrive at the Jeffrey– Kirwan and Guillemin–Kalkman theorems for the case of a circle action, it remains to apply residues to both sides of the ABBV localization formula for the symplectic cut.

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The goal of this paper is to illustrate that an analogous approach works for higherdimensional torus actions as well. Let us outline the main idea. Lerman’s original deﬁnition of symplectic cutting was given for the case of circle actions. However it can be generalized to multidimensional torus actions when the symplectic cut is deﬁned using any rational convex polytope. We will only consider the special case of symplectic cutting with respect to a cone. Let be a convex rational polyhedral cone (centered at the origin) in the dual t∗ of the Lie algebra t. If σ is an open face of (that is, the interior of a face of ), let T σ be the subtorus of T whose Lie algebra is annihilated by σ . Then, as a topological space, the symplectic cut M is µ−1 ()/ ∼, where p ∼ q if µ(p) = µ(q) ∈ σ , for some open face σ of and p, q lying in the same T σ orbit. As shown in [LMTW], for a generic choice of , the cut space M is a symplectic orbifold with a Hamiltonian T action. The moment map image of M is just the intersection µ(M) ∩ . Moreover, any equivariant cohomology class η ∈ HT∗ (M) naturally descends to an equivariant class η on M . Some connected components of the ﬁxed point set MT may be identical to those of M T ; we call them the old connected components (see Deﬁnition 2.1). One of the connected components of MT is always the reduced space M0 . Hence if we apply ABBV localization to the class η on the symplectic cut M , we will get a formula which relates the integration over M0 to the integration over the connected components of MT , some of which are the connected components of M T . We will show that we can apply the iterated residue operation to both sides of this ABBV localization formula, so that the term corresponding to M0 simpliﬁes. More speciﬁcally, this term becomes a constant times the integral of κ0 (η) over M0 . The major difference with the circle case is that in the formula just described, besides a contribution from the term corresponding to M0 and the terms corresponding to the old connected components, there will be contributions coming from the new connected components of MT , which are neither part of M T nor part of M0 (see Deﬁnition 2.1). However, using the ideas of [GK] we can iterate this process to get rid of these terms. Namely for each new connected component F , we can symplectically cut a certain submanifold M of M with respect to some cone , so that F is a connected component of the T action on M . Then we can apply the ABBV theorem and the residue operation again to M to express the integral over F in terms of integrals over the connected components of the ﬁxed point set M T . As we will show this process can be iterated until all the terms coming from the new ﬁxed points disappear. So the integration of the Kirwan map over M0 can be expressed as a sum of terms which involve integration only over the connected components of M T . The objects that carry the information about which cones are chosen in this process are called dendrites, and are deﬁned in Section 4.3. Every dendrite gives a localization formula. If all the cones used in a dendrite are one-dimensional, then we recover the Guillemin–Kalkman [GK] localization theorem. However, if we choose higher-dimensional cones, in other words multidimensional dendrites, we get new localization formulas. Notice that the Jeffrey–Kirwan localization theorem does not involve any iteration. Nevertheless, it ﬁts into the framework just described. While for actions of tori

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of dimension greater than one, any symplectic cut with respect to a cone always has new connected components of MT (the connected components which are neither in M T nor in M0 ), it is plausible that their contribution to the ABBV localization theorem becomes zero after taking residues. In this case the iterative process described above would stop at step one. We were not able to use precisely this argument to give a proof of the Jeffrey–Kirwan localization theorem. However, we will show that for a good choice of the cone , a very similar argument which involves taking residues of the ABBV formula for symplectic cuts yields the Jeffrey–Kirwan formula given in Theorem A. The paper is organized as follows. In Section 2 we carefully review well-known objects from the theory of Hamiltonian group actions, such as symplectic reductions, symplectic cuts, equivariant cohomology and the Kirwan map. We also recall the orbifold version of the ABBV localization theorem. Section 3 is devoted to the residue operation. The deﬁnitions of residues are based on the theory of complex variables, do not involve any symplectic geometry and are independent of the material summarized in Section 2. In Section 4 we present the generalization of the Guillemin–Kalkman localization theorem to the case of multidimensional dendrites. Finally, using an analogous approach, in Section 5 we give a new proof of the Jeffrey–Kirwan localization theorem.

2 Symplectic cuts and other preliminaries In this section we recall the construction of symplectic cuts with respect to cones and other results in symplectic geometry. All of the results in this section (with only one exception: Proposition 2.2) have appeared in the literature, so we state them without proof. 2.1 Symplectic reduction Let (M, ω) be a symplectic manifold with the action of a Hamiltonian torus T and a moment map µ : M → t∗ . For p ∈ t∗ the symplectic reduction Mp = M//p T of M at p is deﬁned to be µ−1 (p)/T . Whenever p is a regular value of the moment map, the symplectic reduction Mp is an orbifold. For a subtorus H ⊆ T , denote by M H the ﬁxed point set of the H action on M and by MiH the connected components of M H . It is well known that every MiH is a symplectic manifold with a Hamiltonian action of the torus T /H . The convexity theorem of Atiyah [A] and Guillemin–Sternberg [GS] states that if M is compact and connected then µ(M) is a convex polytope. In particular, every µ(MiH ) is a convex polytope inside µ(M); we call it a wall of µ. Let T be a product of two subtori H × S. Denote by π the natural projection t∗ → s∗ . Then the composition π ◦ µ is a moment map for the S action on M. Moreover, the action of S on M restricts to a Hamiltonian action on M H whose moment map is again given by π ◦ µ. Because of this, for p ∈ µ(M H ) we call the space

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@ @ M H //p S = (µ−1 (p) ∩ M H ) T = (µ−1 (p) ∩ M H ) S the symplectic reduction of M H at p. If M is compact and q ∈ s∗ then the set π −1 (q) ∩ µ(M H ) contains ﬁnitely many points pi . It is easy to see that the ﬁxed point set of the H action on M//q S is the union of spaces M H //pi S. 2.2 Cutting with respect to a cone Given a linearly independent set β = {β1 , . . . , βk } of weights of T , deﬁne the cone = β ⊂ t∗ to begiven by all nonnegative linear combinations of the weights of β, namely = { si βi |si ≥ 0}. (Note that we allow k to be less than dim T .) For a set I of indices between 1 and k, denote by I the open face of given by positive linear combinations of weights indexed by the elements of I , that is, I = { i∈I si βi |si > 0}. It will be convenient to denote the subtorus of T perpendicular to I by T I . Assume that (M, ω) carries a Hamiltonian T action with moment map µ. For simplicity assume that the T action is effective so that µ(M) is a polytope of dimension equal to dim T . As a topological space the symplectic cut with respect to a cone = β is the space M = µ−1 ()/ ∼, where x ∼ x if µ(x) = µ(x ) ∈ βI and x and x lie in the same T I orbit. Clearly, the torus action on M descends to an action on M . Notice that the action on M is not effective unless dim = dim T , since the subtorus T of T perpendicular to acts trivially on M . In the case of a circle action, when every cone is just a ray, Lerman [Le] realized that under certain mild conditions this space is an orbifold, it carries a natural symplectic form, and the residual torus action is Hamiltonian. As shown in [LMTW] similar results hold for symplectic cuts with respect to cones. Even more generally, symplectic cutting has been extended to cutting with respect to polytopes [LMTW], cutting for nonabelian groups [Me] and Kähler cutting with respect to polytopes [BGL]. For the purposes of this paper we only need to consider symplectic cutting with respect to cones. To recall the results of [LMTW] let us give another construction of M . Let Cβi be a complex line on which T acts with weight βi . Let Cβ = Cβ1 × · · · × Cβk . Then the T action on Cβ is Hamiltonian with respect to the symplectic form ωC = √ −1 dzi ∧ d z¯ i , where zi is the standard complex coordinate on Cβi . Its moment map is ψ(z) = βi |zi |2 , so that the image of Cβ under the moment map is the cone . Consider the symplectic form (ω, −ωC ) on M × Cβ . Then the diagonal torus T ⊂ T × T acts on this space in a Hamiltonian fashion and its moment map φ is given by φ(x, z) = µ(x) − ψ(z). As shown in [LMTW] the symplectic cut M is homeomorphic to the symplectic reduction (M × Cβ )//0 T . Hence, to guarantee that M is a symplectic orbifold it is enough to assume that zero is a regular value of φ. It is easy to see that this is equivalent to the following:

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(*) Every I is transverse to every wall W of µ(M), that is, dim( I ∩ W ) ≤ |I | + dim W − dim T . If (*) holds, we say that is transverse to µ and from now on we consider only the cones satisfying (*). We can write T × T as the product of the ﬁrst copy of T (that is, T × e) and T . Hence after reducing M × Cβ with respect to T , the action of the ﬁrst copy of T descends to an action on the reduction M . As shown in [LMTW], this action is Hamiltonian and there exists a moment map µ on M whose image is the intersection µ(M) ∩ . Let us describe the connected components of the ﬁxed point set MT . We will separate them into three sets: the old ﬁxed points, the new ﬁxed points, and the ﬁxed points at zero. Deﬁnition 2.1. The old ﬁxed points exist only when dim = dim T ; they are all the connected components Fi of MT for which µ (Fi ) is in the interior of . The set of T ﬁxed points at zero is deﬁned to be the connected component µ−1 (0) of M , which is just the symplectic reduction M0 . The new ﬁxed points are all the other connected components Fi of MT . It is not difﬁcult to see that every Fi is also a connected component of the ﬁxed point set M T , while Fi do not correspond to any ﬁxed points on M. 2.3 Kirwan map We recall the deﬁnition of equivariant cohomology HT∗ (M) with complex coefﬁcients using the Cartan model. Denote

∗T (M) = S(t∗ ) ⊗ ∗ (M)T , where S(t∗ ) denotes the algebra of polynomials on t and ∗ (M)T denotes all T invariant differential forms. So, if f ∈ S(t∗ ), α ∈ ∗ (M)T and X ∈ g we set (f ⊗ α)(X) = f (X)α. The T -equivariant differential on ∗T (M) which deﬁnes the equivariant cohomology is given by dT (f ⊗ α) = f ⊗ dα − xi f ⊗ ıξ i α, M

i is the vector where ξ1 , . . . , ξd is a basis of g; x1 , . . . , xd is the dual basis of t∗ ; and ξM ∗ i ﬁeld generated by the action of ξ . If the equivariant form β ∈ T (M) is closed, that is dT β = 0, we denote by [β] the cohomology class it represents. If T acts locally freely on M, then it is well known that

HT∗ (M) = H ∗ (M/T ), and if the action is trivial, then HT∗ (M) = H ∗ (M) ⊗ S(t∗ ) = H ∗ (M) ⊗ HT∗ (pt).

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For p ∈ t∗ let ip be the inclusion µ−1 (p) → M. If p is a regular value of the moment map, then T acts locally freely on µ−1 (p) so that HT∗ (µ−1 (p)) ∼ = H ∗ (µ−1 (p)/T ) = H ∗ (M//p T ).

(2.1)

Composition of the pullback ip∗ with the isomorphism (2.1) deﬁnes the Kirwan map κp : HT∗ (M) → H ∗ (M//p T ). Kirwan [K] showed that if M is compact this map is surjective. In the presence of another Hamiltonian action on M by a torus T which commutes with the action of T , the Kirwan map generalizes to its equivariant version: κp : HT∗×T (M) → HT∗ (M//p T ). Kirwan surjectivity was generalized to this case in [Go]. Analogously, in the case when T = H × S and p ∈ µ(M H ) we can deﬁne the Kirwan map κpH : HS∗ (M H ) → H ∗ (M H //p S). We will mostly be interested in the equivariant version of this map. Let us take account of the trivial action of H on both M H and M H //p S. So in the rest of the paper κpH will be the map κpH : HT∗ (M H ) → HH∗ (M H //p S). (2.2) Let us now apply the equivariant version of the Kirwan map to symplectic cuts with respect to cones. Given a cone = β , the product T × T acts on M × Cβ and the symplectic cut M is produced by reducing M × Cβ at 0 with respect to the T action. So if we think of T × T as the product of T × e and T , then the equivariant version of the Kirwan map produces the map κ : HT∗×T (M × Cβ ) → HT∗×e (M ) = HT∗ (M ). As mentioned before, the action of T on M might not be effective, since the torus T orthogonal to the cone acts trivially on M . In particular, HT∗ (M ) = HT∗/T (M ) ⊗ HT∗ (pt). For η ∈ HT∗ (M) denote by η the class κ (η ⊗ 1) ∈ HT∗ (M ). It is an easy exercise to see that η ∈ HT∗/T (M ) ⊗ 1. Let us also notice that if Fi is a connected component of the ﬁxed point set of both M and M with µ(Fi ) = µ (Fi ) ∈ Int(), then ι∗Fi η = ι∗Fi η , where by abuse of notation we denote by ιFi both the inclusion of Fi into M and into M . (Because is transverse to µ(M), the existence of such Fi implies that the dimension of the cone is at least the dimension of µ(M).) It is also easy to see that ιM0 η = κ0 (η) ⊗ 1 ∈ H ∗ (M0 ) ⊗ 1 ⊂ HT∗ (M0 ),

(2.3)

where, since T acts trivially on M0 , we know that HT∗ (M0 ) = H ∗ (M0 ) ⊗ S(t∗ ).

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2.4 ABBV localization theorem Suppose that M is compact, oriented and carries the action of a torus T . The pushfor ward map from HT∗ (M) to HT∗ (pt) is the integration over M and is denoted by . The theorem of Atiyah–Bott and Berline–Vergne (or just the ABBV theorem) [AB, BV] states that for η ∈ HT∗ (M) η= M

Fi

ι∗Fi (η) Fi

e(ν(Fi ))

.

(2.4)

Here Fi are the connected components of the ﬁxed point set M T , ιFi : Fi → M are their inclusions, ν(Fi ) are the normal bundles of Fi , and e(ν(Fi )) are their T equivariant Euler classes. We need to know more about these T -equivariant Euler classes. Because of the splitting principle [BT] we can assume without loss of generality that e(ν(Fi )) splits as a sum of line bundles L1 ⊗ · · · ⊗ Lk . Assume T acts on the ﬁbers of Li with weight λi . Then the T -equivariant Euler class is 1 (λi + c1 (Li )), (2.5) e(ν(Fi )) = where c1 (Li ) is the ﬁrst Chern class of Li . TheABBV localization formula was generalized to orbifolds by Meinrenken [Me]. We refer to [Me] for details. Let us just mention that the only difference with (2.4) is the appearance of constants before each term of the formula: 1 ι∗Fi (η) 1 η= , (2.6) dM M dFi Fi e(ν(Fi )) Fi

where for a connected orbifold X, the size of the ﬁnite stabilizer at a generic point of X is equal to dX . Moreover (2.5) is still a valid formula for orbifolds, where the λi (when properly interpreted) are rational weights of the action on the normal bundle. Another important property of the equivariant Euler classes is the following generalization of [GK, Proposition 3.1]. Proposition 2.2. Assume T = S × H , π : t∗ → s∗ is the natural projection, p ∈ µ(M H ), and q = π(p). Then κpH : HT∗ (M H ) → HH∗ (M H //p S) takes the T -equivariant Euler class of the normal bundle of M H onto the H -equivariant Euler class of the normal bundle of M H //p S in M//q S. Proof. The argument is almost identical to the one used in [GK]. Namely, let Z = µ−1 (p) ∩ M H and let ρ be the projection from Z to M H //p S and i : Z → M H . Then i ∗ ν(M H ) = ρ ∗ ν(M H //p S). The proposition follows from functoriality of the Euler class as a map from oriented vector bundles to cohomology.

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3 Residues In this section we deﬁne the residue operations and discuss their basic properties. Most of the deﬁnitions and results of this section have appeared before in either [JK1] or [JK2]. However, we repeat all the deﬁnitions of different kinds of residues (equations (3.1), (3.2), (3.8), (3.11), (3.12), (3.15), and (3.16)), so that these deﬁnitions can be understood without referring to the above sources. At the same time, for shortness, we do not repeat the arguments of [JK1] or [JK2], which prove the signiﬁcance of these deﬁnitions, and do not repeat the proofs of certain properties of residues. 3.1 Residues of meromorphic 1-forms in one variable Think of the Riemann sphere as the one point compactiﬁcation C∪{∞} with the complex coordinate z on C. Let f (z) be a meromorphic function on the Riemann sphere with values in a topological vector space V which can be written as the ﬁnite sum f (z) = gj (z)eiλj z , j

where gj (z) are rational functions of z and λj ∈ R. Then in the case when all λj = 0 we deﬁne Res(f dz) = res(gj (z)eiλj z ; z = b) (3.1) λj >0 b∈C

as was done in [JK2]. The other case we will be interested in is when λj = 0 for all j ; then we deﬁne Res(f dz) = lim Res(f (z)eisλz dz) (3.2) s→0+

for some λ > 0. It is easy to see that in this case Res(f dz) is just the sum of all residues on C and since the sum of all residues of a meromorphic function is zero we conclude that Res(f dz) = −resz=∞ (f dz). Given a linear map ψ : V → W between two topological spaces, the residue commutes with it ψ(Res(f )dz) = Res(ψ(f )dz). (3.3) In the case when V carries an algebra structure, an example which will be important for us is the residue of the function of the form f =

g(z) , cz + a

where a ∈ V , c ∈ C − {0}, and g(z) is a polynomial in z with values on V . For A = az use 1 = 1 − A + A2 − . . . , (3.4) 1+A 0 a j to rewrite f as a sum m j =−∞ γj z for γj ∈ V , which converges for | z | < 1.

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Let w =

1 z

be another coordinate on the Riemann sphere. Then ∞

γ−j w j

j =−m0

−dw w2

is the Taylor expansion of f dz at w = 0. Hence ⎛ ⎛ ⎞ ⎞ m0 ∞ γj zj dz⎠ = resw=0 ⎝ γ−j w j −2 dw⎠ = γ−1 . Res(f dz) = −resz=∞ ⎝ j =−∞

j =−m0

(3.5) In the case when g(z) is just a constant g0 ∈ V , we get Res(f dz) =

g0 . c

(3.6)

We emphasize that the residue is well deﬁned only as a function of meromorphic 1-forms, not functions; the residue at 0 of the 1-form f (z)dz is independent of the choice of coordinate z. (The residue is invariant under a change of variables z → g(z) provided that g(z) is a meromorphic function of z and dg(0) = 0, whereas this is not true of usual deﬁnition of the residue at 0 of a meromorphic function.) 3.2 Residues of functions of several variables Let us now consider a function f of several complex variables with values in a topological space V . More precisely, we assume f is deﬁned on the complexiﬁed Lie algebra tC = t ⊗ C of the torus T , and f is a linear combination of functions of the form q(X)eiλ(X) h(X) = 8k (3.7) j =1 αj (X) for some polynomials q(X) of X ∈ tC , with values in V , λ ∈ t∗ and some α1 , . . . , αk ∈ t∗ − {0}. Choose a coordinate system X1 , . . . , Xm on t, and denote by the same letters the complexiﬁed coordinates which provide a coordinate system of tC . Let t be the subspace of t given by zeros of X1 , . . . , X . Deﬁne Resm (hdXm ) = Res(hdXm ),

(3.8)

where the variables X1 , . . . , Xm−1 are held constant while calculating this residue. As explained in [JK2, Remark 3.5(1)], in the case λ = 0 the residue Resm is well deﬁned only for a generic choice of coordinates X1 , . . . , Xm (the precise condition on the coordinates being speciﬁed in this remark). Moreover, by [JK2, Remark 3.5(2)], Resm (f dXm ) is a linear combination of functions of the form iλ(X) q(X)e ˜ ˜ h(X) = 8k−1 , ˜ j (X) j =1 α

(3.9)

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where q(X) ˜ is a polynomial on the complexiﬁcation of t/tm−1 and α˜ j are in the dual of t/tm−1 . To consider the case when λ = 0, in other words h0 (X) = 8k

q(X)

(3.10)

,

j =1 αj (X)

we may make a choice of λ0 ∈ t∗ for which λ0 (X) = deﬁne Resm (h0 dXm ) as

m

0 j =1 λj Xj

with λ0m > 0 and

0

Resm (h0 dXm ) = lim Resm (h0 eisλ dXm ).

(3.11)

s→0+

We can easily check that the residue Resm is a continuous function of s ∈ R+ and this limit exists, so we can deﬁne Resm (hdXm ) even when λ = 0. It is still true in this situation that Resm (f dXm ) is a linear combination of functions of the form given in (3.10) with λ = 0. In the case when V is an algebra, an important generalization of (3.6) is the following.

Lemma 3.1. Resm

g0 dXm c i i Xi + v

=

g0 , cm

where g0 , v ∈ V and ci ∈ C. Proof. The proof follows from (3.5) after setting c = cm and a = + v.

m−1 i=1

ci Xi

3.3 Iterated residues We again consider functions f which are linear combinations of functions of the form (3.7). As was just explained, Resm (f dXm ) is a linear combination of functions of the form (3.9), which allows us to take the residue of this function again. So we set Resm m = Resm and by induction deﬁne the iterated residue m m m Resm (f [dX] ) = Res(Res+1 (f [dX]+1 )dX ),

(3.12)

where [dX]m stands for the form dX ∧ · · · ∧ dXm and, as above, the coordinates m X1 , . . . , X−1 are held constant while calculating this residue. Clearly Resm (f [dX] ) is a function on the complexiﬁcation of t/t−1 . In the case V is an algebra, a generalization of Lemma 3.1 states the following. Lemma 3.2. For a generic choice of coordinates X1 , . . . , Xm , ' * g0 g 0 Resm = , [dX]m 8m−+1 det( α) ˜ (αi (X) + vi ) i=1 where g0 , vi ∈ V , αi ∈ t∗ −{0}, and α˜ is the the matrix {aij }1≤i≤m−+1;≤j ≤m , where αi (X) = aij Xj .

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Proof. The proof is analogous to the proof of property (iv) for the iterated residue in [JK2, Proposition 3.4]. Notice that for h0 deﬁned in (3.10) the deﬁnition of m ) is given in [JK2] as the limit as s → 0+ of Resm (h0 eisλ(X) [dX]m ) 0 Resm (h [dX] where λ(X) = j λj Xj and we choose λ so that all the λj satisfy λj > 0.

Given a linear map ψ : V → W between two topological spaces, the iterated residue commutes with this map: m m m ψ(Resm (f [dX] )) = Res (ψ(f )[dX] ).

(3.13)

3.4 Residues with respect to cones Iterated residues depend on the choice of coordinates on t. Fix an inner product on t. Let us deﬁne residues which depend only on this inner product and a choice of a certain cone in t. We introduce a function f which is a linear combination of functions of the form (3.7). We consider the set where none of the functions αj appearing in the denominators of functions h become zero, namely the set {X ∈ t : αj (X) = 0 for all αj }.

(3.14)

Let be an open cone which is a connected component of this set. Then for a generic choice of coordinate system X = (X1 , . . . , Xm ) on tC for which (0, . . . , 0, 1) ∈ deﬁne the residue with respect to the cone by Res (h[dX]) = /Resm 1 (h[dX]),

(3.15)

[dX]m 1

where [dX] = and / is the determinant of any (m) × (m) matrix whose columns are the coordinates of an orthonormal basis of t deﬁning the same orientation on t as the chosen coordinate system. To guarantee that Res (h[dX]) is well deﬁned (where h is of the form (3.7)), we need to make one additional assumption: we assume that λ is not in any proper subspace of t spanned by some αi ’s. It was shown in [JK2] that under the above assumptions Res (h[dX]) is well deﬁned, does not depend on the choice of the coordinates but only on the choice of the cone and the inner product on t. Originally, the residue Res was introduced in [JK1] as a generalization of a certain integral over a vector space. In [JK2, Proposition 3.4], it was shown that the deﬁnition (3.15) coincides with the original deﬁnition of Res from [JK1]. In [JK2, Proposition 3.2] it was shown that certain properties together with linearity uniquely deﬁne Res . Let us recall these properties: 1. Let α1 , . . . , αv ∈ ∗ be vectors in the dual cone. Suppose that λ is not in any cone of dimension m − 1 or less spanned by a subset of the {αi }. If J = (j1 , . . . , jm ) j j is a multiindex and XJ = X11 . . . Xmm , then ' * J iλ(X) [dX] X8e =0 Res v i=1 αi (X) unless all of the following properties are satisﬁed:

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315

a) {αi }vi=1 span t∗ as a vector space, b) v − (j1 + · · · + jm ) ≥ m, c) λ ∈ α1 , . . . , αv + , the positive span of the vectors {αi }. 2. If properties (1)(a)–(c) above are satisﬁed, then ' ' * * J iλ(X) [dX] J k eisλ(X) [dX] X8e X (iλ(X)) 8 Res = , lim Res v k! vi=1 αi (X) s→0+ i=1 αi (X) k≥0

and all but one term in this sum are 0 (the nonvanishing term being that with k = v − (j1 + · · · + jm ) − m). 3. The residue is not identically 0. If properties (1)(a)–(c) are satisﬁed with α1 , . . . , αm linearly independent in t∗ , then ' * iλ(X) [dX] e 1 Res 8m , = det(α) α (X) i=1 i where α is the nonsingular matrix whose columns are the coordinates of α1 , . . . , αm with respect to any orthonormal basis of t deﬁning the same orientation. When λ is of the form ki=1 si αi , where fewer than m of the si are nonzero, then we deﬁne ' ' * * i(λ(X)+sρ(X)) [dX] eiλ(X) [dX] 8 e 8 = lim Res , (3.16) Res v v s→0+ i=1 αi (X) i=m αi (X) where ρ ∈ t∗ is chosen so that ρ(ξ ) > 0 for all ξ ∈ , and for small s, λ + sρ does not lie in any cone of dimension m − 1 or less spanned by a subset of the {βj }. We will need another property of residues. Lemma 3.3. Let f (X1 , . . . , Xm ) be a function on t given by a linear combination of functions of the form (3.7) with λ = 0. Moreover, assume that for every set of values a1 , . . . , am−1 of the variables X1 , . . . , Xm−1 the function g(z) = f (a1 , . . . , am−1 , z) is holomorphic. Let be an appropriate choice of cone such that Res (f [dX]) is deﬁned, in particular contains the point (0, . . . , 0, 1). Then Res (f [dX]) = Res− (f [dY ]), where Y1 , . . . , Ym is a set of coordinates such that (0, . . . , 0, 1) ∈ −. Proof. For ﬁxed a1 , . . . , am−1 , let g(z) =

gj (z)eiλj z .

Then Res(g(z)dz) = res+ (g(z)dz) =

λj >0 b∈C

res(gj (z)eiλj z ; z = b).

(3.17)

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Deﬁne

res− (g(z)dz) =

res(gj (z)eiλj z ; z = b).

λj <0 b∈C

Since g(z) is holomorphic, we have res+ (g(z)dz) + res− (g(z)dz) = 0.

(3.18)

Since Res− does not depend on the choice of coordinates as long as (0, . . . , 0, 1) ∈ −, we may choose Yi = Xi for 1 ≤ i ≤ m − 1, and Ym = −Xm . Then Resm (f (Y1 , . . . , Ym )dYm ) = Resm (−f (X1 , . . . , Xm−1 , −Xm )dXm ) = res− (−f (X1 , . . . , Xm )dXm ) =† res+ (f (X1 , . . . , Xm )dXm ) = Resm (f (X1 , . . . , Xm )dXm ), where (†) holds because of (3.18). Now (3.17) follows immediately from the deﬁnition of Res using iterated residues.

4 A generalization of Guillemin–Kalkman localization In this section we discuss how to obtain the localization formula of Guillemin and Kalkman [GK] by applying ABBV localization and then residue operations on symplectic cuts along certain cones of dimension one. The same approach but with cones of higher dimension and iterated residues provides a generalization of Guillemin– Kalkman localization. 4.1 Guillemin–Kalkman localization Let (M, ω) be a compact symplectic manifold with an effective Hamiltonian T action. Let µ : M → t∗ be the moment map. Pick a one-dimensional cone transverse to µ. Assume the cone is generated by a single weight β. If dim T = m, consider all (m − 1)-dimensional walls of µ which intersects. Every such wall Wi is an image µ(Mi ) of a connected component Mi of M Hi for some one-dimensional subtorus Hi of T . If pi is the intersection Wi ∩, then Gi = Mi //pi T is a connected component of the ﬁxed point set of the T /T action on M . The only other connected component of this ﬁxed point set is M0 , the reduction of M at zero by T . (If dim T = 1, then Gi are also connected components of the ﬁxed point set M T , and using the notation introduced in Deﬁnition 2.1, Gi would be called the old connected components of the ﬁxed point set. If dim T > 1, then in terms of the same notation Gi are the new ﬁxed points.) Recall that any equivariant cohomology class η ∈ HT∗ (M) descends to a class η on M : η = κ (η ⊗ 1) ∈ HT∗/T (M ) ⊗ 1 ⊂ HT∗ (M ).

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Think of η as a class in HT∗/T (M ) and apply the orbifold version of the ABBV localization theorem to it: 1 ι∗G (η ) 1 κ0 (η) 1 i + , (4.1) η = dM M dM0 M0 e(ν(M0 )) dGi Gi e(ν(Gi )) Gi

where ιGi are the inclusions of Gi into M and the Kirwan map at zero κ0 appears because of (2.3). Apply the residue operation to both sides of (4.1). For this we have to specify a coordinate X1 on the one-dimensional Lie algebra of T /T . Clearly the dual of this Lie algebra can be identiﬁed with the line in t∗ passing through the cone . So β deﬁnes a coordinate on the Lie algebra of T /T , and we deﬁne Xβ = X1 to be this coordinate. Using X1 we get the formula ι∗ (η ) dXβ dXβ dXβ κ0 (η) Gi Res1 + . η = Res1 Res1 dM M dM0 M0 e(ν(M0 )) dGi Gi e(ν(Gi )) Gi

(4.2) Since M η is just a polynomial function on the Lie algebra of T /T , by (3.5) the left-hand side of (4.2) is zero. The normal bundle ν(M0 ) is just a line bundle whose equivariant Euler class is given by e(ν(M0 )) = Xβ + c1 (ν(M0 )), (4.3) where c1 (ν(M0 )) is the ﬁrst Chern class of ν(M0 ). Thus by (3.13) and (3.6) dXβ κ0 (η)dXβ κ0 (η) 1 Res1 =(††) Res1 dM0 M0 e(ν(M0 )) dM0 M0 Xβ + c1 (ν(M0 )) 1 = κ0 (η). dM0 M0 κ0 (η) Remark 4.1. To show that (††) follows from (3.13) think of e(ν(M as a function on 0 )) ∗ ∗ t with values in H (M). If we deﬁne ψ : H (M) → C to be the usual integration on M, then (3.13) applied to this ψ yields (††).

Hence (4.2) yields a formula for the integral of κ0 (η) over M0 of the Kirwan map of η in terms of the residues of certain integrals over the Gi . To put this formula in the form in which it appeared in [GK] let us transform the terms corresponding to the Gi in (4.2) by commuting Res1 with the integration and the Kirwan map. By (3.13) and Proposition 2.2, we have ' ' ' ∗ * * ι∗ (η ) * ιMi η dXβ 1 Gi Hi Res1 = dXβ , Res1 κpi dGi Gi e(ν(Gi )) dGi Gi e(ν(Mi )) where ιMi is just the inclusion of Mi into M, and as deﬁned in (2.2) κpHii is the Kirwan map from HT∗ (Mi ) to HT∗/T (Gi ).

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Notice that each Mi is ﬁxed by the one-dimensional torus Hi , so that the Lie i on t such that algebra hi is a subspace of t. Pick a coordinate system X1i , . . . , Xm i i i Xm = β and X1 , . . . , Xm−1 vanish on hi , so that tm−1 = hi . Then it is not difﬁcult to see that Res1 ◦ κpHii = κpHii ◦ Resm,i , where Resm,i is Resm deﬁned using the coordinate system {Xji } (note that the deﬁnition of Resm was given in (3.11)). This allows us to obtain the following restatement of [GK, Theorem 3.1]. Theorem B. For η ∈ HT∗ (M) κ0 (η) = − M0

'

dM 0

Gi

dGi

Gi

κpHii Resm,i

ι∗Mi η e(ν(Mi ))

* dβ .

(4.4)

Guillemin and Kalkman iterated this result using certain combinatorial objects called dendrites to produce a formula for M0 κ0 (η) in terms of the integration over the connected components of the ﬁxed point set M T of the original T action on M. We will discuss dendrites and their generalizations in Section 4.3. 4.2 Higher-dimensional generalization of Guillemin–Kalkman localization We now generalize the formula of the previous section to the case when we cut with a cone of any dimension. As before, let (M, ω) be a compact symplectic manifold with an effective Hamiltonian T action. Let µ : M → t∗ be the moment map. Pick any cone transverse to µ. Assume is generated by the linearly independent weights β1 , . . . , βk . Recall that in Deﬁnition 2.1 we distinguished three kinds of ﬁxed points of M : the old ﬁxed points (whose connected components are denoted by Fi ), the new ﬁxed points (whose connected components are denoted by Fi ), and the symplectic reduction M0 . As in the case of one-dimensional cones any equivariant cohomology class η ∈ HT∗ (M) descends to a class η on M η = κ (η ⊗ 1) ∈ HT∗/T (M ) ⊗ 1 ⊂ HT∗ (M ). As before, we think of η as a class in HT∗/T (M ) and apply the ABBV localization theorem to it: 1 ι∗F (η ) κ0 (η) 1 1 i + η = dM M dM0 M0 e(ν(M0 )) dFi Fi e(ν(Fi )) Fi (4.5) ∗ 1 ιF (η ) i + . dFi Fi e(ν(Fi )) Fi

Here, κ0 is the Kirwan map at zero and ιFi , ιFi are the inclusions of Fi , Fi into M .

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Let us apply the iterated residue operation to both sides of (4.5). As in the onedimensional case the dual of the Lie algebra of T /T can be identiﬁed with the subspace in t∗ passing through the cone . Consider the set of weights {αi } which appear in the isotropy representation of the T /T action at the connected components of the ﬁxed point set of the T /T action on M . Notice that the weights β˜1 , . . . , β˜k of the isotropy representation of T /T at M0 are the images of the weights β1 , . . . , βk under the map (t/t )∗ → t∗ . In particular, the set {αi } contains the weights β˜i . Let {X1 , . . . , Xk } be a generic coordinate system on T /T such that the determinant of the matrix which expresses β˜i as a linear combination of Xj is one. Then by applying Resk1 to both sides of (4.5) we get the formula 0=

[dX]k1 Resk1 dM0

M0

[dX]k1 κ0 (η) + Resk1 e(ν(M0 )) dFi

Fi

+

Fi

[dX]k1 Resk1 dFi

ι∗Fi (η ) Fi

e(ν(Fi )) ι∗F (η ) i

Fi

e(ν(Fi ))

(4.6) ,

where the left-hand side is zero since M η is just a polynomial function on the Lie algebra of T /T . The weights of the isotropy representation of T /T on the normal bundle ν(M0 ) are just β˜i . Hence ν(M0 ) splits as a direct sum of line bundles ⊕i Li , where T /T acts on the ﬁbers of Li by the weights β˜i . Then the Euler class of ν(M0 ) is 1 (β˜i + c1 (Li )), (4.7) e(ν(M0 )) = i

where c1 (Li ) are the ﬁrst Chern classes of Li . Thus by (3.13) (see also Remark 4.1) and Lemma 3.2, we have [dX]k1 1 κ0 (η) = κ0 (η). (4.8) Resk1 dM0 M0 e(ν(M0 )) dM0 M0 By analogy with the one-dimensional case, we should now simplify the second and third terms of the right-hand side of (4.6). But the second term is already in the form we want since it is written in terms of the ﬁxed points of M. To simplify the last term of (4.6), ﬁrst consider the situation when a point pi = µ (Fi ) is an intersection of the interior of with a wall of µ(M) of dimension r > 0. Assume that this wall is the moment map image of Mi , which is stabilized by Hi ⊂ T . Then by (3.13) and Proposition 2.2, we have ' ' ' ∗ ** ι∗ (η ) * ιMi η 1 1 Fi k k k k Hi = Res1 [dX]1 Res1 [dX]1 κp , i dFi dFi Fi e(ν(Mi )) Fi e(ν(Fi )) (4.9) where ιMi is just the inclusion of Mi into M, and as deﬁned before κpH i is the Kirwan map from HT∗ (Mi ) to HT∗/T (Fi ).

i

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For the more general situation when pi is an intersection of any face of with a wall of µ(M) of complementary dimension, formula (4.9) becomes a little more complicated. Namely the Euler class of Fi is no longer a Kirwan map image of the Euler class of Mi , but rather this is multiplied by another class νi . The class νi can be understood as follows. Take the symplectic cut M,i of Mi with respect to the cone . This symplectic cut sits naturally inside M . Then νi is the Euler class of the normal bundle of Fi inside M,i . While this class can be quite complicated, by the generalized Kirwan surjectivity theorem of [Go] we know that there exists a class τi on Mi such that κpH i (τi ) = νi . So we conclude that i

'

1 Resk1 [dX]k1 dFi

ι∗F (η )

*

i

Fi

e(ν(Fi ))

1 = dFi

'

Fi

Resk1

'

[dX]k1 κpH i i

ι∗Mi η e(ν(Mi ))τi

** .

i } To simplify this term even further, let us choose a set of coordinates {X1i , . . . , Xm on t as follows. Each Mi is ﬁxed by a subtorus Hi of T , so that its Lie algebra hi lies inside t. (Notice that dim Hi ≤ dim and this inequality might be strict.). i Choose any generic coordinates X1i , . . . Xm−k which vanish on hi and coordinates i i Xm−k+1 , . . . , Xm , which are the images of X1 , . . . , Xk under the map (t/t )∗ → t∗ . Then we conclude that

Resk1 ◦ κpH i = κpH i ◦ Resm m−k+1 , i

i

i i where Resm m−k+1 on the right-hand side is taken with respect to coordinates X1 , . . . , Xm . Thus we get the following generalization of [GK, Theorem 3.1] to the case of cones of arbitrary dimension.

Theorem C. For η ∈ HT∗ (M), κ0 (η) = − M0

dM 0

Fi

−

dFi

dM

Fi

0

Fi

dFi

Fi

Resk1 [dX]k1

ι∗Fi (η ) e(ν(Fi ))

i m κpH i Resm m−k+1 [dX ]m−k+1 i

ι∗Mi η e(ν(Mi ))τi

(4.10) .

4.3 Dendrites and their generalizations In addition to Theorem B, Guillemin and Kalkman [GK] proved a formula which uses Theorem B iteratively to express M0 κ0 (η) in terms of integration over the connected components of M T . They used the term “dendrites’’ to designate the combinatorial objects responsible for how the iteration is performed. Consider a ﬁnite collection D of tuples ((q), W ), where (q) is a onedimensional shifted cone (that is, if ⊂ t∗ is a cone centered at the origin and q ∈ t∗ , then (q) = {x + q | x ∈ }) and W is a wall of µ(M) such that (q) lies inside the afﬁne space spanned by W . Notice that according to our conventions

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every W uniquely deﬁnes a subtorus HW of T and a connected component MW of M HW such that W = µ(MW ).) Assume that D contains a tuple (0 (0), µ(M)) (we will treat 0 as the cone used in Theorem B). Then we say that D is a dendrite if the following conditions hold: 1. For ((q), W ) ∈ D, the cone (q) is transverse to µ(MW ), 2. For ((q), W ) ∈ D, if (q) intersects a codimension one wall W of µ(MW ) at a point q , then there is a unique cone such that ( (q ), W ) ∈ D. Obviously it is possible to construct a dendrite which contains a tuple (0 (0), µ(M)) as long as 0 is transverse to µ(M). Now consider formula (4.4) of Theorem B produced using the cone 0 . Assume we are given a dendrite D containing (0 (0), µ(M)). Every term in the summation on the right-hand side of (4.4) corresponds to an intersection pi = µ (Gi ) of 0 with a codimension one wall Wi . By deﬁnition of the dendrite, there exists a unique cone i such that (i (pi ), Wi ) ∈ D. Set Hi = HWi and Mi = MWi and notice that Gi = Mi //pi (T /Hi ). Moreover, let us shift the moment map on Mi by deﬁning µi = µ − pi . Then, we can apply Theorem B to the T /Hi action on Mi and the cone i to compute ' * ι∗Mi η Hi κpi Resm dXm e(ν(Mi )) Mi //p (T /Hi ) i

as a sum of integrals over symplectic reductions of the form N //(T /H ), where H is a two-dimensional subtorus of T and N isa connected component of M H . Repeating this process we can express M0 κ0 (η) as a summation of integrals over the connected components of the ﬁxed point set M T . More speciﬁcally, for a dendrite D we say that the sequence of tuples from D ((0 (q0 ), W0 ), (1 (q1 ), W1 ), . . . , (k (qk ), Wk )) is a path P if (0 , µ(M)) = (0 (q0 ), W0 ); Wk is zero-dimensional, so that there is a connected component F of M T with µ(F ) = Wk (we will denote this F by FP ); each qj is an intersection of j −1 (qj −1 ) with a codimension one wall Wj of µ(MWj −1 ). For a path P , let kj = dim Wj . Then for γ ∈ HT∗/HW (MWj −1 ) and an appropriately j −1

chosen set of coordinates on the Lie algebra of T /HWj −1 deﬁne ' * ι∗MW γ j Qj (γ ) = −Reskj dXkj . e(ν(MWj )) For a path P , deﬁne QP = Qk ◦ · · · ◦ Q1 . The above discussion proves the following. Theorem D. For a dendrite D and η ∈ HT∗ (M), κ0 (η) = dM0 QP (η), M0

P

FP

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where the above sum is taken over all possible paths in D. Now consider a collection D of tuples ((q), W ), where (q) is a shifted cone of any dimension and W is a wall of µ(M) such that (q) lies inside the afﬁne space spanned by W . Again, every W uniquely determines a subtorus HW of T and a connected component MW of M HW such that W = µ(MW ). Assume that D contains a tuple (0 (0), µ(M)). Then we say that D is a multidimensional dendrite if the following conditions hold 1. For ((q), W ) ∈ D the cone (q) is transverse to µ(MW ), 2. For ((q), W ) ∈ D if a face of (q) of dimension r ≤ dim W intersects a wall W of µ(MW ) of dimension dim W − r at a point q , then there is a unique cone such that ( (q ), W ) ∈ D. In analogy with the case of dendrites of one-dimensional cones, consider formula (4.10) of Theorem C produced using the cone 0 . Assume we are given a multidimensional dendrite D containing (0 (0), µ(M)). Every term in the second summation on the right-hand side of (4.10) corresponds to a new connected component Fi of MT . Moreover, its moment map image pi = µ (Fi ) is always an intersection of a face of with a wall Wi of complementary dimension. By deﬁnition of the multidimensional dendrite, there exists a unique cone i such that (i (pi ), Wi ) ∈ D. (Notice that MWi and HWi are just Mi and Hi in the notation of Theorem C.) As in the case of one-dimensional cones, we know that Fi = Mi //pi (T /Hi ), so we can apply Theorem C to the T /Hi action on Wi and the cone i (pi ) to compute ' * ι∗Mi η Hi m i m κp Resm−k+1 [dX ]m−k+1 i e(ν(Mi ))τi Mi //p (T /Hi ) i

as a sum of integrals over symplectic reductions of the form N //(T /H ), where H is a subtorus of T of dimension at least 2 and N is a connected component of M H . Using the multidimensional dendrite we can iterate the process and express T M0 κ0 (η) as a summation of integrals over the connected components of M as follows. For a multidimensional dendrite D we say that the sequence of tuples ((0 (q0 ), W0 ), (1 (q1 ), W1 ), . . . , (k (qk ), Wk )) is a path P if (0 , µ(M)) = (0 (q0 ), W0 ); Wk is zero-dimensional, so that there exists a connected component F of M T with µ(F ) = Wk (again we denote FP = F ); each qj is the intersection of the interior of a face σ of j −1 (qj −1 ) with a wall Wj of µ(MWj −1 ) such that dim σ + dim Wj = dim Wj −1 . For a path P , let kj = dim Wj and mj = dim j . Then for γ ∈ HT∗/HW (MWj −1 ) and an appropriately chosen set j −1

of coordinates on the Lie algebra of T /HWj −1 deﬁne ' Qj (γ ) = For a path P deﬁne

k −Reskjj −mj +1

* ι∗MW γ kj j [dX]kj −mj +1 . e(ν(MWj ))

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QP = Qk ◦ · · · ◦ Q1 . It is clear that Theorem D holds in the case when D is a multidimensional dendrite.

5 New proof of the Jeffrey–Kirwan localization formula As explained in the previous section, it is possible to iterate the formulas of Theorems B and C using (multidimensional) dendrites to express integration of κ0 (η) over M0 as a sum of integrals of certain forms over the connected components of M T . In this section we present another way of writing such a formula. The rough idea is to choose a very wide cone , whose dual is inside a certain cone , then apply the residue operation with respect to to the ABBV formula for ηei ω˜ (where ω˜ is the equivariant symplectic form) in such a way that the terms corresponding to the new ﬁxed points Fi of M are zero. As a result we will obtain a new proof of the Jeffrey–Kirwan localization theorem [JK1]. 5.1 Choice of the cone In this section we explain how to choose the cone . It is convenient to think of as a very wide cone, in other words a cone whose boundary is close to being a hyperplane. Given an effective Hamiltonian action of T on a compact manifold M, let {αi } be the set of all weights appearing in the isotropy representation of T at the ﬁxed points. Pick a connected component of the set {ξ ∈ t|αi (ξ ) = 0 for all i}. Consider the dual cone ∗ = {X ∈ t∗ |X(ξ ) ≥ 0 for all ξ ∈ }. Now pick a cone transverse to µ(M) spanned by m = dim t weights β1 , . . . , βm such that satisﬁes 1. ∗ ⊆ , or in other words ∗ ⊆ , 2. For every wall W of µ(M), if W ∩ is not empty, then there exists ξW ∈ ∗ such that the maximum of q(ξW ) for q ∈ W ∩ is attained at a vertex of W . Let us explain why such a cone always exists. Pick a rational vector ξ0 ∈ and let H be the hyperplane in t∗ annihilated by ξ0 . For a weight p in the interior of ∗ deﬁne Hp = H + p. Then Hp ∩ ∗ is a rational polytope. In particular, there exists a simplex S ⊂ Hp with rational vertices β˜1 , . . . , β˜m which contains the polytope Hp ∩ ∗ . Take the weights β1 , . . . , βm , which deﬁne , to be multiples of β˜1 , . . . , β˜m . Clearly satisﬁes (1). Moreover, by increasing the size of S, that is by making wider, we can always guarantee that satisﬁes (2). Since there are inﬁnitely many choices of simplices S, there is a choice of S for which is transverse to µ(M). Now consider the symplectic cut M and the set {γi } of all the weights which appear as weights of isotropy representations at ﬁxed points of M . Consider the set ∗ ∩ {ξ ∈ t|γi (ξ ) = 0 for all i},

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¯ be a connected component of this set. and let the cone For each connected component F of MT , let {γjF } be the set of weights of the ¯ that is if isotropy representation of T at F . Polarize the weights {γjF } using , F F F F F ¯ γj (ξ ) < 0 for all ξ ∈ then set γ¯j = γj , otherwise let γ¯j = −γj . Deﬁne CF to be the cone containing all the points of the form µ (F ) + sj γ¯jF with sj being nonnegative real numbers. If F is an old connected component of the ﬁxed point set, then the polarization ¯ is the same as using , since ¯ ⊂ . If F = M0 , then of the weights {γjF } using CF = −. For the new ﬁxed points let us prove the following important fact. Lemma 5.1. Let F be a new connected component of the ﬁxed point set MT . Then the cone CF does not intersect the interior of . Proof. Let p = µ (F ). Since intersects µ(M) transversely, there is a subtorus H of T and a connected component M of M H for which F is the symplectic reduction of M at p. Moreover, p is just the intersection of the wall W = µ(M ) of dimension k and an open face σ of of dimension m − k. Let us order the weights {γjF }nj=1 of the isotropy representation of T at F in a special way. Assume that each of the ﬁrst k weights is parallel to an intersection W ∩ σ , where σ is an open face of such that dim σ = dim σ + 1 and the closure of σ contains σ . Assume also that each of the last n − k weights is parallel to an intersection W ∩ σ , where W is a wall with dim W = dim W + 1 and W ⊂ W . (It is easy to see that every γjF falls into one of these categories.) Let CF1 be the cone spanned by the ﬁrst k polarized weights centered at the origin and let CF2 be the cone spanned by the other weights, also centered at the origin. Then CF = p + CF1 + CF2 . Since the cone p + CF2 lies in the afﬁne space passing through σ , it remains to show that p + CF1 does not intersect the interior of . If two of the weights γjF are parallel, then we can remove one of them without changing the polarized cone F . Since the ﬁrst k weights point in exactly k different directions (there are precisely k faces σ of with σ ⊂ σ¯ and dim σ = dim σ + 1), we can assume without loss of generality that k = k. Let the ﬁrst k weights span a cone C˜ F centered at the origin. It is clear that in a small neighborhood of p the polytope ∩ W is equal to the cone p + C˜ F . Moreover, since the ﬁrst k weights are linearly independent, the cones C˜ F and CF1 are either the same or do not have common points in the interior. So it remains to show that C˜ F and CF1 are different, which is the same as showing that during polarization at least one of the ﬁrst k weights changes sign. So, we must show that it is impossible to have γjF (ξ ) < 0 for 1 ≤ j ≤ k for ¯ If this happens then the maximum of q(ξ ) for q ∈ W ∩ is attained at every ξ ∈ . µ (F ), which is impossible, by property (2) of the cone .

5.2 Jeffrey–Kirwan localization ¯ and be deﬁned as in the previous section. Consider the symplectic Let cones , cut M . Let p ∈ be a point close to the origin. Then µε = µ − εp for ε > 0 is

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325

a moment map on M . Deﬁne ω˜ ε = ω + iµε to be an equivariant symplectic form on M , where ω is the symplectic form on M . For a form η ∈ HT∗ (M), there corresponds a form η ∈ HT∗ (M ). Apply the ABBV localization theorem to get ε η eω˜ ε = IM + IFεi + IFε , (5.1) 0 i

M

where for F = M0 , Fi , or Fi , ∗ ∗ ιF (η eω ) ιF (η eω ) 1 i(µ (F )−εp) 1 i(µε (F )) = . e e IFε = dF dF F e(ν(F )) F e(ν(F )) ¯ and −: ¯ Let us now apply Lemma 3.3 to the function M η eω˜ ε and cones ¯

¯ η eω˜ ε ) = Res− [dX]

Res ([dX] M

η eω˜ ε . M

Hence by (5.1) we get

¯ ε ε + I + IFε Res [dX] IM Fi 0 i

¯ − ε ε = Res I Fi + IFε . [dX] IM0 +

(5.2)

i

Let us show that four of the six terms of (5.2) are zero. Indeed by Lemma 5.1 the cones CFi do not contain εp. Hence by property (1) of the residue, we get ¯ Res (IFε ) = 0. i

Analogously, since the cone CM0 does not contain εp ¯

ε Res (IM ) = 0. 0

Similarly, for small enough ε the cones −CFi and −CFj do not contain εp. Hence by property (1) of the residue, we have ¯

Res− (IFε ) = 0, i

¯

Res− (IFεj ) = 0

for all i and j . ¯ and deﬁne the Hence only two terms of (5.2) are not zero. Moreover, since same polarization at each Fi one of these terms can be modiﬁed using ¯

Res (IFεi ) = Res (IFεi ). Hence these computations transform (5.2) into ¯ ε )= Res (IFεi ). Res− (IM 0

(5.3)

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Let us now take the limit of both sides of (5.3) as ε → 0. By property (3) of the residue map, and by equations (3.16) and (4.8), we have ¯ ε lim Res− (IM ) = c κ0 (ηeω ), 0 ε→0+

for some constant

M0

c .

M0

Hence the limit of (5.3) gives ' ω iµ (Fi ) e κ0 (ηe ) = c Res i

ι∗Fi (η eω ) Fi

*

e(ν(Fi ))

for some constant c. To ﬁnish the proof of Theorem A, remember that Fi are the old connected components of the ﬁxed point set MT , so that Fi ⊂ M T and µ(Fi ) = µ (Fi ). In particular, ι∗Fi (η eω ) is the same as the restriction of ηeω ∈ HT∗ (M) to Fi . Moreover, if for a connected component F of M T its moment map image µ(F ) is not inside , then the cone at F polarized with respect to does not contain the origin, and by property (1) of the residue the term which corresponds to F in the formula (1.1) is zero.

References [A]

M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1–15. [AB] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology, 23-1 (1984), 1–28. [BV] N. Berline and M. Vergne, Classes caractéristiques équivariantes: Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris Sér. I Math., 295-9 (1982), 539–541. [BT] R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York, 1982. [BGL] D. Burns, V. Guillemin, and E. Lerman, Kähler cuts, arXiv:math.DG/ 0212062. [Go] R. F. Goldin, An effective algorithm for the cohomology ring of symplectic reductions, Geom. Functional Anal., 12 (2002) 567–583 [GK] V. Guillemin and J. Kalkman, The Jeffrey–Kirwan localization theorem and residue operations in equivariant cohomology, J. Reine Angew. Math., 470 (1996), 123–142. [GS] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping I, Invent. Math., 67 (1982), 491–513. [JK1] L. C. Jeffrey and F. C. Kirwan, Localization for nonabelian group actions, Topology, 34-2 (1995), 291–327. [JK2] L. C. Jeffrey and F. C. Kirwan, Localization and the quantization conjecture, Topology, 36-3 (1997), 647–693. [K] F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, Princeton, NJ, 1984. [Le] E. Lerman, Symplectic cuts, Math. Res. Lett., 2-3 (1995), 247–258. [LMTW] E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward, Nonabelian convexity by symplectic cuts, Topology, 37-2 (1998), 245–259. [Me] E. Meinrenken, Symplectic surgery and the Spin-c Dirac operator, Adv. Math., 134 (1998), 240–277

Reﬁnements of the Morse stratiﬁcation of the normsquare of the moment map∗ Frances Kirwan Mathematical Institute Oxford University Oxford OX1 3LB UK [email protected] This paper is dedicated to Alan Weinstein with many thanks and best wishes on the occasion of his 60th birthday. Abstract. Let X be any nonsingular complex projective variety on which a complex reductive group G acts linearly, and let Xss and Xs be the sets of semistable and stable points of X in the sense of Mumford’s geometric invariant theory GIT [17]. Then X has a G-equivariantly perfect stratiﬁcation by locally closed nonsingular G-invariant subvarieties with Xss as an open stratum, which can be obtained as the Morse stratiﬁcation of the normsquare of a moment map for the action of a maximal compact subgroup K of G [9]. In this paper this stratiﬁcation is reﬁned to obtain stratiﬁcations of X by locally closed nonsingular G-invariant subvarieties with X s as an open stratum. The strata can be deﬁned inductively in terms of the sets of stable points of closed nonsingular subvarieties of X acted on by reductive subgroups of G and their projectivized normal bundles. When G is abelian, another way to obtain a reﬁned stratiﬁcation is by perturbing the moment map; the reﬁnement is then itself equivariantly perfect, and its strata can be described in terms of the sets of stable points of linear actions of reductive subgroups of G for which semistability and stability coincide. This is useful even when G is nonabelian, since important questions about the cohomology of the GIT quotient (or Marsden–Weinstein reduction) X//G can be reduced to questions about the quotient of X by a maximal torus of G.

Let X be any nonsingular complex projective variety with a linear action of a complex reductive group G, and let Xss and X s be the sets of semistable and stable points of X in the sense of Mumford’s geometric invariant theory [17]. We can choose a maximal compact subgroup K of G and an inner product on the Lie algebra k of K which is invariant under the adjoint action. Then X has a G-equivariantly perfect stratiﬁcation {Sβ : β ∈ B} by locally closed nonsingular G-invariant subvarieties with X ss as an ∗ The author is a member of VBAC (Vector Bundles on Algebraic Curves), which is par-

tially supported by EAGER (EC FP5 contract HPRN-CT-2000-00099), and acknowledges with gratitude the hospitality of the University of Melbourne and the Australian National University during the writing of part of this paper.

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open stratum, which can be obtained as the Morse stratiﬁcation for the normsquare of a moment map µ : X → k∗ for the action of K on X [9]. In this note the Morse stratiﬁcation {Sβ : β ∈ B} is reﬁned to obtain stratiﬁcations of X by locally closed nonsingular G-invariant subvarieties with X s as an open stratum. The strata can be deﬁned inductively in terms of the sets of stable points of closed nonsingular subvarieties of X acted on by reductive subgroups of G, and their projectivized normal bundles. These reﬁnements of the Morse stratiﬁcation are not in general equivariantly perfect; that is, the associated equivariant Morse inequalities are not necessarily equalities. However, when G is abelian, we can modify the moment map (or equivalently modify the linearisation of the action) by the addition of any constant since the adjoint action is trivial. Perturbation of the moment map by adding a small constant then provides an equivariantly perfect reﬁnement of the stratiﬁcation {Sβ : β ∈ B}, and a generic perturbation gives us a reﬁned stratiﬁcation whose strata can be described inductively in terms of the sets of stable points of linear actions of reductive subgroups of G for which stability and semistability coincide. This is useful even when G is not abelian, since important questions about the cohomology of the Marsden– Weinstein reduction µ−1 (0)/K (or equivalently the geometric invariant theoretic quotient X//G) can be reduced to questions about the quotient of X by a maximal torus of G. The same constructions work when X is a compact Kähler manifold with a Hamiltonian action of K. Even when X is symplectic but not Kähler, reﬁnements of the Morse stratiﬁcation for µ2 can be obtained by choosing a suitable almost complex structure and Riemannian metric. The moduli spaces M(n, d) of holomorphic bundles of rank n and degree d over a Riemann surface of genus g ≥ 2 can be constructed as quotients of inﬁnitedimensional spaces of connections in a way that is analogous to the construction of quotients in geometric invariant theory; the role of the moment map is played by curvature and the role of the normsquare of the moment map is played by the Yang– Mills functional. In [13] reﬁnements of the Morse stratiﬁcation of the Yang–Mills functional are studied using the ideas of this paper. The motivation for this study was the search for a complete set of relations among the standard generators for the cohomology of these moduli spaces M(n, d) when n and d are coprime and n > 2 [4]. The layout of this paper is as follows. Section 1 reviews some background material and Section 2 uses the partial desingularisation construction of [10] to deﬁne a stratiﬁcation {γ : γ ∈ } of Xss with X s as an open stratum. Section 3 gives an inductive description of the strata γ in X ss in terms of the stable and semistable points of linear actions of reductive subgroups and subquotients of G on nonsingular subvarieties of X and their projectivized normal bundles. Section 4 reﬁnes the stratiﬁcation {γ : γ ∈ } to obtain strata which are described inductively purely in terms of the stable points (not the semistable points) of the linear actions appearing in Section 3, and in Section 5 this stratiﬁcation is used to reﬁne the Morse stratiﬁcation {Sβ : β ∈ B} of X. In Section 6 an alternative reﬁnement of this stratiﬁcation is obtained when G is abelian by perturbing the moment map; in this case the inner product on the Lie algebra of K can also be perturbed to give a reﬁned stratiﬁcation.

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Finally Section 7 discusses applications to the study of the cohomology ring of the Marsden–Weinstein reduction µ−1 (0)/K (or equivalently the geometric invariant theoretic quotient X//G), and in particular the relationship between this cohomology ring and the corresponding cohomology ring when G is replaced by a maximal torus.

1 Background In this section we shall review brieﬂy the material we need from [9] (see also [6, 17, 18]). Let X be a connected nonsingular projective variety embedded in projective space Pn and let G be a complex reductive group acting linearly on X via a homomorphism ρ : G → GL(n + 1; C). Then G is the complexiﬁcation of a maximal compact subgroup K, and by rechoosing the coordinates on Pn if necessary, we can assume that K acts unitarily on Pn via ρ : K → U (n + 1). The geometric invariant theoretic quotient X//G is the projective variety whose homogeneous coordinate ring is the G-invariant part of the homogeneous coordinate ring of X. A point x of X is called semistable if there exists an invariant homogeneous polynomial f which does not vanish on x, and x is called stable if in addition the orbit Gx is a closed subset of the set of semistable points X ss and has dimension dim G. There is a rational map from X to X//G which restricts to a G-invariant surjective morphism φ : Xss → X//G from the open subset Xss of X, and every ﬁbre of φ which meets the set X s consisting of the points of X which are stable for the action of G is a single G-orbit in Xss . The image of Xs is an open subset of X//G which can thus be identiﬁed via φ with the quotient X s /G. We shall assume that X s = ∅ (but see Remark 2.1). X has a Kähler structure given by the restriction of the Fubini–Study metric on Pn , and the Kähler form ω is a K-invariant symplectic form in X. There is a moment map µ : X → k∗ , where k is the Lie algebra of K, deﬁned by µ(x) = ρ ∗ ((2π ix ∗ 2 )−1 x ∗ x¯ ∗t ) for x ∈ X ⊆ Pn represented by x ∗ ∈ Cn+1 \{0}, when the Lie algebra of U (n + 1) and its dual are both identiﬁed with the space of skew-Hermitian (n + 1) × (n + 1) matrices in the standard way. Then µ−1 (0) is a subset of Xss , and the inclusion induces a homeomorphism from the Marsden–Weinstein reduction (or symplectic quotient) µ−1 (0)/K to the geometric invariant theoretic quotient X//G. In fact, X ss is the set of points x in X such that µ−1 (0) meets the closure of the G-orbit of x, and X s is the set of points x in X such that µ−1 (0) meets the G-orbit of x at a point which is regular for µ. In the good case when Xss = Xs then X//G = X ss /G and its rational cohomology is isomorphic to the equivariant cohomology of X ss . If we ﬁx an invariant inner product on the Lie algebra k, then we can consider the function µ2 as a Morse function on X. It is not in general a Morse function in the classical sense, or even a Morse–Bott function, since the connected components of

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its set of critical points may not be submanifolds of X, but nonetheless it induces a Morse stratiﬁcation {Sβ : β ∈ B} of X such that the stratum to which x ∈ X belongs is determined by the limit set of its path of steepest descent for µ2 with respect to the Fubini–Study metric. This stratiﬁcation can also be deﬁned purely algebraically and has the following properties [9]. Proposition 1.1. (i) Each stratum Sβ is a G-invariant locally closed nonsingular subvariety of X. (ii) The unique open stratum S0 is the set X ss of semistable points of X. (iii) The stratiﬁcation is equivariantly perfect over the rationals, so that i−λ(β) dim HG (Sβ ), dim HGi (X) = dim HGi (X ss ) + β=0

where λ(β) is the real codimension of Sβ in X. (iv) If β = 0, then there is a proper nonsingular subvariety Zβ of X acted on by a reductive subgroup Stab β of G such that ∗ (Zβss ), HG∗ (Sβ ) ∼ = HStab(β)

where Zβss is the set of semistable points of Zβ with respect to an appropriate linearisation of the action of Stab(β). Remark 1.2. (i) All cohomology in this paper has rational coefﬁcients. (ii) We shall assume that the invariant inner product chosen on k is rational, and we shall use it to identify k∗ with k throughout. (iii) Note that G-equivariant cohomology is the same as K-equivariant cohomology, since G retracts onto its maximal compact subgroup K. If we choose a positive Weyl chamber t+ in the Lie algebra t of a maximal torus T of K, then we can identify the indexing set B with a ﬁnite subset of t+ (or equivalently with the set of adjoint orbits of the points in this ﬁnite subset of t+ , since each adjoint orbit meets t+ at exactly one point). More precisely, let α0 , . . . , αn be the weights of the representation of T on Cn+1 and use the restriction to t of the ﬁxed invariant inner product on k to identify t∗ with t. Then β ∈ t+ belongs to B if and only if β is the closest point to 0 of the convex hull in t of some nonempty subset of {α0 , . . . , αn }. Moreover, Stab(β) is the stabiliser of β under the adjoint action of G, and Zβ is the intersection of X with the linear subspace {[x0 : · · · : xn ] ∈ Pn : xi = 0 if αi .β = β2 } of Pn . Equivalently, Zβ is the union of those connected components of the ﬁxed point set of the subtorus Tβ of T generated by β on which the constant value taken by the real-valued function x → µ(x).β is β2 . If we use the ﬁxed inner product to identify k with its dual, then the image µ(Zβ ) of Zβ is contained in the Lie algebra of

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the maximal compact subgroup StabK (β) = K ∩ Stab(β) of Stab(β). Since moment maps are only unique up to the addition of a central constant, we can take µ − β as our moment map for the action of StabK (β) on Zβ . This corresponds to a modiﬁcation of the linearisation of the action of Stab(β) on Zβ whose restriction to the complexiﬁcation Tβc of Tβ is trivial, and we deﬁne Zβss to be the set of semistable points of Zβ with respect to this modiﬁed linear action. Equivalently, Zβss is the stratum labelled by 0 (the minimum stratum) for the Morse stratiﬁcation of the function µ − β2 on Zβ . Then A Sβ = GY¯β \ GY¯γ = GYβss , (1.1) γ >β

where Y¯β = {x ∈ X : xi = 0 if αi .β < β2 } and Yβ = {x ∈ Y¯β : xi = 0 for some i such that αi .β = β2 }, while

Yβss = pβ−1 (Zβss ),

(1.2)

where pβ : Yβ → Zβ is the obvious projection given by pβ (x) = lim exp(−itβ)x. t→∞

(1.3)

Remark 1.3. In [9] Zβss is deﬁned as above to be the set of semistable points of Zβ with respect to the action of Stab(β) with the linearisation modiﬁed in a way which corresponds to replacing the moment map µ by (a positive integer multiple of) µ − β. However, we can, if we wish, regard µ − β as a moment map for the action of StabK (β)/Tβ on Zβ , and then Zβss is the set of semistable points for the induced linear action of its complexiﬁcation Stab(β)/Tβc . The advantage of this description is that the analogous deﬁnition of Zβs is a useful one, whereas Zβ has no points that are stable with respect to the action of Stab(β) when β = 0. Therefore, we shall deﬁne Zβs to be the set of stable points for the action of Stab(β)/Tβc on Zβ , linearized so that the corresponding moment map is a positive integer multiple of µ − β. Equivalently, Zβss and Zβs are the sets of semistable (respectively, stable) points of Zβ under the action of the subgroup of Stab(β) whose Lie algebra is the complexiﬁcation of the orthogonal complement to β in the Lie algebra of StabK (β). In fact, if B is the Borel subgroup of G associated to the choice of positive Weyl chamber t+ and if Pβ is the parabolic subgroup B Stab(β), then Yβ and Yβss are Pβ -invariant and we have Sβ ∼ (1.4) = G ×Pβ Yβss . Moreover, Yβ is a nonsingular subvariety of X and pβ : Yβ → Zβ is a locally trivial ﬁbration whose ﬁbre is isomorphic to Cmβ for some mβ ≥ 0.

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Now an element g of G lies in the parabolic subgroup Pβ if and only if exp(−itβ)g exp(itβ) tends to a limit in G as t → ∞, and this limit deﬁnes a surjection qβ : Pβ → Stab(β). Since the surjections pβ : Yβss → Zβss and qβ : Pβ → Stab(β) are retractions satisfying pβ (gx) = qβ (g)pβ (x) (1.5) for all g ∈ Pβ and x ∈ Yβss , this gives us the isomorphism ∗ (Zβss ) HG∗ (Sβ ) ∼ = HStab(β)

of Proposition 1.1(iv). Moreover, since G = KB and B ⊆ Pβ we have GY¯β = K Y¯β , which is compact, and hence A S¯β ⊆ GY¯β ⊆ Sβ ∪ Sγ . (1.6) γ >β

Note that x = [x0 , . . . , xn ] ∈ X is semistable (respectively stable) for the action of the complex torus T c if and only if 0 belongs to the convex hull in t of the set of weights {αj : xj = 0} (respectively to the interior of this convex hull), and that x is semistable (respectively stable) for the action of G if and only if every element gx of its G-orbit is semistable (respectively stable) for the action of T c . In particular this tells us that S0 = Xss . It also tells us that if β ∈ B and y ∈ Y¯β \Y ss , then there is a subset S of {αj : αj .β ≥ β

β2 } such that where

β

y ∈ Stab(β)Yβ ,

is the closest point to 0 of the convex hull of S and

(1.7) β

= β.

Remark 1.4. Of course, the deﬁnitions of the subvarieties Zβ , Zβss and so on in this section depend on the action of G on X ⊆ Pn and its linearisation via the representation ρ : G → GL(n + 1; C). When it is necessary to make these explicit in the notation we shall write Zβ (X, ρ), Zβss (X, ρ) and so on, but we shall omit X if X = Pn , so that, for example, Zβ (X, ρ) = X ∩ Zβ (ρ). Remark 1.5. In the special case when every semistable point is stable, the quotient variety X//G is topologically the ordinary quotient Xss /G, where G acts with only ﬁnite stabilisers on Xss , which means that its Betti numbers dim H i (X//G) are the same as the equivariant Betti numbers dim HGi (X ss ) of Xss . These can be calculated inductively using Proposition 1.1 together with the fact that the equivariant cohomology of a nonsingular complex projective variety is isomorphic as a vector space to the tensor product of its ordinary cohomology and the equivariant cohomology of a point. Remark 1.6. If X is any compact symplectic manifold with a Hamiltonian action of a compact Lie group K and a moment map µ : X → k∗ , then the Morse stratiﬁcation of µ2 has essentially the properties described here, with G replaced by K and Pβ replaced by K ∩ Stab(β) [9].

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2 Stratifying the set of semistable points Suppose now that X has some stable points but also has semistable points that are not stable. In [10, 11] it is described how one can blow X up along a sequence of nonsingular G-invariant subvarieties to obtain a G-invariant morphism X˜ → X, where X˜ is a complex projective variety acted on linearly by G such that X˜ ss = X˜ s . ˜ The induced birational morphism X//G → X//G of the geometric invariant theoretic ˜ quotients is then a partial desingularisation of X//G in the sense that X//G has only orbifold singularities (it is locally isomorphic to the quotient of a nonsingular variety by a ﬁnite group action) whereas the singularities of X//G are in general much worse. In this section we shall review the construction of the partial desingularisation ˜ X//G and use it ﬁrstly to stratify Xss and subsequently (in Section 5) to reﬁne the stratiﬁcation {Sβ : β ∈ B} of X described in Section 1. The set X˜ ss can be obtained from X ss as follows. There exist semistable points of X that are not stable if and only if there exists a nontrivial connected reductive subgroup of G ﬁxing a semistable point. Let r > 0 be the maximal dimension of a reductive subgroup of G ﬁxing a point of X ss and R(r) a set of representatives of conjugacy classes of all connected reductive subgroups R of dimension r in G such that ZRss = {x ∈ Xss : R ﬁxes x} is nonempty. Then

A

GZRss

R∈R(r)

is a disjoint union of nonsingular closed subvarieties , of X ss . The action of G on ss ss X lifts to an action on the blowup X(1) of X along R∈R(r) GZRss which can be ss in X linearized so that the complement of X(1) (1) is the proper transform of the subset φ −1 (φ(GZRss )) of Xss , where φ : X ss → X//G is the quotient map (see [10, 7.17]). ss is ﬁxed by a reductive subgroup of G of dimension at Moreover, no point of X(1) ss least r, and a point in X(1) is ﬁxed by a reductive subgroup R of dimension less than r in G if and only if it belongs to the proper transform of the subvariety ZRss of X ss . ss to obtain X ss such that no reductive We can now apply the same procedure to X(1) (2) ss . If we repeat this process subgroup of G of dimension at least r −1 ﬁxes a point of X(2) ss = X ss , X ss , X ss , . . . , X ss such that no reductive enough times, we obtain X(0) (1) (2) (r) ss , and we set X ˜ ss = Xss . subgroup of G of positive dimension ﬁxes a point of X(r) (r) Equivalently, we can construct a sequence ss ss ss X(R = X ss , X(R , . . . , X(R = X˜ ss , τ) 0) 1)

where R1 , . . . , Rτ are connected reductive subgroups of G with r = dim R1 ≥ dim R2 ≥ · · · dim Rτ ≥ 1, ss along its closed nonsingular and if 1 ≤ l ≤ τ , then X(Rl ) is the blow up of X(R l−1 ) ss ss subvariety GZRl ∼ = G ×Nl ZRl , where Nl is the normaliser of Rl in G. Similarly,

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F. Kirwan

˜ X//G = X˜ ss /G can be obtained from X//G by blowing up along the proper transforms of the images ZR //N in X//G of the subvarieties GZRss of Xss in decreasing order of dim R. If 1 ≤ l ≤ τ , then there is a G-equivariant stratiﬁcation {Sβ,l : β ∈ Bl } of X(Rl ) by nonsingular G-invariant locally closed subvarieties such that one of the ss of X . This stratiﬁcation strata, indexed by 0 ∈ Bl , coincides with the open subset XR Rl l is constructed exactly as the stratiﬁcation {Sβ : β ∈ B} of X was constructed in the last section; note that X(Rl ) is in general only quasi-projective rather than projective, but it is shown in [10] that the construction of the stratiﬁcation still works for X(Rl ) and the properties given in Proposition 1.1 still hold. There is a partial ordering on Bl with 0 as its minimal element such that if β ∈ Bl , then the closure in X(Rl ) of the stratum Sβ,l satisﬁes Sβ,l ⊆

A

Sγ ,l .

γ ∈Bl ,γ ≥β

If β ∈ Bl and β = 0, then the stratum Sβ,l retracts G-equivariantly onto its (tranverse) ss intersection with the exceptional divisor El for the blowup X(Rl ) → X(R . This l−1 ) ss ˆ exceptional divisor is isomorphic to the projective bundle P(Nl ) over GZ , where Rl

ss Zˆ Rssl is the proper transform of ZRssl in X(R and Nl is the normal bundle to GZˆ Rssl l−1 ) ss in XRl−1 . The stratiﬁcation {Sβ,l : β ∈ Bl } is determined by the action of Rl on the ﬁbres of Nl over ZRssl (see [10, Section 7]). The composition ss ss ss → X(R → · · · X(R → X ss X˜ ss = X(R τ) τ −1 ) 1)

is an isomorphism over the set X s of stable points of X, and the complement of X s in X˜ ss is just the union of the proper transforms in X˜ ss of the exceptional divisors ss E1 , . . . , Ek for the blowups XRl → X(R for l = 1, . . . , τ . l−1 ) ss We can now stratify X as follows. We take as the highest stratum the nonsingular closed subvariety GZRss1 whose complement in Xss can be naturally identiﬁed with the complement X(R1 ) \E1 of the exceptional divisor E1 in X(R1 ) . Recall that GZRss1 ∼ = G ×N1 ZRss1 , where N1 is the normaliser of R1 in G, and ZRss1 is equal to the set of semistable points for the action of N1 , or equivalently for the induced action of N1 /R1 , on ZR1 , which is a union of connected components of the ﬁxed point set of R1 in X (see [10, Section 5]). Since R1 has maximal dimension among those reductive subgroups of G with ﬁxed points in Xss , we have ZRss1 = ZRs 1 , where ZRs l is the set of stable points for the action of Nl /Rl on ZRl for 1 ≤ l ≤ τ . Next, we take as strata the nonsingular locally closed subvarieties

Morse stratiﬁcation of the normsquare of the moment map

335

{Sβ,1 \E1 : β ∈ B1 , β = 0} ss \E of X(R1 ) \E1 = X ss \GZRss1 , whose complement in X(R1 ) \E1 is X(R 1 = 1) ss ss ss ss ss \E X(R1 ) \E1 , where E1 = X(R1 ) ∩ E1 , and then we take the intersection of X(R 1 1) with GZRss2 . This intersection is GZRs 2 , where ZRs 2 is the set of stable points for the ss \E can be naturally identiﬁed action of N2 /R2 on ZR2 , and its complement in X(R 1 1) with the complement in X(R2 ) of the union of E2 and the proper transform Eˆ 1 of E1 . Our next strata are the nonsingular locally closed subvarieties

{Sβ,2 \(E2 ∪ Eˆ 1 ) : β ∈ B2 , β = 0} ss \(E ∪ E ˆ 1 ), of X(R2 ) \(E2 ∪ Eˆ 1 ), whose complement in X(R2 ) \(E2 ∪ Eˆ 1 ) is X(R 2 2) s and the stratum after these is GZR3 . Repeating this process gives us strata that are all nonsingular locally closed G-invariant subvarieties of X ss indexed by the disjoint union {R1 } ∪ {R1 } × (B1 \{0}) ∪ · · · ∪ {Rτ } ∪ {Rτ } × (Bτ \{0}),

and the complement in Xss of the union of these strata is just the open subset X s . We take Xs as our ﬁnal stratum indexed by 0, so that the indexing set for our stratiﬁcation of Xss is the disjoint union = {R1 } ∪ {R1 } × (B1 \{0}) ∪ · · · ∪ {Rτ } ∪ {Rτ } × (Bτ \{0}) ∪ {0}.

(2.1)

Moreover, the given partial orderings on B1 , . . . , Bτ together with the ordering in the expression (2.1) above for induce a partial ordering on , with R1 as the maximal element and 0 as the minimal element, such that the closure in Xss of the stratum γ indexed by γ ∈ satisﬁes A γ ⊆ δ . (2.2) δ∈,δ≥γ

Thus this process gives us a stratiﬁcation {γ : γ ∈ }

(2.3)

of Xss such that the stratum indexed by the minimal element 0 of coincides with the open subset X s of X ss . Remark 2.1. We have been assuming that X s = ∅, but this procedure gives us a stratiﬁcation of X ss even when X s is empty. The only difference when X s is empty is that the procedure terminates at some stage l when ss X(R = GZˆ Rssl ∼ = G ×Nl Zˆ Rssl l−1 )

and gives us a stratiﬁcation indexed by = {R1 } ∪ {R1 } × (B1 \{0}) ∪ · · · ∪ {Rl−1 } ∪ {Rl−1 } × (Bl−1 \{0}) ∪ {Rl } such that the stratum indexed by the minimal element Rl of is the open subset GZRs l of X ss . Note also that ZRs l is nonempty, since otherwise

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F. Kirwan

ZRssl = Nl ZRss for some R containing Rl with dim R > dim Rl , and then GZRss = GZRss = X ss , so the procedure would have terminated at an earlier stage.

3 Inductive description of the strata γ in Xss The last section described a stratiﬁcation {γ : γ ∈ } of X ss such that the stratum indexed by the minimal element 0 of coincides with the open subset X s of X ss . In this section we shall study the strata γ in more detail. Note that the strata γ with γ = 0 fall into two classes. Either γ = Rl for some l ∈ {1, . . . , τ }, in which case the stratum γ is GZRs l , or else γ = (Rl , β), where β ∈ Bl \{0} for some l ∈ {1, . . . , τ } and the stratum γ is Sβ,l \(El ∪ Eˆ l−1 ∪ · · · ∪ Eˆ 1 ). In the latter case, we know from (1.4) that ss ∼ ss Sβ,l = GYβ,l , = G ×Pβ,l Yβ,l

(3.1)

ss ﬁbres over Z ss via p : Y ss → Z ss with ﬁbre Cmβ,l for some m where Yβ,l β β,l > β,l β,l β,l 0, and ss ss Sβ,l ∩ El = G(Yβ,l ∩ El ) ∼ ∩ El ), (3.2) = G ×Pβ,l (Yβ,l ss ∩ E ﬁbres over Z ss with ﬁbre Cmβ,l −1 (see [10, Lemmas 7.6 and 7.11]). where Yβ,l l β,l Thus ss Sβ,l \El ∼ \El ), (3.3) = G ×Pβ,l (Yβ,l ss \E ﬁbres over Z ss with ﬁbre Cmβ,l −1 × (C\{0}). Let where Yβ,l l β,l

πl : El ∼ = P(Nl ) → GZˆ Rssl denote the projection. [10, Lemma 7.9] tells us that if x ∈ Zˆ Rssl , then the intersection of Sβ,l with the ﬁbre πl−1 (x) = P(Nl,x ) of πl at x is the union of those strata indexed by points in the adjoint orbit Ad(G)β in the stratiﬁcation of P(Nl,x ) induced by the representation ρl of Rl on the normal Nl,x to GZˆ Rssl at x. Note that we can assume that Rl ∩ K is a maximal compact subgroup of Rl and that Rl ∩ T is a maximal torus for Rl ∩ K, and then Ad(G)β meets a positive Weyl chamber for Rl in Lie(Rl ∩ T ) in a ﬁnite number of points

Morse stratiﬁcation of the normsquare of the moment map

β = β1 = Ad(w1 )β,

β2 = Ad(w2 )β,

...,

337

βrβ,l = Ad(wrβ,l )β,

where w1 = 1, w2 , . . . , wrβ,l ∈ G represent elements of the Weyl group of G. ss and π (y) = gx, where g ∈ G and x ∈ Z ˆ ss , then x is ﬁxed by Now if y ∈ Zβ,l l Rl −1 −1 Ad(g )β and so Ad(g )β lies in the Lie algebra of Rl . Since Rl ∩ T is a maximal compact torus of Rl , there exists r ∈ Rl such that Ad(rg −1 )β ∈ Lie(Rl ∩ T ). Then Ad(rg −1 )β = Ad(wj )β for some j ∈ {1, . . . , rβ,l }, and hence wj−1 rg −1 ∈ Stab(β),

so g ∈ Stab(β)wj−1 Rl . Conversely if g = hwj−1 r, where h ∈ Stab(β) and r ∈ Rl , ss if and only if h−1 y lies in Z ss , where π (h−1 y) = w −1 rx ∈ w −1 Z ˆ ss . then y ∈ Zβ,l l β,l Rl j j Thus

A ss ss = Stab(β) Zβ,l ∩ wj−1 πl−1 (Zˆ Rssl ) Zβ,l 1≤j ≤rβ,l

=

A

1≤j ≤rβ,l

−1 ˆ ss ss Stab(β)wj−1 ZAd(w ∩ π ( Z ) . Rl l j )β,l

ss ∩ w −1 π −1 (Z ˆ ss )) meets Stab(β)(Z ss ∩ w −1 π −1 (Zˆ ss )), then Also, if Stab(β)(Zβ,l Rl β,l Rl j l i l Stab(β)wj−1 Zˆ Rssl meets wi−1 Zˆ Rssl , and since GZˆ Rssl ∼ = G ×Nl Zˆ Rssl , this means that there is some h ∈ Stab(β) and n ∈ Nl such that wi h = nwj , so that βi = Ad(wi )β ∈ Ad(Nl )βj . Conversely, if βi ∈ Ad(Nl )βj then wi h = nwj for some h ∈ Stab(β) and n ∈ Nl , and so ss ss Stab(β)(Zβ,l ∩ wj−1 πl−1 (Zˆ Rssl )) = Stab(β)(Zβ,l ∩ h−1 wi−1 nπl−1 (Zˆ Rssl )) ss = Stab(β)(Zβ,l ∩ wi−1 πl−1 (Zˆ Rssl )). ss is a disjoint union Thus Zβ,l ss Zβ,l =

B

ss Stab(β) Zβ,l ∩ wj−1 πl−1 (Zˆ Rssl ) ,

1≤j ≤sβ,l

where Ad(w1 )β = β, . . . , Ad(wsβ,l )β form a set of representatives for the Ad(Nl ) ss and S orbits in Ad(G)β, and Yβ,l β,l can be expressed similarly as disjoint unions. In fact, since by (1.5) the ﬁbration ss ss pβ : Yβ,l → Zβ,l ss , where q : P → satisﬁes pβ (gy) = qβ (g)pβ (y) for all g ∈ Pβ and y ∈ Yβ,l β β Stab(β) is the projection, we have

B ss ss Yβ,l = Pβ pβ−1 Zβ,l ∩ wj−1 πl−1 (Zˆ Rssl ) 1≤j ≤sβ,l

and

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F. Kirwan

B

Sβ,l =

ss Gpβ−1 Zβ,l ∩ wj−1 πl−1 (Zˆ Rssl ) .

1≤j ≤sβ,l

This means that we could, if we wished, replace the indexing set Bl \{0}, whose elements correspond to the G-adjoint orbits Ad(G)β of elements of the indexing set for the stratiﬁcation of P(Nl,x ) induced by the representation ρl , by the set of their Nl -adjoint orbits Ad(Nl )β. Then we would still have (3.1)–(3.3), but now ss ss ss = Stab(β)(Zβ,l ∩ πl−1 (Zˆ Rssl )) ∼ ∩ πl−1 (Zˆ Rssl )) (3.4) Zβ,l = Stab(β) ×Nl ∩Stab(β) (Zβ,l

and

ss ∼ ss ∩ wj−1 πl−1 (Zˆ Rssl )), Yβ,l = Pβ ×Qβ pβ−1 (Zβ,l

where

(3.5)

Qβ = qβ−1 (Nl ∩ Stab(β))

is a subgroup of Pβ , and hence

ss ∼ ss Sβ,l ∼ ∩ wj−1 πl−1 (Zˆ Rssl ) . = G ×Pβ Yβ,l = G ×Qβ pβ−1 Zβ,l

(3.6)

Furthermore, πl now restricts to a ﬁbration ss ∩ πl−1 (Zˆ Rssl )) → Zˆ Rssl πl : Zβ,l

(3.7)

whose ﬁbre at x ∈ Zˆ Rssl is Zβss (ρl ) deﬁned as in Remark 1.4, where ρl is the representation of Rl on the normal Nl,x to GZˆ ss at x. Rl

ss of the exceptional If 1 ≤ j ≤ l − 1, then the proper transform Eˆ j in X(R l) ss meets the exceptional divisor E ∼ P(N ) transversely, divisor Ej in X(R and their = l l ) j intersection is the restriction P(Nl |Eˆ j ∩GZˆ R ) l

ss of the projective bundle P(Nl ) over GZˆ Rssl to the intersection in X(R of GZˆ Rssl with l−1 ) ss the proper transform of Ej in X(Rl−1 ) (which by abuse of notation we shall also denote by Eˆ j ). Moreover, the complement in GZˆ ss of its intersection with the exceptional

divisors Eˆ 1 , . . . , Eˆ l−1 is GZRs l . Thus

Rl

\E \E γ = GYβ,l ∼ = G ×Pβ Yβ,l ,

where

(3.8)

\E

ss \(El ∪ Eˆ l−1 ∪ · · · ∪ Eˆ 1 ) Yβ,l = Yβ,l

ss ∩ π −1 (Z s )) with ﬁbre Cmβ,l −1 × (C\{0}), and ﬁbres over Stab(β) ×Nl ∩Stab(β) (Zβ,l Rl l

ss ∩ π −1 (Z s ) ﬁbres over Z s ss Zβ,l Rl l (Rl ) with ﬁbre Zβ (ρl ). In addition, if we set

\E \E ss Yβ = Yβ,l ∩ pβ−1 Zβ,l ∩ πl−1 (Zˆ Rssl ) ,

Morse stratiﬁcation of the normsquare of the moment map

339

we have from (3.5) and (3.6) that

and hence

\E \E Yβ,l ∼ = Pβ ×Qβ Yβ

(3.9)

\E γ ∼ = G ×Qβ Yβ ,

(3.10)

\E

where Qβ = qβ−1 (Nl ∩ Stab(β)) and pβ : Yβ with ﬁbre Cmβ,l −1 × (C\{0}).

ss ∩ π −1 (Z ˆ ss ) is a ﬁbration → Zβ,l Rl l

Remark 3.1. The moduli space M(n, d) of semistable holomorphic bundles of rank n and degree d over a ﬁxed Riemann surface of genus g ≥ 2 can be constructed as a quotient of an inﬁnite-dimensional afﬁne space of connections C by a complexiﬁed gauge group Gc , in an inﬁnite-dimensional version of the construction of quotients in geometric invariant theory, or, equivalently, as an inﬁnite-dimensional symplectic reduction with curvature as a moment map. When n and d are coprime, semistability coincides with stability and M(n, d) is the topological quotient of the semistable subset C ss of C by the action of Gc . The role of the normsquare of the moment map is played by the Yang–Mills functional, which was studied by Atiyah and Bott in their fundamental paper [1]. Atiyah and Bott studied the stratiﬁcation of C deﬁned using the Harder–Narasimhan type of holomorphic bundle, which they expected to be the Morse stratiﬁcation of the Yang–Mills functional (this was later shown to be the case [2]). The methods of this paper can be used to provide a stratiﬁcation {γ : γ ∈ } of C ss with C s as the unique open stratum. This stratiﬁcation of C ss and induced reﬁnements of the Yang–Mills stratiﬁcation of C are studied in detail in [13], where they are related to natural reﬁnements of the notion of the Harder–Narasimhan type of a holomorphic bundle.

4 A reﬁned stratiﬁcation of Xss We can now iterate the construction of the stratiﬁcation (2.3) described in Section 2 and use induction on the dimension of G to deﬁne a stratiﬁcation ˜ ˜ γ˜ : γ˜ ∈ } {

(4.1)

of Xss by G-equivariant nonsingular subvarieties which reﬁnes the stratiﬁcation (2.3). When the dimension of G is zero so that X s = X ss = X, then ˜ = and the stratiﬁcation has one stratum which is X itself. When dim G > 0, we shall reﬁne the stratiﬁcation {γ : γ ∈ } deﬁned in (2.3) as follows. If γ ∈ \{0, R1 , . . . , Rτ } then γ = (Rl , β), where β ∈ Bl \{0} for some l ∈ {1, . . . , τ }, and by (3.8) we have \E \E γ = GYβ,l ∼ = G ×Pβ Yβ,l , \E

ss ∩ π −1 (Z s )) with ﬁbre Cmβ,l −1 × where Yβ,l ﬁbres over Stab(β) ×Nl ∩Stab(β) (Zβ,l Rl l

ss ∩ π −1 (Z s ) ﬁbres over Z s ss (C \ {0}), and Zβ,l Rl l (Rl ) with ﬁbre Zβ (ρl ). We have a linear

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action of Rl ∩ Stab(β)/Tβc on Zβ (ρl ) which corresponds (up to multiplication by a positive integer) to the moment map µ − β. Therefore, by induction on dim G, we can ˜ γ : β˜ ∈ ˜ γ } of Z ss (ρl ) by nonsingular assume that we have deﬁned a stratiﬁcation { β β˜ Rl ∩ Stab(β)-invariant subvarieties. In fact, since the stabiliser in G of any x ∈ ZRs has connected component Rl , we can assume that we have a stratiﬁcation of Zβss (ρl ) by nonsingular Stab(x) ∩ Stab(β)-invariant subvarieties. Since Stab(x) ⊆ Nl and since the ﬁbration ss πl : Zβ,l ∩ πl−1 (ZRs l ) → ZRs l is Nl ∩ Stab(β)-equivariant with ﬁbre Zβss (ρl ), this gives us a stratiﬁcation of

ss ∩ π −1 (Z s ) by nonsingular N ∩ Stab(β)-invariant subvarieties, and hence a Zβ,l l Rl l stratiﬁcation of ss Stab(β) ×Nl ∩Stab(β) (Zβ,l ∩ πl−1 (ZRs l ))

by nonsingular Stab(β)-invariant subvarieties. We also have a ﬁbration \E ss ss \(El ∪ Eˆ l−1 ∪ · · · ∪ Eˆ 1 ) → Stab(β) ×Nl ∩Stab(β) (Zβ,l ∩ πl−1 (ZRs l )) pβ : Yβ,l = Yβ,l

with ﬁbre Cmβ,l −1 × (C\{0}), which satisﬁes pβ (gx) = qβ (g)pβ (x) for all g ∈ Pβ \E \E and x ∈ Yβ,l (see (1.5)). Thus we get an induced stratiﬁcation of Yβ,l by Pβ -invariant subvarieties, and ﬁnally an induced stratiﬁcation of \E γ ∼ = G ×Pβ Yβ,l

˜ for β˜ ∈ ˜ γ . In particular γ has an open by nonsingular G-invariant subvarieties β˜ stratum ˜γ γs = (4.2) 0 γ

corresponding to the open stratum Zβs (ρl ) of Zβss (ρl ) consisting of stable points for the action of Rl ∩ Stab(β)/Tβc . ˜ γ˜ : γ˜ ∈ } ˜ of Xss indexed by In this way, we obtain a stratiﬁcation { ˜ = {γ˜ = (γ ) : γ ∈ {0, R1 , . . . , Rτ }} ∪ {γ˜ = (Rl , β) : 1 ≤ l ≤ τ and β ∈ Bl \{0}} ∪ {γ˜ = (Rl , β, γ1 , . . . , γt ) : t ≥ 1 and 1 ≤ l ≤ τ and β ∈ Bl \{0} (4.3) and (γ1 , . . . , γt ) ∈ ˜ (Rl ,β) \{0}}, ˜ γ˜ are given by where ˜ (Rl ,β) is deﬁned inductively as above, and the strata ˜ (0) = X s . If 1 ≤ l ≤ τ and β ∈ Bl \{0}, then ˜ (Rl ) = GZRs l

and

˜ (Rl ,β) = s (Rl ,β) ,

while if γ˜ = (Rl , β, γ1 , . . . , γt ), then ˜ (Rl ,β) . ˜ γ˜ = (γ1 ,...,γt )

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5 The reﬁned Morse stratiﬁcation In Section 2 a stratiﬁcation {γ : γ ∈ } of the set X ss of semistable points of X was deﬁned, and in Section 4 this stratiﬁcation was reﬁned to give a stratiﬁcation ˜ γ˜ : γ˜ ∈ } ˜ of X ss . Via the inductive description of the strata Sβ of the Morse {

stratiﬁcation of µ2 in terms of the semistable points of nonsingular subvarieties of X given in Proposition 1.1, we can use these stratiﬁcations of X ss to reﬁne the Morse stratiﬁcation. ˜ γ˜ : γ˜ ∈ } ˜ of X ss conFor simplicity we shall just discuss the stratiﬁcation { structed in Section 4. The construction of this stratiﬁcation can be applied for each β ∈ B to the action of Stab(β) on the nonsingular projective subvariety Zβ of X that appeared in Proposition 1.1(iv) (or more precisely to the action of the quotient Stab(β)/Tβc of Stab(β) by its complex subtorus Tβc which acts trivially on Zβ ) to give a stratiﬁcation ˜ [β] : γ˜ ∈ ˜ [β] } { γ˜

of Zβss by nonsingular Stab(β)-invariant subvarieties, with Zβs as the stratum indexed by (0). Since Sβ = GYβss satisﬁes (1.4) and we have a retraction pβ : Yβss → Zβss satisfying (1.5), we can stratify Sβ as the disjoint union of strata [β]

[β]

˜ )∼ ˜ ) Gpβ−1 ( = G ×Pβ pβ−1 ( γ˜ γ˜

(5.1)

for γ ∈ ˜ [β] . This gives us a stratiﬁcation ˜ {S˜β˜ : β˜ ∈ B} of X indexed by

B˜ = ˜ ∪

A

(5.2)

{β} × ˜ [β] ,

β∈B\{0}

˜ ˜ deﬁned as in Section 4 if β˜ ∈ , ˜ and if β˜ = (β1 , . . . , βt ), where where S˜β˜ = β β1 ∈ B\{0} and (β2 , . . . , βt ) ∈ ˜ [β] , then ˜ [β1 ] S˜β˜ = Gpβ−1 ( (β2 ,...,βt ) ). 1 ˜ reﬁnes the original stratiﬁcation {Sβ : β ∈ B} and This stratiﬁcation {S˜β˜ : β˜ ∈ B} has the following properties. Proposition 5.1. (i) Each stratum S˜β˜ is a G-invariant locally closed nonsingular subvariety of X. (ii) The unique open stratum S˜(0) is the set X s of stable points of X. ˜ then the closure S˜ ˜ in X (iii) There is a partial ordering > on B˜ such that if β˜ ∈ B, β ˜ of the stratum Sβ˜ satisﬁes A S˜β˜ ⊆ S˜γ˜ . γ˜ ≥β˜

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˜ ˜ and (iv) B˜ has a subset ˜ such that γ˜ < β˜ for all γ˜ ∈ ˜ and β˜ ∈ B\ A S˜γ˜ = X ss . γ˜ ∈˜

(v) If β˜ ∈ B˜ and β˜ = (0), then the stratum S˜β˜ can be described inductively in terms of the sets of stable points of certain nonsingular linear sections Z of X and projectivized normal bundles of nonsingular subvarieties of X, acted on by reductive subgroups of G and their quotients. Remark 5.2. If X is a compact Kähler manifold which has a Hamiltonian action of a compact group K with moment map µ : X → k∗ , then the Morse stratiﬁcation for µ2 can be reﬁned just as in Proposition 5.1. Even when X is symplectic but not Kähler we can construct a similar reﬁnement by choosing a suitable almost complex structure and Riemannian metric on X (see [10, 16]). Example 5.3. Consider the action of G = SL(2; C) and its maximal compact subgroup K = SU (2) on X = (P1 )n , with the moment map given by the center of gravity in R3 when P1 is identiﬁed suitably with the unit sphere in R3 and R3 is identiﬁed with the Lie algebra of SU (2). An element (x1 , . . . , xn ) of (P1 )n is semistable (respectively stable) for the action of G if and only if at most n/2 (respectively strictly fewer than n/2) of the points xj coincide anywhere on P1 . The Morse stratiﬁcation for the normsquare of the moment map on X has strata S0 = X ss and S2j −n for n/2 < j ≤ n. If n/2 < j ≤ n, then the elements of S2j −n correspond to sequences of n points on P1 such that exactly j of these points coincide somewhere on P1 , and S2j −n retracts equivariantly onto the subset of X, where j points coincide somewhere on P1 and the remaining n−j points coincide somewhere else on P1 , which is a single G-orbit with stabilizer C∗ , for j < n, and with stabiliser a Borel subgroup of G when j = n (see [9, Section 16.1] for more details). If n is odd, then semistability coincides with stability and the reﬁned stratiﬁcation coincides with the Morse stratiﬁcation of X. that n is even, so that semistability and stability do not coincide. The semistable elements of X which are ﬁxed by nontrivial connected reductive subgroups of G are those represented by sequences (x1 , . . . , xn ) of points of P1 such that there exist distinct p and q in P1 with exactly half of the points x1 , . . . , xn equal to p and the rest equal to q. They form n!/2((n/2)!)2 Gorbits, and their stabilisers are all conjugate to the maximal torus Tc = C∗ of G. These stabilisers act with weights 2 and −2, each with multiplicity (n/2) − 1, on ˜ the normals to the orbits. We obtain the partial desingularization X//G by blowing up X//G at the points corresponding to these orbits, or equivalently by blowing up X ss along these orbits, removing from the blowup the unstable points (which form the proper transform of the set of (x1 , . . . , xn ) ∈ X ss such that exactly half of the points x1 , . . . , xn coincide somewhere on P1 ) and ﬁnally quotienting by G. The ˜ of X thus has as its strata the set S˜(0) = X s of reﬁned stratiﬁcation {S˜β˜ : β˜ ∈ B} stable points, the set S˜(T ) consisting of points represented by sequences (x1 , . . . , xn ) in P1 such that there exist distinct p and q in P1 with exactly half of x1 , . . . , xn

Morse stratiﬁcation of the normsquare of the moment map

343

equal to p and the rest equal to q, the set S˜(T ,2) consisting of points represented by sequences (x1 , . . . , xn ) in P1 such that there exists p in P1 with exactly half of the points x1 , . . . , xn equal to p and the rest different from p and not all equal to each other, and ﬁnally the strata S2j −n (for n/2 < j ≤ n) of the Morse stratiﬁcation. Example 5.4. A very similar example is given by the action of G = SL(2; C) on X = Pn identiﬁed with the space of unordered sequences of n points in P1 ; that is, with the projectivized symmetric product P(S n (C2 )) (see [9, Section 16.2] for more details). The diagonal subgroup T ∼ = S 1 of K = SU (2) acts with weights n 2 n, n − 2, n − 4, . . . , 2 − n, −n on S (C ) = Cn+1 . An element [a0 , . . . , an ] of Pn corresponds to the n roots in P1 of the polynomial in one variable t with coefﬁcients a0 , . . . , an ; it is semistable (respectively stable) for the action of G if and only if at most n/2 (respectively strictly fewer than n/2) of these roots coincide anywhere on P1 , and the Morse stratiﬁcation for the normsquare of the moment map on X is essentially the same as in Example 5.3 above. Again when n = 2m is even, the semistable elements of X that are ﬁxed by nontrivial connected reductive subgroups of G are those represented by polynomials such that there exist distinct p and q in P1 with exactly half the roots equal to p and the rest equal to q. They form a single Gorbit, and the stabiliser C∗ acts with weights ±4, ±6, . . . , ±n = ±2m on the normal ˜ of X this time has as its strata the to the orbit. The reﬁned stratiﬁcation {S˜β˜ : β˜ ∈ B} s set S˜(0) = X of stable points, the set S˜(T ) represented by polynomials with exactly two distinct roots each with multiplicity m = n/2, for 2 ≤ k ≤ m the set S˜(T ,2k) represented by polynomials in the orbit of one of the form am t m + am+k t m+k + am+k+1 t m+k+1 + · · · + a2m t 2m , where am and am+k are nonzero, and ﬁnally the strata S2j −n (for n/2 < j ≤ n) of the Morse stratiﬁcation. Example 5.5. For a more complicated example consider the action of G = SL(3; C) and its maximal compact subgroup K = SU (3) on X = (P2 )n . A sequence (x1 , . . . , xn ) of points in P2 is semistable if and only if there is no projective line L in P2 such that |{j : xj ∈ L}| n > (5.3) 2 3 and no point p ∈ P2 such that |{j : xj = p}| >

n , 3

(5.4)

and is stable if we can replace > with ≥ in (5.3) and (5.4) (see for example [9] (16.5)). The stratiﬁcation {Sβ : β ∈ B} can be described as follows (see [9] Proposition 16.9). Any (x1 , . . . , xn ) ∈ (P2 )n which is not semistable determines a unique ﬂag 0 = M0 ⊂ M1 · · · ⊂ Ms = C3 in C3 with s = 2 or 3, such that if 1 ≤ i ≤ s, then

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k1 ks > ··· > , m1 ms where ki = |{j : xj ∈ Mi \Mi−1 }|

and mi = dim(Mi /Mi−1 );

moreover, if mi = 2, then those xj lying in Mi \Mi−1 determine a semistable element of (P1 )ki after projection into P(Mi /Mi−1 ) ∼ = P1 . Then (x1 , . . . , xn ) lies in the stratum labelled by the projection into Lie(SU (3)) of the vector k1 ks β= ,..., ∈ Lie(U (3)) m1 ms in which each ki /mi appears mi consecutive times. Thus B is the projection into the Lie algebra Lie(SU (3)) of / n n n 0 k k 2n , , , ,n − k :
2 2

2 2

2 2

where T1 = {(t, t, t −2 ) : t ∈ C∗ }. Here (x1 , . . . , xn ) belongs to the stratum S˜(k/2,k/2,n−k) if and only if there is a line L in P2 containing exactly k of the points xj and at most k/2−1 of these points coincide anywhere on L, while (x1 , . . . , xn ) belongs to S˜(k/2,k/2,n−k,T1 ) if exactly k/2 of these points coincide somewhere on L and the remaining k/2 coincide somewhere else, and (x1 , . . . , xn ) belongs to S˜(k/2,k/2,n−k,T1 ,3)

Morse stratiﬁcation of the normsquare of the moment map

345

if exactly k/2 of these points coincide somewhere on L and the remaining k/2 on L do not all coincide. If n − k is even, we have S(k, n−k , n−k ) = S˜(k, n−k , n−k ) S˜(k, n−k , n−k ,T2 ) S˜(k, n−k , n−k ,T2 ,3) , 2

2

2

2

2

2

2

2

where T2 = {(t −2 , t, t) : t ∈ C∗ }. Here (x1 , . . . , xn ) belongs to the stratum S˜(k,(n−k)/2,(n−k)/2) if and only if there is some point p in P2 where exactly k of the points xj coincide, and no line through p contains at least (n − k)/2 of the remaining points xj , while (x1 , . . . , xn ) belongs to S˜(k,(n−k)/2,(n−k)/2,T2 ) if there are lines L1 and L2 meeting at p and each containing exactly (n − k)/2 of the remaining points xj , and (x1 , . . . , xn ) belongs to S˜(k,(n−k)/2,(n−k)/2,T2 ,3) if there is a line L through p containing exactly (n − k)/2 of the remaining points xj but no other line through p contains at least (n − k)/2 of these remaining points. Finally, if n is divisible by 3, then we have X ss = Xs S˜(T ) {S˜(T ,β) : β ∈ BT \ {0}} S˜(T1 ) S˜(T1 ,3) S˜(T1 ,−3) , where T is the standard maximal torus of SU (3) and BT \ {0} can be identiﬁed with the set of four vectors 1 1 1 1 1 1 , 0, − , , −1 . , 1, − , − , (1, 0, −1) , 2 2 2 2 2 2 Here (x1 , . . . , xn ) belongs to S˜(T ) if there are three points p1 , p2 , p3 ∈ P2 such that exactly n/3 of the points xj coincide at pi for i = 1, 2, 3, while (x1 , . . . , xn ) belongs to S˜(T1 ) if there is a point p ∈ P2 and a line L not containing p such that exactly n/3 of the points xj coincide at p and the remaining 2n/3 lie on L with strictly fewer than n/3 coinciding anywhere on L. Also, (x1 , . . . , xn ) belongs to S˜(T1 ,3) if there is a line L in P2 containing exactly 2n/3 of the points xj and strictly fewer than n/3 points coincide anywhere on P2 , while (x1 , . . . , xn ) belongs to S˜(T1 ,−3) if there is a point p ∈ P2 where exactly n/3 of the points xj coincide, and strictly fewer than 2n/3 of the points xj lie on any line in P2 . Finally, if β ∈ BT and (x1 , . . . , xn ) ∈ S˜(T ,β) , then there is some p ∈ P2 and a line L through p such that exactly n/3 of the points xj coincide at p and exactly 2n/3 lie on L, and (a) when β = (1/2, 0, −1/2), then p and L are unique; (b) when β = (1/2, 1/2, −1), then L is unique but there is another point p on L where n/3 of the points xj coincide; (c) when β = (1, −1/2, −1/2), then p is unique but there is another line L through p containing exactly 2n/3 of the points xj ; (d) when β = (1, 0, −1), then there is another point p ∈ P2 \ L where exactly n/3 of the points xj coincide, but the n/3 points xj lying on L \ {p} do not all coincide.

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Note that this stratiﬁcation of X ss is a proper reﬁnement of the pullback to X ss of the inﬁnitesimal orbit type stratiﬁcation of X//G which is described, for example, in [16]. Example 5.6. We can generalise Examples 5.3 and 5.5 by considering the action of SL(m; C) on a product of Grassmannians X=

r 1

Grass(lj , Cm ),

j =1

where Grass(l, Cm ) denotes the Grassmannian of l-dimensional linear subspaces of Cm and we linearise the action by using the Plücker embedding. The Morse stratiﬁ˜ cation {Sβ : β ∈ B} is described in [9, Section 16.3], and the reﬁnement S˜β˜ : β˜ ∈ B} can be calculated by adapting the methods used in [13] (especially [13, Section 5]) to reﬁne the Yang–Mills stratiﬁcation.

6 Reﬁnements when G is abelian The reﬁnements considered so far of the Morse stratiﬁcation {Sβ : β ∈ B} for µ2 are unfortunately unlikely to be equivariantly perfect. When G = T c is abelian there is another way to reﬁne this stratiﬁcation which does lead to equivariantly perfect stratiﬁcations. If G is abelian, then there is a partial desingularisation X// G of X//G obtained by perturbing the linearisation of the action of G on X, or equivalently by replacing the moment map µ : X → k∗ by µ − , where ∈ k∗ is a generic constant close to 0. Since G is abelian the coadjoint action of K = T on k∗ = t∗ is trivial and µ − is an equivariant moment map for the action of K on X. Recall from Section 1 that the Morse stratiﬁcation {Sβ : β ∈ B} for µ2 is indexed by the closest points to 0 in the convex hulls of nonempty subsets of the set of weights {α0 , . . . , αn }, and x = [x0 : · · · : xn ] ∈ X ⊆ Pn lies in the stratum Sβ indexed by the closest point β to 0 in the convex hull of {αj : xj = 0}. Similarly, the Morse strata {Sβ : β ∈ B } for µ − 2 correspond to the closest points to in such convex hulls. More precisely, x = [x0 : · · · : xn ] ∈ X lies in the stratum Sβ indexed by the closest point β to 0 in the convex hull of {αj − : xj = 0}; then β + is the closest point to in the convex hull of {αj : xj = 0}. If β ∈ B does not lie in the convex hull of a proper subset of {αj : αj .β = β2 }, then for sufﬁciently small , we have Sβ = Sβ()

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347

for some small perturbation β() of β, where β() + is the closest point to of the convex hull of {αj : αj .β = β2 }. Otherwise for sufﬁciently small the stratum Sβ in the Morse stratiﬁcation for µ2 is the union of Sβ() and other strata in the 2 Morse stratiﬁcation for µ − , which correspond to the closest points to of those subsets of {αj : αj .β = β2 } whose closest point to 0 is β. Moreover, if β ∈ B and is chosen generically, we can assume that β does not lie in the convex hull of any proper subset of {αj : (αj − ).β = β2 }. This means that every point of Zβ = {x ∈ X : xj = 0 unless (αj − ).β = β2 } which is semistable for the action of the subgroup of G whose Lie algebra is spanned by {αj − − β : (αj − ).β = β2 } is also stable for the action of this subgroup. Moreover, the set of semistable points for the action of this subgroup is the same as the set Zβ,ss of semistable points of Zβ under the action of the subgroup of G whose Lie algebra is the complexiﬁcation of the orthogonal complement to β in t (see Remark 1.3). Thus we have a reﬁnement of the Morse stratiﬁcation for µ2 which is both equivariantly perfect and has the property that all strata can be described in terms of the sets of semistable points of nonsingular closed subvarieties Zβ of X under suitable linear actions of reductive subgroups of G for which semistability coincides with stability. Proposition 6.1. If G = T c is abelian and ∈ k∗ = t∗ is chosen generically and sufﬁciently close to 0, then the Morse stratiﬁcation {Sβ : β ∈ B } for µ − 2 is an equivariantly perfect reﬁnement of the Morse stratiﬁcation {Sβ : β ∈ B} for µ2 . In addition if β ∈ B , then we have Sβ = Yβ,ss = pβ−1 (Zβ,ss ) and HG∗ (Sβ ) = HG∗ (Zβ,ss ), where Zβ = {x ∈ X : xj = 0 unless (αj − ).β = β2 }, Yβ = {x ∈ X : xj = 0 if (αj − ).β < β2 and xj = 0 for some j such that (αj − ).β = β2 }, and pβ (x) = lim exp(−itβ)x. t→∞

Furthermore, Zβ,ss

is the set of semistable points for the action on Zβ of the subgroup of G whose Lie algebra is spanned by

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{αj − − β : (αj − ).β = β2 }, and every semistable point for this action is stable. Remark 6.2. When G is abelian and X ss = Xs we can obtain the same result by perturbing the inner product on k = t instead of perturbing the moment map from µ to µ − . Remark 6.3. Yet another possible procedure when G = (C∗ )r is abelian is to use induction on r to reduce to the simple case when G = C∗ . Example 6.4. Consider the action of the maximal torus T c = {(t, t −1 ) : t ∈ C∗ } ∼ = C∗ of G = SL(2; C) acting on X = Pn or X = (P1 )n as in Examples 5.3 and 5.5. An element of X represented by a sequence (x1 , . . . , xn ) of points of P1 is semistable (respectively stable) for T c if and only if at most n/2 (respectively strictly fewer than n/2) of the points xj coincide at 0 or ∞, and the Morse stratiﬁcation {SβT : β ∈ B T } for the normsquare of the moment map µT : X → t is indexed by {0} ∪ {2j − n : 0 ≤ j ≤ n} T with (x1 , . . . , xn ) representing a point of S2j −n when exactly j of the points xi coincide at 0 if j > n/2, and exactly n − j of the points xi coincide at ∞ if j < n/2. When n is odd this stratiﬁcation is unchanged if we perturb the linearisation slightly. When n is even, the only change is that S0T becomes two strata: the subset represented by (x1 , . . . , xn ) with exactly n/2 of the points coinciding at 0 (or at ∞, depending on the sign of the perturbation), and its complement in S0T . This contrasts with the reﬁnement {S˜ T˜ : β˜ ∈ B˜ T } in which S0T is subdivided into three strata when X = (P1 )n β and more when X = Pn (see Examples 5.3 and 5.5).

Example 6.5. When the maximal torus T c ∼ = (C∗ )2 of SL(3; C) acts on X = (P2 )n , T T the Morse stratiﬁcation {Sβ : β ∈ B } for µT 2 and its reﬁnement {S˜ T˜ : β˜ ∈ B˜ T } β deﬁned as in Section 5 can be described in a way closely analogous to the stratiﬁcations ˜ associated to the action of G = SL(3; C) which {Sβ : β ∈ B} and {S˜β˜ : β˜ ∈ B} were described in Example 5.6; the difference is that the points p and lines L which appear in the description are [1 : 0 : 0], [0 : 1 : 0] or [0 : 0 : 1] or one of the lines joining these points. A generic perturbation of the linearisation provides a coarser reﬁnement of {SβT : β ∈ B T } than the stratiﬁcation {S˜ T˜ : β˜ ∈ B˜ T }. If k or n − k is β

even, then the strata SβT with indices β in the Weyl group orbit of (k/2, k/2, n − k) or (k, (n − k)/2, (n − k)/2) decompose as the disjoint union of two reﬁned strata instead of the three in the reﬁnement {S˜ T˜ : β˜ ∈ B˜ T }, in a way which is directly analogous β

to the decomposition of S0T in Example 6.4. Similarly, if n is divisible by 3, then S0T decomposes into a smaller number of strata than in the stratiﬁcation {S˜ T˜ : β˜ ∈ B˜ T }; β in particular the analogues of the strata S˜(T ) and S˜(T1 ) described in Example 5.6 are amalgamated with higher-dimensional strata.

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349

7 Reduction to a maximal torus The Morse stratiﬁcation {Sβ : β ∈ B} of the normsquare of the moment map is useful for studying the cohomology of the GIT quotient X//G because it is equivariantly perfect over the rationals; that is, if λ(β) is the real codimension of Sβ in X and if Uβ is the open subset A Uβ = Sβ ∪ Sβ , β <β

which contains Sβ as a closed subset, then the Thom–Gysin long exact sequences ∗−λ(β)

· · · → HG

T Gβ

(Sβ ) → HG∗ (Uβ ) → HG∗ (Uβ \ Sβ ) → · · ·

break up into short exact sequences ∗−λ(β)

0 → HG

T Gβ

(Sβ ) → HG∗ (Uβ ) → HG∗ (Uβ \ Sβ ) → 0

of equivariant cohomology with rational coefﬁcients. Remark 7.1. This happens because the composition of the Thom–Gysin map T Gβ with the restriction map HG∗ (Uβ ) → HG∗ (Sβ ) is given by multiplication by the equivariant Euler class eβ of the normal bundle to Sβ in Uβ , and it follows from a criterion of Atiyah and Bott [1, Section 13] that eβ is not a zero-divisor in HG∗ (Sβ ). Note that the restriction maps from HG∗ (X) to HG∗ (X ss ) and also to HG∗ (Uβ ) for β ∈ B \ {0} are compositions of restriction maps from HG∗ (Uβ ) to HG∗ (Uβ \ Sβ ) which are all surjective. In particular the cohomology ring HG∗ (X ss (which in the good case when Xss = X s is isomorphic to the rational cohomology ring of the geometric invariant theoretic quotient X//G or equivalently the Marsden–Weinstein reduction µ−1 (0)/K) is isomorphic to the quotient of HG∗ (X) by the kernel of the restriction map ρ : HG∗ (X) → HG∗ (X ss ). ˜ described in Section 5 of {Sβ : β ∈ B} Unfortunately the reﬁnement {S˜β˜ : β˜ ∈ B} is not in general equivariantly perfect. It is still the case that we have Thom–Gysin long exact sequences for equivariant cohomology, and the kernels of the restriction maps from HG∗ (X) to HG∗ (X ss ) and to HG∗ (X s ) can be described in terms of the images of the associated Thom–Gysin maps, but the description is not as clean as in the equivariantly perfect case, and the restriction map to HG∗ (X s ) is not necessarily surjective (see Example 7.16 below). However, when G is abelian, the alternative reﬁnement {Sβ : β ∈ B } described in Section 6 is equivariantly perfect. We can exploit this fact even when G is not abelian by making use of the close relationship between the restriction maps ρ : HG∗ (X) → HG∗ (X ss ) and

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ρT : HT∗ (X) → HT∗ (X ss,T ), where T c is a maximal torus of G (we can assume that it is the complexiﬁcation of T = T c ∩ K which is a maximal torus of K) and X ss,T is the set of T c -semistable points of X. Therefore, in this ﬁnal section, we shall describe this relationship. The kernel of ρ can be described in several different ways. The ﬁrst follows immediately from the fact that the stratiﬁcation {Sβ : β ∈ B} is equivariantly perfect. ∗−λ(β) Lemma 7.2. For β ∈ B \ {0} let TC Gβ : HG (Sβ ) → HG∗ (X) be any lift to X of ∗−λ(β) (Sβ ) → HG∗ (Uβ ). Then the Thom–Gysin map T Gβ : HG D ker ρ = imTC Gβ . β∈B\{0}

Remark 7.3. Note that such lifts always exist by Remark 7.1. Another closely related description is given in [3, 4, 12]. Lemma 7.4. Let R be a subset of ker ρ such that for every β ∈ B \ {0} and every η ∈ HG∗ (Sβ ) there exists ζ ∈ R such that ζ |Sγ = 0 unless γ ≥ β and ζ |Sβ = ηeβ , where eβ is the equivariant Euler class of the normal bundle to Sβ . Then R spans ker ρ. In the good case when Xss = Xs a rather different description follows from the formulas for the intersection pairings in H ∗ (X//G) given in [8] (see also [7, 14, 15, 20]). j

Lemma 7.5. If Xss = Xs and η ∈ HG (X), then η ∈ ker ρ if and only if ' * (ηζ )| F res D2 =0 eµF eF F F ∈F

for all ζ ∈ HGk (X) with j + k = dimR (X//G). Here F is the set of connected components F of the ﬁxed point set of T in X and eF ∈ HT∗ (X) is the T -equivariant Euler class of the normal bundle to F in X. Also µF : t → R denotes the linear function on t given by the constant value in t∗ taken by the moment map µ on F ∈ F, and the polynomial D : t → R is deﬁned by 1 D= γ, γ >0

where γ runs over the positive roots of K.

Morse stratiﬁcation of the normsquare of the moment map

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Proof. This follows immediately from Poincaré duality for the orbifold X//G and the surjectivity of ρ : HG∗ (X) → HG∗ (X ss ) ∼ = H ∗ (X//G) together with [8, Theorem 8.1], which tells us that, up to multiplication by a nonzero constant, the formula in Lemma 7.5 is the intersection pairing ρ(η)ρ(ζ ) X//G

of ρ(η) and ρ(ζ ) in X//G. Remark 7.6. The multivariable residue res which appears in Lemma 7.5 above is a linear map deﬁned on a class of meromorphic differential forms on t ⊗ C. In order to apply it to the individual terms in the residue formula it is necessary to make some choices which do not affect the residue of the whole sum. Once the choices have been made, many of the terms in the sum contribute zero and the formula can be rewritten as a sum over a subset F+ of the set F of components of the ﬁxed point set X T , consisting of those F ∈ F on which the constant value taken by µT lies in a certain cone with its vertex at 0. When dim T = 1 and t is identiﬁed with R, we can take F+ = {F ∈ F : µT (F ) > 0}, and if we replace F with F+ in the formula in Lemma 7.5, then we can interpret res as the usual residue at 0 of a rational function in one variable. Remark 7.7. When G is abelian we have analogues of Lemmas 7.2 and 7.4 for the reﬁnement {Sβ : β ∈ B } of the stratiﬁcation {Sβ : β ∈ B} obtained by perturbing the moment map µ to µ − (see Section 6). We also have the following description of ker ρ due to Tolman and Weitsman [19]. We can use these results even when G is not abelian via Lemma 7.10 below. Lemma 7.8. When G = T c is abelian and X ss = Xs , then ker ρ = {η ∈ HT∗ (X) : η|F ∩Xξ = 0 for all F ∈ F}, ξ ∈t

where Xξ = {x ∈ X : µ(x)(ξ ) ≤ 0} if ξ ∈ t. Recall that the Weyl group W of K acts faithfully on the Lie algebra t of T and is generated by reﬂections. For any w ∈ W we denote by (−1)w the determinant of w regarded as an automorphism of t. If W acts on a module M, then M W = {m ∈ M : wm = m for all w ∈ W }

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is the set of W -invariants in M and M antiW = {m ∈ M : wm = (−1)w m for all w ∈ W } is the set of anti-invariants for the action of W on M. There is a natural identiﬁcation of HG∗ (X) with [HT∗ (X)]W , which in the case when X is a point becomes a natural identiﬁcation of H ∗ (BG) with [H ∗ (BT )]W , where H ∗ (BT ) is the Q-algebra of polynomial functions on t. There is (up to sign) one fundamental anti-invariant in H ∗ (BT ), which is the product D of the positive roots of K, and we have the following well-known facts (see, for example, [5, Lemma 1.2]). Lemma 7.9. (i) [H ∗ (BT )]antiW is a free [H ∗ (BT )]W -module of rank one generated by D. (ii) [H ∗ (BT )]antiW is a direct summand of H ∗ (BT ) as an [H ∗ (BT )]W -module, with splitting given by 1 η → (−1)w wη. |W | w∈W

Note that HT∗ (X) has analogous properties since it is isomorphic to H ∗ (BT ) ∗ H (X) as an H ∗ (BT )-module (see [9, Proposition 5.8]), although not as a ring.

⊗

Lemma 7.10. Suppose that X ss = X s . If η ∈ HG∗ (X) ∼ = [HT∗ (X)]W , then the following are equivalent: (i) η ∈ ker ρ; (ii) ηD ∈ ker ρT ; (iii) ηD2 ∈ ker ρT . Moreover, multiplication by D induces a bijection ker ρ → ker ρT ∩ [HT∗ (X)]antiW with inverse η →

(7.1)

1 (−1)w wη. D|W | w∈W

Remark 7.11. The corresponding results are true when X is a compact symplectic manifold with a Hamiltonian action of a compact group K, provided that 0 is a regular value of the moment map. Remark 7.12. Martin [14, 15] gave a direct proof of the equivalence (i) ⇔ (iii) when X ss = X s , while Guillemin and Kalkman [7] observed that it follows immediately from Lemma 7.5. The equivalence (i) ⇔ (ii) and the bijection (7.1) goes back in essence at least to Ellingsrud and Strømme [5, Section 4]. Direct proofs of the equivalences (i) ⇔ (ii) and (ii) ⇔ (iii) and the bijection (7.1) are given below; the assumption that X ss = Xs is not needed for the equivalence (i) ⇔ (ii) or for the existence of the bijection (7.1), and, in fact, the most convenient assumption for the proof below

Morse stratiﬁcation of the normsquare of the moment map

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of the equivalence (ii) ⇔ (iii) is not that X ss = X s but rather the corresponding assumption Xss,T = Xs,T for the action of the maximal torus. Thus Lemma 7.10 is true when either X ss = X s or X ss,T = X s,T , and if (iii) is omitted, then it is true without either of these hypotheses. Note also that the direct proofs of the equivalences (i) ⇔ (iii) and (ii) ⇔ (iii) do not require X to be compact; it sufﬁces that µ−1 (0) or equivalently X//G should be compact, and for example they can be applied to the study of moduli spaces of bundles over a compact Riemann surface (see [8]). Before giving a proof of Lemma 7.10 we shall relate the stratiﬁcations {Sβ : β ∈ B} and {SβT : β ∈ BT } and the associated Thom–Gysin maps T Gβ and T GTβ for the actions of G and its maximal torus T c on X. Recall from Section 1 that the indexing set BT consists of the closest points to 0 of the convex hulls in t∗ ∼ = t of the weights α0 , . . . , αn for the linear action of T c on X ⊆ Pn , and that B = BT ∩ t+

and

BT = {wβ : β ∈ B, w ∈ W },

where t+ is a positive Weyl chamber in t. If β ∈ B, then in the notation of Section 1, we have Sβ = GYβss and SβT = Yβss,T , where Yβss,T = pβ−1 (Zβss,T ) and Zβss,T is the set of semistable points for an appropriate linearisation of the action of T c on Zβ . There are induced isomorphisms ∗ HG∗ (Sβ ) ∼ (Zβss ) = HStab(β)

ss,T and HT∗ (SβT ) ∼ = HT∗ (Zβ )

with surjective restriction maps ∗ ∗ HStab(β) (Zβ ) ∼ (Zβss ) = [HT∗ (Zβ )]Wβ → HStab(β)

and HT∗ (Zβ ) → HT∗ (Zβss,T ), where Wβ is the Weyl group of Stab(β). Notice that the boundary S¯βT \ SβT of SβT is contained in A A SβT ⊆ Sβ . β ∈BT :β >β

β ∈B:β >β

Let Dβ ∈ H ∗ (BT ) be the product of the positive roots of Stab(β), and if β ∈ B let T TC Gβ be any lift to HT∗ (X) of the Thom–Gysin map T GTβ associated to the inclusion of SβT in A UβT = SβT ∪ SβT . β ∈BT :β <β ss,T If η ∈ HT∗ (Zβ ), then η restricts to an element of HT∗ (Zβss,T ) ∼ = HT∗ (Yβ ) = HT∗ (SβT ) and

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T Gβ (η) (−1)w w Dβ TC

w∈W

is a W -anti-invariant element of HT∗ (X). Thus we have a well-deﬁned element

1 T Gβ (η) (−1)w w Dβ TC D w∈W

of [HT∗ (X)]W ∼ = HG∗ (X) (see Lemma 7.9), whose restriction to A U β = Sβ ∪ Sβ β ∈B:β <β T is independent of the choice of lift TC Gβ of the Thom–Gysin map T GTβ and thus can be expressed as

1 1 (−1)w w Dβ T GTβ (η) = (−1)w Dwβ T GTwβ (wη). D D w∈W

w∈W

∗ Lemma 7.13. If β ∈ B and η ∈ HStab(β) (Zβ ) ∼ = [HT∗ (Zβ )]Wβ , then

T Gβ (η) =

1 1 (−1)w w Dβ T GTβ (η) = (−1)w Dwβ T GTwβ (wη). |W |D |W |D w∈W

w∈W

∗ ∗ Proof. Note that η ∈ HStab(β) (Zβ ) represents an element of HStab(β) (Zβss ) ∼ = HG∗ (Sβ ) ss by restriction from Zβ to Zβ . Since the equivariant Euler class eβ of the normal bundle Nβ to Sβ is not a zero-divisor in HG∗ (Sβ ), it sufﬁces to show that

1 1 (−1)w w Dβ T GTβ (η) = (−1)w Dwβ T GTwβ (wη) |W |D |W |D w∈W

w∈W

A

restricts to 0 on

Sβ

β =β,β ≤β ∗ and restricts to ηeβ on Sβ . Since HStab(β) (Zβss ) ∼ = HG∗ (Sβ ) it sufﬁces to check that the ss restriction to Zβ is ηeβ . But T = w(S T ) ⊆ S ∪ Swβ β β

A β >β

so T GTwβ (wη) restricts to 0 on A β =β,β ≤β

Sβ

Sβ

Morse stratiﬁcation of the normsquare of the moment map

355

as required. Also the composition of T GTβ with restriction to SβT is multiplication by the equivariant Euler class eβT of the normal bundle NβT to SβT , and Zβss ⊆ Zβss,T ⊆ SβT . Hence the restriction of Dβ T GTβ (η) to Zβss is Dβ ηeβT . Now Sβ ∼ = G ×Pβ Yβss

and

SβT = Yβss,T ,

where Yβss is an open subset of Yβss,T , so their normal bundles Nβ and NβT are related by NβT |Yβss ∼ = g/pβ ⊕ Nβ |Yβss , where g and pβ are the Lie algebras of G and Pβ . Therefore, on restriction to Yβss (or to Zβss ), we have Deβ eβT = Dβ ∗ so the restriction of Dβ T GTβ (η) to HStab(β) (Zβss ) is

Dβ ηeβT = Dηeβ . Hence the restriction to HG∗ (Sβ ) of

1 1 (−1)w w Dβ T GTβ (η) = (−1)w Dwβ T GTwβ (wη) |W |D |W |D w∈W

is

w∈W

1 (−1)w w Dηeβ ) = ηeβ |W |D w∈W

as required, since eβ and η ∈ HG∗ (Sβ ) are W -invariant. Proof of Lemma 7.10. First recall from Lemma 7.9 that the map p : [HT∗ (X)]antiW → [HT∗ (X)]W ∼ = HG∗ (X) deﬁned by p(ζ ) =

1 (−1)w wζ |W |D w∈W

is a bijection whose inverse is given by multiplication by D, since if ζ is W -invariant, then p(Dζ ) = ζ , and if ζ is anti-invariant, then Dp(ζ ) = ζ (see [5, (4.3)]). Suppose that η ∈ HG∗ (X). If η ∈ ker ρ, then by Lemma 7.2 we can write η=

β∈B\{0}

TC Gβ (ηβ )

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F. Kirwan

∗ ∗ for some ηβ ∈ HStab(β) (Zβ ) representing an element of HG∗ (Sβ ) ∼ (Zβss ), = HStab(β) ∗−λ(β) (Sβ ) → HG∗ (X) is any lift to X of the Thom–Gysin map where TC Gβ : HG ∗−λ(β) (Sβ ) → HG∗ (Uβ ). By Lemma 7.13 we can choose TC Gβ so that T G β : HG

TC Gβ (ηβ ) =

1 (−1)w Dwβ T GTwβ (wηβ ), |W |D w∈W

where TC Gβ is a lift of the Thom–Gysin map T GTβ and the choice of these lifts TC Gβ respects the action of W . Then T

Dη =

T

β∈B\{0}

1 (−1)w Dwβ T GTwβ (wηβ ) ∈ |W | w∈W

D

imTC Gβ = ker ρT . T

β∈BT \{0}

Conversely, suppose that Dη ∈ ker ρT . Then by Lemma 7.2 T Dη = TC Gβ (ηβ )

(7.2)

β∈BT \{0}

∼ H ∗ (Z ss,T ), where for some ηβ ∈ HT∗ (Zβ ) representing an element of HT∗ (SβT ) = β T T TC Gβ is any lift to HG∗ (X) of the Thom–Gysin map T GTβ . But η is W -invariant, so Dη is anti-invariant, and so Dη =

1 (−1)w1 w1 (Dη) |W | w1 ∈W

1 = |W |

β∈BT \{0}

Gw1 β (w1 ηβ ) w1 ∈ W (−1)w1 TC T

if the lifts TC Gβ are chosen to respect the action of W . We have T

BT = {wβ : w ∈ W, β ∈ B}, and if w ∈ W and β ∈ B, then wβ = β if and only if w ∈ Wβ . Thus Dη =

1 |W |

w1 ∈ W

β∈BT \{0} w2 ∈W

(−1)w1 C T T Gw1 w2 β (w1 ηw2 β ) |Wβ |

(−1)w (−1)w2 C T T Gwβ (ww2−1 ηw2 β ) |Wβ | β∈BT \{0} w2 ∈W 1 (−1)w2 −1 T = Gβ w2 ∈ W w2 ηw2 β (−1)w w TC . |W | |Wβ |

=

1 |W |

β∈BT \{0} w∈W

If w˜ ∈ Wβ , then

w∈W

Morse stratiﬁcation of the normsquare of the moment map

⎛ w˜ ⎝

357

⎞

(−1)w2 (−1)w2 w2−1 ηw2 β ⎠ = ww ˜ 2−1 ηw2 β |Wβ | |Wβ |

w2 ∈W

w2 ∈W

=

(−1)w3 (−1)w˜ w3−1 ηw3 wβ ˜ |Wβ |

w3 ∈W

= (−1)w˜

(−1)w3 w−1 ηw3 β . |Wβ | 3

w3 ∈W

Since elements of HT∗ (Zβ ) which are anti-invariant under the action of Wβ are multiples of Dβ , it follows that (−1)w2 w−1 ηw2 β = Dβ ζβ |Wβ | 2

w2 ∈W

∗ for some ζβ ∈ [HT∗ (Zβ )]Wβ ∼ (Zβ ), and hence = HStab(β)

Dη =

1 |W |

T Gβ (ζβ ) (−1)w w Dβ TC

β∈BT \{0} w∈W

so that η∈

D

imTC Gβ = ker ρ

β∈B\{0}

by Lemmas 7.2 and 7.13. This proves the equivalence (i) ⇔ (ii), and the same argument shows that the bijection HG∗ (X) → [HT∗ (X)]antiW given by multiplication by D restricts to a bijection from ker ρ to ker ρT ∩[HT∗ (X)]antiW . The observation that the equivalence (i) ⇔ (iii) follows directly from Lemma 7.5 when Xss = Xs now completes the proof of Lemma 7.10. However, it is also easy to show directly that if X ss,T = Xs,T , then (ii) ⇔ (iii) for any η ∈ HG∗ (X) ∼ = [HT∗ (X)]W ; that is, Dη ∈ ker ρT if and only if D2 η ∈ ker ρT . This follows from Lemma 7.5 applied to the action of T c , together with Poincaré duality on X//T c and the surjectivity of ρT , as Dη ∈ ker ρT ⇔ ρT (Dη)ρT (ζ ) = 0 for all ζ ∈ HT∗ (X). X//T c

Since Dη is anti-invariant, this holds for all ζ ∈ HT∗ (X) if and only if it holds for all anti-invariant ζ ; that is, for all ζ of the form Dξ , where ξ ∈ [HT∗ (X)]W . Thus Dη ∈ ker ρT ⇔ 0 = ρT (Dη)ρT (Dξ ) = ρT (D2 ηξ ) X//T c

X//T c

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F. Kirwan

for all invariant ξ ∈ HT∗ (X), or equivalently (since D2 η is invariant) for all ξ ∈ HT∗ (X). But by Poincaré duality again we have ρT (D2 ηξ ) = 0 X//T c

for all ξ ∈ HT∗ (X) if and only if ρT (D2 η) = 0, as required. Remark 7.14. These ideas are used in [4] to obtain a complete set of relations between the standard generators of the moduli space M(n, d) of stable holomorphic vector bundles of rank n and degree d over a ﬁxed compact Riemann surface of genus g ≥ 2 when n and d are coprime. There the role of X//G is played by the moduli space M(n, d), and the role of X//T c is played by the corresponding moduli space of parabolic bundles where the parabolic structure is associated to a full ﬂag. The generic perturbation of the linearisation used in Section 6 to obtain a reﬁned stratiﬁcation is played by a generic perturbation of the parabolic weights. Example 7.15. Suppose that X = Pn and that as usual the maximal torus T c of G acts with weights α0 , . . . , αn . In this case the closure S¯βT = Yβ of any T c -stratum SβT is a linear subspace of Pn and hence is nonsingular. Thus there is an obvious choice T of lift TC Gβ to X of the Thom–Gysin map T GTβ which is given by the Thom–Gysin map associated to the inclusion of S¯βT in X. We have HT∗ (X) ∼ = H ∗ (BT )[ζ ]/I, where I is the ideal in the polynomial ring H ∗ (BT )[ζ ] generated by the polynomial (ζ + α0 ) · · · (ζ + αn ), while HT∗ (S¯βT ) ∼ = H ∗ (BT )[ζ ]/Iβ , where Iβ is the ideal generated by 1

(ζ + αj ),

αj .β≥β2

and

1

T TC Gβ (Iβ + η) = I + η αj

1

where

(ζ + αj ),

.β<β2

(ζ + αj )

αj .β<β2

represents the equivariant Euler class eβT of the normal bundle to S¯βT in X. Also, HT∗ (Zβ ) ∼ = H ∗ (BT )[ζ ]/Jβ , where Jβ is the ideal generated by

Morse stratiﬁcation of the normsquare of the moment map

1

359

(ζ + αj ).

αj .β=β2

Hence by Lemma 7.13 there are lifts TC Gβ to X of the Thom–Gysin maps ∗−λ(β) ∗−λ(β) T Gβ : HStab(β) (Zβss ) ∼ (Sβ ) → HG∗ (Uβ ) = HG

represented by ⎛ 1 TC Gβ (Jβ + η) = I + (−1)w w ⎝Dβ η |W |D w∈W

1

⎞ (ζ + αj )⎠ ,

αj .β<β2

and by Lemma 7.2 the equivariant cohomology ring HG∗ (X ss ) is isomorphic to the quotient of the polynomial ring H ∗ (BT )[ζ ] by the ideal generated by (ζ +α0 ) · · · (ζ + αn ) and all polynomials in ζ with coefﬁcients in H ∗ (BT ) of the form ⎛ ⎞ 1 1 (−1)w w ⎝Dβ η (ζ + αj )⎠ |W |D 2 w∈W

αj .β<β

for some β ∈ B \ {0} and η ∈ H ∗ (BT )[ζ ]. When G = SL(2; C) acts on X = Pn as in Example 5.4, the weights are nα, (n − 2)α, . . . , −nα, where α is a basis vector for t and B = {(2j − n)α : j >

n } ∪ {0}. 2

Moreover, |W | = 2 and D = 2α, and if β = (2j − n)α ∈ B \ {0}, then Dβ = 1 and TC Gβ sends Jβ + η(ζ, α) to ⎛ ⎞ 1 1 1 ⎝ I+ η(ζ, α) (ζ + (n − 2k)α) − η(ζ, −α) (ζ − (n − 2k)α)⎠ . 4α k>j

k>j

Thus when n is odd H ∗ (X//G) is generated as a Q-algebra by ζ and α with relations given by ⎛ ⎞ 1 1 1⎝ η(ζ, α) (ζ + (n − 2k)α) − η(ζ, −α) (ζ − (n − 2k)α)⎠ α k>j

for all polynomials η in ζ and α.

k>j

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F. Kirwan

Example 7.16. When G = SL(2; C) acts on X = (P1 )n the stratiﬁcation {S˜β˜ : β˜ ∈ ˜ is described in Example 5.3; it differs from the Morse stratiﬁcation {Sβ : β ∈ B} B} for µ2 only in that when n is even the open stratum S0 = Xss is decomposed into the union of three strata, which are S˜(0) = X s together with S˜(T ) and S˜(T ,2) . Here H ∗ (BT ) ∼ = Q[α], where α has degree two and the nontrivial element of the Weyl group sends α to −α, while HT∗ (X) is generated by n + 1 elements ζ1 , . . . , ζn , α of degree two subject to the relations (ζ1 )2 = · · · = (ζn )2 = α 2

(7.3)

and HG∗ (X) is generated by ζ1 , . . . , ζn and α 2 subject to the same relations. The connected components SJ of the strata Sβ for β ∈ B \ {0} are indexed by subsets J of {1, . . . , n} of size |J | > n/2, and their elements are sequences (x1 , . . . , xn ) ∈ (P1 )n for which there is some p ∈ P1 satisfying xj = p if and only if j ∈ J . The connected components SJT of the T -strata SβT are deﬁned in the same way with p = 0 or p = ∞. We have SJT ∼ = (P1 )n−|J | and if {1, . . . , n} \ J = {i1 , . . . , in−|J | }, then the associated Thom–Gysin map T ∗−2|J | T TC GJ : HT (SJ ) → HT∗ (X)

sends a polynomial p(ζi1 , . . . , ζin−|J | , α) ∈ HT∗ ((P1 )n−|J | ) to p(ζi1 , . . . , ζin−|J | , α)

1

(ζj + α) ∈ HT∗ ((P1 )n ).

j ∈J

Thus by Lemma 7.13 a lift ∗−2|J |+2 TC GJ : HG (SJ ) ∼ = H ∗−2|J |+2 (BT ) → HG∗ (X)

of the Thom–Gysin map T GJ is given by ⎞ ⎛ 1 1 1 ⎝ TC GJ (p(α)) = p(α) (ζj + α) − p(−α) (ζj − α)⎠ . 4α j ∈J

j ∈J

It follows that HG∗ (X ss ) is generated by ζ1 , . . . , ζn and α 2 subject to the relations (7.3) and ⎞ ⎛ 1 1 1 1 1⎝ (ζj + α) − (ζj − α)⎠ = 0 = (ζj + α) + (ζj − α) (7.4) α j ∈J

j ∈J

j ∈J

j ∈J

for all subsets J of {1, . . . , n} with |J | > n/2. When n is even, the components of S˜(T ) are indexed by partitions of {1, . . . , n} into J1 J2 , where |J1 | = |J2 | = n/2, and their elements are sequences (x1 , . . . , xn ) ∈

Morse stratiﬁcation of the normsquare of the moment map

361

(P1 )n for which there are p1 = p2 in P1 satisfying xj = p1 if j ∈ J1 and xj = p2 if j ∈ J2 . The components of S˜(T ,2) are indexed by subsets J1 of {1, . . . , n} with |J1 | = n/2, and their elements are sequences (x1 , . . . , xn ) ∈ (P1 )n for which there is some p1 ∈ P1 satisfying xj = p1 if and only if j ∈ J1 , but no p2 ∈ P1 satisfying xj = p2 if j ∈ {1, . . . , n} \ J . We have HG∗ (S{J1 ,J2 } ) ∼ = H ∗ (BT ) ∼ = HG∗ (SJ1 ∪ S{J1 ,J2 } ) and the restriction map from HG∗ (SJ1 ∪ S{J1 ,J2 } ) to HG∗ (SJ1 ) is surjective. Lifts to HG∗ (X) of the Thom–Gysin maps T G{J1 ,J2 } : HG∗−2n+4 (S{J1 ,J2 } ) → HG∗ (X ss )

(7.5)

and T GJ1 : HG∗−n+2 (SJ1 ) → HG∗ (X ss \ S{J1 ,J2 } )

(7.6)

are given by T G{J1 ,J2 } (p(α)) ⎞ ⎛ 1 1 1 ⎝ n/2−1 n/2−1 = p(α)α (ζj + α) − p(−α)(−α) (ζj − α)⎠ 4α j ∈J1

and

j ∈J1

⎛ ⎞ 1 1 1 ⎝ T GJ1 (p(α)) = p(α) (ζj + α) − p(−α) (ζj − α)⎠ . 4α j ∈J1

j ∈J

Thus the kernel of the restriction map HG∗ (X) → HG∗ (X s ) is generated by the relations (7.4) for all subsets J of {1, . . . , n} with |J | ≥ n/2. Note, however, that although the Thom–Gysin maps (7.5) and (7.6) are injective, the Thom–Gysin map associated to the inclusion of SJ2 in X ss \ (S{J1 ,J2 } ∪ SJ1 ) (which is just the composition of T GJ2 as at (7.6) with restriction from X ss \ S{J1 ,J2 } to Xss \ (S{J1 ,J2 } ∪ SJ1 )) is not injective, and the restriction map from HG∗ (X) to HG∗ (X s ) ∼ = H ∗ (X s /G) is not surjective when n ≥ 4 is even. For example, when n = 4, then X s /G ∼ = P1 \ {0, 1, ∞} and so dim H 1 (X s /G) = 2, whereas the equivariant cohomology HG∗ (X) of X is all in even degrees.

References [1] M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A, 308 (1982), 523–615.

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[2] G. Daskalopoulos, The topology of the space of stable bundles on a Riemann surface, J. Differential Geom., 36 (1992), 699–746. [3] R. A. Earl, The Mumford relations and the moduli of rank three stable bundles, Comp. Math., 109 (1997), 13–48. [4] R. A. Earl and F. C. Kirwan, Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface, Proc. London Math. Soc., 89-3 (2004), 570–622. [5] G. Ellingsrud and S. Strømme, On the Chow ring of a geometric quotient, Ann. Math. (2), 130 (1989), 130–159. [6] D. Gieseker, Geometric Invariant Theory and Applications to Moduli Problems: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Academiae Scientarium Fennica, Helsinki, 1980. [7] V. Guillemin and J. Kalkman, A new proof of the Jeffrey–Kirwan localization theorem, J. Reine Agnew. Math., 470 (1996), 123–142. [8] L. C. Jeffrey and F. C. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. Math., 148 (1998), 109–196. [9] F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematics Notes, Vol. 31, Princeton University Press, Princeton, NJ, 1985. [10] F. C. Kirwan, Partial desingularisations of quotients of non-singular varieties and their Betti numbers, Ann. Math., 122 (1985), 41–85. [11] F. C. Kirwan, Rational intersection cohomology of quotient varieties, Invent. Math., 86 (1986), 471–505. [12] F. C. Kirwan, Cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 853–906. [13] F. C. Kirwan, Moduli spaces of bundles over Riemann surfaces and the Yang–Mills stratiﬁcation revisited, preprint, 2003; math.AG/0305346. [14] S. K. Martin, Transversality theory, cobordisms and invariants of symplectic quotients, 1999; Ann. Math., to appear; math.SG/0001001. [15] S. K. Martin, Symplectic quotients by a nonabelian group and by its maximal torus, 1999; Ann. Math., to appear; math.SG/0001002 (2001). [16] E. Meinrenken and R. Sjamaar, Singular reduction and quantization, Topology, 38 (1999), 699–762. [17] D. B. Mumford, J. Fogarty, and F. C. Kirwan, Geometric Invariant Theory, 3rd ed., Modern Surveys in Mathematics, Vol. 34, Springer-Verlag, New York, 1994. [18] P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay, 1978. [19] S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom., 11-4 (2003), 751–773. [20] E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys., 9 (1992), 303–368.

Quasi, twisted, and all that… in Poisson geometry and Lie algebroid theory Yvette Kosmann-Schwarzbach Centre de Mathématiques (U.M.R. 7640 du C.N.R.S.) École Polytechnique F-91128 Palaiseau France [email protected] Dedicated to Alan Weinstein. Abstract. Motivated by questions from quantum group and ﬁeld theories, we review structures on manifolds that are weaker versions of Poisson structures, and variants of the notion of Lie algebroid. We give a simple deﬁnition of the Courant algebroids and introduce the notion of a deriving operator for the Courant bracket of the double of a proto-bialgebroid. We then describe and relate the various quasi-Poisson structures, which have appeared in the literature since 1991, and the twisted Poisson structures studied by Ševera and Weinstein.

Introduction In 1986, Drinfeld introduced both the quasi-Hopf algebras that generalize the Hopf algebras deﬁning quantum groups and their semiclassical limits, the Lie quasibialgebras. This naturally led to the notion of quasi-Poisson Lie groups, which I introduced in [31, 32]. A quasi-Hopf algebra is a bialgebra in which the multiplication is associative but the comultiplication is only coassociative up to a defect measured by an element in the triple tensor product of the algebra. Similarly, the deﬁnitions of the ELie quasi-bialgebras and the quasi-Poisson Lie groups involve a given element in 3 g, where g is the underlying Lie algebra, which Drinfeld denoted by ϕ. In a Lie quasibialgebra, the bracket is a Lie bracket because it satisﬁes the Jacobi identity, but the compatible cobracket is not a true Lie bracket on the dual of g because it only satisﬁes the Jacobi identity up to a defect measured by the element ϕ. On a quasi-Poisson Lie group, there is a multiplicative bivector ﬁeld, π , whose Schouten bracket, [π, π], does not vanish, but is also expressed in terms of ϕ. The desire to understand the group-valued moment maps and the quasi-Hamiltonian spaces of Alekseev, Malkin, and Meinrenken [3] in terms of Poisson geometry led to the study of the action of quasi-Poisson Lie groups on manifolds equipped with a bivector ﬁeld [1]. A special

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case of a quasi-Poisson structure on a Lie group occurs when the bivector vanishes and only ϕ remains, corresponding to a Lie quasi-bialgebra with a trivial cobracket. The quasi-Poisson manifolds studied in [2] are manifolds equipped with a bivector, on which such a quasi-Poisson Lie group acts. Closed 3-form ﬁelds appeared in Park’s work on string theory [46], and in the work on topological ﬁeld theory of Klimcˇ ik and Strobl, who recognized the appearance of a new geometrical structure which they called WZW-Poisson manifolds [29]. They chose this name because the role of the background 3-form is analogous to that of the Wess–Zumino term introduced by Witten in a ﬁeld theory with target a group, and more recently they proposed to shorten the name to WZ-Poisson manifolds. Shortly after these publications circulated as preprints, Ševera and Weinstein studied such structures in the framework of Courant algebroid theory, calling them Poisson structures with a 3-form background . They are deﬁned in terms of a bivector ﬁeld π and a closed 3-form, denoted by ϕ in [52], but which we shall denote by ψ to avoid confusion with the above. Again π is not a Poisson bivector—unless ψ vanishes, in which case the Poisson structure with background reduces to a Poisson structure—its Schouten bracket is the image of the 3-form ψ under the morphism of vector bundles deﬁned by π, mapping forms to vectors. Ševera and Weinstein also called the Poisson structures with background ψ-Poisson structures, or twisted Poisson structures. This last term has since been widely used [48, 51, 14, 13], hence the word “twisted’’ in the title of this paper. It is justiﬁed by a related usage in the theory of “twisted sheaves,’’ and we shall occasionally use this term but we prefer Poisson structure with background because, in Drinfeld’s theory of Lie quasi-bialgebras, “twist’’and “twisting’’ have a different and now standard meaning. Section 4.1 of this paper is a generalization of Drinfeld’s theory to the Lie algebroid setting. The theory of Lie bialgebras, on the one hand, is a special case of that of the Lie bialgebroids, introduced by Mackenzie and Xu [43]. It was shown by Roytenberg [48] that the “quasi’’ variant of this notion is the framework in which the Poisson structures with background appear naturally. Lie algebras, on the other hand, are a special case of the Loday algebras. Combining the two approaches, we encounter the Courant algebroids of Liu, Weinstein, and Xu [39], or rather their equivalent deﬁnition in terms of non-skew-symmetric brackets. We shall present these a priori different notions, and shall show how they can be related. In Section 1, we give a brief overview of the various theories just mentioned. In particular we deﬁne the proto-bialgebroids and the Lie quasi-bialgebroids, which generalize the Lie quasi-bialgebras, as well as their duals, the quasi-Lie bialgebroids. In Section 2, we give a simple deﬁnition of the Courant algebroids, which we prove to be equivalent to the usual deﬁnition [39, 47] (Theorem 2.1). Liu, Weinstein, and Xu [39] showed that the construction of the double of Lie bialgebroids can be accomplished in the framework of Courant algebroid theory by introducing Manin triples for Lie bialgebroids. Along the lines of [48], we extend these considerations to the case of proto-bialgebroids and, in particular, to both “quasi’’ cases. Thus, we study the more general Manin pairs for Lie quasi-bialgebroids. This is the subject of Section 3, where we also introduce the notion of a deriving operator (in the spirit of [34] and [36]) for the double of a proto-bialgebroid, and we prove an existence

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theorem (Theorem 3.2). Section 4 is devoted to the study of examples. The twisting of Lie quasi-bialgebroids by bivectors generalizes Drinfeld’s twisting of Lie quasibialgebras and leads to the consideration of the quasi-Maurer–Cartan equation, which generalizes the quasi-Poisson condition. One can twist a quasi-Lie bialgebroid with a closed 3-form background by a bivector, and the Poisson condition with background appears as the condition for the twisted object to remain a quasi-Lie bialgebroid. The world of the “quasi’’ structures explored here is certainly nothing but a small part of the realm of homotopy structures, L∞ , G∞ , etc. See, in particular, [57] and the articles of Stasheff [53], Bangoura [7], and Huebschmann [23]. We hope to show that these are interesting objects in themselves.

1 A review Before we mention the global objects such as the generalizations of the Poisson Lie groups, we shall recall their inﬁnitesimal counterparts. 1.1 Lie quasi-bialgebras, quasi-Lie bialgebrasm and proto-bialgebras We shall not review all the details of the structures that are weaker versions of the Lie bialgebra structure, but we need to recall the deﬁnition of Lie quasi-bialgebras. It is due to Drinfeld [17], while in [32] and [8] the dual case, that of a quasi-Lie bialgebra, and the more general case of proto-bialgebras (called “proto-Lie-bialgebras’’ there) are treated. A proto-bialgebra structure on a vector space F is deﬁned by a quadruple E of elements in • (F ⊕F ∗ ) ! C ∞ T ∗ ($F ), where $ denotes the change of parity. We E E denote such a quadruple by (µ, γ , ϕ, ψ), with µ : 2 F → F , γ : 2 F ∗ → F ∗ , E3 E3 ∗ ϕ ∈ F, ψ ∈ F . This quadruple deﬁnes a proto-bialgebra if and only if {µ + γ + ϕ + ψ, µ + γ + ϕ + ψ} = 0, where { , } is the canonical Poisson E bracket of the cotangent bundle T ∗ ($F ), which coincides with the big bracket of • (F ⊕ F ∗ ) [32]. This condition is equivalent to the ﬁve conditions which we shall write below in the more general case of the proto-bialgebroids (see Section 1.5). If either ψ or ϕ vanishes, there remain only four nontrivial conditions. When ψ = 0, the bracket is a Lie bracket, while the cobracket only satisﬁes the Jacobi identity up to a term involving ϕ, and we call the proto-bialgebra a Lie quasi-bialgebra. When ϕ = 0, the bracket only satisﬁes the Jacobi identity up to a term involving ψ, while the cobracket is a Lie cobracket, and we call the proto-bialgebra a quasi-Lie bialgebra. Clearly, the dual of a Lie quasi-bialgebra is a quasi-Lie bialgebra, and conversely. Drinfeld only considered the case ψ = 0. In the English translation of [17], what we call a Lie quasi-bialgebra in this paper was translated as a quasi-Lie bialgebra, a term which we shall reserve for the object dual to a Lie quasi-bialgebra. In fact, it is in the dual object, where ϕ = 0 and ψ = 0 that the algebra structure is only “quasi-Lie.’’ As another E potential source of confusion, we mention that in [47] and [48], the element in 3 F ∗ , which we denote by ψ, is denoted by ϕ, and vice versa. Any proto-bialgebra ((F, F ∗ ), µ, γ , ϕ, ψ) has a double which is d = F ⊕ F ∗ , with the Lie bracket,

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[x, y] = µ(x, y) + ix∧y ψ, ∗γ

[x, ξ ] = −[ξ, x] = −adξ x + adx∗µ ξ, [ξ, η] = iξ ∧η ϕ + γ (ξ, η). Here x and y ∈ F , and ξ and η ∈ F ∗ . Any Lie bialgebra has, associated with it, a pair of Batalin–Vilkovisky algebras in duality. The extension of this property to Lie quasi-bialgebras, giving rise to quasi-Batalin–Vilkovisky algebras in the sense of Getzler [20], has been carried out by Bangoura [5]. There is a notion of quasi-Gerstenhaber algebra (see [48]), and Bangoura has further proved that quasi-Batalin–Vilkovisky algebras give rise to quasi-Gerstenhaber algebras [6]. For a thorough study of bigraded versions of these structures in the general algebraic setting, with applications to foliation theory, see Huebschmann [23]. These “quasi’’ algebras are the simplest examples of G∞ - and BV∞ -algebras, in which all the higher-order multilinear maps vanish except for the trilinear map. 1.2 Quasi-Poisson Lie groups and moment maps with values in homogeneous spaces The global object corresponding to the Lie quasi-bialgebras just presented was introduced in [32] and called a quasi-Poisson Lie group. It is a Lie group with a multiplicative bivector, πG , whose Schouten bracket does not vanish (so that it is not a Poisson bivector), but is a coboundary, namely 1 [πG , πG ] = ϕ L − ϕ R , 2 R where ϕ L (resp., E3 ϕ ) are the left- (resp., right-)invariant trivectors on the group with value ϕ ∈ g at the identity. In [1], we considered the action of a quasi-Poisson Lie group (G, πG , ϕ) on a manifold M equipped with a G-invariant bivector π . When the Schouten bracket of π satisﬁes the condition

1 [π, π] = ϕM , 2

(1.1)

we say E that (M, π ) is a quasi-Poisson G-space. Here ϕM is the image of the element ϕ in 3 g under the inﬁnitesimal action of the Lie algebra g of G on M. The quasiPoisson G-space (M, π ) is called a Hamiltonian quasi-Poisson G-space if there exists a moment map for the action of G on M, which takes values in D/G, where D is the simply connected Lie group whose Lie algebra is the double d = g ⊕ g∗ of the Lie quasi-bialgebra g. See [1] for the precise deﬁnitions. Two extreme cases of this construction are of particular interest. The ﬁrst corresponds to the case where the Lie quasi-bialgebra is actually a Lie bialgebra (ϕ = 0), i.e., the Manin pair with a chosen isotropic complement deﬁning the Lie quasibialgebra is in fact a Manin triple. Then G is a Poisson Lie group and D/G can be identiﬁed with a dual group G∗ of G. The moment maps for the Hamiltonian

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quasi-Poisson G-spaces reduce to the moment maps in the sense of Lu [41] that take values in the dual Poisson Lie group, G∗ . The second case is that of a Lie quasibialgebra with vanishing cobracket (γ = 0), to be described in the next subsection. 1.3 Quasi-Poisson manifolds and group-valued moment maps Assume that G is a Lie group acting on a manifold M, and that g is a quadratic Lie algebra, i.e., a Lie algebra with an invariant nondegenerate symmetric bilinear form. We consider the bilinear form in g ⊕ g deﬁned as the difference of the copies of the given bilinear form on the two terms of the direct sum. Let g be diagonally embedded into g ⊕ g. Then (g ⊕ g, g) is a Manin pair, and we choose the antidiagonal, {(x, −x)|x ∈ g}, as a complement of g ⊂ g ⊕ g. The corresponding Lie quasi-bialgebra has vanishing cobracket because the bracket of two elements in the anti-diagonal is in the diagonal, and therefore the bivector of the quasi-Poisson structure of G is trivial. With this choice of a complement, ϕ is the Cartan trivector of g. In this way, we obtain the quasi-Poisson G-manifolds described in [2]. They are pairs, (M, π ), where π is a G-invariant bivector on M that satisﬁes equation (1.1) with ϕ the Cartan trivector of g. The group G acting on itself by means of the adjoint action is a quasi-Poisson G-manifold, and so are its conjugacy classes. The bivector πG on G is a eaR ∧ eaL , where ea is an orthonormal basis of g. Because the homogeneous space D/G of the general theory is the group G itself in this case, the moment maps for the Hamiltonian quasi-Poisson manifolds are group-valued. Those Hamiltonian quasi-Poisson manifolds for which the bivector π satisﬁes a nondegeneracy condition are precisely the quasi-Hamiltonian manifolds of Alekseev, Malkin and Meinrenken [3]. (Their deﬁnition had been preceded by related constructions on moduli spaces introduced by Weinstein [58] and by Huebschmann and Jeffrey [24].) In [12] Bursztyn and Crainic prove many results that relate the geometry of Hamiltonian quasi-Poisson manifolds and the geometry of the “twisted Dirac structures,’’ deﬁned in terms of the “twisted Courant bracket,’’ which we shall call the “Courant bracket with background’’ and discuss in Section 4.2. 1.4 Lie bialgebroids and their doubles Lie bialgebroids were ﬁrst deﬁned by Mackenzie and Xu [43]. We state the deﬁnition as we reformulated it in [33]. To each Lie algebroid A are associated E • a Gerstenhaber bracket, E [ , ]A , on ( • A), • ∗ A ). • a differential, dA , on ( A Lie bialgebroid is a pair, (A, A∗ ), of Lie algebroids in duality such that dA∗ is a derivation of [ , ]A , or, equivalently, dA is a derivation of [ , ]A∗ . Extending the construction of the Drinfeld double of a Lie bialgebra to the case of a Lie bialgebroid is a nontrivial problem, and several solutions have been offered, by Liu, Weinstein, and Xu [39] in terms of the Courant algebroid A ⊕ A∗ , by Mackenzie [42] in terms of the double vector bundle T ∗ A ! T ∗ A∗ , and by Vaintrob (unpublished) and Roytenberg [47, 48] in terms of supermanifolds. We shall describe some properties of the ﬁrst and third constructions in Section 3.

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1.5 Lie quasi-bialgebroids, quasi-Lie bialgebroids, proto-bialgebroids and their doubles We call attention to the fact that we shall deﬁne here both “Lie quasi-bialgebroids’’ and “quasi-Lie bialgebroids’’ and that, as we explain below, these terms are not synonymous. We extend the notations of [17, 32, 8] to the case of Lie algebroids. A proto-bialgebroid (A, E A∗ ) is deﬁned by anchors ρA and ρA∗ , brackets [ , ]A and E [ , ]A∗ , and elements ϕ ∈ ( 3 A) and ψ ∈ ( 3 A∗ ). By deﬁnition, • • •

the case ψ = 0 is that of Lie quasi-bialgebroids (A is a true Lie algebroid, while A∗ is only “quasi’’); the case ϕ = 0 is that of quasi-Lie bialgebroids (A∗ is a true Lie algebroid, while A is only “quasi’’); the case where both ϕ and ψ vanish is that of the Lie bialgebroids.

While the dual of a Lie bialgebroid is itself a Lie bialgebroid, the dual of a Lie quasi-bialgebroid is a quasi-Lie bialgebroid, and conversely. Whenever A is a vector bundle, the space of functions T ∗ $A, where $ denotes Eon • the change of parity, contains the space of sections of A, the A-multivectors. In particular, the sections of A can be considered as functions on T ∗ $A. Given the ∗ ∗ ∗ canonical same conclusion holds for the sections E• ∗ isomorphism, T $A ! T $A, the A , in particular for the sections of A∗ . of A Lie algebroid bracket [ , ]A on a vector bundle A over a manifold M is deﬁned, together with an anchor ρA : A → T M, by a function µ on the supermanifold T ∗ $A [47, 48, 55, 56]. Let { , } denote the canonical Poisson bracket of the cotangent bundle. The bracket of two sections x and y of A is the derived bracket, in the sense of [34], [x, y]A = {{x, µ}, y}, and the anchor satisﬁes ρA (x)f = {{x, µ}, f }, C ∞ (M).

for f ∈ When (A, A∗ ) is a pair of Lie algebroids in duality, both [ , ]A together with ρA , and [ , ]A∗ together with ρA∗ correspond to functions, denoted by µ and γ , on the same supermanifold T ∗ $A, taking into account the identiﬁcation of T ∗ $A∗ with T ∗ $A. The three conditions in the deﬁnition of a Lie bialgebroid are equivalent to the single equation {µ + γ , µ + γ } = 0. More generally, the ﬁve conditions for a proto-bialgebroid deﬁned by (µ, γ , ϕ, ψ) are obtained from a single equation. By deﬁnition, a proto-bialgebroid structure on (A, A∗ ) is a function of degree 3 and of Poisson square 0 on T ∗ $A. As in the case E of a proto-bialgebra, such a function can be written µ + γ + ϕ + ψ, where ϕ ∈ 3 A E and ψ ∈ 3 A∗ , and satisﬁes {µ + γ + ϕ + ψ, µ + γ + ϕ + ψ} = 0.

(1.2)

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The deﬁnition is equivalent to the conditions ⎧ 1 ⎪ ⎪ {µ, µ} + {γ , ψ} = 0, ⎪ ⎪2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ {µ, γ } + {ϕ, ψ} = 0, 1 {γ , γ } + {µ, ϕ} = 0, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ {µ, ψ} = 0, ⎪ ⎪ ⎩ {γ , ϕ} = 0. • •

When ((A, A∗ ), µ, γ , ϕ, 0) is a Lie quasi-bialgebroid, ((A, A∗ ), µ, γ ) is a Lie bialgebroid if and only if {µ, ϕ} = 0. Dually, when ((A, A∗ ), µ, γ , 0, ψ) is a quasi-Lie bialgebroid, ((A, A∗ ), µ, γ ) is a Lie bialgebroid if and only if {γ , ψ} = 0.

Remark. In the case of a proto-bialgebra, (F, F ∗ ), the operator {µ, ·} generalizes E the Chevalley–Eilenberg coboundary operator on cochains on F with values in • F . In E the term {µ, ϕ}, ϕ should be viewed as a 0-cochain on F with values in 3 F , and E {µ, ϕ} is an element in F ∗ ⊗ 3 F . So is {γ , γ }, which is a trilinear form on F ∗ with values in F ∗ whose vanishing is equivalent to the Jacobi identity for γ . In the term {µ, ψ}, ψ should be viewed as a 3-cochain on F with scalar values, and {µ, ψ} is an E element in 4 F ∗ . Reversing the roles of F and F ∗ , one obtains the interpretation of the other terms in the above formulas. 1.6 Poisson structures with background (twisted Poisson structures) The WZW-Poisson structures introduced by Klimcˇ ik and Strobl [29] were studied by Ševera and Weinstein in 2001 [52], who called them Poisson structures with background , and also twisted Poisson structures. Roytenberg has subsequently shown that they appear by a twisting of a quasi-Lie bialgebroid by a bivector [48]. We shall review this approach in Section 4. The integration of Poisson structures with background into presymplectic groupoids is the subject of recent work of Bursztyn, Crainic, Weinstein and Zhu [13] and of Cattaneo and Xu [14] (who used the term quasi-symplectic groupoids). In addition, Xu [60] has very recently extended the theory of momentum maps to this setting. 1.7 Other structures: Loday algebras, omni-Lie algebras There are essentially two ways of weakening the properties of Lie algebras. One possibility is to introduce a weakened version of the Jacobi identity, e.g., an identity up to homotopy: this is the theory of L∞ -algebras. The relationship of the Courant algebroids to L∞ -algebras was explored in [50]. Another possibility is to consider non-skew-symmetric brackets: this is the theory of Loday algebras, which Loday introduced and called Leibniz algebras. A Loday

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algebra is a graded vector space with a bilinear bracket of degree n satisfying the Jacobi identity, [x, [y, z] = [[x, y], z] + (−1)(|x|+n)(|y|+n) [y, [x, z]],

(1.3)

for all elements x, y and z, where |x| is the degree of x. In Section 2, we shall describe the Loday algebra approach to Courant algebroids, in which case there is no grading. The “omni-Lie algebras’’ introduced by Weinstein in [59] provide an elegant way of characterizing the Lie algebra structures on a vector space V in terms of the graph in V ×gl(V ) of the adjoint operator. In the same paper, he deﬁned the (R, A) C-algebras, the algebraic analogue of Courant algebroids, which generalize the (R, A) Lie algebras (also called Lie–Rinehart algebras or pseudo-Lie algebras), and he posed the question of how to determine the global analogue of an “omni-Lie algebra.’’ In [27], he and Kinyon explored this problem and initiated the search for the global objects associated to generalized Lie algebras that would generalize Lie groups. They proved new properties of the Loday algebras, showing in what sense they can be integrated to a homogeneous left loop, i.e., to a manifold with a nonassociative composition law, and they showed that the Courant brackets of the doubles of Lie bialgebroids can be realized on the tangent spaces of reductive homogeneous spaces. These global constructions are inspired by the correspondence between generalized Lie triple systems and nonassociative multiplications on homogeneous spaces. (Some of the results of Bertram [9] might prove useful in the search for global objects integrating generalized Lie algebras.) For recent developments, see Kinyon’s lecture [26]. 1.8 Generalized Poisson brackets for nonholonomic mechanical systems Brackets of the Poisson or Dirac type that do not satisfy the Jacobi identity appear in many geometric constructions describing nonholonomic mechanical systems. There is a large literature on the subject; see, for instance, [25, 15, 11] and the many references cited there. It would be very interesting to study how these constructions relate to the various structures which we are now considering. In his lecture [42], Marsden showed how to state the nonholonomic equations of Lagrangian mechanics in terms of isotropic subbundles in the direct sum of the tangent and cotangent bundles of the phase space, T ∗ Q, of the system under consideration. He calls such subbundles Dirac structures on T ∗ Q. Yet, it is only when an integrability condition is required that these structures become examples of the Dirac structures to be mentioned in the next section.

2 Courant algebroids The construction of the double of a Lie bialgebra with the structure of a Lie algebra does not extend into a construction of the double of a Lie bialgebroid with the structure of a Lie algebroid because the framework of Lie algebroid theory is too narrow to permit it. While it is not the only solution available, the introduction of the new notion of Courant algebroid permits the solution of this problem.

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The deﬁnition of Courant algebroids, based on Courant’s earlier work [16], is due to Liu, Weinstein and Xu [39]. It was shown by Roytenberg [47] that a Courant algebroid can be equivalently deﬁned as a vector bundle E → M with a Loday bracket on E, an anchor ρ : E → T M and a ﬁeld of nondegenerate, symmetric bilinear forms ( | ) on the ﬁbers of E, related by a set of four additional properties. It was further observed by Uchino [54] and by Grabowski and Marmo [21] that the number of independent conditions can be reduced. We now show that it can be reduced to two properties which are very natural generalizations of those of a quadratic Lie algebra. In fact, (i) and (ii) below are generalizations to algebroids of the skew-symmetry of the Lie bracket, and of the condition of ad-invariance for a bilinear form on a Lie algebra, respectively. Deﬁnition 2.1. A Courant algebroid is a vector bundle E → M with a Loday bracket on E, i.e., an R-bilinear map satisfying the Jacobi identity, [x, [y, z]] = [[x, y], z] + [y, [x, z]], for all x, y, z ∈ E, an anchor, ρ : E → T M, which is a morphism of vector bundles, and a ﬁeld of nondegenerate symmetric bilinear forms ( | ) on the ﬁbers of E, satisfying (i) (ii)

ρ(x)(u|v) = (x | [u, v] + [v, u]), ρ(x)(u|v) = ([x, u] | v) + (u | [x, v]),

for all x, u, and v ∈ E. Remark. Property (i) is equivalent to (i )

1 ρ(x)(y|y) = (x|[y, y]) 2

(which is [47, Deﬁnition 2.6.1, property 4] and [52, Section 1, property 5]). The conjunction of properties (i) and (ii) is equivalent to property (ii) together with (i )

(x|[y, y]) = ([x, y]|y)

(which is [51, Appendix A, property 5]). We now prove two important consequences of properties (i) and (ii) which have been initially considered to be additional, independent deﬁning properties of Courant algebroids. Theorem 2.1. In any Courant algebroid, (iii) the Leibniz rule is satisﬁed, i.e., [x, fy] = f [x, y] + (ρ(x)f )y, for all x and y ∈ E and all f ∈ C ∞ (M);

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(iv) the anchor, ρ, induces a morphism of Loday algebras from E to (T M), i.e., it satisﬁes ρ([x, y]) = [ρx, ρy], for all x and y ∈ E. Proof. The proof of (iii), adapted from [54], is obtained by evaluating ρ(x)(fy|z) in two ways. We ﬁrst write, using the Leibniz rule for vector ﬁelds acting on functions, ρ(x)(fy|z) = (ρ(x)f )(y|z) + fρ(x)(y|z). Then, using property (ii) twice, we obtain ([x, fy]|z) + (fy|[x, z]) = (ρ(x)f )(y|z) + f ([x, y]|z) + f (y|[x, z]). and (iii) follows by the nondegeneracy of ( | ). The proof of (iv) is that of the analogous property for Lie algebroids (see, e.g., [37]). It is obtained by evaluating [x, [y, f z]], for z ∈ E, in two ways, using both the Jacobi identity for the Loday bracket [ , ] and (iii).

It follows from the remark above together with Theorem 2.1 and from the arguments of Roytenberg in [47] that our deﬁnition of Courant algebroids is equivalent to that of Liu, Weinstein, and Xu in [39]. A Dirac subbundle (also called a Dirac structure) in a Courant algebroid is a maximally isotropic subbundle whose space of sections is closed under the bracket. Courant algebroids with base a point are quadratic Lie algebras. More generally, Courant algebroids with a trivial anchor are bundles of quadratic Lie algebras with a smoothly varying structure. The notion of a Dirac subbundle in a Courant algebroid with base a point reduces to that of a maximally isotropic Lie subalgebra in a quadratic Lie algebra, in other words, to a Manin pair. We shall show that a Courant algebroid together with a Dirac subbundle is an appropriate generalization of the notion of a Manin pair from the setting of Lie algebras to that of Lie algebroids. A deep understanding of the nature of Courant algebroids is provided by the consideration of the nonnegatively graded manifolds. This notion was deﬁned and used by Kontsevich [30], Ševera [51] (who called them N -manifolds), and T. Voronov [56]. In [49], Roytenberg showed that the nonnegatively graded symplectic manifolds of degree 2 are the pseudo-euclidean vector bundles, and that the Courant algebroids are deﬁned by an additional structure, that of a homological vector ﬁeld, associated to a cubic Hamiltonian ' of Poisson square 0, preserving the symplectic structure. The bracket and the anchor of the Courant algebroid are recovered from this data as the derived brackets, [x, y] = {{x, '}, y} and ρ(x)f = {{x, '}, f }. T. Voronov [56] studied the double of the nonnegatively graded QP -manifolds which are a generalization of the Lie bialgebroids.

3 The double of a proto-bialgebroid We shall now explain how to generalize the construction of a double with a Courant algebroid structure from Lie bialgebroids to proto-bialgebroids.

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3.1 The double of a Lie bialgebroid Liu, Weinstein, and Xu [39] have shown that complementary pairs of Dirac subbundles in a Courant algebroid are in one-to-one correspondence with Lie bialgebroids: If E is a Courant algebroid, if E = A⊕B, where A and B are maximally isotropic subbundles, and if A and B are closed under the bracket, then • • •

A and B are in duality, B ! A∗ , the bracket of E induces Lie algebroid brackets on A and B ! A∗ , with respective anchors the restrictions of the anchor of E to A and A∗ , the pair (A, A∗ ) is a Lie bialgebroid.

Conversely, if (A, A∗ ) is a Lie bialgebroid, the direct sum A⊕A∗ is equipped with a Courant algebroid structure such that A and A∗ are maximally isotropic subbundles, and A and (A∗ ) are closed under the bracket, the bilinear form being the canonical one, deﬁned by (x + ξ |y + η) = ξ, y + η, x, for x and y ∈ A, ξ and η ∈ (A∗ ). 3.2 The case of proto-bialgebroids The construction which we just recalled can be extended to the proto-bialgebroids [48]. Let A be a vector bundle. Recall that a proto-bialgebroid structure on (A, A∗ ) is a function of degree 3 and of square 0Eon T ∗ $A, which can be written EPoisson 3 µ + γ + ϕ + ψ, where ϕ ∈ ( A) and ψ ∈ ( 3 A∗ ), and µ (resp., γ ) deﬁnes a ∗ bracket and anchor on A (resp., A ). The Courant bracket of the double, A ⊕ A∗ , of a proto-bialgebroid, (A, A∗ ), deﬁned by (µ, γ , ϕ, , ψ), is the derived bracket, [x + ξ, y + η] = {{x + ξ, µ + γ + ϕ + ψ}, y + η}. Here x and y are sections of A, ξ and η are sections of A∗ , and [x + ξ, y + η] is a section of A ⊕ A∗ . (The right-hand side makes sense more generally when x and y are A-multivectors, and E ξ and η are A∗ -multivectors, but the resulting quantity is not necessarily a section of • (A ⊕ A∗ ).) The anchor is deﬁned by {{x + ξ, µ + γ + ϕ + ψ}, f } = {{x, µ}, f } + {{ξ, γ }, f } = (ρA (x) + ρA∗ (ξ ))(f ), for f ∈ C ∞ (M). We set [x, y]µ = {{x, µ}, Ey} and [ξ, η]γE= {{ξ, γ }, η}. The associated quasi-differentials, dµ and dγ , on ( • A∗ ) and ( • A) are dµ = {µ, ·} and dγ = {γ , ·}, which satisfy (dµ )2 + {dγ ψ, ·} = 0,

(dγ )2 + {dµ ϕ, ·} = 0.

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We denote the interior product of a form α by a multivector x by ix α, with the sign convention, ix∧y = ix ◦ iy , and we use an analogous notation for the interior product of a multivector by a form. γ µ The Lie derivations are deﬁned by Lx = [ix , dµ ] and Lξ = [iξ , dγ ]. We ﬁnd, for x and y ∈ A, ξ and η ∈ (A∗ ), [x, y] = [x, y]µ + ix∧y ψ,

(3.1)

[x, ξ ] = −iξ dγ x

(3.2)

[ξ, x] =

γ Lξ x

+ Lµ x ξ,

− ix dµ ξ,

[ξ, η] = iξ ∧η ϕ + [ξ, η]γ ,

(3.3) (3.4)

that is, γ

[x + ξ, y + η] = [x, y]µ + Lξ y − iη dγ x + iξ ∧η ϕ + [ξ, η]γ + Lµ x η − iy dµ ξ + ix∧y ψ. These formulas extend both the Lie bracket of the Drinfeld double of a proto-bialgebra [8], recalled in Section 1.1, and the Courant bracket of the double of a Lie bialgebroid [39]. 3.3 Deriving operators If ξ isE a section of A∗ , by eξ we denote the operation of exterior multiplication by ξ on ( • A∗ ). In this subsection, the square brackets E [ , ] without a subscript denote the graded commutators of endomorphisms of ( • A∗ ). E Deﬁnition 3.1. We say that a differential operator D on ( • A∗ ) is a deriving operator for the Courant bracket of A ⊕ A∗ if it satisﬁes the following relations: [[ix , D], iy ] = i[x,y]µ + eix∧y ψ ,

(3.5)

[[ix , D], eξ ] = −iiξ dγ x + eLµx ξ ,

(3.6)

[[eξ , D], ix ] = iLγ x − eix dµ ξ ,

(3.7)

ξ

[[eξ , D], eη ] = iiξ ∧η ϕ + e[ξ,η]γ . (3.8) E• ∗ If we x ∈ A with ix ∈ End(( A )), and ξ ∈ (A∗ ) with eξ ∈ Eidentify • ∗ End(( A )), the preceding relations become [[x, D], y] = [x, y]µ + ix∧y ψ, [[x, D], ξ ] = −iξ dγ x + Lµ x ξ, γ

[[ξ, D], x] = Lξ x − ix dµ ξ , [[ξ, D], η] = iξ ∧η ϕ + [ξ, η]γ , so that the Courant bracket deﬁned in Section 3.2 can also be written as a derived bracket [34, 36].

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Remark. With the preceding identiﬁcation, the relation ix eξ + eξ ix = ξ, x implies that (x + ξ )(y + η) + (y + η)(x + ξ ) = (x + ξ |y + η). E This shows that ( • A∗ ) is a Clifford module of the Clifford bundle of A ⊕ A∗ , the point of departure of Alekseev and Xu in [4]. Does the Courant bracket of a proto-bialgebroid admit E a deriving operator? We ﬁrst treat the case of a Lie bialgebroid. The space ( • A∗ ) has the structure of a Gerstenhaber algebra deﬁned by γ . We shall assume that this Gerstenhaber algebra admits a generator in the following sense [38]. Deﬁnition 3.2. Let [ , ]A be any Gerstenhaber bracket on an associative, graded commutative algebra (A, ∧). An operator, ∂, on A is a generator of the bracket if [u, v]A = (−1)|u| (∂(u ∧ v) − ∂u ∧ v − (−1)|u| u ∧ ∂v),

(3.9)

for all u and v ∈ A. In particular, a Batalin–Vilkovisky algebra is a Gerstenhaber algebra which admits a generator of square 0. Remark. For the Schouten–Nijenhuis bracket of multivectors on a manifold M and the operator on multivectors constructed from the de Rham differential and a volume form on M, formula (3.9) is to be found in an early reference, Kirillov [28]. Lemma 3.1. If ∂ is a generator of bracket [ , ]A , then, for all u and v ∈ A, [eu , ∂] = e∂u − [u, ·]A ,

(3.10)

and [[eu , ∂], ev ] = −e[u,v]A ,

(3.11)

where eu is left ∧-multiplication by u ∈ A. Proof. The ﬁrst relation follows from the deﬁnitions by a short computation, and the second is a consequence of the ﬁrst since [[eu , ∂], ev ] = [e∂u − [u, ·]A , ev ] = −[[u, ·]A , ev ] = −e[u,v]A , for all u and v ∈ A.

E Theorem 3.1. If ∂∗ is a generator of the Gerstenhaber bracket of ( • A∗ ), then dµ − ∂∗ is a deriving operator for the Courant bracket of A ⊕ A∗ . E Proof. We consider various operators acting on sections of • A∗ . We recall from [35] (see [38] for the case A = T M) that for any x ∈ A, [ix , ∂∗ ] = −idγ x .

(3.12)

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We shall also make use of the following relations:

for ξ ∈ A∗ , E•

for any u ∈ (

[eξ , dµ ] = edµ ξ ,

(3.13)

[iu , eξ ] = (−1)|u|+1 iiξ u ,

(3.14)

[eζ , ix ] = (−1)|ζ |+1 eix ζ ,

(3.15)

A), and

E•

A∗ ). for all ζ ∈ ( µ (1) Let x and y be in A. We compute [ix , dµ − ∂∗ ] = Lx + idγ x , whence [[ix , dµ − ∂∗ ], iy ] = i[x,y]µ + [idγ x , iy ] = i[x,y]µ . This proves (3.5), corresponding to (3.1). (2) Let x be in A and let ξ be in (A∗ ). We compute − iiξ dγ x . [[ix , dµ − ∂∗ ], eξ ] = [Lµ x , eξ ] + [idγ x , eξ ] = eLµ xξ This proves (3.6), corresponding to (3.2). (3) Since ∂∗ is a generating operator of [ , ]γ , (3.10) is valid and therefore [eξ , ∂∗ ] = e∂∗ ξ − [ξ, ·]γ .

(3.16)

Since ∂∗ ξ is of degree 0, e∂∗ ξ commutes with ix . Therefore, [[eξ , dµ − ∂∗ ], ix ] = [edµ ξ , ix ] − [e∂∗ ξ , ix ] + [[ξ, ·]γ , ix ] = −eix dµ ξ + [[ξ, ·]γ , ix ]. E Let us now prove that the derivation [[ξ, ·]γ , ix ] of ( • A∗ ) coincides with the derivation iLγ x . In fact, they both vanish on 0-forms, and on a 1-form α, ξ

[ξ, ix α]γ − ix [ξ, α]γ = ρA∗ (ξ )α, x − [ξ, α]γ , x, while γ

iLγ x (α) = α, Lξ x = α, iξ dγ x + α, dγ ξ, x ξ

= dγ x(ξ, α) + dγ ξ, x(α) = ρA∗ (ξ )α, x − [ξ, α]γ , x. Thus [[eξ , dµ − ∂∗ ], ix ] = −eix dµ ξ + iLγ x , ξ

and (3.7), corresponding to (3.3), is proved. (4) Let ξ and η be sections of A∗ . Then [[eξ , dµ ], eη ] = [edµ ξ , eη ] = 0, while, by (3.11), [[eξ , ∂∗ ], eη ] = −e[ξ,η]γ , proving (3.8), corresponding to (3.4).

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E We now turn to the case of a proto-bialgebroid, deﬁned by ϕ ∈ ( 3 A) and E3 ∗ ψ ∈ ( A ). The additional terms in the four expressions to be evaluated are (1) (2) (3) (4)

[[ix , iϕ ], iy ] = 0, and [[ix , eψ ], iy ] = [eix ψ , iy ] = eix∧y ψ . [[ix , iϕ ], eξ ] = 0, and [[ix , eψ ], eξ ] = [eix ψ , eξ ] = 0. [[eξ , iϕ ], ix ] = [iiξ ϕ , ix ] = 0, and [[eξ , eψ ], ix ] = 0. [[eξ , iϕ ], eη ] = [iiξ ϕ , eη ] = iiξ ∧η ϕ , and [[eξ , eψ ], eη ] = 0.

Therefore, we can generalize Theorem 3.1 as follows. E Theorem 3.2. If ∂∗ is a generator of the Gerstenhaber bracket of ( • A∗ ), then dµ −∂∗ +iϕ +eψ is a deriving operator for the Courant bracket of the double, A⊕A∗ , E E of the proto-bialgebroid (A, A∗ ) deﬁned by ϕ ∈ ( 3 A) and ψ ∈ ( 3 A∗ ). It is clear that the addition to a deriving operator of derivations ix0 and eξ0 of E the associative, graded commutative algebra ( • A∗ ) will furnish a new deriving operator. The importance of the notion of a deriving E operator comes from the fact that if we can modiﬁy dµ and ∂∗ by derivations of ( • A∗ ) in such a way that the deriving operator has square 0, then the Jacobi identity for the resulting non-skewsymmetric bracket follows from the general properties of derived brackets that were proved in [34]. Let (A, µ) be a Lie E algebroid, let (A, A∗ ) be the triangular Lie bialgebroid deﬁned by a bivector π ∈E( 2 A) satisfying [π, π]µ = 0, and let dπ = [π, ·]µ be the differential on ( • A) (see Section 4.1.1 below). We assume that there exists a nowhere vanishing section, ν, of the top exterior power of the dual. Let ∂ν be the E generator of the Gerstenhaber bracket of ( • A) deﬁned by ν, which is a generator of square 0. We set xν = ∂ν π. Then, xν is a section of A, which is called the modular ﬁeld of (A, A∗ ) associated with ν [35]. We shall now give a short proof of the existence of a deriving operator of square 0 for the Courant bracket of the dual of (A, A∗ ). Theorem 3.3. The operator dπ − ∂ν + exν is a deriving operator of square 0 of the Courant bracket of the double of the Lie bialgebroid (A∗ , A). Proof. ByEdeﬁnition, the Laplacian of the strong differential Batalin–Vilkovisky algebra (( • A), ∂ν , dπ ) is [dπ , ∂ν ], and we know that it satisﬁes the relation [dπ , ∂ν ] = Lµ xν . (See [35] and [38] for the case of a Poisson manifold.) Since, by Theorem 3.2, the operator dπ − ∂ν is a deriving operator, and since this property is not modiﬁed by the addition of the derivation exν , it is enough to prove that the operator dπ − ∂ν + exν is of square 0. In fact, since both dπ and ∂ν are of square 0, 1 [dπ − ∂ν , dπ − ∂ν ] = −Lµ xν = −[xν , ·]µ . 2

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Therefore, 1 [dπ − ∂ν + exν , dπ − ∂ν + exν ] = −[xν , ·]µ + [exν , dπ ] − [exν , ∂ν ]. 2 By (3.13), [exν , dπ ] = edπ xν , which vanishes since xν leaves π invariant, while by (3.16), [exν , ∂ν ] = e∂ν xν − [xν , ·]µ . In addition, ∂ν xν = 0, since xν = ∂ν π and ∂ν is of square 0. Therefore, the square of dπ − ∂ν + exν vanishes.

In particular, if (M, π ) is a Poisson manifold, we obtain a deriving operator of square 0 of the Courant algebroid, double of the Lie bialgebroid (T ∗ M, T M), dual to the triangular Lie bialgebroid (T M, T ∗ M). More generally, Alekseev and Xu [4] consider deriving operators of the Courant bracket of a Courant algebroid whose square is a scalar function, which they call “generating operators’’ (but which should not be confused with the generating operators of Batalin–Vilkovisky algebras). They show that there always exists such a generating operator for the double of a Lie bialgebroid, (A, A∗ ), and that its square is expressible in terms of the modular ﬁelds of A and A∗ (see [4, Theorem 5.1 and Corollary 5.9]). It is easily seen that the case of a triangular Lie bialgebroid is a particular case of their theorem and corollary, in which the generating operator is equal to the deriving operator of Theorem 3.3, and the square of the generating operator actually vanishes. In fact, in the case of a triangular Lie bialgebroid (A, A∗E ), the Laplacian [dµ , ∂π ] of the strong differential Batalin–Vilkovisky algebra (( • A∗ ), ∂π , dµ ) vanishes because ∂π = [iπ , dµ ], and therefore the modular ﬁeld of A vanishes. In addition, xν = 12 X0 , where X0 is the modular ﬁeld of A∗ [18] and ∂ν xν = 0. Hence, in the expression for the square of the generating operator given in [4], both terms vanish.

4 Examples We shall ﬁrst analyze various constructions of Lie bialgebroids, Lie quasi-bialgebroids and quasi-Lie bialgebroids, then we shall consider the Courant brackets in the theory of Poisson structures with background. 4.1 Twisting by a bivector 4.1.1 Triangular Lie bialgebroids E Let (A, µ) be a Lie algebroid, and let π be a section of 2 A. On the one hand, such sections generalize the r-matrices and twists of Lie bialgebra theory, and on the other hand, when A = E T M, such sections are bivector ﬁelds on the manifold M. By extension, a section of 2 A is called an A-bivector, or simply a bivector. Let π be the vector bundle map from (A∗ ) to A deﬁned by π (ξ ) = iξ π , for ξ ∈ (A∗ ). Consider the bracket on A∗ depending on both µ and π deﬁned by µ

µ

[ξ, η]µ,π = Lπ ξ η − Lπ η ξ − dµ (π(ξ, η)),

(4.1)

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for ξ and η ∈ (A∗ ). The following relation generalizes the equation γ = −dµ r which is valid in a coboundary Lie bialgebra. Theorem 4.1. Set γµ,π = {π, µ} = −{µ, π}. Then

E•

(i) the associated quasi-differential on (

(4.2)

A) is

dπ = [π, ·]µ ;

(4.3)

(ii) bracket [ξ, η]µ,π , deﬁned by formula (4.1), is equal to the derived bracket, {{ξ, γµ,π }, η}; (iii) if, in addition, 1 ϕ = − [π, π]µ , 2 then ((A, A∗ ), µ, γµ,π , ϕ, 0) is a Lie quasi-bialgebroid.

(4.4)

Proof. The proof of (i) is a straightforward application of the Jacobi identity. To prove (ii) it sufﬁces to prove that the quasi-differential dπ is given by the usual Cartan formula in terms of the anchor π and the Koszul bracket (4.1). This now classic result was ﬁrst proved by Bhaskara and Viswanath in [10] in the case of a Poisson bivector on a manifold when A = T M and [π, π]µ = 0. We proved it independently, and in the general case, in [37]. To prove (iii), use the relations {µ, µ} = 0, and {µ, γµ,π } = 0 which follows from (4.2) and the Jacobi identity. Moreover, {{µ, π}, {µ, π}} = {µ, [π, π]µ }, whence 21 {γµ,π , γµ,π }+{µ, ϕ} = 0, and −2{γµ,π , ϕ} = {{π, µ}, [π, π]µ } = [π, [π, π]µ ]µ = 0. Thus the four conditions equivalent to (1.2) are satisﬁed.

The square of dπ does not vanish in general, (dπ )2 + [ϕ, ·]µ = 0. A necessary and sufﬁcient condition for ((A, A∗ ), µ, γµ,π ) to be a Lie bialgebroid is the generalized Poisson condition, dµ ([π, π]µ ) = 0,

(4.5)

which includes, as a special case, the generalized classical Yang–Baxter equation, and which is equivalent to the conditions to be found in [37, p. 74] and [40, Theorem 2.1]. A sufﬁcient condition is that π satisfy the Poisson condition, [π, π]µ = 0,

(4.6)

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which generalizes both the classical Yang–Baxter equation and the deﬁnition of Poisson bivectors. This condition is satisﬁed if and only if the graph of π is a Dirac subbundle of the standard Courant algebroid, A ⊕ A∗ , the double of the Lie bialgebroid with trivial cobracket, ((A, A∗ ), µ, 0). (See [16] for the case where A = T M, and [39].) The Lie bialgebroid deﬁned by (A, π), where π satisﬁes (4.6) is called a triangular Lie bialgebroid [40]. By Theorem 3.2, a deriving operator for the Courant bracket of the double of the Lie quasi-bialgebroid ((A, A∗ ), µ, γµ,π , − 12 [π, π]µ , 0) is dµ − ∂π + iϕ , where ∂π is the graded commutator [iπ , dµ ], and ϕ = − 12 [π, π]µ . In fact, [38, 33], ∂π generates the bracket [ , ]µ,π of A∗ . If π satisﬁes the Poisson condition (4.6), then dµ − ∂π is a deriving operator. Dually, ((A∗ , A), γµ,π , µ, 0, ψ), with ψ = − 12 [π, π]µ , is a quasi-Lie bialgebroid, and ((A∗ , A), γµ,π , µ) is a Lie bialgebroid if and only if π satisﬁes equation (4.5). 4.1.2 Twisting of a proto-bialgebroid The Lie quasi-bialgebroid (A, A∗ ) and the dual quasi-Lie bialgebroid (A∗ , A) are the result of the twisting by the bivector π of the Lie bialgebroid with trivial cobracket, ((A, A∗ ), µ, 0). The operation of twisting, in this general setting of the theory of Lie algebroids, was deﬁned and studied by Roytenberg in [48]. He showed that one can also twist a proto-bialgebroid, ((A, A∗ ), µ, γ , ϕ, ψ), by a bivector π . The result is a proto-bialgebroid deﬁned by (µ π , γπ , ϕπ , ψπ ), where µ π = µ + π ψ, γπ

(4.7)

= γ + γµ,π + (∧ π )ψ, 1 ϕπ = ϕ − dγ π − [π, π]µ + (∧3 π )ψ, 2 ψπ = ψ. 2

(4.8) (4.9) (4.10)

Here π ψ is the A-valued 2-form on A such that (π ψ)(x, y)(ξ ) = ψ(x, y, π ξ ), E for all ξ ∈ (A∗ ), and ( 2 π )ψ is the A∗ -valued 2-form on A∗ such that ((∧2 π )ψ)(ξ, η)(x) = ψ(π ξ, π η, x), E E for all x ∈ A, while ( 3 π )ψ is the section of 3 A such that for ξ, η and ζ ∈ (A∗ ), ((∧3 π )ψ)(ξ, η, ζ ) = ψ(π ξ, π η, π ζ ). A computation shows that the tensors introduced above satisfy the relations

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π ψ = {π, ψ}, 1 (∧2 π )ψ = {π, {π, ψ}}, 2 1 3 (∧ π )ψ = {π, {π, {π, ψ}}}. 6 These relations are used to prove that ((A, A∗ ), µ π , γπ , ϕπ , ψ) is a proto-bialgebroid. This proto-bialgebroid is a Lie quasi-bialgebroid if and only if ψ = 0, that is, if the initial object itself was a Lie quasi-bialgebroid. It is a quasi-Lie bialgebroid if and only if ϕπ = 0, that is, 1 ϕ − dγ π − [π, π]µ + (∧3 π )ψ = 0. 2

(4.11)

We now list the particular cases of this construction that lead to the various integrability conditions to be found in the literature. (a) Twist of a Lie bialgebroid. (µ, γ , 0, 0) → (µ, γ + γµ,π , −dγ π − 12 [π, π]µ , 0). The result is a Lie quasi-bialgebroid; furthermore, it is a Lie bialgebroid if and only if the bivector π satisﬁes the Maurer–Cartan equation, 1 dγ π + [π, π]µ = 0. 2

(4.12)

This condition is satisﬁed if and only if the graph of π is a Dirac subbundle of the Courant algebroid, A ⊕ A∗ , the double of the Lie bialgebroid ((A, A∗ ), µ, γ , 0, 0) [39]. A necessary and sufﬁcient condition for ((A, A∗ ), µ, γ + γµ,π ) to be a Lie bialgebroid is the weaker condition, dµ (dγ π + 12 [π, π]µ ) = 0. If the cobracket γ of (A, A∗ ) is trivial, to (µ, 0, 0, 0) there corresponds the quadruple (µ, γµ,π , − 12 [π, π]µ , 0): this is the case studied in Section 4.1.1. We know that the result is a Lie quasi-bialgebroid, and it is a Lie bialgebroid if and only if π satisﬁes the Poisson condition (4.6), and that ((A, A∗ ), µ, γµ,π ) is a Lie bialgebroid if and only if the bivector π satisﬁes the generalized Poisson condition (4.5). If the bracket µ of (A, A∗ ) is trivial, to (0, γ , 0, 0) there corresponds the quadruple (0, γ , −dγ π, 0), which gives rise to a Lie bialgebroid if and only if dγ π = 0,

(4.13)

which means that the bivector π on A is closed, when considered as a 2-form on A∗ . (b) Twist of a Lie quasi-bialgebroid. (µ, γ , ϕ, 0) → (µ, γ + γµ,π , ϕπ , 0), where ϕπ = ϕ − dγ π − 12 [π, π]µ . The result is a Lie quasi-bialgebroid; furthermore, it is a Lie bialgebroid if and only if the bivector π and the 3-vector ϕ satisfy the quasiMaurer–Cartan equation 1 dγ π + [π, π]µ = ϕ. (4.14) 2 A necessary and sufﬁcient condition for the pair ((A, A∗ ), µ, γ + γµ,π ) to be a Lie bialgebroid is the weaker condition, dµ ϕπ = 0.

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Assume that the cobracket γ of (A, A∗ ) is trivial. Then, in order for (µ, 0, ϕ, 0) to deﬁne a Lie quasi-bialgebroid, the 3-vector ϕ must satisfy {µ, ϕ} = 0. In this case, condition (4.14) reduces to 1 [π, π]µ = ϕ, (4.15) 2 which is a quasi-Poisson condition, analogous to (1.1). (c) Twist of a quasi-Lie bialgebroid. (µ, γ , 0, ψ) → (µ π , γπ , ϕπ , ψ), where µ π = E E µ + π ψ, γπ = γ + γµ,π + ( 2 π )ψ and ϕπ = −dγ π − 12 [π, π]µ + ( 3 π )ψ. The result is a proto-bialgebroid; furthermore, it is a quasi-Lie bialgebroid if and only if the bivector π and the 3-form ψ satisfy the Maurer–Cartan equation with background ψ or ψ-Maurer–Cartan equation, 1 dγ π + [π, π]µ = (∧3 π )ψ. 2

(4.16)

Assume that the cobracket γ of (A, A∗ ) is trivial. Then, in order for (µ, 0, 0, ψ) to deﬁne a quasi-Lie bialgebroid, the 3-form ψ must be dµ -closed. In this case, condition (4.16) reduces to the Poisson condition with background ψ or ψ-Poisson condition, 1 [π, π]µ = (∧3 π )ψ, 2

(4.17)

to be found in [46, 29, 52]. We shall now consider in greater detail two particular cases of the above construction of a Lie quasi-bialgebroid from a given Lie quasi-bialgebroid equipped with a bivector. 4.1.3 Lie quasi-bialgebras and r-matrices When the base manifold ofE a Lie algebroid is a point,Eit reduces to a Lie algebra, g = (F, µ). An element in 2 F can be viewed as a 2 F -valued 0-cochain on g. E The triangular r-matrices are those elements r in 2 F that satisfy [r, r]µ = 0. Let us explain why the twisting deﬁned by a bivector generalizes the operation of twisting deﬁned on Lie bialgebras, and more generally, on Lie quasi-bialgebras, by Drinfeld [17], and further studied in [32] and [8]. In this case, formula (4.1) reduces to [ξ, η]µ,r = −(dµ r)(ξ, η).

(4.18)

Here dµ r is the Chevalley–Eilenberg coboundary of r, a 1-cochain on g with values E in 2 g. This formula is indeed that of the cobracket on F , obtained by twisting a Lie E bialgebra with vanishing cobracket by an element r ∈ 2 F (see [17, 32]). Formulas (4.3) and (4.4) also reduce to the known fomulas. Then ((F, F ∗ ), µ, −dµ r) is a Lie bialgebra if and only if dµ [r, r]µ = 0, i.e., if and only if r satisﬁes the generalized classical Yang–Baxter equation. A sufﬁcient

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condition is that r satisfy the classical Yang–Baxter equation, [r, r]µ = 0, in which case r is a triangular r-matrix. In this purely algebraic case, the Courant bracket of F ⊕ F ∗ is skew-symmetric, and therefore is a true Lie algebra bracket. It satisﬁes ∗γ

[x, ξ ] = −[ξ, x] = −iξ dγ x + ix dµ ξ = −adξ x + adx∗µ ξ, and therefore coincides with the bracket of the Drinfeld double. A deriving operator for the Lie bracket of the Drinfeld double of a Lie protobialgebra ((F, F ∗ ), µ, γ , ϕ, ψ) is dµ − ∂γ + iϕ + eψ , where dµ (resp., ∂γ ) is the generalization of the Chevalley–Eilenberg cohomology (resp., homology) operator of (F, µ) (resp., (F ∗ , γ )) to the case where the bracket µ (resp., γ ) does not necessarily satisfy the Jacobi identity. 4.1.4 Tangent bundles and Poisson bivectors E When A = T M, the tangent bundle of a manifold M, a section π of 2 A is a bivector ﬁeld on M. Let µLie be the function deﬁning the Lie bracket of vector ﬁelds, and more generally the Schouten bracket of multivector ﬁelds. The associated differential is the de Rham differential of forms, which we denote by d. In this case, we denote the bracket of forms, deﬁned by formula (4.1) above, simply by [ , ]π and the function γµ,π simply by γπ . Thus ((T M, T ∗ M), µLie , γπ , ϕ, 0), with ϕ = − 12 [π, π], is a Lie quasi-bialgebroid, and if [π, π] = 0, i.e., π is a Poisson bivector, then ((T M, T ∗ M), µLie , γπ ) is a Lie bialgebroid. The bracket [ , ]π is then the Fuchssteiner–Magri–Morosi bracket [19, 44], its extension to forms of all degrees being the Koszul bracket [38]. A deriving operator for the Courant bracket of the double, T M ⊕ T ∗ M, of the Lie bialgebroid of a Poisson manifold is d − ∂π , where ∂π = [iπ , d] is the Poisson homology operator, often called the Koszul–Brylinski operator, deﬁned by Koszul in [38] and studied by Huebschmann in [22], in the more general framework of the Lie– Rinehart homology which he introduced. Indeed, it is well known that the operator ∂π generates the Koszul bracket of forms. This was in fact the original deﬁnition given by Koszul in [38]. This deriving operator is of square 0. We can also consider the dual object. Whenever π is a bivector ﬁeld on M, ((T ∗ M, T M), γπ , µLie , 0, ψ), with ψ = − 12 [π, π], is a quasi-Lie bialgebroid, which, when π is a Poisson bivector, is the Lie bialgebroid dual to (T M, T ∗ M). If M is orientable with volume form ν, a deriving operator for the Courant bracket of the double, T ∗ M ⊕T M, is dπ −∂ν , where ∂ν = −∗−1 d∗ (here, ∗ is the operator on multivectors deﬁned by ν). In fact, the operator ∂ν generates the Schouten bracket of multivector ﬁelds [38, 35]. To obtain a deriving operator of square 0, we must add to d−∂ν the derivation eXν , where Xν is the modular vector ﬁeld of the Poisson manifold (M, π ) associated with the volume form ν. In the nonorientable case, one should introduce densities as in [18]. If π is invertible, with inverse , then ∂ = [i , dπ ] generates the Schouten bracket [33] and therefore dπ − ∂ is a deriving operator of square 0 for the Courant bracket of T ∗ M ⊕ T M.

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4.2 Courant bracket of Poisson structures with background 4.2.1 Courant bracket with background E Let (A, µ) be a Lie algebroid and let ψ be a 3-form on A, a section of 3 A∗ . Then, as we remarked in Section 4.1.2, ((A, A∗ ), µ, 0, 0, ψ), is a quasi-Lie bialgebroid if and only if the 3-form ψ is dµ -closed, dµ ψ = 0. This is the most general quasi-Lie bialgebroid with trivial cobracket. By deﬁnition, the functions µ and ψ satisfy {µ, µ} = 0 and {µ, ψ} = 0, so that µ deﬁnes a Lie algebroid bracket, but we obtain a Lie bialgebroid if and only if ψ = 0. The bracket of the double A ⊕ A∗ (in the case of T M ⊕ T ∗ M) was introduced by Ševera and Weinstein [52] who called it the modiﬁed Courant bracket or the Courant bracket with background ψ. This bracket satisﬁes [x, y] = [x, y]µ + ix∧y ψ,

[x, ξ ] = Lµ x ξ,

[ξ, x] = −ix dµ ξ,

[ξ, η] = 0,

that is, [x + ξ, y + η] = [x, y]µ + Lµ x η − iy dµ ξ + ix∧y ψ. By Theorem 3.2, dµ + eψ is a deriving operator of the Courant bracket with background ψ. ∗µ In the case of a Lie algebra, (F, µ), [x, ξ ] = −[ξ, x] = adx ξ . Remark. In [8], we considered the case of the most general Lie quasi-bialgebra with trivial cobracket. Similarly, one can consider the Lie quasi-bialgebroids of the form (µ, 0, ϕ, 0), with {µ, µ} = 0 and {µ, ϕ} = 0, and the Courant bracket with background ϕ, a 3-vector in this case, [x, y] = [x, y]µ ,

[x, ξ ] = Lµ x ξ,

[ξ, x] = −ix dµ ξ,

[ξ, η] = iξ ∧η ϕ,

so that [x + ξ, y + η] = [x, y]µ + iξ ∧η ϕ + Lµ x η − iy dµ ξ. This case is not dual to the preceding one. 4.2.2 Twisting of the Courant bracket with background Let ((A, A∗ ), µ, 0, 0, ψ) be a quasi-Lie bialgebroid with trivial cobracket, where ψ is the background dµ -closed 3-form. For the corresponding Courant bracket with E background, we shall describe the twisting deﬁned as above by a section π of 2 A. The twisting of this quasi-Lie bialgebroid, a special case of that described in Section 4.1.2, yields a proto-bialgebroid whose structural elements depend on µ, ψ, and π , ), and which we shall denote by ( µ, γ, ϕ, ψ µ = µ + π ψ,

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γ = γµ,π + (∧2 π )ψ, 1 ϕ = − [π, π]µ + (∧3 π )ψ, 2 = ψ. ψ We have seen in Section 4.1.2(c) that the resulting twisted object is a quasi-Lie bialgebroid if and only if ϕ = 0, i.e., π satisﬁes the ψ-Poisson condition (4.17), 1 [π, π]µ = (∧3 π )ψ. 2 It was shown in [52] that this condition is satisﬁed if and only if the graph of π is a Dirac subbundle in the Courant algebroid with background, A ⊕ A∗ , the double of the quasi-Lie bialgebroid ((A, A∗ ), µ, 0, 0, ψ). This constitutes a generalization of the property valid in the usual case, reviewed in Section 4.1.1, where ψ = 0 and condition (4.17) reduces to the usual Poisson condition.E E The associated derivations, on ( • A∗ ) and on ( • A), are d µ = dµ + iπ ψ , d γ = [π, ·]µ + i(E2 π )ψ . }, the derivation d µ, µ} = −{ γ,ψ Because 12 { µ does not have vanishing square in general. On the other hand, whenever π satisﬁes the ψ-Poisson condition, d γ is a ∗ ), and a true Gerstenhaber true differential and γ deﬁnes a true Lie bracket on (A E bracket on ( • A∗ ), the modiﬁed Koszul bracket. We now consider the Courant bracket of the associated double, the π -twisted µ Courant bracket with background ψ. The mixed terms are [x, ξ ] = −iξ d γ x + Lx ξ γ and [ξ, x] = Lξ x − ix d µ ξ ; therefore, [x, y] = [x, y]µ + (π ψ)(x, y),

(4.19)

[x, ξ ] = −iξ [π, x]µ − (π ψ)(x, π ξ ) + ix dµ ξ + ix∧π ξ ψ + dµ ξ, x,

(4.20)

[ξ, x] = iξ [π, x]µ + (π ψ)(x, π ξ ) + [π, ξ, x]µ − ix dµ ξ − ix∧π ξ ψ,

(4.21)

[ξ, η] = [ξ, η]µ,π + iπ ξ ∧π η ψ.

(4.22)

In particular, for ψ = 0, we obtain the Courant bracket of the double of the twist by π of the Lie bialgebroid with trivial cobracket, ((A, A∗ ), µ, 0), considered in Section 4.1.1. Therefore, whenever π satisﬁes the generalized Poisson condition (4.5), the above formulas yield the Courant bracket of the double of the Lie bialgebroid ((A, A∗ ), µ, γµ,π ). In the purely algebraic case, we recover the Drinfeld double of a coboundary Lie bialgebra, deﬁned by r, an r-matrix solution of the generalized ∗µ Yang–Baxter equation. Setting r(ξ ) = iξ r, and using the relation ix dµ ξ = adx ξ , we obtain [x, y] = [x, y]µ ,

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[x, ξ ] = −[ξ, x] = −r(adx∗µ ξ ) + adxµ (rξ ) + adx∗µ ξ, ∗µ

∗µ ξ. [ξ, η] = adrξ η − adrη

To conclude, we prove a property of the Lie bracket deﬁned by γ on (A∗ ). Proposition 4.1. If π satisﬁes the ψ-Poisson condition, the mapping π is a morphism of Lie algebroids fom A∗ with the Lie bracket (4.22) to A with the Lie bracket [ , ]µ . Proof. It is clear that the anchor of A∗ is ρA ◦ π . To prove that π satisﬁes π [ξ, η] = [π ξ, π η]µ ,

(4.23)

for all ξ and η ∈ (A∗ ), we recall the relation, π [ξ, η]µ,π − [π ξ, π η]µ =

1 [π, π]µ (ξ, η), 2

(4.24)

proved in [37]. In view of (4.22), where the bracket of A∗ is expressed in terms of [ , ]µ,π and ψ, and of the equality, ((∧3 π )ψ)(ξ, η) = −π (iπ ξ ∧π η ψ), we see that when π satisﬁes the ψ-Poisson condition (4.17), equation (4.23) follows from equation (4.24).

Conclusion In the preceding discussion, we have encountered various weakenings and generalizations of the usual notions of Lie bialgebra, Lie algebroid, and Poisson structure that have appeared in the literature, starting with Drinfeld’s semiclassical limit of quasiHopf algebras, and up to the recent developments due in great part to Alan Weinstein, his coworkers and his former students. We hope to have clariﬁed the relationships and properties of these structures. Acknowledgments It is a pleasure to thank Henrique Bursztyn, Johannes Huebschmann, James D. Stasheff and Thomas Strobl for their comments on an earlier version of this text.

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[49] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in T. Voronov, ed., Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemporary Mathematics, Vol. 315, American Mathematical Society, Providence, RI, 2002, 169–185. [50] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys., 46 (1998), 81–93. [51] P. Ševera, Quantization of Poisson families and of twisted Poisson structures, Lett. Math. Phys., 63 (2003), 105–113. [52] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl., 144 (2001), 145–154. [53] J. D. Stasheff, Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraic topics related to Alan’s interests, in J. E. Marsden and T. S. Ratiu, eds., The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein (this volume), Progress in Mathematics, Vol. 232, Birkhäuser, Boston, 2005, 583–602. [54] K. Uchino, Remarks on the deﬁnition of a Courant algebroid, Lett. Math. Phys., 60 (2002), 171–175. [55] A. Vaintrob, Lie algebroids and homological vector ﬁelds, Uspekhi Mat. Nauk, 52-2 (1997), 161–162 (in Russian); Russian Math. Surveys, 52-2 (1997), 428–429 (in English). [56] T. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids 1 in T. Voronov, ed., Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemporary Mathematics, Vol. 315, American Mathematical Society, Providence, RI, 2002, 131–168. [57] T. Voronov, Higher derived brackets and homotopy algebras, preprint, 2003; math.QA/ 0304038. [58] A. Weinstein, The symplectic structure on moduli space, in H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, eds., The Floer Memorial Volume, Progress in Mathematics, Vol. 133, Birkhäuser, Basel, 1995, 627–635. [59] A. Weinstein, Omni-Lie algebras, in Microlocal Analysis of the Schrödinger Equation and Related Topics (Kyoto 1999), Sûrikaisekikenkyûsho Kôkyûroku (RIMS), Vol. 1176, RIMS, Kyoto University, Kyoto, 2000, 95–102. [60] P. Xu, Momentum maps and Morita equivalence, preprint, 2003; math.SG/0307319.

Minimal coadjoint orbits and symplectic induction∗ Bertram Kostant Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 [email protected] To Alan Weinstein, with admiration and respect. Abstract. Let (X, ω) be an integral symplectic manifold and let (L, ∇) be a quantum line bundle, with connection, over X having ω as curvature. With this data one can deﬁne an ω ), where dim X = 2 + dim X. It is then shown that induced symplectic manifold (X, X We consider the possibility that prequantization on X becomes classical Poisson bracket on X. is the coadjoint orbit of some larger Lie if X is the coadjoint orbit of a Lie group K, then X group G. We show that this is the case if G is a noncompact simple Lie group with a ﬁnite center and K is the maximal compact subgroup of G. The coadjoint orbit X arises (Borel–Weil) from the action of K on p, where g = k + p is a Cartan decomposition. Using the Kostant– Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic ω ) ∼ isomorphism (X, X = (Z, ωZ ), where Z is a nonzero minimal “nilpotent’’ coadjoint orbit of G. This is applied to show that the split forms of the ﬁve exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

0 Introduction 0.1 Let (X, ω) be a connected symplectic manifold and let Ham(X) be the Lie algebra of all smooth Hamiltonian vector ﬁelds on X. The space C ∞ (X) of all smooth C-valued functions on X is a Lie algebra under Poisson bracket. To any ϕ ∈ C ∞ (X) there corresponds ξϕ ∈ Ham(X) and ϕ → ξϕ realizes C ∞ (X) as a Lie algebra central extension 0 −→ C −→ C ∞ (X) −→ Ham(X) −→ 0 (0.1) of Ham(X) by the constant functions. Prequantization, when it exists, is a speciﬁc representation of the Poisson Lie algebra C ∞ (X) which does not descend to Ham(X) (Heisenberg-like—it is nontrivial on the constant functions). A necessary and sufﬁcient condition for prequantization is that the deRham class [ω] ∈ H 2 (X, R) lies in the image of the natural map ∗ This research was supported in part by NSF contract DMS-0209473 and the KG&G Foun-

dation.

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B. Kostant

H 2 (X, Z) → H 2 (X, R). In such a case we will say (Xω) (or just X if ω is understood) is integral. If (Xω) is integral, there exists a complex line bundle L (the quantum line bundle) with connection ∇ over X such that ω = curv(L, ∇).

(0.2)

Using the connection, one deﬁnes the covariant derivative ∇ξ s of any smooth section s of L by any vector ﬁeld (v.f.) on X. Furthermore, the connection can (and will) be chosen so that there exists a Hilbert space structure in each ﬁber of L which is invariant under parallelism. By considering only the unit circles in each ﬁber of L one obtains a principal U (1)-bundle 1 U (1) −→ L ⏐ ?τ

(0.3)

X. The connection deﬁnes a real U (1)-invariant 1-form α on L1 which on each ﬁber corresponds to dθ/2π on U (1), and one has dα = τ ∗ (ω).

(0.4)

Let S be the linear space of all smooth sections of L. Then prequantization is the Lie algebra representation π of C ∞ (X) on S given by π(ϕ)s = (∇ξϕ + 2π iϕ)s.

(0.5)

Let ζ be the vertical vector ﬁeld on L1 (generating the U (1)-action) such that α, ζ = −1. One has a linear isomorphism S → S ⊂ C ∞ (L1 ),

s → s,

where S = {f ∈ C ∞ (L1 ) | ζf = 2π if } and an (associative) algebra isomorphism ⊂ C ∞ (L1 ), C ∞ (X) → C

ϕ → ϕ,

= {f ∈ C ∞ (L1 ) | ζf = 0}. where C = L1 × R+ so that Now let X = 2 + dim X. dim X

(0.6)

Let r ∈ C ∞ (R + ) be the natural coordinate function on R+ so that if t ∈ R+ , then by putting ωX r(t) = t. One deﬁnes a symplectic form ωX on X = d(rα) so that ω, ωX = dr ∧ α + r

(0.7)

where we have put ω = dα. One notes that ξr = ζ and hence (X, ω) arises from ωX (X, ) by (Marsden–Weinstein) symplectic reduction on the hypersurface r = 1. ωX Reversing the direction we refer to the construction of (X, ) from (X, ω) as symplectic induction. Among the statements in the following theorem is the result that In a word, quantization, at least at the prequantization in X is Poisson bracket in X. prequantized level, is classical mechanics two dimensions higher.

Minimal coadjoint orbits and symplectic induction

393

Theorem 0.1. The map C ∞ (X) → C ∞ (X),

ϕ → r ϕ

(0.8)

is a monomorphism of Poisson Lie algebras. Moreover, for any s ∈ S and ϕ ∈ C ∞ (X), one has = [r π(ϕ)(s) ϕ , s]. (0.9) See Theorem 1.6. 0.2 If (X, ω) is the coadjoint orbit of some Lie group K, where ω is the KKS-symplectic ωX form, consider the possibility that (X, ) is the coadjoint orbit of some larger Lie group G. It will be the main theorem of this paper to show that this indeed is true ωX in an important specialized case and that in this case (X, ) is a minimal coadjoint orbit of G. Assume that K is a compact connected Lie group. If V is a ﬁnite-dimensional, complex irreducible module for K (and hence for its complexiﬁcation KC ), then by the Borel–Weil theorem there corresponds to V an integral coadjoint orbit X(V ) ⊂ k∗ , where k = Lie K and k∗ is the dual to k. In fact, X = X(V ) is isomorphic to the unique closed KC -orbit in the projective space Proj V and L is deﬁned by considering the cone over this orbit in V . The orbit is the projective image of the afﬁne variety of extremal weight vectors in V . Now assume that G is a noncompact Lie group with ﬁnite center such that g = Lie G is simple and that K is a maximal compact subgroup of G. Then G/K is a noncompact symmetric space and one has a Cartan decomposition g = k + p. The complexiﬁcation pC is a K (and hence KC ) module via the adjoint representation. There are two cases to be considered. We will say that g is of non-Hermitian type if G/K is non-Hermitian and g is of Hermitian type if G/K is Hermitian. Let I be an index parameterizing the irreducible K-submodules V i , i ∈ I of pC . In the non-Hermitian case I has 1 element (i.e., pC is irreducible ) and I has 2 elements in the Hermitian case. In any case let Xi = X(V i ), i ∈ I , so that X i is an integral coadjoint orbit of K. Let X = Xi , i ∈ I . In the Hermitian case X = −X and {X, −X} = {Xi }, i ∈ I . In the non-Hermitian case X = −X. The Kostant–Sekiguchi correspondence is a correspondence between nilpotent KC -orbits in pC and G-nilpotent orbits in g (or, equivalently, G-coadjoint “nilpotent’’ orbits in g∗ ). See [S] or [V]. Now the afﬁne variety of extremal weight vectors E i in V i is a KC -orbit and, by the Kostant–Sekiguchi correspondence, corresponds to a “nilpotent’’ G-coadjoint orbit Z i in the dual space g∗ . If Z = Z i , then Z = −Z in the Hermitian case and {Z, −Z} = {Z i }, i ∈ I . In the non-Hermitian case Z = −Z and in any case if Y is any nonzero coadjoint G-orbit, then dim Y ≥ dim Z.

(0.10)

On the other hand, C ∞ (Z) is a Lie algebra under Poisson bracket with respect to the KKS symplectic form ωZ on Z, and since Z is a coadjoint orbit, one has a Lie algebra embedding

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B. Kostant

g → C ∞ (Z).

(0.11)

It follows from (0.10) and the orbit covering theorem that dim Z is the smallest possible dimension of a symplectic manifold which has g as a subalgebra of functions under Poisson bracket. A remarkable theorem of Michèle Vergne asserts that Kostant– Sekiguchi corresponding orbits in general are K-diffeomorphic. In the present case because of the minimality (0.10) Vergne’s diffeomorphism can be given very simply and it leads to a K-diffeomorphism → Z. X

(0.12)

dim Z = 2 + dim X.

(0.13)

In particular, The following is our main theorem. In effect, it says by symplectically inducing the K-coadjoint orbit X “sees’’ the noncompact simple Lie algebra g. Theorem 0.2. The map (0.12), now written ωX β : (X, ) → (Z, ωZ ),

(0.14)

is a symplectic diffeomorphism so that one has a (minimal) Lie algebra injection g → C ∞ (X).

(0.15)

Furthermore, if µ is the moment map with respect to the action of K on Z, then µ(Z) = R+ X.

(0.16)

See Theorems 3.10, 3.13, and 3.16. Remark 0.3. Note that (0.15) points to an interesting difference between (X, ω) and ωX the induced symplectic manifold (X, ) in that one cannot have a nontrivial Lie algebra homomorphism g → C ∞ (X) since dim X = dim X−2. Indeed, the statement is the smallest possible dimension of a above concerning Z now implies that dim X symplectic manifold W which admits an embedding of g as a Lie algebra of functions on W under Poisson bracket. 0.3 We say that g is Omin -split if gC is a simple complex Lie algebra and g ∩ Omin = ∅, where Omin is the minimal nilpotent orbit in gC . It is easy to see that g is Omin -split if and only if Cent n ∩ Omin = ∅, where n is the nilpotent component of an Iwasawa decomposition of g (see the argument in the proof of Theorem 3.13). The following result is Theorem 3.17. Theorem 0.4. The simple Lie algebra g is Omin -split if and only if dim Cent n = 1.

(0.17)

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395

Remark 0.5. Another criterion for the Omin -split condition was cited on the top of [B-K, p. 19]. Proposition 3.19 is stated here as follows. Proposition 0.6. If g is Omin -split (e.g., if g is split), then X is not only a K-symmetric space but, in fact, X is a Hermitian symmetric space. Assume that g is Omin -split and g is of non-Hermitian type so that kC is semisimple. Let K be a noncompact real form of KC having Kν as a maximal compact subgroup so that K /Kν = X is the noncompact symmetric dual to the compact symmetric space X. One can then show that not only is X a complex bounded domain but, in fact, X is a tube domain. In particular, by the Kantor–Koecher–Tits theory, X corresponds to a formally real Jordan algebra J (X). If g is a split form of any one of ﬁve exceptional simple Lie algebras, g is nonHermitian so that the statement above applies to g. But a minimal (dimensional) symplectic realization of g as functions on a symplectic manifold is achieved when of a coadjoint orbit of a compact Lie group the manifold is the induced symplectic X K. The group K turns out to be classical in all ﬁve cases so that the exceptional Lie algebras g emerge symplectically from the symplectic induction of a classical coadjoint orbit. In Section 3.4 we present a table which contains the relevant information. The cases of E6 , E7 , and E8 are taken from [B-K]. 0.4 We wish to thank Ranee Brylinski for many conversations on related matters at an earlier time. A number of ideas in this paper evolved when [B-K] was written. This is particularly true of Theorem 1.7, which therefore should be considered collaborative.

ω ) 1 Symplectic Induction and the construction of (X, X 1.1 In this section we recall the theory of prequantization. See [K-1]. Let (X, ω) be a connected symplectic manifold. For any ϕ ∈ C ∞ (X) one deﬁnes a Hamiltonian vector ﬁeld ξϕ on X so that for any vector ﬁeld (v.f.) η, one has ηϕ = ω(ξϕ , η). Poisson bracket in C ∞ (X) is deﬁned by putting [ϕ, ψ] = ξϕ ψ for any ϕ, ψ ∈ C ∞ (X). If Ham(X) = {ξϕ | ϕ ∈ C ∞ (X)} is the Lie algebra of all Hamiltonian vector ﬁelds on X, then C ∞ (X), under Poisson bracket, is a Lie algebra central extension 0 −→ C −→ C ∞ (X) −→ Ham(X) −→ 0 (1.1) of Ham(X), by the constant functions on X, via the map ϕ → ξϕ .

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B. Kostant

We now assume that the de Rham class [ω] ∈ H 2 (X, R) is integral (i.e., [ω] is in the image of the natural map H 2 (X, Z) → H 2 (X, R)). Then one knows that there exists a complex line bundle L with connection ∇, over X, such that ω = curv(L, ∇).

(1.2)

Using the connection one deﬁnes the covariant derivative ∇ξ s of any section s of L (either local or global, but always assumed to be inﬁnitely smooth) by any v.f. ξ on X. If L∗ is L minus the zero section, then L∗ is a principal C∗ bundle C∗ −→ L⏐∗ ? X.

(1.3)

In addition there exists a C∗ -invariant 1-form α on L∗ which, on any ﬁber of (1.3), ∗ pulls back to 2π1 i dz z on C and is such that for any local section s of (1.3), and any v.f. ξ on X, one has ∇ξ s = 2π is ∗ (α), ξ s (1.4) on the domain of s and on this domain d(s ∗ (α)) = ω.

(1.5)

The connection ∇ may be (and will be) chosen so that there exists a Hilbert space structure on each (one-dimensional) ﬁber of L which is invariant under parallel transport. See [K-1, Proposition 2.1.1]. One then deﬁnes a principal U (1) = {eiθ | θ ∈ R} bundle L1 over X with bundle projection τ , 1 U (1) −→ L ⏐ ?τ

(1.6)

X by taking the ﬁbers of L1 to be the unit circles in the corresponding ﬁbers of L. The restriction of α to L1 is real and on each ﬁber of L1 , pulls back, via (1.6), to the 1-form dθ 2π on U (1). Henceforth we will identify α with this restriction, and (1.5) implies dα = τ ∗ (ω). Obviously, dim L1 = 1 + dim X. 1.2 Let S be the space of all smooth global sections of L. Let π : C ∞ (X) → End S be deﬁned by putting, for any ϕ ∈ C ∞ (X) and any s ∈ S,

Minimal coadjoint orbits and symplectic induction

π(ϕ)(s) = (∇ξϕ + 2π iϕ)s.

397

(1.7)

The correspondence ϕ → π(ϕ) is called prequantization and it is a result (see [K-1, Theorem 4.3.1]) that π is a Lie algebra representation of C ∞ (X) on S. ([K-1, Theorem 4.3.1] is stated for real-valued functions on X. It trivially extends to C ∞ (X) by complex linearity.) Recalling (1.1), one notes that π does not descend to a representation of Ham(X). Now with respect to the multiplication action of U (1) on C, we may write L = L1 ×U (1) C so that L is associated to the principal bundle L1 . If one wishes, S may then be identiﬁed with the subspace S ⊂ C ∞ (L1 ) deﬁned by putting S = {f ∈ C ∞ (L1 ) | f (qc) = c−1 f (q) ∀q ∈ L1 , c ∈ U (1)}.

(1.8)

The image of s ∈ S under the linear isomorphism S → S

(1.9)

will be denoted by s. A vector ﬁeld on L1 will be called vertical if it is tangent to the ﬁbers of L1 . One notes that there there exists a unique (real) vertical ﬁeld ζ on L1 such that α, ζ = −1.

(1.10)

With respect to an isomorphism of U (1) with a ﬁber of L1 , note that the restriction of d ζ to that ﬁber corresponds to −2π dθ on U (1). Clearly, the subspace S may be also given by S = {f ∈ C ∞ (L1 ) | ζf = 2π if }. (1.11) Now for any ϕ ∈ C ∞ (X), let ϕ ∈ C ∞ (L1 ) be given by the pullback ϕ = τ ◦ ϕ and ∞ = { let C ϕ | ϕ ∈ C (X)}. Clearly, = {f ∈ C ∞ (L1 ) | ζf = 0}. C

(1.12)

A vector ﬁeld on L1 will be called horizontal if it is orthogonal to α. If ξ is any v.f. on X it is clear that there exists a unique horizontal vector ﬁeld ξ on L1 such that ∞ τ∗ ( ξ ) = ξ . Now for any ϕ ∈ C (X) and vector ﬁeld ξ on X let η(ϕ,ξ ) be the v.f. on L1 deﬁned by putting ξ + ϕ ζ. (1.13) η(ϕ,ξ ) = One readily establishes the following (see [K-1, Proposition 2.9.1] for the case where C∗ is used instead of U (1)). Proposition 1.1. The vector ﬁeld η(ϕ,ξ ) on L1 is U (1)-invariant and any U (1)invariant v.f. on L1 is uniquely of this form. Now consider the question as to whether or not θ (η(ϕ,ξ ) )(α) = 0,

(1.14)

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B. Kostant

where θ (η(ϕ,ξ ) ) is Lie differentiation by η(ϕ,ξ ) ). The following result interprets prequantization as the ordinary action on S by those vector ﬁelds on L1 which (1) are U (1)-invariant and (2) annihilate α by Lie differentiation. For any ϕ ∈ C ∞ (X), let ηϕ = η(ϕ,ξϕ ) .

(1.15)

Except for the use of C∗ and L∗ instead of U (1) and L1 , the following result is [K-1, Theorem 4.2.1]. Theorem 1.2. Let ϕ ∈ C ∞ (X). Then the vector ﬁeld ηϕ on L1 (1) is U (1)-invariant and (2) annihilates α by Lie differentiation. Furthermore, any v.f. on L1 which satisﬁes (1) and (2) is uniquely of this form. Moreover, for any s ∈ S, one has = ηϕ π(ϕ)(s) s.

(1.16)

1.3 Let R+ denote the multiplicative group of positive real numbers. Let

so that

= L1 × R+ X

(1.17)

= dim X + 2. dim X

(1.18)

Since we are dealing here with a direct product of manifolds, functions, forms and and we will freely use the vector ﬁelds on L1 and R+ have obvious extensions to X same notation for the extensions. Let r ∈ C ∞ (R+ ) be the natural coordinate function. That is, r(t) = t for any t ∈ R+ . Let ωX be the real closed (in fact, exact) 2-form on deﬁned by putting ωX X = d(rα) so that ωX ω, = dr ∧ α + r

(1.19)

where we have put ω = τ ∗ (ω). as a sum of a tangent vector to R+ and By decomposing a tangent vector to X 1 a tangent vector to L and then decomposing the latter into a sum of a horizontal tangent vector (i.e., orthogonal to α) and a vertical tangent vector (tangent to a ﬁber of (1.5)), it follows easily that ω is nonsingular so one has the following. ωX Proposition 1.3. (X, ) is a symplectic manifold. It follows immediately from (1.10) and (1.19) that the interior product of ω by ζ equals dr. Consequently, one has In fact, Proposition 1.4. The vector ﬁeld ζ (see (1.10)) is Hamiltonian on X. ζ = ξr .

(1.20)

Let ϕ ∈ C ∞ (X) and s ∈ S. Note then, as a consequence of (1.11) and (1.12), that Proposition 1.14 yields

Minimal coadjoint orbits and symplectic induction

[r, ϕ] = 0, [r, s] = 2π i s.

399

(1.21)

It is obvious from the sentence after (1.10) that the Hamiltonian ﬂow of ζ is just deﬁned by its principal bundle action on L1 . As a the free action of U (1) on X consequence of (1.19), one then has the following. ωX Proposition 1.5. Marsden–Weinstein reduction of (X, ) by the function r at the value r = 1 is isomorphic to the original symplectic manifold (X, ω). ωX Since (X, ω) arises from (X, ) by symplectic reduction we can speak of the above construction of (X, ωX ) from (X, ω) as symplectic induction. 1.4 If η is a vector ﬁeld on a manifold, then ι(η) will denote the operator of interior product. We continue with the assumptions of Section 1.3. Proposition 1.6. Let ϕ ∈ C ∞ (X). Then ξ ϕ = so that if ψ ∈ C ∞ (X), then ] = [ ϕ, ψ

1 ξϕ r

(1.22)

1 [ϕ, ψ]. r

(1.23)

Proof. Since ξϕ is horizontal in L1 , clearly ι( 1r ξϕ ) annihilates dr ∧ α. But from the deﬁnition of ξϕ , obviously ι(ξϕ ) ω = d ϕ. (1.24) Thus

1 1 ι ξϕ ωX ω = ι(ξ ϕ )r r r = d ϕ.

But this establishes (1.22). The equality (1.24) is immediate from the deﬁnition of Poisson bracket.

The main part of the following result, (1.27), says that prequantization in X is ordinary Poisson bracket in the induced symplectic manifold X. Theorem 1.7. The map C ∞ (X) → C ∞ (X),

ϕ → r ϕ

(1.25)

is a monomorphism of Poisson Lie algebras. Moreover, ηϕ (see Theorem 1.2) is a for any ϕ ∈ C ∞ (X). In fact, Hamiltonian vector ﬁeld on X

400

B. Kostant

ηϕ = ξr ϕ.

(1.26)

Finally, for any s ∈ S and ϕ ∈ C ∞ (X), one has = [r π(ϕ)(s) ϕ , s].

(1.27)

Proof. Let ϕ, ψ ∈ C ∞ (X). Then by (1.21), ] = r 2 [ ]. [r ϕ, r ψ ϕ, ψ ] = r [ϕ, But then [r ϕ, r ψ ψ] by (1.23). This proves the ﬁrst statement of the theorem. But now by (1.13) and (1.15), one has ι(ηϕ )ωX ϕ ζ )(dr ∧ α + r ω) = ι(ξ ϕ + = ι(ξϕ )r ω + ι( ϕ ζ )(dr ∧ α)

(1.28)

= rd ϕ+ ϕ dr by (1.24) and (1.10). But the right-hand side of the last line of (1.28) is just d(r ϕ ). This proves (1.26). But then (1.27) follows from (1.16).

ωX Remark 1.8. The construction of the induced symplectic manifold (X, ) of course depends upon the choice of the connection ∇. The set (with an obvious equivalence relation) of all such connections is a principal homogeneous space for the character group π 1 (X) of the fundamental group π1 (X) of X. See [K-1, Theorem 2.5.1]. In ωX particular, the connection, and hence (X, ), is unique if X is simply connected.

2 The coadjoint orbit case and a minimal Kostant–Sekiguchi correspondence 2.1 We now consider the case of Section 1, where X is a coadjoint orbit of a connected Lie group K and ω is the KKS symplectic form. We do not assume now that K is compact but in the main application K will be compact. Let k = Lie K and let k∗ be the dual space to k. Let ν ∈ k∗ and let X be the K-coadjoint orbit of ν so that as K-homogenous spaces X = K/Kν , (2.1) where Kν is the isotropy subgroup at ν. The action of k ∈ K on µ ∈ X is denoted by k · µ and one has k · µ, y = µ, Ad k −1 (y) for any y ∈ k. For any x ∈ k, let ξ x be the v.f. on X deﬁned so that for any f ∈ C ∞ (X) and µ ∈ X, one has d (ξ x f )(µ) = dt ((exp(−tx) · µ)|t=0 . Then x → ξ x deﬁnes the inﬁnitesimal action of k on X corresponding to the group action of K on X. The KKS symplectic form ω on X is such that for x, y ∈ k and µ ∈ X, ω(ξ x , ξ y )(µ) = µ, [y, x].

(2.2)

Minimal coadjoint orbits and symplectic induction

The map

k → C ∞ (X),

x → ϕ x

is a homomorphism of Lie algebras (using Poisson bracket in x ∈ k, the function ϕ x on X is deﬁned by ϕ x (µ) = µ, x

401

(2.3) C ∞ (X)),

where, for (2.4)

for any µ ∈ k∗ . Furthermore, ξ x is a Hamiltonian vector ﬁeld on X, for any x ∈ k, and, in fact (see [K-1, (5.3.5)]), ξ x = ξϕ x . (2.5) Now iR = Lie U (1) and

2π iν : kν → iR

(2.6)

is a homomorphism of Lie algebras, where kν = Lie Kν . Note that Kν may not be connected. By [K-1, Corollary 1 to Theorem 5.7.1], one has [ω] ∈ H 2 (X, R) is integral if there exists a character χ : Kν → U (1)

(2.7)

dχ = 2π iν|kν .

(2.8)

whose differental is given by Remark 2.1. In [K-1, Corollary 1 to Theorem 5.7.1], it is assumed that K is simply connected. In such a case the statement of the corollary is that [ω] ∈ H 2 (X, R) is integral if and only if there exists χ satisfying (2.7) and (2.8). The reference to the corollary is valid in the general case under consideration here since (2.7) and (2.8) for K obviously imply these conditions for the simply connected covering group of K. Furthermore, by the corollary, the choice of χ determines L1 and the connection 1-form α in L1 . See (2.9) below. Henceforth we assume (2.7) and (2.8) so that the assumptions of Section 1 are satisﬁed. Noting that K is a principal Kν -bundle over X, one constructs L1 as the associated bundle given by L1 = K ×Kν U (1), (2.9) where the action of Kν on U (1) is given by the homomorphism χ . See [K-1, Section 5.7], where we have replaced C∗ by U (1). By (2.9) the space L1 is clearly homogeneous for the product group K × U (1) (using left translation for K). In fact, as homogeneous spaces, one has an isomorphism (K × U (1))/H ∼ = L1 , where H = {(k, χ (k)−1 ) | k ∈ Kν }. In particular, this gives rise to a K × U (1)surjection σ : K × U (1) → L1 . The connection 1-form α on L1 may then be given by the equation dθ , (2.10) σ ∗ (α) = αν , 2π where αν is the left invariant 1-form on K whose value at the identity is ν. See [K-1, (5.7.3) and (5.7.4)].

402

B. Kostant

2.2 We are assuming that X is a coadjoint orbit of a connected Lie group K so that one has a Lie algebra homomorphism (2.3). We are assuming (2.7) and (2.8) so that the ωX induced symplectic manifold (X, ) exists. We now consider the possibility that (X, ωX ) may be the coadjoint orbit of some larger Lie group G. In such a case, one would have a Lie algebra homomorphism g → C ∞ (X),

(2.11)

where g = Lie G. Assume now that K is a compact connected Lie group. Let KC be a complex reductive Lie group having K as a maximal compact subgroup so that we can take kC = Lie KC , where kC is the complexiﬁcation of k. Let Bk be a nonsingular symmetric KC -invariant bilinear form on kC which is negative deﬁnite on k. We may regard the complexiﬁcation k∗C of k∗ as the dual to kC . The bilinear form Bk deﬁnes a KC -linear isomorphism γ : k∗C → kC . (2.12) Obviously, γ (k∗ ) = k. It follows that Kν is just the stabilizer of γ (ν) in K under the adjoint representation so that Kν is connected and, as one knows, X is simply connected. In addition, one knows that Kν contains a maximal torus T of K. One has γ (ν) ∈ t,

(2.13)

where t = Lie T since γ (ν) is obviously central in kν . Let h = it. The complexiﬁcation hC = tC is a Cartan subalgebra of kC and, identifying the real dual h∗ of h with γ −1 (h) one has ⊂ h∗ , where is the T weight lattice. But since 2π iν|t exponentiates to a character of T , by (2.7), one has that λ ∈ , by (2.13), where λ = 2π iν. (2.14) Let πλ : KC → Aut Vλ be the irreducible representation of KC with extremal hC -weight λ (i.e., any hC -weight of πλ lies in the convex hull of the Weyl group orbit of λ). Obviously, the complexiﬁcation (kν )C is the centralizer of γ (λ) in kC . Furthermore, γ (λ) ∈ h and hence γ (λ) is a hyperbolic element of kC . Remark 2.2. To any hyperbolic element y of a complex reductive Lie algebra s, one associates a parabolic Lie subalgebra qy (s) of s characterized as follows: the centralizer sy of y in s is a Levi factor of qy (s) and the nilradical of qy (s) is the span of the eigenvectors of ad y belonging to positive eigenvalues. In the notation of Remark 2.2, put qν = qγ (λ) (kC ) and let Qν ⊂ KC be the parabolic subgroup corresponding to qν . One readily has Kν = K ∩ Qν

(2.15)

so that, since X ∼ = K/Kν as K-homogeneous spaces, the action of K on KC /Qν induces a K-isomorphism

Minimal coadjoint orbits and symplectic induction

X → KC /Qν .

403

(2.16)

A nonzero vector in a ﬁnite-dimensional complex irreducible K (and hence KC )module is called an extremal weight vector if it is a weight vector for an extremal weight of some Cartan subalgebra of kC . The Cartan subalgebra can always be chosen so that it is the complexiﬁcation of a Cartan subalgebra of K. One knows that the set of all extremal weight vectors is of the form KC C∗ v and, in fact, is of the form KC∗ v, where v is an extremal weight vector. Let E ⊂ Vλ be the variety of extremal weight vectors in Vλ . Thus if 0 = vλ is a weight vector for the hC extremal weight λ, then E = πλ (KC )C∗ vλ . Let Proj(Vλ ) be the projective space of Vλ and let Proj(E) be the subvariety of Proj(Vλ ) deﬁned by E. One knows that Proj(E) is the unique closed KC orbit in Proj(Vλ ). Furthermore, if pλ ∈ Proj(E) is the point corresponding to C∗ vλ , then Qν is the isotropy group at pλ and hence (Borel–Weil theory) the isomorphism (2.16) deﬁnes a K-isomorphism X → Proj(E).

(2.17)

Let {u, v} be a Hilbert space structure Hλ in Vλ which is invariant under the action of πλ (K) and let E 1 = {v ∈ Vλ | {v, v} = 1}. We may choose vλ so that vλ ∈ E 1 . Now if χ is deﬁned (uniquely since Kν is connected) as in (2.7) and (2.8) so L1 is given by (2.9), let φ : L1 → E 1 (2.18) be the K × U (1)-map given by φ(k, c) = cπλ (k)(vλ ),

(2.19)

where (k, c) ∈ K × U (1). Furthermore, if t ∈ R+ , let →E : X φ

(2.20)

be the K × U (1) × R+ -map given by (q, t) = tφ(q), φ

(2.21)

where (q, t) ∈ L1 × R+ . The following is an immediate consequence of (2.17). (see (2.18) and (2.20)) are, respectively, K × Proposition 2.3. The maps φ and φ + U (1) and K × U (1) × R isomorphisms. Remark 2.4. Note that if ν = 0, then KC operates transitively on E. This is clear since if (Kν )C is the subgroup of KC corresponding to (kν )C , then obviously (Kν )C operates transitively on C∗ vλ . 2.3 We now assume that g is a real simple Lie algebra of noncompact type and G is a corresponding Lie group with ﬁnite center. We now choose K and k of Section 2.2 so

404

B. Kostant

that K is a maximal compact subgroup of G. Consequently, there is a space p ⊂ g of hyperbolic elements such that g=k+p (2.22) is a Cartan decomposition of g. Let a ⊂ p be a maximum abelian subalgebra. Let ⊂ a∗ be the set of restricted roots, and for each β ⊂ , let gβ ⊂ g be the corresponding restricted be a choice of a positive root root space. Let + ⊂ system and let n = β∈ + gβ so that g=k+a+n is an Iwasawa decomposition of g. Let n− = β∈ of a in k so that one has the direct sum

(2.23) +

g−β and let m be the centralizer

g = m + a + n + n− .

(2.24)

If β, φ ∈ , we will put β ≥ φ if β − φ is a sum of elements in + . Obviously, β ≥ φ and φ ≥ β if and only if β = φ. Let ψ ∈ + be a maximal element with respect to this ordering. Obviously, gψ ⊂ Cent n. But since Cent n is clearly stable under the action of ad a, it follows that Cent n is spanned by g β = gβ ∩ Cent n over all β ∈ + . Proposition 2.5. The restricted root ψ is the unique maximal element in

+

and

Cent n = gψ so that, in particular, Cent n is a restricted root space. Proof. Assume g β = 0 for some β ∈ + . Since m normalizes n and commutes with a, it follows that g β is stable under the adjoint action of m+a+n. Thus, corresponding to the adjoint action, for the real enveloping algebras one has UR (n− )g β = UR (g)g β by (2.24) and the PBW theorem. But by simplicity one has UR (g)g β = g. Thus β ≥ ψ. But similarly ψ ≥ β. Hence ψ = β.

Let Wa be the restricted Weyl group operating in a and let sψ ∈ Wa be the reﬂection deﬁned by ψ. Thus sψ is the identity on the hyperplane aψ = {x ∈ χ | ψ(x) = 0}. Let xψ be the unique element in a such that sψ xψ = −xψ and ψ(xψ ) = 2.

(2.25)

The Killing form on the complexiﬁcation gC of g is real on g and positive deﬁnite on p. Let B be the positive multiple (x, y) of the Killing form normalized by the condition that (xψ , xψ ) = 2. (2.26) Let Bk of Section 2.2 be deﬁned so that Bk = B|k. Let θ be the complex Cartan involution on gC corresponding to the complexiﬁed Cartan decomposition gC = kC + pC . Let σ and σu be, respectively, the conjugate linear involutions of gC deﬁned

Minimal coadjoint orbits and symplectic induction

405

by the real forms g and gu = k + ip. One notes that gu is a compact form of gC so that B is negative deﬁnite on gu . Consequently, H is an Ad gu -invariant Hilbert space structure {x, y} on gC , where {x, y} = −(x, σu y).

(2.27)

It is immediate that σ commutes with θ and that σu = θ σ . In particular, the restriction of H|g deﬁnes a K-invariant real Hilbert space structure on g and that for x, y ∈ g, {x, y} = −(x, θy).

(2.28)

Since θ is minus the identity on a, one immdiately has θ (gψ ) = g−ψ .

(2.29)

Let e ∈ gψ be such that {e, e} = 1. That is, (e, θ e) = −1

(2.30)

by (2.28). Proposition 2.6. (xψ , e, −θ e) is an S-triple. Proof. One has [xψ , e] = 2e by (2.25). But −θ e ∈ g−ψ by (2.29) so that [xψ , −θ e] = −2(−θ e). Let y = [e, −θ e]. Then [y, a] = 0 by (2.29). On the other hand, clearly θ (y) = −y so that y ∈ p. Since a is maximally commutative in p, this implies that y ∈ a. But if x ∈ a, then (x, [e, −θ e]) = ([x, e], −θ e) = ψ(x)(e, −θ e) = ψ(x)

(2.31)

by (2.30). This implies that y is B-orthogonal to aψ so that y = rxψ for some r ∈ R. But putting x = xψ in (2.31), one has (xψ , y) = 2 by (2.25). However, (xψ , xψ ) = 2 by (2.26). Thus r = 1 so that [e, −θ e] = xψ .

Let uψ be the complex TDS spanned by xψ , e, and −θ e over C. 2.4 Let the notation be as in Section 2.3. Let d = dim Cent n so that d = dim gψ

(2.32)

by Proposition 2.5. Proposition 2.7. The maximal eigenvalue of ad xψ on gC is 2 and (gψ )C is the corresponding eigenspace. In particular, the multiplicity of the eigenvalue 2 of ad xψ is d. Furthermore, if β ∈ − {ψ}, then β(xψ ) ∈ {0, 1} so that the spectrum of ad xψ in nC is nonnegative and, in fact, the spectrum is contained in the set {0, 1, 2}. The spectrum of ad xψ on gC is contained in the set {2, 1, 0, −1, −2}.

406

B. Kostant

Proof. From the representation theory of a TDS (e.g., uψ ), one has β(xψ ) ∈ Z for any β ∈ + . On the other hand, if β ∈ + , then since β + ψ cannot be a restricted root one has [e, gβ ] = 0. Thus from the representation theory of uψ , one has β(xψ ) ∈ Z+ .

(2.33)

But now ψ(xψ ) = 2 by (2.25) so that (gψ )C is contained in the eigenspace of ad xψ for the eigenvalue 2. But now using notation and the argument in the proof of Proposition 2.4, one has gC = (UR (n− )gψ )C (2.34) so that 2 is the maximal eigenvalue of ad xψ by (2.33). On the other hand, if β ∈ + and [g−β , gψ ] = 0, we assert that β(xψ ) > 0. Note that the assertion implies all the statements of the proposition, by (2.33) and (2.34). Assume the assertion is false so that β(xψ ) = 0. But β = ψ and hence ψ − β ∈ . But then β − ψ ∈ and (β − ψ)(x−ψ ) = −2. From the representation theory of a TDS, one has that [e, [e, gβ−ψ ]] = 0. But this implies that β + ψ ∈ , contradicting the maximality of ψ.

Let ei , i = 1, . . . , d, be an orthonormal basis of gψ with respect to H|gψ . We assume that the basis is chosen so that ed = e. Under the adjoint action of uψ the element ei clearly generates a three-dimensional irreducible representation ui of uψ since [e, ei ] = 0. Of course, ud = uψ . (2.35) Remark 2.8. Note that, as a consequence of Proposition 2.7, if u=

d

(2.36)

ui ,

i=1

then u is the primary component in gC for the three-dimensional irreducible representation of uψ under the adjoint action and that any irreducible component in gC /u has dimension 1 or 2. The following lemma is well known and is readily established using the commutation relations of an S-triple. Lemma 2.9. Assume that v is a complex TDS and (x , e , f ) is an S-triple whose elements span v. Then (h, v, w) is an S-triple also spanning v, where h = i(e − f ), v = 1/2(ix + e + f ),

(2.37)

w = 1/2(−ix + e + f ). Furthermore, one recovers (x , e , f ) from (h, v, w) by x = −i(v − w), e = 1/2(−ih + v + w), f = 1/2(ih + v + w).

(2.38)

Minimal coadjoint orbits and symplectic induction

407

We apply Lemma 2.9 for the case where v = uψ and (x , e , f ) = (xψ , e, −θ e). Let h = i(e + θ e), v = 1/2(ixψ + e − θ e), w = 1/2(−ixψ + e − θ e)

(2.39)

so that (h, v, w) is an S-triple whose elements span uψ . Obviously, there exists an automorphism of uψ which carries (xψ , e, −θ e) to (h, v, w). Since any automorphism of a complex TDS is inner, it follows that xψ and h are conjugate in gC by an element in Ad(gψ )C . In particular, then, by Proposition 2.7, the multiplicity of the eigenvalue 2 of ad h in gC is d. Lemma 2.10. One has h ∈ kC so that we may write d = dk + dp ,

(2.40)

where dk is the multiplicity of the eigenvalue 2 of ad h|kC and dp is the multiplicity of the eigenvalue 2 of ad h|pC . One has dp ≥ 1. In fact, v, w ∈ pC and [h, v] = 2v. Proof. Obviously, θ h = h by (2.39) so that h ∈ kC . Similarly, v, w ∈ pC since θ v = −v and θ w = −w by (2.39), noting that xψ ∈ a ⊂ pC . One has [h, v] = 2v since (h, v, w) is an S-triple.

We can strengthen Lemma 2.10. Theorem 2.11. Let the notation be as in Lemma 2.10. Then dk = d − 1 and dp = 1. That is, the one-dimensional subspace Cv is the eigenspace of ad h|pC corresponding to the eigenvalue 2. Also 2 is the highest eigenvalue of ad h|pC . Proof. The three-dimensional uψ -modules ui (see (2.36)) are, of course, equivalent to the adjoint representation of uψ . For j = 1, . . . , d, let δj : uψ → uj be the uψ -equivalence normalized so that δj e = ej . Note that δd is the identity map. Let vj = δj v. It is then immediate from Proposition 2.7 that {vj }j = 1, . . . , d, is a basis of the ad h eigenspace in gC for the eigenvalue 2. To prove the ﬁrst statement of the theorem it sufﬁces by Lemma 2.10 to show that vj ∈ k C

for i = j, . . . , d − 1.

(2.41)

But now by the S-triple commutation relations ixψ = i[θ e, e], −θ e = −1/2[θ e, xψ ], and e = −1/2[e, xψ ]. Thus if xj , fj ∈ uj are deﬁned by xj = [θ e, ej ] and fj = −1/2[θ e, xj ], one has (2.42) vj = 1/2(ixj + ej + fj ) by (2.39). On the other hand, [e, xj ] = [e, [θ e, ej ]] = [[e, θe], ej ] (since [e, ej ] = 0) = −[xψ , ej ] = −2ej .

(2.43)

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B. Kostant

Let j ∈ {1, . . . , d − 1}. To establish (2.41), we will ﬁrst prove that xj ∈ kC . In fact, we will prove that xj ∈ m. (2.44) Since θ|a is minus the identity, one has θ e ∈ g−ψ . But then xj clearly commutes with a. But the centralizer of a in g is m + a. To prove (2.44), it obviously sufﬁces to prove that xj is B-orthogonal to a. But {ej , e} = {ej , ed } = 0. Thus −(ej , σu e) = −(ej , θe) = 0. But then if y ∈ a, one has (y, xj ) = (y, [θ e, ej ]) = ([ej , y], θe). But [ej , y] = −ψ(y)ej . Thus (y, xj ) = 0 establishing (2.44). To prove (2.41) it now sufﬁces, by (2.42), to prove that θfj = ej . But θfj = −1/2[e, xj ] since θ xj = xj by (2.44). But then θfj = ej by (2.43). This proves (2.41). Since h and xψ are conjugate, the ﬁnal statement of Theorem 2.11 follows from Proposition 2.7.

2.5 We retain the notation of Section 2.4, and we will the apply the results of Section 2.4 to the symplectic considerations of Section 2.2. Proposition 2.12. One has (v, w) = 1.

(2.45)

{v, v} = 1.

(2.46)

Furthermore, w = −σu v so that

Proof. Since h and xψ are conjugate, one has (h, h) = 2

(2.47)

by (2.26). But since (h, v, w) is an S-triple, one has (v, w) = 1/2([h, v], w). But ([h, v], w) = (h, [v, w]) = (h, h). Thus (2.47) implies (2.45). But now v = 1/2(ixψ + e − θ e) and w = 1/2(−ixψ + e − θ e) by (2.39). Recall σu = θ σ . But clearly σ v = 1/2(−ixψ + e − θ e) and hence θ σ v = 1/2(ixψ − e + θ e). Hence −σu v = w. But then (2.46) follows from (2.45) and (2.27).

Of course h is a hyperbolic element in kC and qh (kC ) is the parabolic subalgebra of kC deﬁned by h. See Remark 2.2. Theorem 2.13. Under the adjoint action of kC on pC the one-dimensional subspace Cv is stable under qh (kC ). In fact, for any x ∈ qh (kC ), one has [x, v] = (h, x)v.

(2.48)

Proof. If x is contained in the nilradical of qh (kC ), then [x, v] = 0 by the last line in Theorem 2.11. On the other hand, if x ∈ (kC )h (a Levi factor of qh (kC )), then Cv is stable under ad x by the multiplicity one statement in Theorem 2.11 of the eigenvalue 2 of ad h in pC . This proves the ﬁrst statement of Theorem 2.13. For x ∈ qh (kC ) let f be the linear functional on qh (kC ) deﬁned so that [x, v] = f (x)v. But then f (x) = ([x, v], w) by (2.45). However, ([x, v], w) = (x, [v, w]) = (x, h). Thus f (x) = (h, x).

Minimal coadjoint orbits and symplectic induction

409

Let z = −ih so that z ∈ k and z = e + θ e by (2.39). Let T ⊂ K be a maximal torus such that z ∈ t. Then h ∈ h, where, as in Section 2.2, h = it. Let γ : k∗C → kC be as in (2.12), and let λ ∈ h∗ (recalling the identiﬁcation h∗ = γ −1 (h)) be such that γ (λ) = h.

(2.49)

Recall that ⊂ h∗ is the T -weight lattice. Note that hC ⊂ (kC )h so that hC ⊂ qh (kC ). As an immediate consequence of Theorem 2.13, one has the following. Proposition 2.14. One has λ ∈ . Furthermore, the KC -module V generated by v with respect to the adjoint action of KC on pC is irreducible and is equivalent to Vλ with v corresponding to vλ . Henceforth we will identify V with Vλ and v with vλ . One has πλ (k)y = Ad k(y), where y ∈ V and k ∈ KC . Let ν = λ/2π i so that ν ∈ k∗ . As in Section 2.2, let X be the K-coadjoint orbit of ν so that (X, ω) is a symplectic K-homogeneous space, where ω is the KKS symplectic form. One readily notes that Lie Kν = k ∩ (kC )h

(2.50)

so that k ∩ (kC is a compact form of the Levi factor (kC of qh (kC ), where Kν is the isotropy group at ν. Furthermore, one knows (as a general fact about coadjoint orbits of compact connected Lie groups) that Kν is connected so that (2.7) and (2.8) are satisﬁed, where χ is the character on Kν deﬁned by the action of Kν on Cvλ . Thus ωX as in Section 2.2, we can construct the induced symplectic manifold (X, ). As in Section 2.2 let E ⊂ Vλ be the variety of extremal weight vectors so that E = πλ (KC )C∗ vλ . Let Hλ be the K-invariant Hilbert space structure in Vλ given the restriction H|Vλ . As in Section 2.2, E 1 is the space of vectors in E having length 1 with respect to Hλ . Note that vλ ∈ E 1 (2.51) )h

)h

by (2.46). We recall that Proposition 2.3 sets up a K × U (1) isomorphism L1 → E 1

(2.52)

→E X

(2.53)

and a K × U (1) × R+ isomorphism

Let Kv be the isotropy group at v = vλ for the action of K on E 1 and let Ke be the isotropy group at e ∈ g for the adjoint action of K on g. Theorem 2.15. The following three subgroups of K are equal: (1) Kv ; (2) Ke ; (3) the centralizer of the TDS uλ in K.

410

B. Kostant

Proof. Let K be the subgroup of K deﬁned by (3). Since v, e ∈ uψ , one obviously has K ⊂ Kv and K ⊂ Ke . But clearly Ad k commutes with θ for any k ∈ K. Thus θ (e) is ﬁxed by Ad k for any k ∈ Ke . But then all three elements of the S-triple in Proposition 2.6 are ﬁxed by Ad k. Hence Ke = K by the deﬁnition of uψ (see Section 2.3). But σu also commutes with the adjoint action of K since k ⊂ gu . But w = −σu v by Proposition 2.12. Thus w is ﬁxed by Ad k for any k ∈ Kv . Hence any such k ﬁxes the three elements of the S-triple (h, v, w). But these elements also span uψ . Thus K = Kv .

Let O be the Ad G orbit of e in g and let O 1 = {f ∈ O | {f, f } = 1}. Note that e ∈ O 1 by the choice of e in Section 2.3. Since K operates unitarily with respect to H, the adjoint action of k stabilizes E 1 and O 1 . As a corollary of Theorem 2.15, one has the following. Theorem 2.16. The compact group K operates transitively on E 1 and on O 1 . Furthermore, these spaces are isomorphic as K-homogeneous spaces. In fact, b is such an isomorphism, where for any k ∈ K, b(Ad k(vλ )) = Ad k(e).

(2.54)

Proof. Recall that z = e + θ e = −ih ∈ k. But if s ∈ R, one has πλ (exp sz)vλ = e−2si vλ

(2.55)

by (2.47) and (2.48). In particular, k → φ(k, 1) in (2.19) surjects k onto E 1 by Proposition 2.3. Consequently, K operates transitively on E 1 . On the other hand, if A and N are the subgroups of G which correspond, respectively, to a and n then one has the group Iwasawa decomposition G = KAN . But e is ﬁxed under the adjoint action of N since e ∈ Cent n (see Proposition 2.5). On the other hand, Ad A(e) = R+ e. Thus O = Ad KR+ e. (2.56) However, clearly R+ e∩O 1 = {e}. Hence K operates transitively on O 1 . The theorem then follows from the equality of (1) and (2) in Theorem 2.15.

The variety E is KC homogeneous by Remark 2.4. The KC orbit E in pC corresponds to the G-orbit O in g by the Kostant–Sekiguchi correspondence (a correspondence of KC nilpotent orbits in pC and G nilpotent orbits in g; see [S]). Michèle Vergne has proved the corresponding orbits are K-diffeomorphisms (see [V]). The proof is highly nontrivial. However, by Theorem 2.16, in the special case of E and O the diffeomorphism is transparent. Obviously, by (2.21) and (2.56), one has K × R+ diffeomorphisms E 1 × R+ → E, (f, t) → tf, (2.57) 1 + (u, t) → tu. O × R → O, We may therefore extend the domain of deﬁnition of b so that, using the notation of (2.57), one has a K × R+ -diffeomorphism b : E → O,

where b(tf ) = tb(f ).

(2.58)

Minimal coadjoint orbits and symplectic induction

411

ω ) ∼ 3 The symplectic isomorphism (X, X = (Z, ωZ ) 3.1 We continue with the notation of Section 2. Recalling (2.12), one notes that since Bk = B|kC and since kC and pC are B-orthogonal, the isomorphism g∗C to gC deﬁned by B is an extension of (2.12). The extension will also be denoted γ . Let ε ∈ g∗ be deﬁned so that γ (ε) = e/π. (3.1) Let Z ⊂ g∗ be the G-coadjoint orbit of ε and let ωZ be the KKS-symplectic form on Z. To avoid confusion with the vector ﬁeld ξ x on X (see Section 2.1) deﬁned by any x ∈ k, with respect to the coadjoint action of K on X, we will denote by +y the vector ﬁeld on Z deﬁned by any y ∈ g with respect to the coadjoint action of G on Z. The analogue of (2.2) is the formula ωZ (+x , +y )(ρ) = ρ, [y, x]

(3.2)

for any ρ ∈ Z. In particular, if ρ = ε, one has ωZ (+x , +y )(ε) = (1/π)(e, [y, x]).

(3.3)

Recall that ω is the KKS form on the K-coadjoint orbit X of ν = λ/2π i. Lemma 3.1. Let x, y ∈ k. Then ωZ (+x , +y )(ε) = ω(ξ x , ξ y )(ν).

(3.4)

Proof. [y, x] is ﬁxed by θ since [y, x] ∈ k. Thus (1/π )(e, [y, x]) = ((e + θ e)/2π, [y, x]). But e + θ e = (1/ i)h by (2.39) and γ (λ) = h. But then γ (ν) = (e + θ e)/2π.

(3.5)

Thus (1/π )(e, [y, x]) = ν, [y, x]. But then (3.4) follows from (2.2) and (3.3).

Now recalling (2.9), the circle bundle L1 is given by L1 = K ×Kν U (1), where the action of Kν on U (1) is given by the character (recall that Kν is connected) χ (exp x) = eλ(x) = e(h,x)

(3.6)

= L1 × R+ so that for any x ∈ kν (see (2.48)). Now by (1.17) one has X = K ×Kν U (1) × R+ . X

(3.7)

be the point whose components are the identity in K, 1 in U (1), and 1 in Let o ∈ X + R with respect to (3.7). Extend the domain of the bundle projection τ (see (1.6)) to so that τ (q, t) = τ (q) for (q, t) ∈ L1 × R+ . One notes that X τ (o) = ν.

(3.8)

412

B. Kostant

Proposition 3.2. There exists a K × R+ diffeomorphism → Z, β:X

(3.9)

β(tk · o) = t (Coad k(ε)).

(3.10)

where . The result then follows from Proposition 2.3, Proof. Put β = (1/π )γ −1 ◦ b ◦ φ Theorem 2.16, (2.58), and the invariance of B.

3.2 ωX Our main objective will be to prove that β : (X, ) → (Z, ωZ ) is an isomorphism of symplectic manifolds. → T (Z) be the diffeomorphism of tangent bundles deﬁned by the Let β∗ : T (X) so that differential of β. Let βo be the restriction of β∗ to the tangent space To (X) → Tε (Z) βo : To (X)

(3.11)

deﬁned by the is a linear isomorphism. For any x ∈ k, let ηx be the vector ﬁeld on X Since β is a K-map, one has action of K on X. β∗ (ηx ) = +x .

(3.12)

One can be very explicit about ηx . Proposition 3.3. Let x ∈ k. Then using the notation of (1.13) and (2.4), one has Gx ζ. ηx = ξx + ϕ

(3.13)

ηx is characterized by the property Proof. As a vector ﬁeld on L1 (and hence on X), that (1) it commutes with the U (1)-action, (2) it annihilates α by Lie differentiation, and (3) τ∗ (ηx ) = ξ x . (3.14) The result then follows from (1.13), (2.5), and Theorem 1.2.

Let k⊥ ν be the B-orthocomplement of kν in k so that the map k⊥ ν → Tν (X)

(3.15)

given by x → (ξ x )ν is a linear isomorphism. Let Ro be the space of horizontal tangent vectors (i.e., orthogonal to α) to L1 at o so that Ro has codimension 1 in To (L1 ) and codimension 2 in To (X). Lemma 3.4. One has (ηx )o ∈ Ro for any x ∈ k⊥ ν and the map k⊥ ν → Ro , is a linear isomorphism.

x → (ηx )o

(3.16)

Minimal coadjoint orbits and symplectic induction

413

Proof. Let x ∈ kν⊥ . Then ϕ x (ν) = 0 by (2.4) since γ (ν) ∈ kν . Thus (ηx )o = ξx

(3.17)

by (3.13). This proves that (ηx )o ∈ Ro . But since τ∗ : Ro → Tν (X) is clearly an isomorphism the remaining statements of the proposition follow from (3.14) and the isomorphism (3.15).

Let Rε = βo (Ro ) so that (3.11) restricts to the linear isomorphism βo : Ro → Rε . induced by the retriction of ωX Let ωo be the symplectic bilinear form on To (X) to To (X), and let ωε be the symplectic bilinear form on Tε (Z) induced by the retriction of ωZ to Tε (Z). We wish to prove that βo : (To (X), ωo ) → (Tε (Z), ωε )

(3.18)

is an isomorphism of symplectic vector spaces. We ﬁrst establish the following. Lemma 3.5. The restrictions ωo |Ro and ωε |Rε are nonsingular and βo : (Ro , ωo |Ro ) → (Rε , ωε |Rε )

(3.19)

is an isomorphism of symplectic vector subspaces. ⊥ Proof. Recalling the deﬁnition of ωX (see (1.19)) it is clear that if x, y ∈ kν , then x y x y x y x y ωX ω(ξ , ξ )(o) by (3.17). But ωX (η , η )(o) = (η , η )(o) = ω(ξ , ξ )(ν). Thus ωo |Ro is nonsingular by the linear isomorphism (3.15). But then ωε |Rε is nonsingular and (3.19) is an isomorphism of symplectic vector subspaces by (3.4) and (3.12).

with respect Now let Ro⊥ be the two-dimensional orthocomplement of Ro in To (X) to ωo . It is clear from (1.19) and Section 1.2 that Ro⊥ is spanned by ζo and (d/dr)o . It will be convenient for us to modify this basis of Ro⊥ . Recall that z ∈ k is given by z = e + θ e. Lemma 3.6. One has (ηz )o = −ζo /π so that

(ηz )

o

and (−2rd/dr)o are a basis of

Ro⊥ .

(3.20) Furthermore,

ωo ((−2rd/dr)o , (ηz )o ) = −2/π.

(3.21)

Proof. By (2.49) one has γ (ν) = z/2π (3.22) z z z ζ )o by (3.13). But ϕ (o) = ϕ z (ν) so that z ∈ kν . Thus ν = 0. Hence (η )o = (ϕ z z and ϕ (ν) = (γ (ν), z) by (2.4). Hence ϕ (o) = 1/2π(z, z) by (3.22). But since z = −ih one has (z, z) = −2 by (2.47). This proves (3.20). But now by (1.19), since r(o) = 1, (ξ z )

ωo ((−2rd/dr)o , (ηz )o ) = (dr ∧ α)(−2rd/dr, ηz )(o) = −2/π by (3.20) and (1.10). This proves (3.21).

414

B. Kostant

Let κ be the Euler vector ﬁeld on Z. Thus if f ∈ C ∞ (Z) and µ ∈ Z, then (κf )(µ) = d/dtf (µ + tµ)|t=0 . Clearly, β∗ (rd/dr) = κ.

(3.23)

βo ((−2rd/dr)o ) = (+xψ )ε

(3.24)

Lemma 3.7. One has (see Proposition 2.6). Proof. One has [xψ , e] = 2e by Proposition 2.6. But then coad xψ (ε) = 2ε

(3.25)

since π γ is an equivalence of g-modules (see (3.1)). But (+xψ )ε = −2κε . (See the beginning of Section 2.1 to explain the minus sign.) But then (3.24) follows from (3.23).

Lemma 3.8. One has ωε ((+xψ )ε , (+z )ε ) = −2/π.

(3.26)

Proof. ωε ((+xψ )ε , (+z )ε ) = ωZ (+xψ , +z )(ε) = ε, [z, xψ ] = (1/π)(e, [z, xψ ]) = (1/π)([e, e + θ e], xψ ) = −(1/π)(xψ , xψ ) = −2/π

by Proposition 2.6

by (2.26).

We can now prove the following. Theorem 3.9. The map βo : (To (X), ωo ) → (Tε (Z), ωε )

(3.27)

is an isomorphism of symplectic vector spaces. Proof. Let Rε⊥ be the two-dimensional orthocomplement of Rε in Tε (Z) with respect to ωε . We assert (Assertion A) that (+z )ε and (+xψ )ε is a basis of Rε⊥ . We ﬁrst show that (+z )ε ∈ Rε⊥ . To do this it sufﬁces to show that ωZ (+z , +x )(ε) = 0

(3.28)

for all x ∈ k since Rε is spanned by (+x )ε for x ∈ k⊥ ν , by Lemmas 3.4 and 3.5. But if x ∈ k, then ωZ (+z , +x )(ε) = 1/π(e, [x, z])

Minimal coadjoint orbits and symplectic induction

415

= (1/2π)((e + θ e), [x, z]) since [x, z] ∈ k = (1/2π )([z, z], x) = 0. Thus (+z )ε ∈ Rε⊥ . Let y ∈ k⊥ ν . Then ωZ (+xψ , +y )(ε) = (1/π)(e, [y, xψ ]) = (1/π)([xψ , e], y) = (2/π)(e, y) by Proposition 2.6 = (1/π)(e + θ e, y) since y ∈ k = 0 since z ∈ kν and y ∈ k⊥ ν. But then (+xψ )ε ∈ Rε⊥ by Lemmas 3.4 and 3.5. This proves Assertion A since the left side of (3.26) is nonzero. But now we assert (Assertion B) that βo (Ro⊥ ) ⊂ Rε⊥ and βo : (Ro⊥ , ωo |Ro⊥ ) → (Rε⊥ , ωε |Rε⊥ )

(3.29)

is an isomorphism of symplectic subspaces. Indeed, βo ((ηz )o ) = (+z )ε by (3.12) and βo ((−2rd/dr)o ) = (+zψ )ε by (3.24). But then Assertion B follows from Assertion A together with Lemma 3.6 and the fact that −2/π occurs on the right side of (3.21) and on the right side of (3.26). But then the theorem follows from the symplectic isomorphism (3.29) together with the symplectic isomorphism (3.19).

The following is our main result. Theorem 3.10. Let G be a connected noncompact Lie group with ﬁnite center such that Lie g is simple. Let K be a maximal compact subgroup and let X be the coadjoint ωX orbit of K deﬁned as in Section 2.5. Let (X, ) be the symplectic manifold obtained = dim X + 2. See Section 1.3. Let from X by symplectic induction so that dim X (Z, ωZ ) be the coadjoint orbit of G, together with its KKS symplectic form, deﬁned as in Section 3.1, and let ωX β : (X, (3.30) ) → (Z, ωZ ) be the K-diffeomorphism (a special case of M. Vergne’s theorem) deﬁned as in Section 3.1 using the Kostant–Sekiguchi correspondence. Then (3.30) is an isomorphism of symplectic manifolds. Proof. Note that L1 (see (2.9)) is K-homogeneous since (2.7) is surjective (ν = 0 is K × R+ homogeneous by (1.17). For any (k, s) ∈ since G is noncompact). Thus X + deﬁned by the action of (k, s) on X. K × R , let k,s be the diffeomorphism of X Thus if q ∈ X, then k,s (q) = s(k · q). Let mk,s be the diffeomorphism of Z deﬁned so that if µ ∈ Z, then mk,s (µ) = s(Coad k(µ)). It follows from (2.56) that O is K × R+ homogeneous. Consequently, Z is K × R+ homogeneous since (see (3.1)) π γ : Z → O obviously commutes with the action of K × R+ . From the deﬁnition of β (see (3.10)), it is immediate that one has the commutative diagram equality of maps

416

B. Kostant

β ◦ k,s = mk,s ◦ β.

(3.31)

and put p = k,s q. We assert that the pullback Let q ∈ X (k,s )∗ ((ωX )p ) = s(ωX )q .

(3.32)

But recalling (1.19), it is obvious that α and ω are invariant under k,s . On the other hand, clearly (k,s )∗ r = sr and hence (k,s )∗ dr = sdr. This proves (3.32). Now let µ ∈ Z and let ρ = mk,s (µ). We assert that (mk,s )∗ ((ωZ )ρ ) = s(ωZ )µ .

(3.33)

Since ωZ is obviously K-invariant it sufﬁces to prove (3.33) under the assumption that k is the identity of K. Making that assumption, one has ρ = sµ. But also +y for any y ∈ g is invariant under mk,s . Thus for any y ∈ g, one has (mk,s )∗ ((+y )µ ) = (+y )sµ .

(3.34)

But then for any x, y ∈ g, one has (mk,s )∗ ((ωZ )sµ )((+x )µ , (+y )µ ) = (ωZ )sµ ((+x )sµ , (+y )sµ ) = sµ, [y, x] = s(ωZ )µ ((+x )µ , (+y )µ ). be arbitrary and let µ = βp. To prove Theorem 3.10, But this proves (3.33). Let p ∈ X it sufﬁces to show that β ∗ ((ωZ )µ ) = (ωX (3.35) )p . By transitivity, there exists (k, s) ∈ K × R+ such that k,s p = o. But then mk,s µ = ε by the commutativity equation (3.31) and, in fact, β ∗ ((ωZ )µ ) = (k,s )∗ ((βo )∗ (((mk,s )−1 )∗ (ωZ )µ )). However, (mk,s )−1 )∗ (ωZ )µ = s −1 (ωZ )ε by (3.33). But (βo )∗ (s −1 (ωZ )ε ) = s −1 (ωX )o by Theorem 3.9. Finally, (k,s )∗ (s −1 (ωX ) ) = (ω ) by (3.32). This proves o p X (3.35).

3.3 We wish to characterize the varieties Z and X in more general terms. Recall (E. Cartan’s theory) that the noncompact symmetric space G/K is one of two types: (1) nonHermitian symmetric or (2) Hermitian symmetric. In the non-Hermitian case, pC is KC -irreducible and kC is semisimple. In the Hermitian case, Cent kC is one dimensional and if I is a set indexing the KC -irreducible submodules V i , i ∈ I , of pC , then I is a two-element set and pC = V i. (3.36) i∈I

Also, if i ∈ I , there exists a linear isomorphism δi : Cent kC → C such that V i = {u ∈ pC | ad x(u) = δi (x)u ∀x ∈ Cent kC }. In addition, one has

(3.37)

Minimal coadjoint orbits and symplectic induction

δi = −δi ,

417

(3.38)

{i, i }

where = I. In case (1), we will say that g is of non-Hermitian type, and in case (2), we will say that g is of Hermitian type, Recall that gψ = Cent n, and we have put d = dim gψ . See Sections 2.3 and 2.4. Proposition 3.11. If d > 1, then M is transitive on the unit sphere S (relative to H) in gψ . Furthermore, g is of Hermitian type if and only if (a) d = 1 and (b) M operates trivially on gψ . Proof. The transitivity statement actually is stated in [K-2, Theorem 2.1.7 and the remark that follows it]. However, one need not use that reference. We have already established what is needed to prove transitivity on S in the present paper. Indeed, by (2.43) and (2.44), one has that [m, e] spans the H orthocomplement of Re in gψ . This implies Ad M(e) is open in S. On the other hand, it is also closed since M is compact. Thus Ad M(e) = S if d > 1 since S is connected if d > 1. In any case, one has Ad G(e) = Ad K(R+ e) (3.39) by the Iwasawa decomposition G = KAN . But Ad G(e) and hence also Ad K(e) spans g by the simplicity of g. But if g is of Hermitian type, then certainly Cent k = 0 and if 0 = x ∈ Cent k the function f (k) = (Ad k(e), x) on K is constant, real, and nonzero. But then −e ∈ / Ad Me. Hence d = 1 and M operates trivially on gψ by the transitivity statement. Conversely, if d = 1 and M operates trivially on gψ , then the line Ce is stable under the complex parabolic subalgebra (m + a + n)C . But then e generates a complex irreducible ad gC -module, necessarily containing g, and hence equal to gC . Thus gC is simple and e is the extremal weight vector corresponding to (m + a + n)C . But then gC is a spherical ad gC -module by the Cartan–Helgason theorem. Any nonzero spherical vector must clearly lie in kC (e.g., by (3.37), (3.38), and the simplicity of gC ). But then Cent kC = 0. Thus g is of Hermitian type.

Remark 3.12. Note that as a consequence of Proposition 3.11, one has the statement −e ∈ / Ad M(e) ⇐⇒ g is of Hermitian type.

(3.40)

But one also has the statement −e ∈ / Ad G(e) ⇐⇒ g is of Hermitian type.

(3.41)

Indeed, if g is of non-Hermitian type, then −e ∈ Ad M(e) by (3.40) and hence, of course, −e ∈ Ad G(e). If g is of Hermitian type, let 0 = x ∈ Cent k. If −e ∈ Ad G(e), then the function f in the proof of Proposition 3.11 must change signs by (3.39) and hence cannot be constant. This proves (3.41). Let Proj(g∗ ) be the real projective space deﬁned by g∗ and let P : g∗ − {0} → Proj(g∗ ) be the projectivization map. Let C = γ −1 (Cent n) so that ε ∈ C ⊂ g∗ (see Proposition 2.5 and (3.1)).

418

B. Kostant

= in Proj(g∗ ) so that if Y = is any Theorem 3.13. There exists a unique closed G-orbit Z ∗ = = G-orbit in Proj(g ), then Z is contained in the closure of Y . In particular, = ≥ dim Z = dim Y

(3.42)

= = Z. = and equality occurs in (3.41) if and only if Y ∗ = if Furthermore, if Z ⊂ g is any nonzero G-coadjoint orbit, then Z ⊂ P −1 (Z) and only if Z is of the form Coad G(e ) for e ∈ C − {0}. In fact, using the notation of Theorem 3.10, one has = Z = P −1 (Z) (3.43) in case g is of non-Hermitian type. In particular, Z = −Z in the non-Hermitian case. If g is of Hermitian type, let J be an index set parameterizing all G-coadjoint orbits, = Then J is a two-element set and one has Z j , j ∈ J , in P −1 (Z). {Z, −Z} = {Z j },

j ∈ J,

(3.44)

so that Z = −Z in the Hermitian case. In either case, one has dim Y ≥ dim Z

(3.45)

for any nonzero coadjoint G-orbit Y . Proof. Let 0 = x ∈ g. Then the span g1 of Ad g(x) over all g ∈ G is a nonzero ideal in g and hence g1 = g. Thus there exists g ∈ G such that the component x of Ad g(x) in gψ with respect to the decomposition g=m+a+

gϕ ϕ∈

is not zero. But then by Proposition 2.7, one has lim P (Ad(exp txψ )(x)) = P (x ),

t→+∞

(3.46)

where we also let P denote the projection map of g onto Proj(g). On the other hand, if x ∈ gψ − {0}, then P (Ad G(x )) = P (Ad G(x )) by Proposition 3.11. But P (Ad G(x )) is compact since P (x ) is ﬁxed by the action of AN and hence P (Ad G(x )) = K · P (x ). But P −1 (P (Ad G(x )) is a single Ad G-orbit by Proposition 3.11 and (3.40) in case g is of non-Hermitian type. On the other hand, it decomposes into a union of two distinct Ad G orbits, one the negative of the other, by Proposition 3.11 and (3.41), in case g is of Hermitian type. Since γ : g∗ → g is a

G-equivalence the statements above clearly carry over to g∗ . Remark 3.14. Because g is simple and Z is a G-coadjoint orbit there exists a Lie algebra injective homomorphism g → C ∞ (Z) (of course, with respect to the Poisson algebra structure on C ∞ (Z)). But then the symplectic isomorphism (3.30) implies that there exists a Lie algebra injective homomorphism

Minimal coadjoint orbits and symplectic induction

419

g → C ∞ (X). is the smallest By the coadjoint orbit covering theorem and (3.45) note that dim X possible dimension of a symplectic manifold W which admits an embedding of g as a Lie algebra of functions on W under Poisson bracket. This points to an interesting ωX difference between (X, ω) and the induced symplectic manifold (X, ) in that one cannot have a nontrivial Lie algebra homomorphism g → C ∞ (X) since dim X = dim Z − 2

(3.47)

by (1.18). If V is a complex ﬁnite-dimensional irreducible K (and hence KC module), let X(V ) ⊂ k∗ be the integral K-coadjoint orbit associated to V by the Borel– Weil theorem. Thus if v ∈ V is an extremal weight vector there exists a Cartan subalgebra hC of kC , which is the complexiﬁcation of a Cartan subalgebra of k, such that v is an hC -weight vector for an extremal hC -weight λ. We may regard λ ∈ k∗C , where λ|[hC , kC ] = 0. Then ν = λ/2π i is real on k and X(V ) is the coadjoint orbit of ν. It is straightforward to show that X(V ) is independent of the choice of hC . The embedding k → g induces, by transpose, a surjection µg,k : g∗ → k∗ .

(3.48)

Remark 3.15. Note that if Y is a G-coadjoint orbit, then the restriction µg,k |Y is the moment map for the action of K on (Y, ωY ), where ωY is the KKS-symplectic form on Y . Indeed, this follows immediately from (2.4), where G replaces K and Y replaces X. Theorem 3.16. We use the notation of Theorem 3.10. One has µg,k (Z) = R+ X.

(3.49)

Moreover, if g is of non-Hermitian type, then X = X(pC ). Also, X = −X in the non-Hermitian case. If g is of Hermitian type, let X i = X(V i ) for i ∈ I (see (3.37)). Then X = −X and {X, −X} = {X i }, i ∈ I. (3.50) Furthermore (recalling (3.44)), there exists a unique bijection α : I → J such that for i ∈ I , (3.51) µg,k (Z α(i) ) = R+ X i . If X = Xi , then Z = Z α(i) . Finally, adding to the symplectic isomorphism (3.30), for any i ∈ I , there exists a symplectic K-diffeomorphism Gi , ωG ) → (Z α(i) , ω α(i) ), βi : (X Z Xi where ωZ α(i) is the KKS form on the G-coadjoint orbit Z α(i) .

(3.52)

420

B. Kostant

Proof. Let p : g → k be the projection corresponding to the Cartan decomposition g = k + p. Obviously, p ◦ γ = γ ◦ µg,k (3.53) on g∗ . Recall (see Section 2.5) that O is the G-adjoint orbit of e. But O = Ad KR+ e by (2.56). Thus f ∈ O if and only if there exists t ∈ R+ and k ∈ K such that f = t Ad k(e/π). But clearly p(e/π) = (e + θe)/(2π ). That is p(e/π ) = h/(2π i) by (2.39). Thus p(f ) = t Ad k(h/(2π i)). Applying γ −1 to both sides, one has µg,k (γ −1 (f )) = t Coad k(γ −1 (h)/(2π i)) by (3.53). But O = γ (Z) (see Section 3.1) so that if ρ = γ −1 (f ), then ρ ∈ Z and the most general element in Z is of this form. Furthermore, γ −1 (h) = λ (see (2.49)) and ν = λ/(2π i) (see Section 2.5). Thus µg,k (ρ) = t Coad k(ν)). Since k ∈ K and t ∈ R+ are arbitrary and X is the K-coadjoint orbit of ν, this proves (3.49). Now since B|pC is nonsingular, it follows that pC as a K-module is selfcontragedient. In particular, µ is a weight of pC ⇐⇒ −µ is a weight of pC .

(3.54)

Recall the choice of X in Section 2.5. By deﬁnition X = X(Vλ ),

(3.55)

where Vλ is the K-irreducible submodule of pC deﬁned in Section 2.5 with extremal weight vector v (see (2.48)). Assume g is of non-Hermitian type. Then pC is Kirreducible so that Vλ = pC . This proves X = X(pC ). But then X = −X by (3.54). Now assume that g is of Hermitian type. Then Vλ = V i for some i ∈ I . But µ| Cent k = δi | Cent k for any weight µ of Vλ by (3.37). Thus X = −X and one easily has (3.50) by (3.54). But (3.49) implies µg,k (−X) = R+ (−Z). But then (3.51) follows from (3.44) and (3.49). The choice of e ∈ gψ was arbitrary (see Section 2.3) subject only to the condition that {e, e} = 1. We can therefore replace e by −e. In that case h would be replaced by −h and λ by −λ. Consequently, X is replaced by −X and Z by −Z. Thus the argument which leads to the symplectic K-isomorphism (3.30) yields a symplectic C ω−X K-diffeomorphism (−X,

C ) → (−Z, ω−Z ). This implies (3.52). 3.4 Even though g is simple the complexiﬁcation gC may not be simple. Indeed, this is the case if g itself were complex but its complex structure is ignored. We will say that g is Omin -split if gC is a simple complex Lie algebra and e ∈ Omin , where Omin is the minimal nilpotent orbit in gC . Note that by the transitivity statement in Proposition 3.11 and Theorem 3.13, the deﬁnition of Omin -split depends only on g and is, in particular, independent of the choice of e. If g is Omin -split, then dim Z = 2h∨ − 2,

(3.56)

Minimal coadjoint orbits and symplectic induction

421

where h∨ is the dual Coxeter number of gC . This is clear since one knows dimC Omin = 2h∨ − 2. But by the irreducibility of the adjoint representation of gC one has Omin = Ad gC (e). However, O = Ad g(e) (see Section 2.5). But then (3.56) follows by computing the dimension of the respective tangent spaces at e. See Section 3.1 for the deﬁnition of Z. Theorem 3.17. The simple Lie algebra g is Omin -split if and only if dim Cent n = 1.

(3.57)

Proof. If dim Cent n = 1 then Ce is stabilized, under the adjoint represention, by the complex parabolic subalgebra (m + a + n)C . But then e generates an irreducible gC -module under the adjoint representation of gC . But this module contains g by the simplicity of g. Hence the module equals gC so that gC is simple. Furthermore, e ∈ Omin since e is an extremal weight vector of this module. Thus g is Omin -split. Conversely assume that g is Omin -split. Then uψ (see Section 2.3) is conjugate to the TDS of a highest root vector in gC by [K-3, Corollary 3.6]. Since the root in question is long, the multiplicity of the eigenvalue 2 of ad xψ is 1. Thus d = 1 by Proposition 2.7. That is, dim Cent n = 1. See Section 2.4.

Examples 3.18. By Theorem 3.17, g is Omin -split if (a) g is split (so that all restricted root spaces are one dimensional); (b) gC is simple and g is a quasi-split form of gC (since nC is a maximal nilpotent Lie algebra of gC ); (c) g is of Hermitian type (by Proposition 3.11). Note that if g is Omin -split, then dim X = 2h∨ − 4

(3.58)

by (3.47) and (3.56). In general, X ∼ = K/Kν can be given the structure of a partial complex ﬂag manifold (see (2.16)). If g is Omin -split, much more can be said. Proposition 3.19. If g is Omin -split (e.g., if g is split), then X is not only a Ksymmetric space but, in fact, X is a Hermitian symmetric space. Proof. Recall the notation of Section 2.4. If g is Omin -split then by Theorem 3.17 the number dk in Theorem 2.11 is 0. Thus since xψ and h are conjugate, the eigenvalues of ad h|kC are in the set {1, 0, −1}. Since (kν )C is the ad h|kC -eigenspace for the eigenvalue 0 the result is immediate.

Remark 3.20. Assume that g is Omin -split and g is of non-Hermitian type so that kC is semisimple. Let K be a noncompact real form of KC having Kν as a maximal compact subgroup so that K /Kν = X is the noncompact symmetric dual to the compact symmetric space X. But now X = −X by Theorem 3.16 so that h and −h are K-conjugate. One readily shows that this implies that h lies in a TDS of kC so that not only is X a complex bounded domain but, in fact, X is a tube domain. See [K-W]. In particular, by the Kantor–Koecher–Tits theory, X corresponds to a formally real

422

B. Kostant

Jordan algebra J (X). See [K-S, Section 5] for a classiﬁcation of the simple formally real Jordan algebras and the corresponding tube domains. One then has dim X = 2 dim J (X).

(3.59)

If g is a split form of any one of ﬁve exceptional simple Lie algebras, g is nonHermitian so that Remark 3.20 applies to g. But a minimal (dimensional) symplectic realization of g as functions on a symplectic manifold is achieved when the man of a coadjoint orbit of a compact Lie group K. ifold is the induced symplectic X See Remark 3.14. The group K turns out to be classical in all ﬁve cases so that the exceptional Lie algebras g emerge symplectically from the symplectic induction of a classical coadjoint orbit. The table below contains the relevant information. The cases of E6 , E7 , and E8 are taken from [B-K]. To avoid complicated notation, the compact groups listed below are correct only up to ﬁnite coverings. The symbol H (n)/F denotes the Jordan algebra of n × n Hermitian matrices over the ﬁeld F . The compact form of Sp(2n, C) will be denoted simply by Sp(2n). gC type K X dim X J (X) G2 SU (2) × SU (2) P1 (C) × P1 (C) 4 R⊕R F4 SU (2) × Sp(6) P1 (C) × (Sp(6)/U (3)) 14 R ⊕ (H (3)/R) E6 Sp(8) Sp(8)/U (4) 20 H (4)/R SU (8) SU (8)/(SU (4) × SU (4) × U (1)) 32 H (4)/C E7 E8 Spin(16) Spin(16)/U (8) 56 H (4)/H

References [B-K] R. Brylinski and B. Kostant, Lagrangian models of minimal representations of E6 , E7 , and E8 , in S. Gindikin, J. Lepowsky, and R. L. Wilson, eds., Functional Analysis on the Eve of the 21st Century: In Honor of the Eightieth Birthday of I. M. Gelfand , Progress in Mathematics, Vol. 131, Birkhäuser, Boston, 1995, 13–53. [K-1] B. Kostant, Quantization and unitary representations, in C. T. Taam, ed., Lectures in Modern Analysis III, Lecture Notes in Mathematics, Vol. 170, Springer-Verlag, New York, 1970, 87–207. [K-2] B. Kostant, On the existence and irreducibility of certain series of representations, in I. M. Gelfand, ed., Lie groups and Their Representations, Wiley, New York, 1971, 231–329. [K-3] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., 81-4 (1959), 973–1032. [K-S] B. Kostant and S. Sahi, The Capelli identity, tube domains, and the generalized Laplace transform, Adv. Math., 87 (1991), 71–92. [K-W] A. Koranyi and J. Wolf, Realization of Hermitian symmetric spaces as generalized half-planes, Ann. Math., 81 (1965), 265–288. [S] J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan, 39 (1987), 127–138. [V] M. Vergne, Instantons et correspondance de Kostant–Sekiguchi, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 901–906.

Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces∗ Camille Laurent-Gengoux and Ping Xu Department of Mathematics Pennsylvania State University University Park, PA 16802 USA [email protected], [email protected] Dedicated to Alan Weinstein on the occasion of his 60th birthday. Abstract. We study the prequantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces using S 1 -gerbes. We give a geometric description of the integrality condition. As an application, we study the prequantization of the quasi-Hamiltonian G-spaces of Alekseev– Malkin–Meinrenken.

1 Introduction Quasi-symplectic groupoids are natural generalizations of symplectic groupoids [7, 22]. The main motivation of [22] in studying quasi-symplectic groupoids was to introduce a single, uniﬁed momentum map theory in which ordinary Hamiltonian G-spaces, Lu’s momentum maps of Poisson group actions, and the group-valued momentum maps of Alekseev–Malkin–Meinrenken can be understood under a uniform framework. An important feature of this uniﬁed theory is that it allows one to understand the diverse theories in such a way that techniques in one can be applied to the others. It turns out that much of the theory of Hamiltonian spaces of a symplectic groupoid can be generalized to quasi-symplectic groupoids. In particular, one can perform reduction and prove that J −1 (O)/ is a symplectic manifold, where O ⊂ M is an orbit of the groupoid ⇒ M. More generally, one can introduce the classical intertwiner space X2 × X1 between two Hamiltonian -spaces X1 and X2 , generalizing the notion studied by Guillemin–Sternberg [10] for ordinary Hamiltonian G-spaces. One shows that this is a symplectic manifold (whenever it is a smooth manifold). In particular, when is the AMM quasi-symplectic groupoid [6, 22], this reduced space describes the symplectic structure on the moduli space of ﬂat connections on a surface [3]. ∗ This research was partially supported by NSF grant DMS03-06665.

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As in the case of symplectic groupoids [20], one can introduce Morita equivalence for quasi-symplectic groupoids. In particular, it has been proved [22] that (i) Morita-equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories in the sense that there is a bijection between their Hamiltonian spaces; (ii) the classical intertwiner space X2 × X1 is independent of the Morita equivalence class of . This Morita invariance principle accounts for various well-known results concerning the equivalence of momentum maps, including the Alekseev–Ginzburg– Weinstein linearization theorem [1, 9] and the Alekseev–Malkin–Meinrenken equivalence theorem for group-valued momentum maps [3] (see [22] for details). One important feature of Hamiltonian G-spaces is the Guillemin–Sternberg theorem which states that “[Q, R] = 0’’: quantization commutes with reduction [10, 13]. One expects that “[Q, R] = 0’’ should be a general guiding principle for all momentum map theories. To carry out such a quantization program, the ﬁrst important step is the construction of prequantum line bundles. In this paper, we study the prequantization of Hamiltonian spaces for quasi-symplectic groupoids. Our method uses the theory of S 1 -bundles and S 1 -gerbes over a groupoid along with their characteristic classes, as developed in [4, 5]. Roughly, our construction can be described as follows. Aprequantization of a quasi-symplectic groupoid ( ⇒ M, ω+ ) is an S 1 -central extension R → of the groupoid ⇒ M (or an S 1 -gerbe over the groupoid) equipped with a pseudo-connection having ω + as pseudo-curvature. Such a prequantization exists if and only if ω + is a de Rham integral 3-cocycle and is exact (assuming that is a proper groupoid). A prequantization of a Hamiltonian space is then an S 1 -bundle L over R ⇒ M together with a compatible pseudo-connection, where the R-action on L is S 1 -equivariant. A prequantization of the symplectic intertwiner space X2 × X1 can be constructed using these data. Indeed one can show that R\(L1 ×M L2 ) is a prequantization of the symplectic intertwiner space X2 × X1 , and the natural 1-form on L1 ×M L2 induced by the connection forms on L1 and L2 descends to a prequantization connection on the quotient space R\(L1 ×M L2 ). When is not exact, one must pass to a Morita-equivalent quasisymplectic groupoid ﬁrst. Then the Morita invariance principle guarantees that the resulting quantization does not depend on the particular choice of Morita-equivalent quasi-symplectic groupoid. As a special case, when is the AMM quasi-symplectic groupoid, our construction yields the prequantization of quasi-Hamiltonian G-spaces of Alekseev–Malkin–Meinrenken and their symplectic reductions, and our quantization condition coincides with that of Alekseev–Meinrenken [2]. Quantization of Hamiltonian spaces for symplectic groupoids was studied in [21]. Note that in the usual Hamiltonian case, since the symplectic 2-form deﬁnes a zero class in the third cohomology group of the groupoid T ∗ G ⇒ g∗ , which is the equivariant cohomology HG3 (g∗ ), gerbes do not appear explicitly. However, for a general quasi-symplectic groupoid (for instance the AMM quasi-symplectic groupoid), since the 3-cocycle ω + may deﬁne a nontrivial class, gerbes are inevitable in the construction. Also note that no nondegeneracy condition is needed in the quantization construction, so we drop this assumption in the present paper to assure full generality. This paper is organized as follows. In Section 2, we review some basic results concerning pre-quasi-symplectic groupoids and their Hamiltonian spaces. In Sec-

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425

tion 3, we gather some important results on S 1 -bundles and S 1 -central extensions. We give a simple formula for the index of an S 1 -bundle over a central extension in terms of the Chern class. In Section 4, we introduce prequantizations of pre-quasisymplectic groupoids and discuss compatible prequantizations of their Hamiltonian spaces. Section 5 is devoted to the description of a geometric integrality condition of pre-Hamiltonian -spaces. The application to quasi-Hamiltonian G-spaces is discussed. Unless speciﬁed, by a groupoid in this paper, we always mean a Lie groupoid whose orbit space is connected. A remark is in order concerning terminology. In [7], quasi-symplectic groupoids are called presymplectic groupoids, where some “nondegeneracy’’ condition is assumed. Here we choose to use the “quasi’’ part of the terminology to refer to the presence of a 3-form and to use “pre-’’ to mean that “nondegeneracy’’ is ﬂexible. Note that it would be interesting to investigate what notion of polarization would be relevant for the next step of this quantization scheme. Prequantization of symplectic groupoids was ﬁrst studied by Alan Weinstein and the second author in [19], when the second author was his Ph.D. student. In the same paper, S 1 -central extensions of Lie groupoids were also systematically investigated for the ﬁrst time. Undoubtedly,Alan Weinstein’s work and insights have had a tremendous impact on the development of this subject in the past two decades. It is our great pleasure to dedicate this paper to him.

2 Pre-Hamiltonian -spaces and classical intertwiner spaces 2.1 Pre-quasi-symplectic groupoids and their pre-Hamiltonian spaces First, let us recall the deﬁnition of the de Rham double complex of a Lie groupoid. Let ⇒ M be a Lie groupoid. Deﬁne for all p ≥ 0 p = ×M · · · × M , :; < 9 p times

i.e., p is the manifold of composable sequences of p arrows in the groupoid ⇒ M (and 0 = M). We have p + 1 canonical maps p → p−1 (each leaving out one of the p + 1 objects involved in a sequence of composable arrows), giving rise to a diagram / / / 1 (1) · · · 2 / 0 . / • Consider the double complex (• ): · ·O ·

· ·O ·

d

1 ( O 0)

d ∂

d

0 (0 )

· ·O ·

/ 1 ( ) O 1

d ∂

d ∂

/ 0 ( ) 1

(2)

/ 1 ( ) O 2

∂

/ ···

∂

/ ···

d ∂

/ 0 ( ) 2

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Its boundary maps are d : k (p ) → k+1 (p ), the usual exterior derivative of differentiable forms and ∂ : k (p ) → k (p+1 ), the alternating sum of the pullback maps of (1). We denote the total differential by δ = (−1)p d + ∂. The cohomology • groups of the total complex CdR (• ) k HdR (• ) = H k ( • (• ))

are called the de Rham cohomology groups of ⇒ M. Deﬁnition 2.1. A pre-quasi-symplectic groupoid is a Lie groupoid ⇒ M equipped with a 2-form ω ∈ 2 () and a 3-form ∈ 3 (M) such that d = 0,

dω = ∂ ,

and

∂ω = 0.

(3)

In other words, ω + is a 3-cocycle of the total de Rham complex of the groupoid ⇒ M. A pre-quasi-symplectic groupoid ( ⇒ M, ω + ) is said to be exact if is an exact 3-form on M. A quasi-symplectic groupoid is a pre-quasi-symplectic groupoid ( ⇒ M, ω +

), where ω satisﬁes a certain nondegenerate condition [7, 22]. Quasi-symplectic groupoids are natural generalization of symplectic groupoids, whose momentum map theory uniﬁes various momentum map theories, including the ordinary Hamiltonian G-spaces, Lu’s momentum maps of Poisson group actions, and group-valued momentum maps of Alekseev–Malkin–Meinrenken. Deﬁnition 2.2. Given a pre-quasi-symplectic groupoid ( ⇒ M, ω + ), a preHamiltonian -space is a (left) -space J : X → M (i.e., acts on X from the left) with a compatible 2-form ωX ∈ 2 (X) such that (1) dωX = J ∗ ; (2) the graph of the action = {(r, x, rx)|t (r) = J (x)} ⊂ × X × X (where X is the manifold X endowed with the form −ωX ) is isotropic with respect to the 2-form (ω, ωX , −ωX ). To illustrate the intrinsic meaning of the above compatibility condition, let us elaborate it in terms of groupoids. Let ×M X ⇒ X be the transformation groupoid corresponding to the -action, and, by abuse of notation, J : ×M X → the natural projection. It is simple to see that ×M X X

J

/

J

/M

(4)

is a Lie groupoid homomorphism. Therefore, it induces a map, i.e., the pullback map, on the level of the de Rham complex: J ∗ : • (• ) → • (( ×M X)• ).

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Proposition 2.3 ([22]). Let ( ⇒ M, ω + ) be a pre-quasi-symplectic groupoid and J : X → M a left -space. A 2-form ωX ∈ 2 (X) is compatible with the action if and only if J ∗ (ω + ) = δωX . (5) 2.2 Classical intertwiner spaces Consider a pre-quasi-symplectic groupoid ( ⇒ M, ω + ), and pre-Hamiltonian J1

J2

spaces (X1 → M, ω1 ), and (X2 → M, ω2 ). Assume that \(X2 ×M X1 ) is a smooth manifold, and denote by p : X2 ×M X1 → \(X2 ×M X1 ) the natural projection. Note that i ∗ (−ω2 , ω1 ), where i : X2 ×M X1 → X2 × X1 is the natural embedding, is a closed 2-form on X2 ×M X1 . Proposition 2.4. The 2-form i ∗ (−ω2 , ω1 ) descends to a closed 2-form on \(X2 ×M X1 ). Therefore, \(X2 ×M X1 ) is a presymplectic manifold. To prove this proposition, we need a technical lemma. Lemma 2.5 ([11]). Let ⇒ M be a Lie groupoid and X → M a left -space. Assume that \X is a smooth manifold. A differential form ω ∈ ∗ (X) descends to a differential form on the quotient \X if and only if ∂ω = 0, where ∂ is with respect to the transformation groupoid ×M X ⇒ X. Proof of Proposition 2.4. Note that the manifold X2 ×M X1 with the momentum map J : X2 ×M X1 → M, J (x2 , x1 ) = J1 (x1 ) = J2 (x2 ), is naturally a -space, where ⇒ M acts on X2 ×M X1 diagonally. Then ∂[i ∗ (−ω2 , ω1 )] = i ∗ (−∂ω2 , ∂ω1 ) = i ∗ (−J2∗ ω, J1∗ ω)

= ((J2 × J1 ) ◦ i)∗ (−ω, ω)

= 0, where J1 , J2 and i are, respectively, the groupoid homomorphisms: × M Xk Xk

Jk

Jk

/ k = 1, 2.

(6)

/M

and ×M (X1 ×M X2 ) X1 ×M X2

/ ( ×M X1 ) × ( ×M X2 ) ,

i

i

/ X1 × X 2

(7)

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and ∂[i ∗ (−ω2 , ω1 )] and ∂ωk , k = 1, 2, are with respect to the groupoids on the left-hand side of equations (7) and (6), respectively. The conclusion thus follows from Lemma 2.5.

The presymplectic manifold \(X2 ×M X1 ) is called the classical intertwiner space and is denoted by X2 × X1 for simplicity. In particular, if ( ⇒ M, ω + ) is J1

J2

a quasi-symplectic groupoid and (X1 → M, ω1 ) and (X2 → M, ω2 ) are Hamiltonian -spaces, and if J1 : X1 → M and J2 : X2 → M are clean, then X2 × X1 becomes a symplectic manifold. See [22] for details.

3 S 1 -bundles and S 1 -central extensions In this section we recall some basic results concerning S 1 -bundles and S 1 -central extensions over a Lie groupoid. For details, consult [4, 5, 18]. 3.1 Integral de Rham cocycles Let us recall some basic facts concerning singular homology. For any manifold N , we denote by (C• (N, Z), d) the piecewise smooth singular chain complex, and Zk (N, Z) the space of smooth k-cycles. For a smooth map φ : M → N , we denote by φ∗ both the chain map from (C• (M, Z), d) to (C• (N, Z), d) and the morphism of singular homology H∗ (M, Z) → H∗ (N, Z) induced by φ. For any Lie groupoid ⇒ 0 , consider the double complex C• (• , Z): ··· ↓d

··· ↓d

··· ↓d

∂

∂

∂

∂

C1 (0 , Z) ← C1 (1 , Z) ← C1 (2 , Z) ↓d ↓d ↓d C0 (0 , Z) ← C0 (1 , Z) ← C0 (2 , Z), where 0 = M, and ∂ : Ck (p , Z) → Ck (p−1 , Z) is the alternating sum of the chain maps induced by the face maps. We denote the total differential by δ = (−1)p d + ∂. Its homology will be denoted by Hk (• , Z). By Zk (• , Z) we denote the space of k-cycles and by [C] ∈ H∗ (• , Z) the class of a given cycle C. Note that Ck (p , Z) is the free Abelian group generated by the piecewise smooth maps k → p . The construction above can be carried out in exactly the same way replacing Z by R. The corresponding homology groups are denoted by Hk (• , R). According to the universal-coefﬁcient formula (see, for example, [17]), there is a canonical isomorphism Hk (• , R) ! Hk (• , Z) ⊗Z R. • There is a natural pairing between C• (• , R) and CdR (• ) given as follows. For any generator C : k → p in C• (• , Z), + C ∗ ω if ω ∈ k (p ), k C, ω = (8) 0 otherwise.

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For simplicity, we will denote this pairing by C ω. With this notation, the pairing satisﬁes the following identities: ω = dω, dC

C

ω= ∂C

C

ω= δC

(9)

∂ω,

δω. C

Moreover, if φ : G → H is a groupoid homomorphism, then for any C ∈ • C• (G• , R) and ω ∈ CdR (H• ), ω = φ ∗ ω. (10) φ∗ (C)

C

The following result is standard (see, for example, [8, Proposition 6.1]). k ( ) → R, ([C], [ω]) → Proposition 3.1. The pairing Hk (• , R) ⊗ HdR • nondegenerate.

Let k ZdR (• , Z)

C

ω is

= {ω ∈

k ZdR (• )

|

ω ∈ Z for any cycle C ∈ Zk (• , Z)}.

(11)

C k ( , Z) are called integral de Rham cocycles, or simply integral Elements in ZdR • cocycles.

3.2 S 1 -bundles and S 1 -central extensions In this subsection, we recall some basic notations and results concerning S 1 -bundles and S 1 -central extensions over a Lie groupoid. For details, see [4, 5]. Deﬁnition 3.2. Let ⇒ M be a Lie groupoid. A (right) S 1 -bundle over ⇒ M is a (right) S 1 -bundle P over M, together with a (left) action of on P which respects the S 1 -action (i.e., we have (γ · x) · t = γ · (x · t) for all t ∈ S 1 and all compatible pairs (γ , x) ∈ ×M P ). Let Q ⇒ P denote the corresponding transformation groupoid ×M P ⇒ P . There is a natural groupoid homomorphism π from Q ⇒ P to ⇒ M. Of course, Q is an S 1 -bundle over . 1 (Q ), where θ ∈ 1 (P ) is a conA pseudo-connection is a 1-cochain θ ∈ CdR • 2 (Q ) descends to nection 1-form for the S 1 -bundle P → M. One checks that δθ ∈ CdR •

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2 ( ). In other words, there exist unique ω ∈ 1 () and ∈ 2 (M) a 2-cocycle in ZdR • such that δθ = π ∗ (ω + ).

Then ω + is called the pseudo-curvature, which is an integral 2-cocycle. Its class [ω + ] ∈ H 2 (• , Z) is called the Chern class of the S 1 -bundle P . Proposition 3.3 ([4, 5]). Let ⇒ M be a proper Lie groupoid. Assume that ω + ∈ 2 ( ) is an integral 2-cocycle. Then there exists an S 1 -bundle

1 () ⊕ 2 (M) ⊂ CdR • P over ⇒ M and a pseudo-connection θ ∈ 1 (P ) for the bundle P → M whose pseudo-curvature equals ω + . Deﬁnition 3.4. Let ⇒ M be a Lie groupoid. An S 1 -central extension of ⇒ M consists of (1) a Lie groupoid R ⇒ M, together with a morphism of Lie groupoids (π, id) : [R ⇒ M] → [ ⇒ M], (2) a left S 1 -action on R, making π : R → a (left) principal S 1 -bundle. These two structures are compatible in the sense that (s · x)(t · y) = st · (xy) for all s, t ∈ S 1 and (x, y) ∈ R ×M R. Given an S 1 -central extension R of ⇒ M, a pseudo-connection is a 2-cochain 2 (R ), where θ ∈ 1 (R) is a connection 1-form for the bundle R → θ + B ∈ CdR • and B ∈ 2 (M). It is simple to check that δ(θ + B) descends to a 3-cocycle in Z 3 (• ), i.e., δ(θ + B) = π ∗ (η + ω + ) 3 ( , Z) for some η + ω + ∈ Z 3 (• ). Then η + ω + is an integral cocycle in ZdR • 3 and it is called the pseudo-curvature. Its class [η + ω + ] ∈ H (• , Z) is called the Dixmier–Douady class of R.

Proposition 3.5 ([4, 5]). Assume that ⇒ M is a proper Lie groupoid. Given any 3 ( ) such that 3-cocycle η + ω + ∈ ZdR • (1) [η + ω + ] is integral, and (2) is exact, there exists a Lie groupoid S 1 -central extension R ⇒ M of the groupoid ⇒ M and a pseudo-connection θ + B ∈ 1 (R) ⊕ 2 (M) such that its pseudo-curvature equals η + ω + . 3.3 Index of an S 1 -bundle over a central extension π

p

Let R → ⇒ M be an S 1 -central extension, and S 1 → L → M a principal S 1 bundle over the groupoid R ⇒ M with Chern class [L] ∈ H 1 (R• , S 1 ). The example below will be useful in the future. Example 3.6. Consider, for any k ∈ Z, the principal S 1 -bundle Bk : S 1 → · over S 1 ⇒ ·, where the groupoid S 1 ⇒ · acts on Bk by

Quantization of pre-quasi-symplectic groupoids

λ · z = λk z

∀λ ∈ S 1 ⇒ · and

431

∀z ∈ S 1 → ·.

It is well known that H 1 (S•1 , S 1 ) ! Z. Under this isomorphism, the class [Bk ] is simply equal to k. It is also simple to see that the Chern class of Bk can be represented by k where

dt 2π

dt ∈ Z 1 (S 1 ) ⊂ Z 2 ((S 1 )• ), 2π

(12)

is the normalized Haar measure on S 1 .

For any m ∈ M, there exists a groupoid homomorphism fm from S 1 ⇒ · to R ⇒ M deﬁned by fm (λ) = λ · 1m ∀λ ∈ S 1 , (13) where 1m ∈ R is the unit element over m ∈ M. This homomorphism induces a map fm∗ : H 1 (R• , S 1 ) → H 1 (S•1 , S 1 ) ! Z.

(14)

For a principal S 1 -bundle L over R ⇒ M, we deﬁne its index by Ind m (L) = fm∗ ([L]) ∈ H 1 (S•1 , S 1 ) ! Z. We list some of its important properties below. p

π

Proposition 3.7. Let R → ⇒ M be an S 1 -central extension, and S 1 → L → M an (right) principal S 1 -bundle over the groupoid R ⇒ M. Then (1) the index is characterized by the relation fm (λ) · l = l · λIndm (L) ∀λ ∈ S 1 ,

l ∈ p −1 (m),

where the dot on the left-hand side denotes the R-action on L, while the dot on the right-hand side refers to the S 1 -action on L; (2) for any m ∈ M, the pullback fm∗ L is isomorphic to BIndm (L) ; (3) Ind m (L) is constant on the groupoid orbits; (4) Ind m (L) is constant on any connected component of M; and (5) if \M is path connected, then the index Ind m (L) is independent of m ∈ M. Proof. (1), (2) Let l be any point in the ﬁber Lm = p −1 (m). For any λ ∈ S 1 , there exists a unique φ(λ) ∈ S 1 such that fm (λ) · l = l · φ(λ).

(15)

The map λ → φ(λ) does not depend on the choice of l in the ﬁber p−1 (m) and is a group homomorphism from S 1 to S 1 . Therefore, it is of the form φ(λ) = λk for some k ∈ Z.

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Now fm∗ ([L]) ∈ H 1 (S•1 , S 1 ) is the Chern class associated to the pullback of L by fm . On the other hand, according to equation (15), fm∗ L is isomorphic (as a principal S 1 -bundle over S 1 ⇒ ·) to Bk . Therefore, k = Ind m (L) and equation (15) implies fm (λ) · l = l · λIndm (L)

∀l ∈ p−1 (m).

(16)

This proves (1) and (2). (3) For any γ ∈ R with s(γ ) = n and t (γ ) = m, we have γ 1m = 1n γ . It follows from equation (13) that γfm (λ) = fn (λ)γ . Now for any l ∈ p −1 (m), we have (γfm (λ)) · l = (fn (λ)γ ) · l. On one hand, we have (γfm (λ)) · l = (γ · l) · λIndm (L) ,

(17)

(fn (λ)γ ) · l = fn (λ)(γ · l) = (γ · l) · λIndn (L) .

(18)

and From equations (17) and (18), it follows that Ind m (L) = Ind n (L). (4) It is clear from equations (16) that Ind m (L) depends continuously on m ∈ M. Since it is a Z-valued function, we have Ind m (L) = Ind n (L) for any pair of points (m, n) ∈ M × M that are in the same connected component of M. (5) This follows from (3) and (4) immediately.

3.4 Index and Chern class From now on, we will assume that the space of orbits M/ is path-connected, and denote the index of L simply by Ind(L). Therefore, we have a group homomorphism: Ind(L) : H 1 (R• , S 1 ) → Z. From the commutativity of the diagram i

H 1 (R• , S 1 ) → H 2 (R• , Z) ↓ ↓ , i

H 1 (S•1 , S 1 ) → H 2 (S•1 , Z) we see that Ind(L) factors through H 2 (R• , Z) → Z. In the following proposition, we give an explicit formula for Ind(L) in terms of the Chern class. Proposition 3.8. Assume that L → M is a principal S 1 -bundle over R ⇒ M with the π 2 (R , Z), where R → Chern class [θ + ω] ∈ HdR ⇒ M is an S 1 -central extension, • 1 2 and θ + ω ∈ (R) ⊕ (M). Then the index of L is given by Ind(L) = θ, π −1 ((m))

where : M → is the unit map.

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433

Proof. Let L be the pullback of the principal S 1 -bundle L via the homomorphism fm : S 1 → R. The Chern class of L is the pullback of the Chern class of L, i.e., 2 (S 1 ). Since f ∗ ω is a 2-form over a point, it the class deﬁned by fm∗ θ + fm∗ ω ∈ CdR • m vanishes and therefore the Chern class of L is represented by fm∗ θ ∈ 1 (S 1 ). By Proposition 3.7, L is isomorphic to BInd(L) . According to equation (12), the dt dt identity fm∗ θ = Ind(L) 2π + δg = Ind(L) 2π + dg holds for some function g ∈ C ∞ (S 1 , R). Now since fm is a bijection from S 1 to π −1 ((m)), we have θ = fm∗ θ. π −1 ((m))

Therefore,

θ=

π −1 ((m))

fm∗ θ

S1

= Ind(L)

S1

S1

dt + 2π

dg = Ind(L).

S1

Recall that a line bundle L → M over R ⇒ M is called a (, R)-twisted line bundle if Ker π ∼ = M × S 1 acts on L by scalar multiplication, where S 1 is identiﬁed with the unit circle in C [18]. The following corollary is an immediate consequence of Proposition 3.8 and Proposition 3.7. Corollary 3.9. Under the hypotheses of Proposition 3.8, the S 1 -bundle L → M deﬁnes a (, R)-twisted line bundle if and only if θ = 1. π −1 ((m))

4 Prequantization of classical intertwiner spaces 4.1 Compatible prequantizations Deﬁnition 4.1. A prequantization of a pre-quasi-symplectic groupoid ( ⇒ M, ω + π

) consists of an S 1 -central extension R → ⇒ M together with a pseudoconnection θ + B ∈ 1 (R) ⊕ 2 (M) such that δ(θ + B) = π ∗ (ω + ).

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According to Proposition 3.3, if ⇒ M is a proper Lie groupoid, a prequantization exists if and only if ( ⇒ M, ω + ) is exact and ω + is an integral 3-cocycle. A pre-quasi-symplectic groupoid ( ⇒ M, ω + ) is said to be integral if ω + is an integral cocycle. Deﬁnition 4.2. Let ( ⇒ M, ω + ) be an exact pre-quasi-symplectic groupoid and J

(R → ⇒ M, θ + B) a prequantization. Assume that (X → M, ωX ) is a preHamiltonian -space. A compatible prequantization of X consists of an S 1 -bundle φ : L → X with a connection 1-form θL ∈ 1 (L) such that

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(1) J˜ = J ◦ φ : L −→ M is a left R-space and the action satisﬁes (s · κ)(t · x) = st · (κx) for all s, t ∈ S 1 and (κ, x) ∈ R ×M X a compatible pair; (2) the 1-form (θ, θL , −θL ) ∈ 1 (R × L × L) vanishes on the graph of the action + = {(κ, l, κl)|κ ∈ R, l ∈ L compatible pairs}; and (3) dθL = φ ∗ (J ∗ B − ωX ). Note that the second condition above is equivalent to saying that (R × L × p L)/T 2 −→ × X × X with p([κ, l, m]) = (π(κ), φ(l), φ(m)) is a ﬂat S 1 -bundle ¯ which is the 1-form on (R × L × L)/T 2 naturally induced with the connection ', from ' = (θ, θL , −θL ) ∈ 1 (R × L × L) (see [21]). Example 4.3. If is the symplectic groupoid (T ∗ G ⇒ g∗ , ω), where ω ∈ 2 (T ∗ G) is the canonical cotangent symplectic 2-form, a prequantization of can be taken to be R ∼ = T ∗ G × S 1 → T ∗ G, the trivial S 1 -bundle and θ = θT ∗ G + dt, where ∗ θT G ∈ 1 (T ∗ G) is the Liouville 1-form and t is the natural coordinate on S 1 . A Hamiltonian -space is a Hamiltonian G-space J : X → g∗ in the usual sense. It is simple to see that a compatible pre-quantization is a G-equivariant prequantization of X, which always exists when G is connected and simply connected [10]. More generally, the following result was proved in [21]. (The theorem was stated for the symplectic case, but it is valid for the presymplectic case as well.) Proposition 4.4. Let ( ⇒ M, ω) be an s-connected and s-simply connected preJ

symplectic groupoid and (X → M, ωX ) a pre-Hamiltonian space. If both ω and ωX 2 () and H 2 (X), respectively, then represent integral cohomology classes in HdR dR there exists a compatible prequantization. For a given pre-quasi-symplectic groupoid ( ⇒ M, ω+ ) and a prequantization (R → ⇒ M, θ +B), let ×M X ⇒ X be the transformation groupoid as in equation (4). By pulling back the S 1 -central extension R → ⇒ M via J , one obtains an S 1 -central extension of groupoids R ×M X → ×M X ⇒ X. Here R ×M X is again a transformation groupoid, where R acts on X by projecting R to and using the given -action on X. By abuse of notation, we still use J to denote the projection R ×M X → R. Therefore, we have the following homomorphism of S 1 -central extensions of groupoids: J

/R

×M X

J

/

X

J

/M

R ×M X

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Remark 4.5. Note that Proposition 2.3 implies that the Dixmier–Douady class of R ×M X → ×M X ⇒ X vanishes. If ⇒ M is a proper groupoid, so is ×M X ⇒ X. Therefore, R ×M X → ×M X ⇒ X deﬁnes a trivial S 1 -gerbe. According to Proposition 4.2 of [4], there exists an S 1 -bundle E → X such that R ×M X ∼ = s ∗ E ⊗ t ∗ E as a central extension. Proposition 4.6. Let ( ⇒ M, ω + ) be an exact pre-quasi-symplectic groupoid J

and (R → ⇒ M, θ + B) its prequantization. Assume that (X → M, ωX ) is a φ

pre-Hamiltonian -space. Then (L → X, θL ) is a compatible prequantization of X if and only if (the associated line bundle of ) φ : L → X is a twisted line bundle over R ×M X → ×M X ⇒ X with the pseudo-connection and the pseudo-curvature being given by θL and J ∗ θ + (J ∗ B − ωX ) ∈ 1 (R ×M X) ⊕ 2 (X), respectively. φ

Proof. Given a compatible prequantization L → X, deﬁne an action of R×M X ⇒ X on L by (κ, φ(l)) · l = κl, where κ ∈ R and l ∈ L are compatible pairs. It is simple to φ

check that all the compatibility conditions are satisﬁed so that L → X is a twisted line bundle over the S 1 -central extension R ×M X → ×M X ⇒ X. It is simple to see that the corresponding transformation groupoid (R ×M X) ×X L ⇒ L is isomorphic to the transformation groupoid R ×M L ⇒ L. Moreover, it is simple to see that Condition (2) of Deﬁnition 4.2 implies that ∂θL = φ ∗ J ∗ θ,

(21)

where, by abuse of notation, we use φ to denote the Lie groupoid homomorphism: R ×M L L

φ

/ R ×M X

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/X

φ

and ∂θL is with respect to the groupoid R ×M L ⇒ L. Therefore, we have δθL = ∂θL + dθL = φ ∗ (J ∗ θ + J ∗ B − ωX ). The converse can be proved by working backwards.

As an immediate consequence, we have the following. Corollary 4.7. Under the hypotheses of Proposition 4.6 and assuming that ⇒ M is J

proper, for a pre-Hamiltonian -space (X → M, ωX ), a compatible prequantization 2 ((R × X) , Z). exists if and only if J ∗ (θ + B) − ωX is an integral 2-cocycle in ZdR M • Proof. One direction is obvious by Proposition 4.6. For the other direction, note that Proposition 2.3 implies that J ∗ (θ + B) − ωX is always a 2-cocycle since δ(J ∗ (θ + B) − ωX ) = J ∗ δ(θ + B) − π ∗ δωX = J ∗ π ∗ (ω + ) − π ∗ J ∗ (ω + ) = 0.

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Here we have used equations (19) and (6). If J ∗ (θ + B) − ωX is an integral cocycle in 2 ((R × X) , Z), according to Proposition 3.3, there exists an S 1 -bundle L → X ZdR M • over R ⇒ X and a pseudo-connection θL ∈ 1 (L) whose pseudo-curvature equals to J ∗ θ + (J ∗ B − ωX ). According to Corollary 3.9, one sees that (the associated line bundle of) L is indeed a twisted line bundle over R ×M X → ×M X ⇒ X. Then L → X is a compatible prequantization by Proposition 4.6.

4.2 Prequantization of classical intertwiner spaces We are now ready to state the main theorem of this section. Theorem 4.8. Let ( ⇒ M, ω + ) be an exact pre-quasi-symplectic groupoid Jk

and (R → ⇒ M, θ + B) a prequantization. Assume that (Xk → M, ωk ), k = 1, 2, are pre-Hamiltonian -spaces, ⇒ M acts freely on X2 ×M X1 and φk

X2 × X1 = \(X2 ×M X1 ) is a smooth manifold. Let (Lk → Xk , θk ), be a compatible prequantization of Xk for k = 1, 2. Then φ : R\(L2 ×M L1 ) → X2 × X1 ,

φ[l2 , l1 ] = [φ2 (l2 ), φ1 (l1 )],

with the S 1 -action λ · [l2 , l1 ] = [λ · l2 , l1 ], λ ∈ S 1 , is an S 1 -principal bundle. Moreover, i ∗ (θ2 , −θ1 ) descends to a connection 1-form on R\(L2 ×M L1 ), which deﬁnes a prequantization of the classical intertwiner space X2 × X1 . Here i : L2 ×M L1 → L2 × L1 is the natural embedding. Proof. One checks directly that φ : R\(L2 ×M L1 ) → X2 × X1 is an S 1 -bundle. Now let R ⇒ M act on L2 ×M L1 diagonally. We have ∂i ∗ (θ2 , −θ1 ) = i ∗ (∂θ2 , −∂θ1 ) = i ∗ (φ2∗ J2∗ θ, −φ1∗ J1∗ θ ) = 0. Hence i ∗ (θ2 , −θ1 ) descends to a 1-form on the quotient space R\(L2 ×M L1 ), which can be easily seen to be a connection 1-form. Now d(i ∗ (θ2 , −θ1 )) = i ∗ (dθ2 , −dθ1 ) = i ∗ (φ2∗ (J2∗ B − ω2 ), φ1∗ (J1∗ B − ω1 )) = i ∗ (φ2 × φ1 )∗ (−ω2 , ω1 ),

where in the last equality we used the relation J1 ◦ φ1 = J2 ◦ φ2 on L2 ×M L1 . Here φk and Jk , k = 1, 2 are groupoid homomorphisms: R ×M Lk Lk

φk

φk

and i is the groupoid homomorphism:

/ R × M Xk / Xk

Jk

/R

Jk

/M

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Quantization of pre-quasi-symplectic groupoids

R ×M (L2 ×M L1 ) L2 × M L1

i

/ (R ×M L2 ) × (R ×M L1 )

i

/ L2 × L 1

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This completes the proof. 4.3 Morita equivalence

Deﬁnition 4.9. Pre-quasi-symplectic groupoids (G ⇒ G0 , ωG + G ) and (H ⇒ H0 , ωH + H ) are said to be Morita equivalent if there exists a Morita equivalence ρ σ bimodule G0 ← X → H0 between the Lie groupoids G and H , together with a ρ×σ

2-form ωX ∈ 2 (X) such that (X → G0 × H0 , ωX ) is a pre-Hamiltonian G × H space, where the G×H -action on X is given by (g, h)·x = gxh−1 for all compatible triples g ∈ G, h ∈ H , and x ∈ X. One easily checks that this is indeed an equivalence relation among pre-quasisymplectic groupoids. Let Q ⇒ X be the transformation groupoid Q : (G × H ) ×(G0 ×H0 ) X ⇒ X. Then the natural projections pr 1 : Q → G and pr 2 : Q → H are groupoid homomorphisms. As an immediate consequence of Proposition 2.3, we have the following identity pr ∗1 (ωG + G ) − pr ∗2 (ωH + H ) = δωX . Note that the axioms of Morita equivalence of Lie groupoids assure that, as groupoids, Q∼ = G[X] and Q ∼ = H [X] (see the proof of [22, Proposition 4.5]), where G[X] ⇒ X and H [X] ⇒ X are the pullback groupoids of G and H using ρ and σ , respectively. Recall that for a given Lie groupoid ⇒ M, two cohomologous 3-cocycles ωi + i ∈ 2 () ⊕ 3 (M), i = 1, 2, are said to differ by a gauge transformation of the ﬁrst type if (ω1 + 1 ) − (ω2 + 2 ) = δB for some B ∈ 2 (M). By a Morita morphism from the pre-quasi-symplectic groupoid ( ⇒ M , ω +

) to ( ⇒ M, ω+ ), we mean a Morita morphism of the Lie groupoid p : → (i.e., is isomorphic to the pullback groupoid [M ] ⇒ M ) such that ω + and p∗ ω + p ∗ differ by a gauge transformation of the ﬁrst type. The following result gives a more intuitive explanation of Morita equivalence. Proposition 4.10. Two pre-quasi-symplectic groupoids are Morita equivalent if and only if there exists a third pre-quasi-symplectic groupoid together with a Morita morphism to each of them.

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Corollary 4.11. For two Morita-equivalent pre-quasi-symplectic groupoids, if one is integral, so is the other. Therefore, Morita equivalence induces an equivalence relation among integral pre-quasi-symplectic groupoids. One of the most important features of Morita-equivalent pre-quasi-symplectic groupoids is the following. Theorem 4.12. Suppose that (G ⇒ G0 , ωG + G ) and (H ⇒ H0 , ωH + H ) are Morita-equivalent pre-quasi-symplectic groupoids with an equivalence bimodule ρ σ G0 ← X → H0 . Then we have the following: (1) Corresponding to any pre-Hamiltonian G-space JF : F → G0 , there is a unique (up to isomorphism) pre-Hamiltonian H -space JE : E → H0 such that F and E are a pair of related pre-Hamiltonian spaces and vice versa. (2) Let JFi : Fi → G0 , i = 1, 2, be pre-Hamiltonian G-spaces and JEi : Ei → H0 , i = 1, 2, their related pre-Hamiltonian H -spaces. Then F2 ×G F1 and E2 ×H E1 are diffeomorphic as presymplectic manifolds (in the sense that if one is smooth so is the other). Proof. This was proved in [22] for quasi-symplectic groupoids and their Hamiltonian spaces. One can prove this theorem in a similar fashion (in fact in a simpler way by using Proposition 2.4). We will leave the details to the reader.

We now can introduce Morita equivalence for the prequantization of pre-quasisymplectic groupoids. Deﬁnition 4.13. Let (G ⇒ G0 , ωG + G ) and (H ⇒ H0 , ωH + H ) be Moritaequivalent integral exact pre-quasi-symplectic groupoids with an equivalence bimodρ σ ule (G0 ← X → H0 , ωX ). We say their prequantizations (RG → G ⇒ G0 , θG +BG ) and (RH → H ⇒ H0 , θH + BH ) are Morita equivalent if X admits a compatible prequantization (Z → X, θZ ) with respect to the prequantization of the pre-quasisymplectic groupoid (RG ×RH )/S 1 → G×H ⇒ G0 ×H0 , (θG +θH )+(BG +BH )). It is simple to see that G0 ←Z→H0 is an equivalence bimodule of S 1 -central extensions in the sense of [18, Deﬁnition 2.11]. Remark 4.14. 1. Note that prequantizations can be Morita equivalent as S 1 -central extensions, but not Morita equivalent as prequantizations. The former simply means that they correspond to isomorphic S 1 -gerbes, and up to a torsion, are determined by their Dixmier–Douady classes. 2. It would be interesting to investigate the following question: given two Moritaequivalent pre-quasi-symplectic groupoids and a prequantization of one of them, is it possible to construct a Morita-equivalent prequantization for the other prequasi-symplectic groupoid?

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439

A useful feature of Morita equivalence is that it gives a recipe which allows us to construct compatible prequantizations. Theorem 4.15. For Morita-equivalent prequantizations of pre-quasi-symplectic groupoids, there is an equivalence of categories of compatible prequantizations of pre-Hamiltonian spaces. Proof. Let (G ⇒ G0 , ωG + G ) and (H ⇒ H0 , ωH + H ) be Morita-equivalent ρ integral exact pre-quasi-symplectic groupoids with an equivalence bimodule (G0 ← σ X → H0 , ωX ), and (RG → G ⇒ G0 , θG + BG ) and (RH → H ⇒ H0 , θH + BH ) Morita-equivalent prequantizations given by (Z → X, θZ ).Assume that J : F → G0 is a pre-Hamiltonian G-space and (L → F, θL ) a compatible prequantization. It is J

known that the corresponding pre-Hamiltonian H -space is E := X ×G F → H0 , where J : E → H0 and the H -action on E are deﬁned by J ([x, f ]) = σ (x) and h · [x, f ] = [x · h−1 , f ], respectively. Let L = Z ×RG L. Then it is clear that L is an S 1 -bundle over E, and RH acts on L equivariantly. It is simple to check that i ∗ (θZ , −θL ), where i : Z ×G0 L → Z × L, descends to a 1-form on the quotient space Z ×RG L, which is indeed a connection 1-form θL on L . It is routine to check that (L → E, θL ) is a compatible prequantization of the pre-Hamiltonian H -space J : E → H0 . The inverse functor can be constructed in a similar fashion.

Remark 4.16. The above theorem indicates a useful method which enables one to transform prequantizations of Hamiltonian LG-spaces to prequantizations of quasiHamiltonian G-spaces of AMM and vice versa. The latter is understood as a compatible prequantization corresponding to the quasi-symplectic groupoid (G × G)[U] ⇒ H Ui , which is the pullback quasi-symplectic groupoid of the AMM quasi-symplectic groupoid using an open covering U = (Ui )i∈I of G (seeH[14], for instance, for an explicit construction). It is known that (G × G)[U] ⇒ Ui is Morita equivalent to the symplectic groupoid (LG × Lg ⇒ Lg, ωLG×Lg ) according to Proposition 4.26 [22]. The question is therefore boiled down to the construction of a compatible prequantization of the Morita equivalence Hamiltonian bimodule.

5 Integral pre-Hamiltonian -spaces The main purpose of this section is to give a geometric integrality condition which guarantees the existence of a prequantization of a pre-Hamiltonian -space. 5.1 Integrality condition Lemma 5.1. Let J : G → H be a groupoid homomorphism. By Ker(J∗ ), we de2 (G ). The following note the kernel of J∗ : H2 (G• , Z) → H2 (H• , Z). Let ω ∈ ZdR • conditions are equivalent: 2 (H ) such that (1) There exists + ∈ ZdR •

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C. Laurent-Gengoux and P. Xu 2 ω + J ∗ + ∈ ZdR (G• , Z).

(2) For any C ∈ Z2 (G• , Z) with [C] ∈ Ker(J∗ ), we have ω ∈ Z. C

Proof. (1) ⇒ (2): By deﬁnition, we have for any C ∈ Z2 (G• , Z), (ω + J ∗ +) ∈ Z.

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C

From equation (9), we also have ∗ (ω + J +) = ω + C

C

+.

J∗ (C)

If [J∗ (C)] = 0, i.e., J∗ (C) = δD for some D ∈ C3 (H• , Z), then ∗ (ω + J +) = ω + + = ω + δ+ = ω C

C

δD

C

D

C

since δ+ = 0. Therefore, C ω ∈ Z. (2) ⇒ (1): Since there exists a Z-submodule H in H2 (G• , Z) such that H2 (G• , Z) = H ⊕ Ker(J∗ ), the Z-map f : Ker(J∗ ) → Z, f ([C]) = ω ∀[C] ∈ Ker(J∗ ), C

can be extended to a Z-map f˜ : H2 (G• , Z) → Z. According to Proposition 3.1, there 2 (G ) such that exists ω ∈ ZdR • f˜([C]) = ω ∀[C] ∈ H2 (G• , Z). C 2 (G , Z). Moreover, we have By equation (11), ω is an integral cocycle in ZdR • (ω − ω) = 0 ∀C ∈ Z2 (G• , Z) such that [C] ∈ Ker(J∗ ). (26) C 2 (H ) → H 2 (G ), we have Since J∗ : H2 (G• , R) → H2 (H• , R) is dual to J ∗ : HdR • • dR 2 ( ). This ⊥ ∗ ∗ Ker(J∗ ) = Im(J ). Therefore, [ω − ω] = J [+] for some + ∈ ZdR • proves (1).

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Deﬁnition 5.2. Let ( ⇒ M, ω + ) be a pre-quasi-symplectic groupoid. A preHamiltonian -space (X → M, ωX ) is said to satisfy the integrality condition if for any C ∈ Z2 (( ×M X)• , Z) and any D ∈ C3 (• , Z), δD = J∗ (C) ⇒ ωX − (ω + ) ∈ Z. (27) C

D

In this case, we also say that the pair (ωX , ω + ) satisﬁes the integrality condition. Remarks 5.3.

1. By taking C = 0, equation (27) implies that D (ω + ) ∈ Z ∀D ∈ Z3 (• , Z). That is, ω + must be an integral 3-cocycle and therefore ( ⇒ M, ω + ) must be an integral pre-quasi-symplectic groupoid. 2. If ω + is a 3-coboundary δK, then the integrality condition is equivalent to (ωX − J ∗ K) ∈ Z ∀C ∈ Z2 (( ×M X)• , Z) such that J∗ [C] = 0. (28) C

From now on, we shall always assume that ( ⇒ M, ω + ) is an integral pre-quasi-symplectic groupoid. The following lemma indicates that it is sufﬁcient to require that equation (27) holds for a single representative (C, D) in every class of Ker(J∗ ). Lemma 5.4. Let ( ⇒ M, ω + ) be an integral pre-quasi-symplectic groupoid. A pre-Hamiltonian -space (X → M, ωX ) satisﬁes the integrality condition if and only if for any class c ∈ Ker J∗ , there exists C ∈ Z2 (( ×M X)• , Z) and D ∈ C3 (• , Z) with c = [C] and J∗ (C) = δD such that ωX − (ω + ) ∈ Z. C

D

Proof. Let C ∈ Z2 (( ×M X)• , Z) and D ∈ C3 (• , Z) be any pair satisfying J∗ (C ) = δD . Then [C ] ∈ Ker(J∗ ). By assumption, there exists a pair (C, D) such that [C] = [C ], J∗ (C) = δD, and C ωX − D (ω + ) ∈ Z. Assume that C = C + δE for some E ∈ C3 (( ×M X)• , Z). Then we have ⎛ ⎞ ωX − (ω + ) − ⎝ ωX − (ω + )⎠ C

D

=−

ωX +

δE

=−

C

(ω + ) − D

δωX + E

D

(ω + ) D

(ω + ) − D

(ω + ) D

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J ∗ (ω + ) +

=− E

(ω + ) − D

D

=

(ω + )

(ω + ). D−J∗

(E)−D

Since δ(D − J∗(E) − D ) = J∗ (C − C − δE) = 0 and ω + is an integral cocycle,

it follows that D−J∗ (E)−D (ω + ) ∈ Z. This completes the proof. Now assume that ( ⇒ M, ω + ) is an integral exact pre-quasi-symplectic groupoid, and R → ⇒ M is a prequantization. Let θ + B ∈ 1 (R) ⊕ 2 (M) be a pseudo-connection satisfying equation (19). In order to ﬁx the notation, recall that we have the following commutative diagram of groupoid homomorphisms J

R ×M X → R ↓π ↓π

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J

×M X → where the horizontal arrows are projections. Lemma 5.5. Assume that C ∈ Z2 ((R ×M X)• , Z) satisﬁes J∗ (C ) = kZ + δD for some D ∈ C3 (R• , Z). Let C = π∗ (C ) and D = π∗ (D ). Then ωX − (ω + ) = k + (ωX − J ∗ (θ + B)), (30) C

C

D

where Z ∈ Z1 (R• , Z) is the 1-cycle deﬁned by equation (42). Proof. First, since π : R ×M X → ×M X reduces to the identity map when being restricted to the unit spaces, we have ωX = ω X = ωX . (31) C

π∗ (C )

C

Now by equation (9), we have J ∗ (θ + B) = (θ + B) = k (θ + B) + (θ + B). C

J∗ (C )

Z

δD

According to Lemma 6.1, Z (θ + B) = Z θ = 1. Therefore, J ∗ (θ + B) = k + (θ + B) C

δD

=k+

δ(θ + B) (by equation (19)) D

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=k+

π ∗ (ω + )

443

(by equation (9))

D

=k+

(ω + ). D

Hence it follows that ωX − (ω + ) = k + (ωX − J ∗ (θ + )). C

C

D

The following proposition gives a useful characterization of integrality condition. Proposition 5.6. Let ( ⇒ M, ω + ) be an integral pre-quasi-symplectic groupoid, J

and (R → ⇒ M, θ + B) a prequantization. Assume that (X → M, ωX ) is a preHamiltonian -space. Then the following conditions are equivalent: 2 ( ) such that (1) There exists a 2-cocycle + ∈ ZdR • 2 ((R ×M X)• , Z). ωX − J ∗ (θ + B) − J ∗ π ∗ + ∈ ZdR

(2) For any cycle C ∈ Z2 ((R ×M X)• , Z) such that [C ] ∈ Ker(π∗ ◦ J∗ ), we have (ωX − J ∗ (θ + B)) ∈ Z. C

(3) The pair (ωX , ω + ) satisﬁes the integrality condition. Proof. (1) ⇐⇒ (2): follows from Lemma 5.1. (2) ⇒ (3): Any class in Ker J∗ ⊂ H 2 (( ×M X)• , Z) can be represented by a 2-cocycle of the form C = π∗ (C ), where C ∈ Z2 ((R ×M X)• , Z). Then [C ] is in the kernel of J∗ ◦ π∗ = π∗ ◦ J∗ . It thus follows that [J∗ (C )] ∈ Ker(π∗ ). By Lemma 6.1, [J∗ (C )] = k[Z] for some k ∈ Z, where Z ∈ C1 (R• , Z) is deﬁned by equation (42). In other words, there exists D ∈ C3 (R• , Z) such that J∗ (C ) = kZ + δD . Let D = π∗ (D ). One can easily see that δD = π∗ (J∗ (C )) = J∗ (C). Then by Lemma 5.5, we have C ωX − D (ω + ) ∈ Z. By Lemma 5.4, this implies that the pair (ωX , ω + ) satisﬁes the integrality condition. (3) ⇒ (2): Let C ∈ Z2 ((R × X)• , Z) be any cycle whose class is in the kernel of π∗ ◦ J∗ . Since [J∗ (C )] ∈ Ker(π∗ ), Lemma 6.1 implies that there exists k ∈ Z and D ∈ C3 (R• , Z) such that J∗ (C ) = kZ + δD . (33) Therefore, by equation (30), we have C (ωX −J ∗ (θ +B)) = −k+ C ωX − D (ω+ ), where C = π∗ (C ) and D = π∗ (D ). By applying π∗ to equation (33), one ﬁnds that J∗ (C) = δD. Since (ωX , ω + ) satisﬁes the integrality condition, it thus follows that C (ωX − J ∗ (θ + B)) ∈ Z.

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As an immediate consequence, we obtain the following main result of the section. Theorem 5.7. Let ( ⇒ M, ω + ) be an exact proper pre-quasi-symplectic J

groupoid, and (X → M, ωX ) a pre-Hamiltonian -space. Then there exists a compatible prequantization R → ⇒ M and L → X if and only if the pair (ωX , ω + ) satisﬁes the integrality condition of equation (27). Proof. Assume that (R → ⇒ M, θ +B) and (L → X, θL ) are a pair of compatible 2 ((R × X) , Z). prequantizations. By Corollary 4.7, we have ωX − J ∗ (θ + B) ∈ ZdR M • Hence (ωX , ω + ) satisﬁes the integrality condition according to Proposition 5.6. Conversely, assume that (ωX , ω + ) satisﬁes the integrality condition. Then ω + must be an integral cocycle. Let (R → ⇒ M, θ + B) be a prequantization, which always exists since is proper. Again according to Proposition 5.6, there exists 2 ( ) such that ω −J ∗ (θ +B)−J ∗ π ∗ + ∈ Z 2 ((R × X) , Z). a 2-cocycle + ∈ ZdR • X M • dR Since is proper, + is cohomologous to α + B0 , where α ∈ 1 () is a closed 1-form and B0 ∈ 2 (M). Then θ + B := (θ + π ∗ α) + (B + B0 ) is clearly also a pseudo2 ((R × X) , Z). From Corollary 4.7, it connection and ωX − J ∗ (θ + B ) ∈ ZdR M • J

follows that (X → M, ωX ) admits a compatible prequantization (L → X, θL ).

5.2 Integral quasi-Hamiltonian G-spaces In this subsection, G is a connected and simply-connected compact Lie group and 1 denotes the unit of G. We intend to study the case where is the AMM quasisymplectic groupoid. Assume that X is a G-space. There is a natural map i : H2 (X, Z) → H2 ((G × X)• , Z) induced by the inclusion C2 (X, Z) ⊂ C2 ((G×X)• , Z). The following lemma indicates that i is in fact an isomorphism. Lemma 5.8. If G is a connected and simply-connected Lie group, then the map i : H2 (X, Z) → H2 ((G × X)• , Z) is an isomorphism. Proof. This is a standard result. For completeness, we sketch a proof below. Let G → EG → BG be the usual G-bundle over the classifying space BG and XG = G\(EG × X). We have the ﬁbration G → EG × X → XG . 2 The second term of the homology Leray–Serre spectral sequence is Ep,q = Hp (XG , Hq (G, Z)), i.e., the homology of XG with local coefﬁcients in Hq (G, Z) (see [12]). Since G is simply-connected, we have H1 (G, Z) = H2 (G, Z) = 0 and 2 has the following form for 0 ≤ p ≤ 3 and 0 ≤ q ≤ 2: Ep,q .. .. .. .. . . . . 0 0 0 0 ··· 0 0 0 0 ··· H0 (XG , Z) H1 (XG , Z) H2 (XG , Z) H3 (XG , Z) · · ·

(34)

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According to the Leray–Serre theorem, this spectral sequence converges to H∗ (EG × X, Z). It is clear from equation (34) that, in particular, we have H2 (EG × X, Z) ! H2 (XG , Z). Since EG is contractible, we get H2 (XG , Z) ! H2 (X, Z). The lemma now follows from the well-known isomorphism H2 ((G × X)• , Z) ! H2 (XG , Z).

Since H2 (g∗ , Z) = 0, Lemma 5.8 implies that H2 ((T ∗ G)• , Z) = 0.

(35)

Since any simply-connected Lie group G satisﬁes H2 (G, Z) = 0, we also have H2 ((G × G)• , Z) = 0.

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Recall that the AMM quasi-symplectic groupoid is (G × G ⇒ G, ω + ) [6, 22], where G is a compact Lie group equipped with an ad-invariant nondegenerate symmetric bilinear form (·, ·). Here G × G ⇒ G is the transformation groupoid, where G acts on itself by conjugation, and ω and are deﬁned as follows. Following [3], we denote by θ and θ¯ the left and right Maurer–Cartan forms on G, respectively, i.e., θ = g −1 dg and θ¯ = (dg)g −1 . Let ∈ 3 (G) denote the bi-invariant 3-form on G corresponding to the Lie algebra 3-cocycle 1 3 ∗ 12 (·, [·, ·]) ∈ ∧ g , i.e.,

=

1 1 ¯ (θ, [θ, θ]) = (θ¯ , [θ¯ , θ]), 12 12

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and ω ∈ 2 (G × G) the 2-form 1 ω|(g,x) = − [(Adx pr ∗1 θ, pr ∗1 θ ) + (pr ∗1 θ, pr ∗2 (θ + θ¯ ))], 2

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where (g, x) denotes the coordinate in G × G, and pr 1 and pr 2 : G × G → G are natural projections. It is known that ω + is an integral 3-cocycle. A triple (X, ωX , J ), where X is a manifold, ωX is a G-invariant 2-form on X and J : X → G is a smooth map, is a quasi-Hamiltonian G-space in the sense of [3] if (B1) the differential of ωX is given by dωX = J ∗ ; (B2) the map J satisﬁes ξˆ and

ωX =

1 ∗ J (ξ, θ + θ¯ ); 2

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(B3) at each x ∈ X, the kernel of ωX is given by ker ωX = {ξˆ (x)|ξ ∈ ker(Ad J (x) +1)}, where ξˆ is the vector ﬁeld on X associated to the inﬁnitesimal action of ξ ∈ g on X. J

It is known [22] that these conditions are equivalent to (X → G, ωX ) being a Hamiltonian -space, where is the AMM quasi-symplectic groupoid (G × G ⇒ G, ω + ). In this case, the integrality can be described in simpler terms as indicated in the following. Proposition 5.9. Let be the AMM quasi-symplectic groupoid (G × G ⇒ G, ω +

), where G is a connected and simply-connected Lie group equipped with an J

ad-invariant nondegenerate symmetric bilinear form. Let (X → G, ωX ) be a quasi-Hamiltonian G-space. Assume that ω + is an integral 3-cocycle in 3 ((G × G) , Z). Then the pair (ω , ω + ) satisﬁes the integrality condition ZdR • X if and only if ∀C ∈ Z2 (X, Z) and D ∈ C3 (G, Z) such that dD = J∗ (C), ωX − ∈ Z. (39) C

D

Note that such D always exists for any C ∈ Z2 (X, Z). Proof. Note that we have the following commutating diagram of groupoid homomorphisms: i

X• → (G × X)• ↓J ↓J

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J

G• → (G × G)• , where X• and G• are spaces X and G are considered as groupoids, while (G × X)• and (G × G)• are the transformation groupoids. Thus one direction is obvious. Conversely, according to equation (36), we have H2 ((G×G)• , Z) = 0. Therefore, Ker(J∗ ) = H2 ((G × X)• , Z). By Lemma 5.8, for any class C ∈ H2 ((G × X)• , Z), there exists C ∈ Z2 (X, Z) such that C = i∗ [C]. Since H2 (G, Z) = 0, there always exists D ∈ C3 (G, Z) such that J∗ (C) = dD. Hence J∗ (i∗ C) = i∗ (J∗ C) = i∗ dD = δ(i∗ D). Now it is clear that ∗ ∗ ωX − (ω + ) = i ωX − i (ω + ) = ωX − . i∗ C

i∗ D

C

D

The conclusion thus follows from Lemma 5.4.

C

D

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Remark 5.10. Note that equation (39) coincides with the quantization condition of Alekseev–Meinrenken [2]. See also [16]. For the case of conjugacy classes, see [14, 15]. As an immediate consequence of Proposition 5.9, we have the following. Corollary 5.11. Let be the AMM quasi-symplectic groupoid (G × G ⇒ G, ω + ). Then 1 ∈ G, considered as a quasi-Hamiltonian G-space, satisﬁes the integrality condition. Let us consider the case of Example 4.3 where is the symplectic groupoid T ∗ G ⇒ g∗ . In this case, we recover a well-known result of Guillemin–Sternberg [10]. Proposition 5.12. Let be the symplectic groupoid (T ∗ G ⇒ g∗ , ω), where G is a connected and simply-connected Lie group. Let J : X → g∗ be a momentum map for a Hamiltonian G-space (X, ωX ) as in Example 4.3. The pair (ωX , ω) satisﬁes the integrality condition if and only if ωX is an integral 2-form. Proof. According to equation (35), we have H2 ((T ∗ G)• , Z) = 0. Therefore, for any C ∈ Z2 ((G × X)• , Z) there exists D ∈ C3 ((T ∗ G)• , Z) such that J∗ (C) = δD. By Lemma 5.8, we may assume that C ∈ Z2 (X, Z). Since H2 (g∗ , Z) = 0, we can assume that D ∈ C3 (g∗ , Z). Since = 0, the integrality condition of equation (27) thus reads C ωX ∈ Z.

In particular, a coadjoint orbit O ⊂ g∗ , endowed with the Kirillov–Kostant– Souriau symplectic structure ωO , satisﬁes the integrality condition if and only if ωO is an integral 2-form. 5.3 Integrality condition and Morita equivalence In general, a pre-quasi-symplectic groupoid may not be exact, as in the case of the AMM-quasi-symplectic groupoid for instance. In such a case, one must pass to a Morita-equivalent pre-quasi-symplectic groupoid in order to construct a prequantization. According to Theorem 4.12, Morita-equivalent quasi-(pre)symplectic groupoids yield equivalent momentum map theories in the sense that there is a bijection between their (pre)-Hamiltonian -spaces and the classical intertwiner spaces are independent of Morita equivalence [22]. More precisely, given a pre-quasi-symplectic groupoid ( ⇒ M, ω + ), where p

may not be exact, one can choose a surjective submersion N → M and consider the pullback groupoid [N] ⇒ N of ⇒ M via p. Then ([N] ⇒ N, p∗ ω + p∗ ) J

is again a pre-quasi-symplectic groupoid. Moreover, if (X → M, ωX ) is a preJN

Hamiltonian -space, then (XN → N, p∗ ωX ) is a pre-Hamiltonian [N]-space, where XN = X ×M N , and p : XN → X and JN : XN → N are the projections to the ﬁrst and second components, respectively. The following proposition indicates that integrality condition is preserved under this pullback procedure.

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Lemma 5.13. The pair (ωX , ω + ) satisﬁes the integrality condition if and only if (p∗ ωX , p∗ ω + p ∗ ) satisﬁes the integrality condition. Proof. By abuse of notation, we use the same letter p to denote the groupoid homomorphisms from [N ] ×N XN ⇒ XN to ×M X ⇒ X, and from [N] ⇒ N to ⇒ M, both of which are Morita morphisms. For any C ∈ Z2 (([N]×N XN )• , Z) and D ∈ C3 ([N]• , Z) with J∗ (C ) = δD , we have p ∗ ωX − p ∗ (ω + ) = ωX − (ω + ), (41) C

(C )

D

C

D

(D )

and D = p∗ clearly satisfy J∗ (C) = δD. where C = p∗ Assume that the pair (ωX , ω + ) satisﬁes the integrality condition. Then equation (41) implies immediately that so too does the pair (p ∗ ωX , p∗ ω + p ∗ ). Conversely, if (p∗ ωX , p∗ ω + p ∗ ) satisﬁes the integrality condition, then ω + must be an integral cocycle. Now we have the commutative diagram: p∗

H2 (([N] ×N XN )• , Z) → H2 (( ×M X)• , Z) ↓ JN ∗ ↓ J∗ p∗ H2 ([N]• , Z) → H2 (• , Z), where the horizontal arrows are isomorphisms. Therefore, p∗ : H2 (([N] ×N XN )• , Z) → H2 (( ×M X)• , Z) induces an isomorphism from Ker(JN ∗ ) to Ker(J∗ ). This implies that any class in Ker(J∗ ) has a representative of the form C = p∗ (C ), where C = δD for some D ∈ C3 (([N] ×N XN )• , Z). Let D = p∗ (D ). By equation (41), we see that if the pair (p∗ ωX , p∗ ω + p ∗ ) satisﬁes the integrality condition then C ωX − D (ω + ) ∈ Z. By Lemma 5.4, we conclude that (ωX , ω + ) satisﬁes the integrality condition. Corollary 5.14. Let (G ⇒ G0 , ωG + G ) and (H ⇒ H0 , ωH + H ) be Moritaequivalent pre-quasi-symplectic groupoids. Assume that (F → G0 , ωF ) and (E → H0 , ωE ) are a pair of corresponding pre-Hamiltonian spaces. Then (ωF , ωG + G ) satisﬁes the integrality condition if and only if (ωE , ωH + H ) satisﬁes the integrality condition. Proof. It sufﬁces to prove this assertion for a Morita morphism of pre-quasisymplectic groupoids. By Lemma 5.13, it remains to prove that the integrality condition is preserved by gauge transformations of the ﬁrst type, which can be easily checked.

As a consequence, given a pre-quasi-symplectic groupoid ( ⇒ M, ω+ ), where p

may not be exact, one can choose a surjective submersion N → M such that p∗ ∈

3 (N) is exact and replace ( ⇒ M, ω + ) by a Morita-equivalent exact H pre-quasisymplectic groupoid ([N] ⇒ N, p∗ ω+p ∗ ). Usually, one takes N := Ui → M,

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where U = (Ui ) is an H open cover of M. Then the pullback pre-quasi-symplectic groupoid is ([U] ⇒ Ui , ω|[U ] + |Ui ), where [U], as a manifold, can be H U identiﬁed with the disjoint union Uij . Lemma 4.7 guarantees that the integrality condition always holds no matter which surjective submersion (or open covering) N → M is taken as long as the initial pair (ωX , ω + ) satisﬁes the integrality condition, and therefore one can always construct a compatible prequantization. Applying the above discussion to the AMM quasi-symplectic groupoid (G×G ⇒ G, ω + ) and using Theorem 5.7, we are led to the following. J

Corollary 5.15. Let (X → G, ωX ) be a quasi-Hamiltonian G-space. The following are equivalent: H H (1) There exists H H a compatible prequantizationH Rij → (G × G)[U] ⇒ Ui and Li → X|Ui , where (G × G)[U] ⇒ Ui is the pullback quasi-symplectic groupoid of the AMM groupoid using any open covering of G such that ∀i, |Ui is an exact form. (2) The integrality condition of equation (39) holds. 5.4 Strong integrality condition Deﬁnition 5.16. Let ( ⇒ M, ω + ) be a pre-quasi-symplectic groupoid. A preHamiltonian -space (X → M, ωX ) is said to satisfy the strong integrality condition if (1) it satisﬁes the integrality condition; and 2 ( ) → H 2 (( × X) ) vanishes. (2) the map J ∗ : HdR • M • dR The following result follows from Theorem 5.7. Proposition 5.17. Let ( ⇒ M, ω + ) be an exact proper pre-quasi-symplectic J

J

groupoid, and (X → M, ωX ) a pre-Hamiltonian -space. Then (X → M, ωX ) satisﬁes the strong integrality condition if and only if for any prequantization of ( ⇒ M, ω + ), X admits a compatible prequantization. J

Proof. If (X → M, ωX ) satisﬁes the strong integrality condition, it is clear from Theorem 5.7 that X admits a compatible prequantization for any prequantization of ( ⇒ M, ω + ). Conversely, given any prequantization (R → ⇒ M, θ + B), J ∗ (θ + B) − ωX 2 ((R × X) , Z). Note that if θ + B is a pseudomust be an integral 2-cocycle in ZdR M • 2 ( ). Since the subset of integral classes connection, so is θ + B + π ∗ + ∀+ ∈ ZdR • 2 ((R × X) , Z) is discrete, then J ∗ (θ + B) − ω + J ∗ π ∗ + being an integral ZdR M • X cocycle for all + implies that [J ∗ ◦ π ∗ (+)] = 0. In other words, the map J ∗ ◦ π ∗ : 2 ( ) → H 2 ((R × X) ) is the zero map. Since R × X → × X ⇒ X HdR • M • M M dR 2 (( × X) ) → deﬁnes a trivial gerbe according to Proposition 2.3, the map π ∗ : HdR M • 2 ((R × X) ) is injective. From the identity J ∗ ◦ π ∗ = π ∗ ◦ J ∗ and the fact that HdR M • 2 ( ) → H 2 (( × X) ) must vanish.

π ∗ is injective, it follows that J ∗ : HdR • M • dR

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The following proposition is an analogue of Corollary 5.14. Proposition 5.18. Let (G ⇒ G0 , ωG + G ) and (H ⇒ H0 , ωH + H ) be Moritaequivalent pre-quasi-symplectic groupoids. Assume that (F → G0 , ωF ) and (E → H0 , ωE ) are a pair of corresponding pre-Hamiltonian spaces. Then (ωF , ωG + G ) satisﬁes the strong integrality condition if and only if (ωE , ωH + H ) satisﬁes the strong integrality condition. Proof. By Corollary 5.14, we just have to check that Condition (2) in the deﬁnition of strong integrality condition is invariant under Morita equivalence. This follows immediately from the commutativity of the diagram 2 (G ) 2 (H ) ! HdR HdR • • ↓ ↓ 2 ((G × 2 HdR G0 F )• ) ! HdR ((H ×H0 E)• ),

where the horizontal arrows are the natural isomorphism between the de Rham cohomologies of two Morita-equivalent groupoids.

Remark 5.19. 2 ( , Z) = 0, then condition (2) in the deﬁnition of 1. If the groupoid satisﬁes HdR • the strong integrality is satisﬁed for any pre-Hamiltonian -space. In this case, a pre-Hamiltonian space satisﬁes the integrality condition if and only if it satisﬁes the strong integrality condition. 2. If G is a connected and simply-connected Lie group, then H2 ((G × G)• , Z) = 0. Therefore, any quasi-Hamiltonian G-space satisfying the integrality condition must satisfy the strong integrality condition.

The following proposition summarizes the results of this section. Proposition 5.20. Let ( ⇒ M, ω + ) be an exact, proper, pre-quasi-symplectic Jk

groupoid, and let (Xk → M, ωk ), k = 1, 2, be pre-Hamiltonian -spaces. Assume J1

J2

that (X1 → M, ω1 ) satisﬁes the integrality condition while (X2 → M, ω2 ) satisﬁes the strong integrality condition. Then there exists a prequantization of ( ⇒ M, ω +

) and compatible prequantizations of both X1 and X2 . Therefore, the classical intertwiner space X2 × X1 is quantizable. Applying this result to the case of the AMM quasi-symplectic groupoid, we have the following. Corollary 5.21. Let G be a connected and simply-connected compact Lie group J

equipped with an ad-invariant nondegenerate symmetric bilinear form, and (X → G, ωX ) a quasi-Hamiltonian G-space. Assume that ωX satisﬁes the integrality condition as in equation (39). Then the reduced symplectic manifold J −1 (1)/G is prequantizable, and the prequantization can be constructed H using the prequantization of the AMM quasi-symplectic groupoid (G × G)[U] ⇒ Ui (more precisely the pullback groupoid of the AMM quasi-symplectic groupoid ) together with a comH H patible prequantization of the Hamiltonian space ( X|Ui → Ui , ωX |Ui ), where U = (Ui )i∈I is some open covering of G such that |Ui is exact ∀i ∈ I .

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6 Appendix We denote by CS 1 the canonical cycle in C1 (S 1 , Z) that generates H1 (S 1 , Z) = Z. If we consider CS 1 as an element of C2 (S•1 , Z), [CS 1 ] generates H2 (S•1 , Z) ! Z. For any point p in a manifold N , we denote by Cp the constant map from S 1 to {p} and consider it as an element of C1 (N, Z). Assume that R → ⇒ M is an S 1 -central extension of groupoids. For any m ∈ M, let Zm = fm∗ (CS 1 ) ∈ C2 (R• , Z),

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where fm : S 1 → R is deﬁned by equation (13). More generally, for any r ∈ R, let fr : S 1 → R be the map λ → λ · r, and set Zr = fr∗ (CS 1 ) − Cr ∈ C2 (R• , Z). Proposition 6.1. Let R → ⇒ M be an S 1 -central extension. Assume that M/ is connected. (1) The class [Zm ] ∈ H2 (R• , Z) does not depend on the choice of m ∈ M. Because of this, we will drop the subscript m and denote this class simply by [Z]. (2) For any r ∈ R, Zr is a cycle and [Zr ] = [Z]. (3) The natural map π∗ : H2 (R• , Z) → H2 (• , Z) is surjective. (4) Its kernel Ker(π∗ ) is generated by [Z]. (5) The following identity holds: θ = 1. Zm

Before we prove this proposition, we ﬁrst need a lemma. Given any point p ∈ N , we will denote by Cp(k) the chain in Ck (N, Z) deﬁned by the constant path k → {p}. π

Lemma 6.2. Let R → ⇒ M be an S 1 -central extension. (1) Any element E in C0 (R, Z) with π∗ (E) = 0 can be written of the form E = δD , where D ∈ C1 (R, Z) satisﬁes π∗ (D ) = 0. (2) π∗ : C• (R• , Z) → C• (• , Z) is a surjective map. (3) Any element in the kernel of π∗ : Hk (R• , Z) → Hk (• , Z) has a representative C ∈ Zk (R• , Z) with π∗ (C) = 0. (4) Any element C in C0 (R2 , Z) with π∗ (C) = 0 is of the form C = dD , where D ∈ C1 (R2 , Z) satisﬁes π∗ (D ) = 0. (5) For any cycle C ∈ C1 (R, Z) such that δC = 0 and π∗ (C ) = 0, we have [C ] = ki [Zri ] i∈I

for some ﬁnite set I , ki ∈ Z and ri ∈ R.

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Proof. (1) The kernel of π∗ : C0 (R, Z) → C0 (, Z) is generated by elements of the π form p − q, where p and q are two points in the same ﬁbre of R → . Hence, it sufﬁces to prove the claim for such a generator. Let D : 1 → R be a path in the ﬁber π −1 (p) satisfying dD = p − q. Set D = D −Cp(1) . Clearly, the identities dD = p −q and π∗ (D ) = 0 hold. Moreover, from π∗ (D ) = 0, it follows that ∂D = s∗ π∗ (D ) − t∗ π∗ (D ). Hence δD = p − q (2) Since the projections π : Rk → k are surjective submersions with ﬁbers isomorphic to k-dimensional torus, all the maps π∗ : Cl (Rk , Z) → Cl (k , Z) are onto for all k, l ∈ N. (3) Let C ∈ Zk (R• , Z) be a cycle with π∗ [C ] = 0. By deﬁnition, there exists D ∈ Ck+1 (• , Z) such that δD = π∗ (C ). By (2), there exists D in Ck+1 (R• , Z) such that π∗ (D ) = D. Set C := C − δD . We have [C] = [C ] and π∗ (C) = 0. (4) The kernel of π∗ : C0 (R2 , Z) → C0 (2 , Z) is generated by elements of the form p − q, where p and q are two points on the same ﬁber of R2 → 2 . It thus sufﬁces to show this property for such generators. Let D : 1 → R2 be a path in the ﬁber over π(p) such that dD = p − q. Let D = D − Cp(1) . Thus π∗ (D ) = π∗ (D − Cp(1) ) = Cπ(p)(1) − Cπ(p)(1) = 0

and

dD = p − q.

(5) For simplicity, we call those chains in C1 (R, Z) of the form C − Cp(1) ﬁbered 1-chains, where p ∈ R is a point and C : 1 → π −1 (π(p)) is a path in the ﬁber through the point p. Any ﬁbered 1-chain is in the kernel of π∗ and hence lies in the kernel of ∂. If a ﬁbered 1-chain E in a given ﬁber satisﬁes dE = 0, then [E] = k[Zr ] for some r ∈ R and k ∈ Z. As a consequence, if a linear combination of ﬁbered 1-chains F is a cycle in C1 (R, Z), then it is clear that [F ] = i∈I ki [Zri ] for some ﬁnite set I , ki ∈ Z and ri ∈ R. Now the kernel of π∗ : C1 (R, Z) → C1 (, Z) is generated by elements of the form C0 − C1 , where Ci , i = 0, 1, are paths 1 → R satisfying π∗ (C0 ) = π∗ (C1 ). Thus there is a map γ : 1 → S 1 such that C0 (t) = γ (t) · C1 (t) ∀t ∈ 1 . Let γ˜ : [0, 1] × 1 → S 1 be a map with γ˜ (0, t) = 1 and γ˜ (1, t) = γ (t). Let us deﬁne two maps D1 and Dˆ : [0, 1] × 1 → R by D1 (s, t) = γ˜ (s, t) · C0 (t) and ˆ We have π∗ (D) = 0 and therefore ∂D = 0. ˆ t) = C0 (t). Set D := D1 − D. D(s, Moreover, by construction, C0 −C1 +δD = C0 −C1 +dD is the sum of two 1-ﬁbered chains: one in the ﬁber through C0 (0) and another in the ﬁber through C0 (1). The conclusion thus follows.

Proof of Proposition 6.1. (1), (2) It is clear that if m and n are in the same connected component of M, then [Zm ] = [Zn ]. Now by the deﬁnition of Zr , we have dZr = d(fr∗ (CS 1 )) − dCr = 0, s∗ (Zr ) = s∗ (fr∗ (CS 1 ) − Cr ) = Cs(r) − Cs(r) = 0, and

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t∗ (Zr ) = t∗ (fr∗ (CS 1 ) − Cr ) = Ct (r) − Ct (r) = 0. Therefore, δ(Zr ) = 0. Consider the map D : S 1 → R2 deﬁned by λ → (fr (λ), ft (r) (λ−1 )). We have dD = 0 and ∂D = fr∗ (CS 1 ) − Cr − ft (r)∗ (CS 1 ). Hence we have [Zr ] = [Zt (r) ] ∀r ∈ R. Similarly, we have [Zr ] = [Zs(r) ] ∀r ∈ R. Since M/ is connected, (1) and (2) follow. (3) Let C ∈ Z2 (• , Z) be any 2-cycle. According to Lemma 6.2 (2), there exists D ∈ C2 (R• , Z) with π∗ (D) = C. In general, δD = 0. However since the restriction of π to M is the identity map, we have ∂D1 − dD2 = 0 and thus δD = ∂D0 − dD1 , where D = D0 + D1 + D2 , Di ∈ Ci (R2−i , Z). Therefore, δD is an element of C0 (R, Z) and π∗ (δD) = δπ∗ (D) = δC = 0. By Lemma 6.2(1), there exists D ∈ C2 (R• , Z) with π∗ (D ) = 0 and δD = δD. Therefore, it follows that D − D is a cycle in Z2 (R• , Z) and π∗ ([D − D ]) = [π∗ (D)] − [π∗ (D )] = [C] − [0] = [C]. (4) According to Lemma 6.2 (3), any class in Ker(π∗ ) has a representative C such that π∗ (C) = 0 and therefore is of the form C0 + C1 , where C0 ∈ C0 (R2 , Z) and C1 ∈ C1 (R, Z) satisfy π∗ (C0 ) = 0 and π∗ (C1 ) = 0. According to Lemma 6.2 (4), there exists D ∈ C1 (R2 , Z) with π∗ (D ) = 0 such that C0 = dD . Consider now C = C − δD ∈ C1 (R, Z). We have δC = δC − δ 2 D = 0,

[C ] = [C],

According to Lemma 6.2(5), we have ki [Zri ] [C ] =

π∗ (C ) = 0.

(43)

i∈I

for some ﬁnite set I , ki ∈ Z and ri ∈ R. From equation (43), it follows that [C] = [C ] = i∈I ki [Zri ]. By Lemma 6.1 (2), we have [C] = ( i∈I ki )[Z]. (5) This holds because θ is a connection 1-form of the S 1 -principal bundle R → .

Acknowledgments The second author would like to thank the Erwin Schrödinger Institute and the University of Geneva for their hospitality while work on this project was being done. We would like to thank Anton Alekseev, Kai Behrend, Eckhard Meinrenken, and Jim Stasheff for useful discussions.

References [1] Alekseev, A., On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom., 45 (1997), 241–256. [2] Alekseev, A., and Meinrenken, E., private communication. [3] Alekseev, A., Malkin, A., and Meinrenken, E., Lie group valued moment maps, J. Differential Geom., 48 (1998), 445–495.

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[4] Behrend, K., and Xu, P., S 1 -bundles and gerbes over differentiable stacks, C. R. Acad. Sci. Paris Sér. I, 336 (2003), 163–168. [5] Behrend, K., and Xu, P., Differential stacks and gerbes, in preparation. [6] Behrend, K., Xu, P., and Zhang, B., Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris Sér. I, 336 (2003), 251–256. [7] Bursztyn, H., Crainic, M., Weinstein, A., and Zhu, C., Integration of twisted Dirac brackets, Duke Math. J., 123-3 (2004), 549–607. [8] Dupont, L., Curvature and Characteristic Classes, Lecture Notes in Mathematics, Vol. 640, Springer-Verlag, New York, 1978. [9] Ginzburg, V., and Weinstein, A., Lie–Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc., 5 (1992), 445–453. [10] Guillemin, V., and Sternberg, S., Geometric quantization and multiplicities of group representations, Invent. Math., 67 (1982), 515–538. [11] Laurent-Gengoux, C., Tu, J.-L., and Xu, P., Chern–Weil map and characteristic classes of principal G-bundles over groupoids, math.DG/0401420. [12] Mc Cleary, J., A User’s Guide to Spectral Sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, Vol. 2, Cambridge University Press, Cambridge, UK, 2000. [13] Meinrenken, E., Symplectic surgery and the SpinC -Dirac operator, Adv. Math., 134 (1998), 240–277. [14] Meinrenken, E., The basic gerbe over a compact simple Lie group, Enseign. Math. (2), 49 (2003), 307–333. [15] Mohrdieck, S., and Wendt, R., Integral conjugacy classes of compact Lie groups, Manuscripta Math., 113 (2004), 531–547. [16] Shabhazi, Z., Ph.D. thesis, Department of Mathematics, University of Toronto, Toronto, 2004. [17] Spanier, E., Algebraic Topology, Springer-Verlag, New York, 1981. [18] Tu, J.-L., Xu, P., and Laurent-Gengoux, C., Twisted K-theory of differentiable stacks, Ann. Sci. École Norm. Sup., to appear; math.KT/0306138. [19] Weinstein, A., and Xu, P., Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., 417 (1991), 159–189. [20] Xu, P., Morita equivalent symplectic groupoids, in Dazord, P., and Weinstein, A., eds., Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Mathematical Sciences Research Institute Publications, Vol. 20, Springer-Verlag, New York, 1991, 291–311. [21] Xu, P., Classical intertwiner spaces and quantization, Comm. Math. Phys., 164 (1994), 473–488. [22] Xu, P., Momentum map and Morita equivalence, J. Differential Geom., to appear; math.SG/0307319.

Duality and triple structures Kirill C. H. Mackenzie Department of Pure Mathematics University of Shefﬁeld Shefﬁeld S3 7RH UK [email protected] Dedicated to Alan Weinstein on the occasion of his sixtieth birthday. Abstract. We recall the basic theory of double vector bundles and the canonical pairing of their duals, introduced by the author and by Konieczna and Urba´nski. We then show that the relationship between a double vector bundle and its two duals can be understood simply in terms of an associated cotangent triple vector bundle structure. In particular, we show that the dihedral group of the triangle acts on this triple via forms of the isomorphisms R , introduced by the author and Ping Xu. We then consider the three duals of a general triple vector bundle and show that the corresponding group is neither the dihedral group of the square nor the symmetry group on four symbols.

Double structures ﬁrst appeared in Poisson geometry with Alan’s groundbreaking work on symplectic groupoids [Coste, Dazord, and Weinstein 1987], [Weinstein 1987] and Poisson groupoids [Weinstein 1988]. The most fundamental example of a symplectic groupoid, the cotangent groupoid T ∗ G of an arbitrary Lie groupoid G , introduced in [Coste et al. 1987], is a groupoid object in the category of vector bundles. An arbitrary Poisson Lie group can be integrated to a symplectic double groupoid [Lu and Weinstein 1989]. At a simpler level, a Poisson structure on a vector bundle is linear [Courant 1990] if and only if the associated anchor is a morphism of certain double vector bundles. All these phenomena involve doubles in the categorical sense: taking S to denote, for example, “vector bundle’’ or “Lie groupoid,’’ a double S is an S object in the category of all S . (Groupoid objects in the category of vector bundles, named VBgroupoids by Pradines [1988], may be regarded as double groupoids of a special type.) More generally, multiple S structures are the n -fold extension of this notion of double. The key link between Poisson geometry and double structures lies in properties of the Poisson anchor. If a Poisson manifold P is a vector bundle on base M , then the Poisson structure is linear if and only if π # : T ∗ P → T P is a morphism of double vector bundles. If P is instead a Lie groupoid on base M , then the groupoid

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is a Poisson groupoid if and only if π # is a morphism of VB-groupoids. Thus the Poisson anchor naturally appears as a map of double structures, and indeed many of the surprising basic features of Poisson and symplectic groupoids are not really so much consequences of Poisson or symplectic geometry as consequences of the duality properties of the associated double structures. This point of view is developed further in [Mackenzie 2004]; in particular, the theory of Poisson groupoids may be developed entirely in terms of groupoid theory and double structures of various kinds. The present paper is concerned with the duality of double and higher multiple vector bundles. A double vector bundle has two duals which are themselves in duality and we show here that the various combinations of the two dualization operations gives rise to the dihedral (or symmetric) group of order six. We show in Section 5 and Section 6 that a double vector bundle and its two duals form the three lower faces of a triple vector bundle, the opposite vertex of which is the cotangent of the double space. This encapsulates and makes symmetric the relations between a double vector bundle and its duals, which can otherwise seem rather involved. One may think of three double vector bundles with a common vertex and appropriate pairings as constituting a two (sic) dimensional version of the familiar notion of pairing of vector bundles; we call this a cornering. In Section 7 we consider the process of dualizing the structures in a triple vector bundle. This may appear to be a routine extension of the double case, but we show that the group of dualization operations here is not the dihedral group of the square, or the symmetric group on four symbols, but a group of order 72. This appears to demonstrate that the behaviour of duality for n -fold vector bundles may be a less routine extension of the double case than one might have expected. In the ﬁnal Section 8, we formulate some general principles which we believe do hold for general multiple vector bundles. The study of general double vector bundles was begun by Pradines [1974], although the case of the tangent double of an ordinary vector bundle (1.3) had been used in connection theory since the late 1950s. More than a decade later, Pradines [1988] introduced the dualization process for VB-groupoids; in the case of double vector bundles this is the duality construction presented here in Section 3. Theorem 3.2 is from [Mackenzie 1999] and was also found by Konieczna and Urban´ ski [1999]. The idea of deriving the pairing (17) from the cotangent triple was noted in [Mackenzie 2002]. The results of Section 4 ﬁrst appeared in the paper [Mackenzie and Xu 1994] of Ping Xu and myself. They are here obtained as a consequence of the general duality of double vector bundles. An expanded account of the double case may be found in [Mackenzie 2004].

1 Double vector bundles Deﬁnition 1.1. A double vector bundle (D; A, B; M) is a system of four vector bundle structures

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457

qBD D −−−−−−−−→ B ⏐ ⏐ qAD ?

⏐ ⏐ ? qB

(1)

A −−−−−−−−→ M qA in which D has two vector bundle structures, on bases A and B , which are themselves vector bundles on M , such that each of the structure maps of each vector bundle structure on D (the bundle projection, addition, scalar multiplication and the zero section) is a morphism of vector bundles with respect to the other structure. We refer to A and B as the side bundles of D , and to M as the double base. In the two side bundles the addition, scalar multiplication and subtraction are denoted by the usual symbols + , juxtaposition, and − . We distinguish the two zero sections, B B writing 0A : M → A , m → 0A m , and 0 : M → B , m → 0m . We may denote D an element d ∈ D by (d; a, b; m) to indicate that a = qA (d) , b = qBD (d) , m = qB (qBD (d)) = qA (qAD (d)). The notation qAD is clear; when the base of the bundle is the double base we write A. qA , for example, rather than qM In the vertical bundle structure on D with base A the vector bundle operations are denoted + , A. , − , with 0A : A → D , a → 0A a , for the zero section. Similarly, A A in the horizontal bundle structure on D with base B , we write + , B. , − and B B A B 0B : B → D , b → 0B b . For m ∈ M the double zero 00A = 00B is denoted m or m

m

02m . The two structures on D , namely (D, qBD , B) and (D, qAD , A) , will occasionally B and D A , respectively. be denoted D In dealing with general double vector bundles such as (1), we thus usually label objects and operations in the two structures on D by the symbol for the base over which they take place. The words “horizontal’’ and “vertical’’ may be used as an alternative, but need to be referred to the arrangement in the diagram (1) or the sequence in (D; A, B; M). When considering examples in which A = B , the words “horizontal’’ and “vertical’’ become necessary, and we use H and V as labels to distinguish the two structures on D . Although the concept of double vector bundle is symmetric, most examples are not; in the sequel it will be important to distinguish between (1) and its ﬂip in Figure 1(a), in which the arrangement of the two structures is reversed. In such processes it is not the absolute arrangement which is signiﬁcant, but the distinction between whichever arrangement is taken at the start and its ﬂip. The condition that each addition in D is a morphism with respect to the other is (d1 + d2 ) + (d3 + d4 ) = (d1 + d3 ) + (d2 + d4 ) B

A

B

A

B

A

(2)

for quadruples d1 , . . . , d4 ∈ D such that qBD (d1 ) = qBD (d2 ) , qBD (d3 ) = qBD (d4 ) , qAD (d1 ) = qAD (d3 ) , and qAD (d2 ) = qAD (d4 ) . Next,

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D

ﬂip

D qA −→ A

⏐ D ⏐ qB ?

T (q) T E −→ T M ⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

B −→ M

E −→ M

(a)

(b) Fig. 1.

t A. (d1 + d2 ) = t A. d1 + t A. d2 , B B

(3)

for t ∈ R and d1 , d2 ∈ D with qBD (d1 ) = qBD (d2 ) ; similarly, t B. (d1 + d2 ) = t B. d1 + t B. d2 , A

A

(4)

for t ∈ R and d1 , d2 ∈ D with qAD (d1 ) = qAD (d2 ) . The two scalar multiplications are related by t A. (u B. d) = u B. (t A. d), (5) where t, u ∈ R and d ∈ D . Lastly, for compatible a, a ∈ A and compatible b, b ∈ B , and t ∈ R ,

and

A + A 0A a+a = 0a B 0a ,

. A 0A ta = t B 0a ,

(6)

B + 0B 0B b+b = 0b b ,

. B 0B tb = t A 0b .

(7)

A

Equations (2)–(7) are known as the interchange laws. Deﬁnition 1.2. A morphism of double vector bundles (ϕ; ϕA , ϕB ; f ) : (D; A, B; M) → (D ; A , B ; M ) consists of maps ϕ : D → D , ϕA : A → A , ϕB : B → B , f : M → M , such that each of (ϕ, ϕB ), (ϕ, ϕA ), (ϕA , f ), and (ϕB , f ) is a morphism of the relevant vector bundles. If M = M and f = id M , we say that ϕ is over M ; if, further, A = A and ϕA = id A , we say that ϕ is over A or preserves A . If A = A and B = B and both ϕA and ϕB are identities, we say that ϕ preserves the side bundles. Example 1.3. For an ordinary vector bundle (E, q, M) , applying the tangent functor to each of the bundle operations yields the tangent prolongation vector bundle (T E, T (q), T M). The zero section is T (0) : T M → T E . We denote the addition by + + and the scalar multiplication and subtraction by and . Together with the standard structure (T E, pE , E) , we have a double vector bundle (T E; E, T M; M), shown in Figure 1(b). There is no preferred arrangement for the side bundles of T E .

.

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Example 1.4. Let A, B and C be any three vector bundles on the one base M , and write D for the pullback manifold A ×M B ×M C over M . Then D may be regarded as the direct sum qA! B ⊕ qA! C over A , and as the direct sum qB! A ⊕ qB! C over B , and with respect to these two structures, D is a double vector bundle with side bundles A and B . We call this the trivial double vector bundle over A and B with core C . It is tempting, but incorrect, to denote it by A ⊕ B ⊕ C . Example 1.5. A double vector bundle (D; A, B; M) may be pulled back over both of its side structures simultaneously. Suppose given vector bundles (A , qA , M ) and (B , qB , M ) and morphisms ϕ : A → A and ψ : B → B , both over a map f : M → M . Let D denote the set of all (a , d, b ) such that ϕ(a ) = qAD (d) , ψ(b ) = qBD (d) and qA (a ) = qB (b ) . Then, with the evident structures, (D ; A , B ; M ) is a double vector bundle and the projection D → D is a morphism over ϕ , ψ , and f . Further examples follow later in the paper.

2 The core and core sequences Until Example 2.2, consider a ﬁxed double vector bundle (D; A, B; M). Each of the bundle projections is a morphism with respect to the other structure and so has a kernel (in the ordinary sense); denote by C the intersection of the two kernels: D A C = {c ∈ D | ∃m ∈ M such that qBD (c) = 0B m , qA (c) = 0m }.

This is an embedded submanifold of D . We will show that it has a well-deﬁned vector bundle structure with base M , projection qC which is the restriction of qB ◦ qBD = qA ◦ qAD and addition and scalar multiplication which are the restrictions of either of the operations on D . Note ﬁrst that the two additions coincide on C since c + c = (c + m ) + (m + c ) = (c + m ) + (m + c ) = c + c , B

c, c

A

B

A

B

A

B

A

(c ) ,

for ∈ C with qC (c) = qC using (2). From this it follows that t B. c = t A. c for integers t , and consequently for rational t , and hence for all real t by continuity. It will often be helpful to distinguish between c ∈ C , regarding C as a distinct vector bundle, and the image of c in D , which we will denote by c . Given c, c ∈ C with qC (c) = qC (c ) there is a unique c + c ∈ C with c = c + c , c + c = c + B A and given t ∈ R there is a unique tc ∈ C such that tc = t B. c = t A. c. It is now easy to prove that (C, qC , M) is a (smooth) vector bundle, which we call the core of (D; A, B; M).

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Theorem 2.1. There is an exact sequence (q D )! τA A −−B−2 qA! B qA! C >−−−> D

(8)

of vector bundles over A , and an exact sequence (q D )! τB B −−A−2 qB! A qB! C >−−−> D

(9)

+ of vector bundles over B , where the injections are τA : (a, c) → 0A a B c and D D B ! ! τB : (b, c) → 0b + c , respectively, and (qB ) and (qA ) denote the maps induced A D D by qB and qA into the pullback bundles. Proof. Take a ∈ Am , c ∈ Cm , where m ∈ M . Then both 0A a and c project under D D B A + qB to 0m . So 0a c is deﬁned and also projects under qB to 0B m . That τA is linear B over A follows from the interchange laws. D Suppose that d ∈ D has qBD (d) = 0B m for some m ∈ M . Write a = qA (d) . D A A B A − − − Then d B 0a is deﬁned and qB (d B 0a ) = 0m . On the other hand, q (d B 0A a)= A ∈ C . This establishes the exactness of (8). The proof of − 0 a − a = 0A . So d m m a B (9) is similar.

We refer to (8) as the core sequence of D over A , and to (9) as the core sequence of D over B . If (ϕ; ϕA , ϕB ; f ) : (D; A, B; M) → (D ; A , B ; M ) is a morphism of double vector bundles, then ϕ : D → D maps C into C , the core of D . It is clear that the restriction, ϕC : C → C , is a morphism of the vector bundle structures on the cores over f : M → M . Examples 2.2. For E an ordinary vector bundle, consider the tangent double vector bundle (T E; E, T M; M). The kernel of T (q) consists of the vertical tangent vectors and the kernel of pE consists of the vectors tangent along the zero section; their intersection is naturally identiﬁed with E itself. For clarity we distinguish X ∈ E from the core element X ∈ T E . The injection map for T E over E is the map τ which sends (X, Y ) ∈ Em × Em to the vector in Em which has tail at X and is parallel to Y . In terms of the prolongation structure, τ (X, Y ) = 0X + + Y . The injection map over T M is υ : (x, Y ) → T (0)(x) + Y . For ϕ : E → E a morphism of vector bundles over f : M → M , the morphism T (ϕ) of the tangent double vector bundles induces ϕ : E → E on the cores. In the case where ϕ and f are surjective submersions, the vertical subbundles form a double vector subbundle (in an obvious sense) (T ϕ E; E, T f M; M) of T E , the core of which is the kernel (in the ordinary sense) of ϕ . The trivial double vector bundle A ×M B ×M C of 1.4 has core C .

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3 Duals of double vector bundles Throughout this section we consider a double vector bundle as in (1), with core bundle C . We will show that dualizing either structure on D leads again to a double vector bundle; in the case of the dual of the structure over A we denote this by D ×A qA×A

qC×∗A −−−−−−−−→ C ∗

⏐ ⏐ ?

⏐ ⏐ ? qC ∗

(10)

−−−−−−−−→ M. qA

A

Here C ∗ is the ordinary dual of C as a vector bundle over M . We denote the dual of D as a vector bundle over A by D × A . (We will later modify this notation for cases in which A and B are identical.) The vertical structure in (10) is the usual dual of the bundle structure on D with base A , and qC ∗ : C ∗ → M is the usual dual of qC : C → M. The additions and scalar multiplications in the side bundles of (10) will be denoted by the usual plain symbols as before. In the two structures on D × A we write + , A. , − and A A + , . , − . The zero of D × A above a ∈ A is denoted 0 ×A . C∗

C∗

C∗

a

The unfamiliar projection qC×∗A : (D × A) → C ∗ is deﬁned by + c, qC×∗A (), c = , 0A a B

(11)

where c ∈ Cm , : (qAD )−1 (a) → R and a ∈ Am . The addition +∗ in D × A → C ∗ C is deﬁned by +∗ , d + d = , d + , d (12) C

B

That this is well deﬁned depends strongly on the condition qC×∗A () = qC×∗A (#). Similarly, deﬁne t .∗ , t . d = t, d, C

B

qA×A (). for t ∈ R and d ∈ D with qAD (d) = ∗ is denoted The zero above κ ∈ Cm 0κ×A and is deﬁned by + c = κ, c, 0κ×A , 0B b A

(13)

∗ is where b ∈ Bm , c ∈ Cm . The core element ψ corresponding to ψ ∈ Bm

+ ψ, 0B b A c = ψ, c. It is straightforward to verify that (10) is a double vector bundle and that its core is B ∗ . We call (10) the vertical dual or dual over A of (1).

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As for any double vector bundle, there are exact sequences A ! (qC× ∗ ) σA qA! B ∗ >−−−> D × A −−−2 qA! C ∗ ,

(14)

of vector bundles over A and (qA×A )! σC ∗ qC! ∗ B ∗ >−−−> D × A −−−2 qC! ∗ A,

(15)

of vector bundles over C ∗ . Here the two injections are given by ×A

σA (a, ψ) = 0a

×A

σC ∗ (κ, ψ) = 0κ

+ ψ,

C∗

+ ψ, A

where a ∈ A, ψ ∈ B ∗ , κ ∈ C ∗ . It is easily seen that σA (a, ψ), d = ψ, qBD (d) for d ∈ D and so σA is precisely the dual of (qBD )! . It is clear from the deﬁnition of qC×∗A that (qC×∗A )! = τV∗ . Thus (14) is precisely the dual of the core exact sequence (8). For the sequence over C ∗ we have + c = κ, c + ψ, b σC ∗ (κ, ψ), 0B b A

∗ , ψ ∈ B∗ , x ∈ B , c ∈ C . for κ ∈ Cm m m m The proof of the following result is straightforward. In Figure 2 and in similar ﬁgures in future, we omit arrows which are the identity.

Proposition 3.1. Consider a morphism of double vector bundles, as in Figure 2(a), which preserves the horizontal side bundles, and which has core morphism ϕC : C → C , where C is the core of D . Dualizing ϕ as a morphism of vector bundles over A, we obtain a morphism ϕ × A of double vector bundles over A and ϕC∗ , as in Figure 2(b), with core morphism ϕB∗ . This completes the description of the vertical dual of (1). There is of course also a horizontal dual qB×B × D B −−−−−−−−→ B qC×∗B

⏐ ⏐ ? C∗

⏐ ⏐ ? qB

(16)

−−−−−−−−→ M, qC ∗

with core A∗ → M , deﬁned in an analogous way. The following result is an entirely new phenomenon, arising from the double structures.

Duality and triple structures

D × A −→ (C )∗

D −→ B

H H ⏐ HH ⏐ H ϕB ϕH ⏐ ⏐ HH ? ? H j H H j H

A −→ M

⏐ ⏐ ?

D −→ B ⏐ ⏐ ?

A

⏐ ⏐ ?

H

H ⏐

H

H ϕC∗ H HH HH j H j

H⏐ ϕ × A ?H −→ M

463

D × A −→ C ∗ ⏐ ⏐ ?

⏐ ⏐ ?

A −→ M

A

−→ M

(a)

(b) Fig. 2.

Theorem 3.2 ([Mackenzie 1999], [Konieczna and Urbanski 1999]). There is a ´ natural (up to sign) duality between the bundles D × A and D × B over C ∗ given by , # = , dA − #, dB ,

(17)

where ∈ D × A, # ∈ D × B have qC×∗A () = qC×∗B (#) and d is any element of D with qAD (d) = qA×A () and qBD (d) = qB×B (#). Each of the pairings on the RHS of (17) is a canonical pairing of an ordinary vector bundle with its dual, the subscripts there indicating the base over which the pairing takes place. Proof. Let and # have the forms (; a, κ; m) and (#; κ, b; m) . Then d must have the form (d; a, b; m) . If d also has the form (d ; a, b; m) , then there is a + c ∈ Cm such that d = d + ( 0A a B c) , and so A , dA = , d A + κ, c + c) and so ( 0B by (11). By the interchange law (2) we also have d = d + b B A

#, dB = #, d B + κ, c. Thus (17) is well deﬁned. To check that it is bilinear is routine. It remains to prove that it is nondegenerate. Suppose , given as above, is such that , # = 0 for all # ∈ (qC×∗B )−1 (κ). Take any ϕ ∈ A∗m and consider # = 0κ×B + ϕ . Then, taking d = 0A a we ﬁnd B , dA = 0 and #, dB = ϕ, a. Thus ϕ, a = 0 for all ϕ ∈ A∗m and so × a = 0A m . It therefore follows from the horizontal exact sequence for D A that = 0κ×A + ψ A

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K. C. H. Mackenzie

∗ . Now taking any c ∈ C and deﬁning d = + for some ψ ∈ Bm 0B m b A c , we ﬁnd that

, dA = κ, c + ψ, b

and

#, dB = κ, c.

So ψ, b = 0 for all b ∈ Bm , since a suitable # exists for any given b . It follows ∗ and so is indeed the zero element over κ . Thus the pairing (17) that ψ = 0 ∈ Bm is nondegenerate.

Note several special cases: 0κ×B = 0, 0κ×A ,

0b×B = 0, 0a×A ,

0a×A , ϕ = −ϕ, a, ψ, 0b×B = ψ, b, ψ, ϕ = 0, A B × × , 0κ + ϕ = −ϕ, a, 0κ + ψ, # = ψ, b, A

B

(18) (19) (20)

where b ∈ B , a ∈ A , ϕ ∈ A∗ , ψ ∈ B ∗ and we have (#; κ, b; m) ∈ D × B and (; a, κ; m) ∈ D × A . Although we have proved that D × A and D × B are dual as vector bundles over ∗ C , we have not yet considered the relationships between the other structures present. This is taken care of by the following result, the proof of which is straightforward. Proposition 3.3 ([Mackenzie 1999]). Let (D; A, B; M) and (E; A, W ; M) be double vector bundles with a side bundle A in common, and with cores C and L, respectively. Suppose given a nondegenerate pairing , of D over A with E over A, and two further nondegenerate pairings, both denoted , , of B with L and of C with W , such that (i) for all b ∈ B , ∈ L, 0B b , = b, ; 0W (ii) for all c ∈ C , w ∈ W , c, w = c, w ; (iii) for all c ∈ C , ∈ L, c, = 0; (iv) for all d1 , d2 ∈ D , e1 , e2 ∈ E such that qBD (d1 ) = qBD (d2 ), E (e ) = q E (e ), q D (d ) = q E (e ) , q D (d ) = q E (e ), we have qW 1 W 2 A 1 A 1 A 2 A 2 d1 + d2 , e1 + e2 = d1 , e1 + d2 , e2 ; B W (v) for all d ∈ D , e ∈ E such that qAD (d) = qAE (e) and all t ∈ R , we have t B. d, t W. e = t d, e . ( In all the above conditions we assume the various elements of the side bundles lie in compatible ﬁbres over M.) Then the map Z : D → E × A deﬁned by Z(d), eA = d, e is an isomorphism of double vector bundles, with respect to id : A → A and the isomorphisms B → L∗ and C → W ∗ induced by the pairings in (i) and (ii). A pairing , satisfying the conditions of Proposition 3.3 is called a pairing of the double vector bundles. Applying this result to the pairing (17) of D × A and D × B , we ﬁnd that the induced pairing of B with B ∗ is the standard one, but that of A∗ with A is the negative of the standard pairing. Hence the signs in the following result are unavoidable.

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Corollary 3.4. The pairing (17) induces isomorphisms of double vector bundles ZA : D × A → D × B × C ∗ ,

ZA (), #C ∗ = , # ,

ZB : D × B → D × A× C ∗ ,

ZB (#), C ∗ = , #

with (ZA ) × C ∗ = ZB . Both isomorphisms induce the identity on the sides C ∗ → C ∗ . ZA is the identity on the cores B ∗ → B ∗ , and induces − id on the side bundles A → A. ZB is the identity on the side bundles B → B , and induces − id on the cores A∗ → A∗ . Example 3.5. Consider a trivial double vector bundle D = A ×M B ×M C. Let = (a, ψ, κ) be an element of D × A = A×M B ∗ ×M C ∗ and let # = (ϕ, b, κ) be an element of D × B . Then taking any d = (a, b, c) ∈ D , we ﬁnd that , # = ψ, b − ϕ, a. The associated maps are given by ZA : A ×M B ∗ ×M C ∗ → A ×M B ∗ ×M C ∗ , ZB : A∗ ×M B ×M C ∗ → A∗ ×M B ×M C ∗ ,

(a, ψ, κ) → (−a, ψ, κ); (ϕ, b, κ) → (−ϕ, b, κ).

The following result is essentially equivalent to Theorem 3.2, but deserves independent statement. Theorem 3.6. For any double vector bundle (D; A, B; M) there is a canonical isomorphism Q from D to the ﬂip of (D × A × C ∗ × B) which preserves the side bundles A and B and is − id on the cores C . Proof. Let $ = ZA × A be the dualization of ZA over A . Denote by F : D → D d , and deﬁne Q = (F ◦ $)−1 .

the map d → − B There are now three operations on double vector bundles: taking the vertical dual, denoted by V , taking the horizontal dual, denoted H , and the operation V H V which by Theorem 3.6 combines the ﬂip and reversal of the sign on the core; we denote this by P . We have V 2 = H 2 = P 2 = I , the identity operation and, by the same method as Theorem 3.6, H V H = P . The group generated by V , H and P therefore has elements I,

V,

HV,

V HV = P,

and is the dihedral group

3

H V H V = H P = V H,

V H V H V = H, (21) of the triangle, or the symmetric group S3 .

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K. C. H. Mackenzie

4 Duals of T E Consider the tangent prolongation double vector bundle (Figure 1(b)) of a vector bundle (E, q, M) . First consider the horizontal dual. The canonical pairing of E ∗ with E prolongs to a pairing of T (E ∗ ) → T M with T E → T M . Suppose given X ∈ T (E ∗ ) and d d ξ ∈ T E with T (q)(X) = T (q∗ )(ξ ) . Then X = dt ϕt |0 ∈ T (E ∗ ) and ξ = dt et |0 ∈ T E , where et ∈ E and ϕt ∈ E ∗ can be taken so that q∗ (ϕt ) = q(et ) for t near zero. Now deﬁne the tangent pairing , by d ϕt , et . (22) X, ξ = dt 0 To show that this is nondegenerate it is sufﬁcient to work locally. Suppose, therefore, that E = M ×V . Regard ξ as (x0 , v0 , w0 ) ∈ Tm0 M ×V ×V and X as (x0 , ϕ0 , ψ0 ) ∈ Tm0 M × V ∗ × V ∗ . Then d d X= (mt , ϕ0 + tψ0 ) , (mt , v0 + tw0 ) , ξ= dt dt 0 0 where

d dt mt |0

= x = T (q)(X) = T (q∗ )(ξ ) . So d X, ξ = ϕ0 + tψ0 , v0 + tw0 . dt 0

Expanding out the RHS, the constant term and the quadratic term vanish in the derivative, and we are left with X, ξ = ψ0 , v0 + ϕ0 , w0 from which it is clear that , is nondegenerate. We now need to establish that this is a pairing of the double vector bundles. Proposition 4.1. The tangent pairing , of T (E ∗ ) with T E over T M satisﬁes ∗, the conditions of Proposition 3.3. In particular, for m ∈ M and ϕ, ϕ1 , ϕ2 ∈ Em e, e1 , e2 ∈ Em , ϕ, e = 0, 0ϕ , e = ϕ, e,

0ϕ , 0e = 0, ϕ, 0e = ϕ, e

and τ∗ (ϕ1 , ϕ2 ), τ (e1 , e2 ) = ϕ1 , e2 + ϕ2 , e1 , where τ∗ and τ are the injections in the core sequences of T (E ∗ ) and T (E). Proof. These are easily veriﬁed from the deﬁnition. For example, ϕ = d e = dt (te)|0 , so

d dt (tϕ)|0

and

Duality and triple structures

467

d 2 ϕ, e = t ϕ, e = 0, dt 0 whereas 0ϕ =

d dt ϕ|0 ,

so d tϕ, e = ϕ, e. 0ϕ , e = dt 0

The bilinearity conditions are easily veriﬁed and the ﬁnal equation follows.

Thus the pairing of the cores of T (E ∗ ) and T (E) is the zero pairing, and so too is the pairing of the zero sections above E ∗ and E . However the core of T (E ∗ ) and the zero section of T (E) are paired under the standard pairing, and the same is true of the zero section of T (E ∗ ) and the core of T (E). It now follows that there is an isomorphism of double vector bundles from T (E ∗ ) to the dual T E × T M of T E over T M . For convenience we denote this simply by T • E and call it the prolongation dual of T E . The next result follows from the general theory of Section 3. Proposition 4.2 ([Mackenzie and Xu 1994]). The map I : T (E ∗ ) → T • (E) deﬁned by I (X), ηT M = X, η, where X ∈ T (E ∗ ) , η ∈ T E , is an isomorphism of double vector bundles preserving the side bndles E ∗ and T M and the core bundles E ∗ . When a name is needed we call I the internalization map. In the future we will almost always work with T (E ∗ ) and the tangent pairing rather than with T • E and I . Now consider the vertical dual of T E . Since the core of the double vector bundle T E is E , dualizing the structure over E leads to a double vector bundle of the form rE T ∗E − −−→ E ∗ cE

⏐ ⏐ ? E

⏐ ⏐ ? qE ∗

(23)

−−−→ M. qE

We refer to this as the cotangent dual of T E . We will give a detailed description of the structures involved. Although this is a special case of the general construction, this example is so basic to the rest of the paper that it merits a speciﬁc treatment. In (23) the vertical bundle is the standard cotangent bundle of E , and the notation TX∗ (E) will always refer to the ﬁbre with respect to cE . In this bundle we use standard notation, and denote the zero element of TX∗ (E) by 0∗X . We drop the subscripts E from the maps when no confusion is likely. The map r : T ∗ E → E ∗ takes the form

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K. C. H. Mackenzie

r(), Y = , τ (X, Y ) = , 0X + + Y , TX∗ (E) ,

where ∈ E ∗ we have

X ∈ Em and Y ∈ Em . Thus r() ∈

∗. Em

(24)

For the addition over

+∗ #, ξ + + η = , ξ + #, η, E

∗ , and ξ ∈ T (E) , where ∈ TX∗ (E) , # ∈ TY∗ (E) with r() = r(#) ∈ Em X ∗ η ∈ TY (E) with T (q)(ξ ) = T (q)(η) . This deﬁnes +∗ # ∈ TX+Y (E) . Similarly, E we have t E.∗ , t ξ = t, ξ ,

.

∗ , is 0rϕ ∈ for t ∈ R and ξ ∈ TX (E) . The zero element of r −1 (ϕ), where ϕ ∈ Em ∗ T0m (E) , where 0rϕ , T (0)(x) + X = ϕ, X

for x ∈ Tm (M), X ∈ Em . Given ω ∈ Tm∗ (M), the corresponding core element ω is ω, T (0)(x) + X = ω, x for x ∈ Tm (M), X ∈ Em . The injection over E , q ! T ∗ M → T ∗ E,

(X, ω) → 0∗X +∗ ω, E

is precisely the dual of T (q)! ; that is, it is the map corresponding to the lifting of 1-forms from M to E . Thus 0∗X E+∗ ω is the pullback of ω ∈ Tm∗ (M) to E at the point X ∈ Em . The core exact sequence for c is r ! =τ ∗ q ! T ∗ M >−−−> T ∗ E −−−2 q ! E ∗ ,

(25)

and this is the dual of the core exact sequence for T E and pE . The other core exact sequence is c! q∗! T ∗ M >−−−> T ∗ E −−−2 q∗! E, (26) where each bundle here is over E ∗ . The injection q∗! T ∗ M → T ∗ E is (ϕ, ω) → 0rϕ + ω and 0rϕ + ω, T (0)(x) + X = ϕ, X + ω, x. Given ω ∈ T ∗ M , the corresponding core element is ω = (ω, 0, 0). To summarize, the two dual double vector bundles of D = T E are D ×E = T ∗ E −→ E ∗ ⏐ ⏐ ?

⏐ ⏐ ?

E −→ M

D ×T M = T • E −→ T M and

⏐ ⏐ ?

⏐ ⏐ ?

E ∗ −→ M

Duality and triple structures

469

and the pairing , f = , ξ E − f, ξ T M

(27)

for suitable ξ ∈ T E . Composing the isomorphism ZE from 3.4 with the dual over E ∗ of the internalization isomorphism I , we get an isomorphism of double vector bundles (I × E ∗ ) ◦ ZE : T ∗ E → T ∗ (E ∗ ); denote this temporarily by S −1 . For ∈ T ∗ E we have S −1 (), XE ∗ = (I ×E ∗ ) ◦ ZE (), XE ∗ = ZE (), I (X)E ∗ = , I (X) = , ξ E − I (X), ξ T M = , ξ E − X, ξ . Here we used the deﬁnition of ZE , the deﬁnition (27), and the deﬁnition of I . It follows that for F ∈ T ∗ (E ∗ ) , writing = S(F) , we have F, XE ∗ = S(F), ξ E − X, ξ . Recall that I , and hence its dual, preserves both sides and the core, whereas ZE induces − id on the sides E . We therefore deﬁne R : T ∗ (E ∗ ) → T ∗ (E),

R(F) = S( E−∗ F).

To summarize, we have the following. Theorem 4.3 ([Mackenzie and Xu 1994]). The map R just deﬁned is an isomorphism of double vector bundles, preserving the side bundles E and E ∗ , and inducing − id : T ∗ M → T ∗ M on the cores. Further, for all ξ ∈ T E , X ∈ T (E ∗ ) , F ∈ T ∗ (E ∗ ) such that ξ and X have the same projection into T M , X and F have the same projection into E ∗ , and F and ξ have the same projection into E , X, ξ = R(F), ξ E + F, XE ∗ .

(28)

We call R the reversal isomorphism. It is proved in [Mackenzie and Xu 1994] that R is an antisymplectomorphism of the canonical symplectic structures.

5 Triple vector bundles The deﬁnition of a triple vector bundle follows the same pattern as in the double case. There are a number of evident reformulations. Deﬁnition 5.1. A triple vector bundle is a manifold III together with three vector bundle structures, over bases D1 , D2 , D3 , each of which is a double vector bundle with side bundles, respectively, E2 and E3 , E3 and E1 , E1 and E2 , where E1 , E2 , E3 are vector bundles over a shared base M , such that each pair of vector bundle structures on III forms a double vector bundle, the operations of which are vector bundle morphisms with respect to the third vector bundle structure.

470

K. C. H. Mackenzie

We display a triple vector bundle in a diagram such as Figure 3(a). (We always read ﬁgures of this type with (III; D1 , D2 ; E3 ) at the rear and (D3 ; E2 , E1 ; M) coming out of the page toward the reader.) The three structures of double vector bundle on III are the upper double vector bundles, and D1 , D2 , D3 are the lower double vector bundles. We refer to (D1 ; E2 , E3 ; M) as the ﬂoor of III and to (III; D2 , D3 ; E1 ) as the roof of III . III −→ D2 ⏐ ⏐ ?

k 1 −→ d2 H⏐ H ⏐ H j HH j ⏐ ⏐H ? ? d3 −→ e1

H ⏐ H H⏐H j HH j

? D3 −→ E1

⏐ D1 −→ E3 ⏐ ?

H

HH j

H

HH j

⏐ ⏐ ?

d1 −→ e3

H

HH j

E2 −→ M (a)

⏐ ⏐ ?

H

HH j

⏐ ⏐ ?

e2 −→ m (b)

Fig. 3.

We have found that, rather than assembling a notation capable of handling any calculation in a triple vector bundle without ambiguity, it is generally preferable to develop an ad hoc notation for each occasion. The great majority of calculations use only certain parts of the structure, and in such cases a modiﬁcation of the notation of Section 1 is often sufﬁcient. Each of the lower double vector bundles Di has a core, which is denoted Ci . The core of the upper double vector bundle (III; D3 , D2 ; E1 ) is denoted K1 . Consider a 1 . Then the d in Figure 3(b) is core element k 1 ∈ III , where k1 projects to e1 ∈ Em 2 the zero over e1 for D2 → E1 , and d3 is the zero over e1 for D3 → E1 . From the morphism condition we then have that e2 and e3 are zeros over m . So d1 = c is a core element for some c ∈ C1 . This deﬁnes a map K1 → C1 . For k, k ∈ K1 over the same element of C1 , deﬁne k +C1 k = k +D1 k ,

(29)

where each of the three bars refers to the roof double vector bundle. With scalar multiplication deﬁned in a similar fashion, K1 → C1 is a vector bundle and a double vector bundle as shown in Figure 4(a). The cores of the other upper double vector bundles are likewise denoted K2 and K3 and form double vector bundles as in Figure 4(b)(c). These three are the core double vector bundles. Although deﬁned by restrictions of the operations in III , they are not substructures of III . Denote the core of (K1 ; C1 , E1 ; M) by W . In Figure 3(b), let k1 = w , where w ∈ W . Then c1 is the zero of C1 over m , and so d1 = c1 is a double zero of D1 .

Duality and triple structures K2 −→ E2

K1 −→ E1 ⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

471

K3 −→ E3

⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

C1 −→ M

C2 −→ M

C3 −→ M

(a)

(b)

(c)

Fig. 4. 3 Next, since e2 is a zero, (d3 ; e2 , e1 ; m) must be of the form d3 = 0D e1 +E2 c3 . But e1 is zero, since w is in the core, so d3 is a core element. But it is known to be a zero over e1 , so must be a double zero. Similarly, d2 is a double zero. This proves most of the next result, and the remainder is an easy veriﬁcation.

Proposition 5.2. Each of the core double vector bundles has as core the set W of elements w ∈ III for which the projection to each of D1 , D2 , D3 is a double zero of the lower double vector bundle. Further, the vector bundle structures with base M induced on W by the various core double vector bundles coincide. Given w ∈ W , the core elements in III corresponding to w1 ∈ K1 , w 2 ∈ K2 , and w 3 ∈ K3 coincide. We call W the ultracore of III . Example 5.3. For a double vector bundle (D; A, B; M), the tangent prolongation triple vector bundle is as shown in Figure 5(a). Two of the three core double vector bundles are copies of (D; A, B; M) and the third is (T C; C, T M; M). The ultracore of T D is C , the core of D . It is illuminating to verify (29) and 5.2 directly in this example. T ∗ D −→ D × B

T D −→ T B

⏐ HH ⏐ HH j H j H ⏐ ⏐ ?

⏐ ⏐ ?

? T A −→ T M

⏐ D −→ B ⏐ ?

H

HH j

H

HH j

⏐ ⏐ ?

HH ⏐ j⏐ H

?D × A

D −→

H

⏐ B ⏐ ?

HH j

A −→ M (a)

H A

HH j

−→ M

(b) Fig. 5.

The key to understanding the relations between the duals of a double vector bundle and the role of the dihedral group (21) lies in constructing a cotangent form of this

472

K. C. H. Mackenzie

example. Both the left and the rear faces of Figure 5(a) are tangent prolongation double vector bundles of ordinary vector bundles and so we can form the ﬁgure in Figure 5(b). In Figure 6(a) we have added the two double vector bundle duals associated to D . In this diagram each of the four vertical sides, and of course the ﬂoor, is a double vector bundle. We need to prove that the roof is a double vector bundle and that Figure 6(a) is a triple vector bundle. First consider the roof, shown in Figure 6(b). We use a short notation for the projections. T ∗ D −→ D × B

rB T ∗ D −→ D × B

H H ⏐ HH⏐ HH ⏐ ⏐ j j H ? ?H

D × A −→ C ∗

D −→

B

⏐

⏐

A

−→ M

rA

⏐ H H⏐ ?H ? HH H j H j H

⏐ ⏐ ?

⏐ ⏐ ?

q×B

D × A −→ C ∗ q×A (b)

(a) Fig. 6.

In this structure, we know that each side is a vector bundle. Proposition 5.4. The structure in Figure 6(b) is a double vector bundle with core T ∗ C . Proof. First, we must prove that the projections form a commutative square. Take f ∈ Td∗ D , where d has the form (d; a, b; m) . Then, for all (d ; a, b ; m), 0d + +A d . rA (f), d = f, A

Here 0d is the zero of T D → D above d and the subscript on + +A indicates that this is the tangent of the addition in D → A . The superscript A on the bar indicates that D is here the core of (T D; T A, T M; A). Writing ϕ = rA (f) , we next have q ×A (ϕ), c = ϕ, 0D a +B c. Here 0D a is the zero of D → A over a and c is the core element of D corresponding to c . Writing out the corresponding formulas for the other side, we must prove that A

B

+B (0D +A c) . 0d + +A (0D a +B c) = 0d + b Using (29), the LHS becomes

(30)

Duality and triple structures

473

A

+A (0D +B c). 0d + a + Writing in terms of tangents to curves, we have A d d d D D 0d = 0D = 0 ) = , d , (t 0 A a a dt 0 dt dt a 0 0

.

c=

d tc . dt 0

Now, using the interchange rule, D D d +A (0D a +B c) = (d +B 0b ) +A (0a +B c) D D = (d +A 0D a ) +B (0b +A c) = d +B (0b +A c)

and from this (30) follows. The proof that rA preserves the addition and scalar multiplication proceeds in a similar way. Next, we show that the core is T ∗ C . Suppose that f ∈ Td∗ D maps to zero under both rA and rB . Then d = c is a core element and f vanishes on elements ξ ∈ Tc D which are vertical with respect to either qAD or qBD . If ξ is vertical with respect to qAD then, in the notation of Figure 7(a), X = 0A m , and it follows that Z = 0m and D so Y is a core element. Likewise, if ξ is vertical with respect to qB , then Y = 0B m and X is a core element. Adding two such representative elements, it follows that f vanishes on all X, as shown in Figure 7(b).

ξ −→ Y

X −→ b

⏐ HH⏐ HH j H j H ⏐ ⏐ ?

⏐ HH ⏐ HH j H j H ⏐ ⏐ ? ? a −→ 0m

? X −→ Z

⏐ ⏐ c −→ 0B m ?

H

HH j

H

HH j

⏐ ⏐ ?

⏐ ⏐ c −→ 0B m ?

H

H

HH j

0A m −→ m (a)

HH j

⏐ ⏐ ?

0A m −→ m (b)

Fig. 7.

Now take ξ ∈ Tc D as shown in Figure 7(a). Because X ∈ T0Am A , it has the form X = T (0A )(Z) + a for some a ∈ Am ; likewise Y has the form Y = T (0B )(Z) + b for some b ∈ Bm . Now deﬁne X = (T (0D +A +B 0c ) + (T (0D 0c ). B )(b) + A )(a) + D

Then γ = ξ − X has the form shown in Figure 8(a) and is an element of T C . D

It is now possible to extend a given ω ∈ Tc∗ C to ω ∈ Tc∗ D by

474

K. C. H. Mackenzie

D×B

γ −→ T (0B )(Z)

⏐ HH ⏐ HH ⏐ H⏐ H H HH ? ?H j j c −→

H

0B m

T (0A )(Z) −→ Z

H ⏐ H⏐ ?H HH HH j j

HH

0A m

H

⏐ HH j ⏐ × ?D A −→ C ∗ D −→

⏐ ⏐ ?

H

⏐ B⏐ ?

HH j

−→ m

H

A

HH j

⏐ ⏐ ?

−→ M

(b)

(a) Fig. 8.

ω, ξ = ω, ξ − X D

and ω is annulled by both rA and rB . This ω is the core element of the double vector bundle in Figure 6(b). Now that the commutativity of the projections has been established, veriﬁcation that addition and scalar multiplication are preserved is straightforward.

The double vector bundle in Figure 6(b) is of a type not previously encountered. It is now a straightforward matter to complete the proof of the following. Theorem 5.5. The structure in Figure 6(a) is a triple vector bundle. The core double vector bundles are (T ∗ A; A, A∗ ; M), (T ∗ B; B, B ∗ ; M) and and the ultracore is T ∗ M . Observe that the triple vector bundle ∗ T D has a much higher degree of symmetry than T D . (T ∗ C; C, C ∗ ; M),

Example 5.6. Consider seven vector bundles E1 , E2 , E3 , C1 , C2 , C3 , W over a shared base M . Let D1 be the trivial double vector bundle with sides E2 and E3 and core C1 , and likewise form D2 and D3 . Similarly, let K1 be the trivial double vector bundle with sides C1 and E1 and core W , and form K2 and K3 in the same way. Lastly, let III be the pullback of all seven vector bundles over M . Then III can be considered as the trivial double vector bundle with side bundles D1 → E3 and D2 → E3 and core K3 → E3 . Likewise, III can be considered the trivial double vector bundle over D2 and D3 with core K1 and over D3 and D1 with core K2 . With these structures, III is a triple vector bundle, the trivial triple vector bundle determined by the given seven vector bundles. More reﬁned versions of this construction exist. For example, suppose given four vector bundles E1 , E2 , E3 , W on M and three double vector bundles (D1 ; E2 , E3 ; M), (D2 ; E3 , E1 ; M), (D3 ; E1 , E2 ; M). Then there is a triple vector bundle for which D1 , D2 , D3 are the lower double vector bundles and W is the ultracore, and for which each of the core double vector bundles is trivial.

Duality and triple structures

475

6 Cornerings Continue with a double vector bundle D as in the previous section. Since D is a vector bundle over A , we have T ∗ D ∼ = T ∗ (D × A) by 4.3, and similarly ∗ ∗ × T D ∼ = T (D B) . Once it has been shown that these isomorphisms respect the triple structures, we can regard 3 as acting on the cube T ∗ D by rotations about the axis from T ∗ D to M . Theorem 6.1. The map R −1 : T ∗ D → T ∗ (D × A) arising from the vector bundle D → A is an isomorphism of triple vector bundles over ZB : D × B → (D × A × C ∗ ) , the other maps on the side structures being identities. Proof. The main work is to show that R −1 is a morphism of vector bundles over ZB . Take ∈ Td∗ D and denote R −1 () by f . Let d have the form (d; a, b, m) and let the projections of to D × A , D × B , and C ∗ be χ , ψ , and κ , respectively. Since R preserves D and D × A , it follows that f projects to d ∈ D and to χ ∈ D × A . For ψ we have, from (24), B ψ, d1 B = , 0d + + B d 1 D

(31)

for any d1 of the form (d1 ; a1 , b; m). For f and we have, by (28), X, ξ T A = , ξ D + f, XD × A ,

(32)

where X ∈ T (D × A) has the form (X; χ , X; a) for some X ∈ T A , and ξ ∈ T D then has the form (ξ ; d, X; a) . Next, for ZB (ψ) ∈ D × A × C ∗ , we have, for each ϕ ∈ D × A of the form (ϕ; a2 , κ; m) , ZB (ψ), ϕ = ϕ, ψ

C∗

= ϕ, d2 A − ψ, d2 B

(33)

for any d2 ∈ D of the form (d2 ; a2 , b; m). Lastly, for the same ϕ we have (D 0χ rC ∗ (f), ϕ = f,

× A)

+ +C ∗ ϕ (D

(D × A)

× A)

.

(34)

×

is the zero in T (D × A) over χ and ϕ (D A) is the core element of Here 0χ T (D × A) corresponding to ϕ . The addition is in the bundle T (D × A) → T C ∗ . We must prove that the RHSs of (33) and (34) are equal. We substitute (D X = 0χ

× A)

+ +C ∗ ϕ (D

× A)

and

ξ = 0d + +B d 1

B

into (32). Providing a1 = a2 , the relevant projections match. Now applying (iv) of Proposition 3.3 to the double vector bundles (T (D × A); T A, T C ∗ ; T M) and (T D; T A, T B; T M), we have (D × A)

0χ X, ξ T A =

(D × A)

= 0χ

+ +C ∗ ϕ (D

× A)

, 0d + +B d 1 T A

, 0d T A + ϕ (D

B

× A)

B

, d 1 T A .

476

K. C. H. Mackenzie (D × A)

In the ﬁrst term, 0χ

is tangent to the path constant at χ , and 0d is tangent to ×

(D A) , 0d T A is tangent to the path constant at the path constant at d ; therefore, 0χ × χ , dA , and is therefore zero. For the second term, ϕ (D A) is tangent to the path × B B t C.∗ ϕ and d 1 is tangent to the path t B. d1 , so ϕ (D A) , d 1 T A is tangent to the path t C.∗ ϕ, t B. d1 A and by (v) of Proposition 3.3 this is tϕ, d1 A . Altogether, we have that X, ξ T A = ϕ, d1 A . Using (31), we have that the RHS of (34) is

ϕ, d1 A − ψ, d1 B and this is equal to the RHS of (33) by the proof of 3.2. The rest of the proof is now straightforward.

For a single vector bundle E → M , the pairing of E ∗ with its dual E ∗∗ can be identiﬁed in a straightforward way with the pairing of E with its dual. For double vector bundles it is ﬁrst necessary to ensure that pairings are chosen in a consistent way. Consider a double vector bundle (D; A, B; M) and assign signs to the two upper structures as in Figure 9(a) in order to show that we pair the duals according to (17). Now, referring to Figure 8(b), we assign signs in such a way that each of A, B, C ∗ has one positive and one negative arrow approaching it, and each of D , D × A , D × B has one positive and one negative arrow departing from it; see Figure 9(b)(c). We therefore, for example, take the pairing of the duals of D × A to be D, D × A × C ∗

B

= D × A, D × A × C ∗ C ∗ − D, D × AA .

⏐ ⏐ ⊕ ?

⊕ D × B −→ B

⊕ D × A −→ C ∗

3 D −−−→ B ⏐ ⏐ ?

3

A −−−→ M

⏐ ⏐ ?

⏐ ⏐ ?

A

−→ M

(a)

3

⏐ ⏐ ?

⏐ ⏐ ?

C ∗ −→ M (c)

(b)

Fig. 9.

Proposition 6.2. For the isomorphism ZB : D × B → (D × A × C ∗ ) , , #

C∗

= , ZB (#)C ∗ ,

(35)

d, #B = − d, ZB (#)

for compatible ∈ D × A , # ∈ D × B , d ∈ D .

B

Duality and triple structures

477

Proof. The ﬁrst is the deﬁnition of ZB . The second follows from ZB × C ∗ = ZA . If we insert these equations into (35), we get (17). Thus the signing on D × A is compatible with that on D . For ordinary vector bundles E1 and E2 on the same base M , one could take the view that a pairing of E1 with E2 is what enables one to construct a cotangent double vector bundle with sides E1 and E2 . In a similar way, three double vector bundles with suitably overlapping sides can be completed to a cotangent triple vector bundle if and only if any two of them are the duals of the third. Deﬁnition 6.3. Consider three double vector bundles as in Figure 10(a), together with six pairings: , E1 of D2 and D3 over E1 , , E2 of D3 and D1 over E2 , , E3 of D1 and D2 over E3 , and , 1 of C1 and E1 over M , , 2 of C2 and E2 over M , , 3 of C3 and E3 over M , such that each pairing of D bundles is a pairing of double vector bundles ( as deﬁned in Proposition 3.3 ) with respect to the pairing of the relevant cores and sides. Then if D2 , D3 E1 = D1 , D2 E3 − D1 , D3 E2 holds, we say that the system is a cornering of D1 with D2 and D3 .

T (T ∗ E) −→ T (E ∗ )

D3

⏐ HH j H ⏐ ? D2 −→ E1

⏐ D1 −→ E2 ⏐ ?

H

H Hj HH H j

⏐ HH ⏐ HH ?

⏐HH ⏐ HH ?

H j H

⏐ ⏐ ?

TE

⏐ H ⏐ ⏐ HH H⏐ ?H ? HH HH j H j H

H

E3 −→ M

−→ T M

H j H

T ∗ E −→ E ∗

E −→ M

(a)

(b) Fig. 10.

Clearly, choosing any double vector bundle in a cornering, the other two double bundles may be identiﬁed with its duals and the cornering may be identiﬁed with the lower sides of the cotangent triple vector bundle associated with the chosen double. Remark 6.4. For an ordinary vector bundle E one may form the cotangent triple of D = T E . Now the canonical diffeomorphism between T ∗ T E and T T ∗ E [Abraham and Marsden 1985] is, since E is a vector bundle, an isomorphism of double vector bundles, and so the triple T ∗ T E is isomorphic to the tangent prolongation of T ∗ E , as shown in Figure 10(b). Now the pairing of the bundles over E in the left face gives rise to the canonical 1-form on T ∗ E , and the pairing of the bundles in the roof gives rise to the canonical 1-form on T ∗ E ∗ .

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K. C. H. Mackenzie

7 Duals of triple vector bundles Consider now a general triple vector bundle III as in Figure 3(a). Dualize III over the base D1 . Each of the upper double vector bundles of which III → D1 is a side has a dual which is familiar from Section 4. Following the example of 5.5, we complete the cube as in Figure 11(a). III× D1 −→ K3 × E3

IIIPY −→ D1

H H ⏐ HH ⏐ HH ⏐ ⏐ H?H HH ? j j

D1

−→

H

E3

⏐ ⏐ ?

K2 × E2 −→ W ∗

⏐ H ⏐ ⏐ H⏐ ?H ? HH HH j j

HH

E2

−→ M

(a)

H

H HH⏐ HH ⏐ HH ?HH j j

D2 −→ E3

D3P −→ E2

⏐ H H ⏐ ⏐ ⏐ HH H ?H ? HH HH j j E1 −→ M (b)

Fig. 11.

Theorem 7.1. There is a triple vector bundle as shown in Figure 11(a) in which the four vertical sides are dual double vector bundles as just described. We omit the details of this. As with the case III = T D , ﬁve of the six faces are double vector bundles of known types and the main work is to show that the roof— which belongs to a new class of examples—is a double vector bundle, and calculate its core, which is K1 × C1 . The proof follows exactly the same outline as in 5.4, though steps involving derivatives must be replaced with forms of the interchange laws. Notice that in Figure 11(a), two of the three upper double vector bundles are standard duals of the double vector bundles in the corresponding positions in Figure 3(a). Two of the lower double vector bundles are duals of core double vector bundles of III . The core double vector bundles of III × D1 are given in Figure 12. The ultracore is E1∗ , the dual of the bundle which in the original was diagonally opposite III in the plane perpendicular to the axis of dualization. The relationship between the three duals of III is embodied in the cotangent quaternary vector bundle of III , as shown in Figure 13. Denote dualization of III along the three axes by X , Y , and Z . In terms of Figure 3(a), take Z to be dualization along the vertical axis, Y to be along D3 and X to be along D2 . Compositions such as ZXZ are triple versions of the operation P studied in Section 3. Precisely, applying ZXZ to III in Figure 3(a) applies P to the rear face and to the front face; denote this by PY . This operation may also be regarded as reﬂection of III in the plane through III , D3 , M , and E3 ; see Figure 11(b). Notice that each face has been ﬂipped in the sense that it cannot be returned to its original position by a proper rotation of the cube. Further, the core double vector bundle which

Duality and triple structures

479

lies in the plane through III , D3 , M , and E3 is left ﬁxed; the other two are ﬂipped and interchanged. With similar deﬁnitions of PX and PZ , we have PX = Y ZY = ZY Z,

PY = ZXZ = XZX,

PZ = XY X = Y XY,

(36)

each of PX , PY , PZ having order 2. Equivalently, each of XY , Y Z , ZX has order 3. New in the triple case are the products QZ = ZXY Z , QX = XY ZX and QY = Y ZXY and their inverses. It is easily found from (36) that QX , QY , and QZ have order 3 and that QZ QY QX = I . Curiously, the equation QX QY QZ = I or, equivalently, (XY Z)4 = I , is not a consequence of (36), but it may be veriﬁed directly by calculating the effect on III . We now have: Theorem 7.2. The group of operations on III generated by X , Y , and Z satisﬁes the relations X2 = I , Y 2 = I , Z 2 = I , (XY Z)4 = I , (Y ZX)4 = I , (ZXY )4 = I , together with (36). By a calculation with [GAP], the group deﬁned by these relations has order 72. Denote the group of operations generated by X , Y , and Z by VB3 . It is straightforward to ﬁnd more than 36 distinct elements of VB3 and so it must have order 72. It thus cannot be, as one might have expected, a subgroup of the full symmetry group of the hypercube, which has order 384 [Coxeter 1973]. This shows that the situation with double vector bundles, in which the operations generated by dualization can be identiﬁed with symmetries of the cotangent triple, does not extend in the analogous fashion to triple vector bundles and symmetries of the hypercube.

8 General principles On the basis of the duality theory for duals and triples, we may formulate some likely principles for the duality of general multiple vector bundles. It may be that the proofs are mainly a matter of acquiring sufﬁcient motivation and notation, but we cannot rule out the possibility that new phenomena arise with increasing dimension. There are three groups associated with an n -fold vector bundle N . First, the various operations generated by ﬂips of the constituent double vector bundles generate an K1 × C1 −→ W ∗

D2 × E3 −→ C2∗

D3 × E2 −→ C3∗

⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

⏐ ⏐ ?

C1

−→ M

E3

−→ M

E2

−→ M

(a)

(b) Fig. 12.

(c)

480

K. C. H. Mackenzie

T ∗ III −→ III× D2 ⏐ @ ⏐ ? @

⏐ ⏐ ?

@

@

@

×E III× D1 −→ K3@ 3

III −→ D2

⏐ @ ⏐@ ⏐ @⏐ ? ? @

@

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ R @ R @ @ @ × D −→ K × E III@ @ 3 1 1 @ ⏐@ ⏐ ⏐ @ ⏐ @ ? ? @ @ @ @ R× @ R ∗ @

@ @ @ D1 −→ E3 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ R @ @ R @ @ @ @ D3 −→ E1 @ @ ⏐ ⏐ ⏐ @ @⏐ ? ? @ @ @ @ R @ R @

@

K2 E2 −→

W

E2 −→ M Fig. 13.

action of the symmetric group Sn . Secondly, there is an obvious sense in which individual n -fold vector bundles may have more symmetry than others, as we remarked in the case of T ∗ D and T D in Section 5. Of most interest, however, is the group VBn generated by the dualization operations. We have seen that VB2 is S3 and that VB3 has order 72. Further, the subgroup of VB3 generated by XY XZ , Y ZY X , and ZXZY has order 12 and is normal, with quotient isomorphic to S3 . In the general case, the n duals of an n-fold vector bundle N and N itself form the (n + 1) lower n -faces of an (n + 1) -fold vector bundle, which may be completed to be the cotangent (n + 1) -fold vector bundle of N , or of any of the duals of N . The (n + 1) upper n -faces of T ∗ N consist of the n cotangent n-fold vector bundles of the upper (n − 1) -faces of N , together with one n -fold vector bundle of a new type, which incorporates data from all of the structure of N . It is reasonable to conjecture that if X1 , . . . , Xn denote the dualization operations, each of order 2, then we have, for each 1 k n and each string i1 , i2 , . . . , ik of k distinct elements of {1, . . . , n} ,

Duality and triple structures

481

(Xi1 Xi2 · · · Xik )k+1 = 1. Note added in proof It has been pointed out to me by Alfonso Gracia-Saz that the calculation in Theorem 7.2 is incomplete. A complete account will be provided in a subsequent article. Acknowledgments I am grateful to Moty Katzman for introducing me to the GAP software, which was very valuable in Section 7.

References R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Addison–Wesley, Reading, MA, 1985. A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, in Publications du Département de Mathématiques de l’Université de Lyon, I, number 2/A-1987, Département de Mathématiques, Université de Lyon, Lyon, 1987, 1–65. T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631–661. H. S. M. Coxeter, Regular polytopes, Dover, New York, 1973. GAP, GAP: Groups, Algorithms, and Programming, Version 4.3, The GAP Group, St. Andrews, Fife, UK, 2002; www.gap-system.org. K. Konieczna and P. Urban´ ski, Double vector bundles and duality, Arch. Math. ( Brno ) , 35-1 (1999), 59–95. J.-H. Lu and A. Weinstein, Groupoïdes symplectiques doubles des groupes de LiePoisson, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 951–954. K. C. H. Mackenzie, On symplectic double groupoids and the duality of Poisson groupoids, Internat. J. Math., 10 (1999), 435–456. K. C. H. Mackenzie, On certain canonical diffeomorphisms in symplectic and Poisson geometry, in Quantization, Poisson Brackets and Beyond ( Manchester, 2001) , Contemporary Mathematics, Vol. 315, American Mathematical Society, Providence, RI, pages 187–198. K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, Cambridge, UK, 2004. K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73-2 (1994), 415–452. J. Pradines, Fibrés vectoriels doubles et calcul des jets non holonomes, Esquisses Mathématiques 29, Université d’Amiens, U.E.R. de Mathématiques, Amiens, France, 1974. J. Pradines, Remarque sur le groupoïde cotangent de Weinstein–Dazord, C. R. Acad. Sci. Paris Sér. I Math., 306-13 (1988), 557–560. A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. ( N.S.) , 16 (1987), 101–104. A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705–727.

Star exponential functions as two-valued elements∗ Y. Maeda1 , N. Miyazaki2 , H. Omori3 , and A. Yoshioka4 1 Department of Mathematics

Faculty of Science and Technology Keio University Hiyoshi, Yokohama, 223-8522 Japan [email protected] 2 Department of Mathematics Faculty of Economics Keio University Hiyoshi, Yokohama, 223-8521 Japan [email protected] 3 Department of Mathematics Faculty of Science and Technology Tokyo University of Science Noda, Chiba, 278-8510 Japan [email protected] 4 Department of Mathematics Faculty of Science Tokyo University of Science Kagurazaka, Tokyo, 102-8601 Japan [email protected] Abstract. We propose a relatively new notion of two-valued elements, which arise naturally in constructing star exponential functions of the quadratics in the Weyl algebra over the complex number. This notion enables us to describe group-like objects of the set of star exponential functions of quadratics in the Weyl algebra.

1 Introduction Classically, geometries are described within the framework of manifolds which are distinguished among the topological spaces. The question may arise if there are ge∗ This research was partially supported by Grants-in-Aid for Scientiﬁc Research 15204005

(ﬁrst author), 15740045 (second author), 14540092 (third author), and 15540094 (fourth author) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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Y. Maeda, N. Miyazaki, H. Omori, and A. Yoshioka

ometries. In this paper, we attempt to propose a notion of two-valued elements, which seems to renew a geometric concept. A nontrivial example of objects we propose in this paper is given by the HopfH S1 ﬁbering S 3 → S 2 . Viewing S 3 = q∈S 2 Sq1 (disjoint union), we consider the double H covering S˜q1 of each ﬁber S˜q1 , and the disjoint union q∈S 2 S˜q1 which is denoted by S˜ 3 . When S˜ 3 is considered as a point set, we are able to deﬁne local trivializations of S˜ 3 |Vi ∼ = Vi × S˜ 1 naturally through the trivializations S 3 |Vi given on a simple open covering {Vi }i∈ of S 2 . This structure permits us to treat S˜ 3 as a local Lie group, and hence itHlooks like a topological H space. On the other hand, we have a projection π : S˜ 3 = q∈S 2 S˜q1 → S 3 = q∈S 2 Sq1 as the union of ﬁberwise projections, as if it were a nontrivial double covering. S˜ 3 cannot be a manifold, since S 3 is simply connected. This might suggest us to make the notion of points vague. In particular, the “points’’ of S˜ 3 should be regarded as two-valued elements with ± ambiguity. We now consider a one-parameter subgroup S 1 of S 3 and the inverse image −1 S 3 are “two-valued", this simply can be considered both π (S 1 ). Since all points of 1 on S × Z2 and the double covering group, i.e., in some restricted region, this object can be regarded as a point set in several ways. In such a region, the ambiguity only arises if these two pictures of point sets are confused. Similar phenomena appear in constructing star exponential functions of quadratic forms in the suitably extended Weyl system, which leads us to explore a new type of geometry with a noncommutative (quantum) aspect. In [9], we have exhibited strange phenomena which violate the associativity for the Weyl algebra over C generated by two generators u and v. Furthermore, we have shown that the Lie algebra over C of quadratic forms can be exponentiated to a “group’’ which acts like a double covering group of simply connected group SLC (2), or the complexiﬁcation of the metaplectic group Mp(2, R). As a sequel to [9], we continue these resutls to the case of Weyl algebra with 2m-generators u1 , . . . , um , v1 , . . . , vm , and show that similar phenomena occur. We show that star exponential functions can be viewed as two-valued elements. We note that an approach using gerbes may also describe such phenomena (see [3, 4, 9]), which would also give rise to a new geometrical formulation.

2 Weyl algebra and orderings 2.1 Weyl algebra The Weyl algebra W is the algebra over C generated by u1 , . . . , um , v1 , . . . , vm with the following commutation relations: [ui , vj ] = −iδij ,

(1)

where is a positive constant and [a, b] = a ∗ b − b ∗ a. Here the product on W is denoted by ∗. For abbreviation, we set u = (u1 , . . . , um ), and z = (u, v) =

Star exponential functions as two-valued elements

485

(u1 , . . . , um , v1 , . . . , vm ). Let Sym(2m, C) be the set of complex symmetric matrices A = (Aij ). For A ∈ Sym(2m, C), we deﬁne a quadratic form by A∗ (z) =

2m i,j =1

1 Aij (zi ∗ zj + zj ∗ zi ). 2

(2)

Denote by A the set of A∗ (z), where A ∈ Sym(2m, C). It is easily seen that A forms a complex Lie algebra isomorphic to spC (m). 2.2 Orderings Orderings are treated in the physical literature (see [1]) of quantum mechanics as the rule of association from c-number functions to q-number functions. There are typical orderings, called the standard ordering, the antistandard ordering and the Weyl ordering, and the in case of complex variables ζk = uk + ivk , ζl∗ = ul −ivl , the normal ordering, the antinormal ordering, and the Weyl ordering. However, from the mathematical view point, it is better to go back to the original understanding of Weyl which says that the ordering is the problem of realization of the Weyl algebra W . Since the Weyl algebra is the universal enveloping algebra of a Heisenberg Lie algebra, the Poincaré–Birkhoff–Witt theorem shows that this algebra can be viewed as an algebra deﬁned on the space of polynomials. For precise formulations of ordering prescriptions in formal deformation quantization, one can refer to the article [2], but the theory using a formal deformation parameter is only a probe for genuine quantum theory. We emphasize here that the deformation parameter in this note is not a formal parameter, but a parameter in the positive reals. Thus we generalize orderings as follows. Let J be a 2m × 2m matrix deﬁned by J = [ I0m −I0m ]. For every symmetric complex 2m × 2m matrix K = (K ij ), we set the product ⎧ ⎛ ⎞⎫ 2m ⎨ i ⎬ ← − − → ⎝ f (z) ∗K g(z) = f exp (3) ∂zi ij ∂zj ⎠ g, ⎩2 ⎭ i,j =1

where = ( ij ) = (K ij + J ij ). The product formula (3) is well deﬁned for all f, g ∈ C[z], where C[z] = C[z1 , . . . , z2m ], and this satisﬁes zi ∗K zj − zj ∗K zi (= [zi , zj ]∗K ) = iJ ij ,

(4)

giving the same commutation relations (1) as the Weyl algebra W . Proposition 2.1. For every complex symmetric 2m × 2m matrix K, (C[z], ∗K ) forms an associative algebra isomorphic to W . Proposition 2.1 gives a realization of the Weyl algebra W , and at the same time it also gives a way to compute expressions in the Weyl algebra W . For instance, computing ui ∗ uj ∗ uk by (3) gives an expression of ui ∗ uj ∗ uk as a polynomial.

486

Y. Maeda, N. Miyazaki, H. Omori, and A. Yoshioka

Thus the product formula (3) will be referred to as K-ordering, i.e., giving an ordering is necessary to a product formula which gives the Weyl algebra W once the generators are ﬁxed. Note that according to the choice of K: 5 4 5 4 5 4 0 −Im 00 0 Im , , , Im 0 −Im 0 00 the product formulas (3) gives the standard ordering, the antistandard ordering and the Weyl ordering, respectively. By these formulation of orderings, intertwiners between K-orderings are explicitly given as follows. Proposition 2.2. For every pair of complex symmetric 2m × 2m matrices K, K , we have the intertwiner T : (C[z], ∗K ) → (C[z], ∗K ) deﬁned as ⎞ ⎛ (5) (K ij − K ij )∂zi ∂zj ⎠ f. T (f ) = exp ⎝ 2i i,j

Namely, the identity T (f ∗K g) = T (f ) ∗K T (g),

(6)

holds for any f, g ∈ C[z]. With the precise statement given in a forthcoming paper, the intertwiner can be extended to a certain class of functions. However, as has been shown in [9] the intertwiner behaves only as a 2-to-2 mapping in the space of exponential functions of quadratic forms, since a square root appears in the amplitude part of intertwined functions. We think this is a basic phenomena which breaks the associativity of the ∗-product on the closed linear hull of the exponential functions of quadratic forms. Such strange phenomena occur only in the case that the deformation parameter is a nonformal parameter. In spite of this, it is important that one can consider one-parameter subgroups via the theory of ordinary differential equations.

3 Star exponential functions 3.1 Star exponential functions tA (z)

We give the explicit formula for the star exponential function e∗ ∗ for a ﬁxed Kordering. For A ∈ Sym(2m, C), we denote by A[z] the symmetric quadratic function deﬁned by 2m A[z] = Aij zi zj . (7) i,j =1

Set C× = C − {0}, and denote by F the set deﬁned by

Star exponential functions as two-valued elements

F = {F = g exp Q[z] | g ∈ C× , Q ∈ Sym(2m, C)}.

487

(8)

For A ∈ Sym(2m, C), we set A∗K (z) =

2m i,j =1

1 Aij (zi ∗K zj + zj ∗K zi ). 2

(9)

The product formula (3) gives A∗K (z) = A[z] + i Tr KA.

(10)

We realize the star exponential function of A∗K (z), for A ∈ Sym(2m, C) with the tA∗K (z)

help of the K-ordering. Namely, in order to get the formula for e∗K FK (t) =

tA∗ (z) e∗K K ,

and consider the following equation: + ∂t FK (t) = A∗K (z) ∗K FK (t), FK (0) = 1.

, we set

(11)

By the product formulas (3) and (9), the evolution equation (11) can be expressed as a differential equation. Thus the real analytic solution is unique, if it exists. By setting FK (t) = gK (t) exp QK (t)[z],

where QK (t) ∈ Sym(2m, C),

(12)

the evolution equation (11) is reduced to a system of ordinary differential equations on gK (t) and QK (t)[z]. By a direct, although rather complicate computation, we have the following. Theorem 3.1. The evolution equation (11) has a unique analytic solution FK (t) ∈ F explicitly given by FK (t) = gK (t) exp QK (t)[z], (13) where J QK (t) = − (tan tJ A) · (I − iK tan tJ A)−1 , gK (t) = (det(cos tJ A − iK sin tJ A))−1/2 .

(14) (15)

Remark. Maillard [5] also obtained this product formula by solving the successive power series of a Riccati-type equation. Note also that for every t ∈ C there is K-ordering such that FK (t) is well deﬁned. The formula (13) for the star exponential functions of A∗K (z) specializes to the standard ordering and the Weyl ordering by plugging in K = [ I0m Im0 ] and K = 0, respectively. In particular, we have the following.

488

Y. Maeda, N. Miyazaki, H. Omori, and A. Yoshioka tA∗ (z)

Corollary 3.2. For any A ∈ Sym(2m, C), the star exponential function e∗ expressed as −J e∗tA∗ (z) = det(cos tJ A)−1/2 · exp (tan tJ A)[z]

is

(16)

via the Weyl ordering. 3.2 Star exponential functions of rank one quadratics We examine the equation (11) for rank one quadratics. m For x = (x1 , . . . , xm ) and y = (y x, y = m 1 , . . . , ym ) ∈ C , we set i=1 xi yi . m m For a, b ∈ Cm , we consider a, u = i=1 ai ui and b, v = i=1 bi vi as elements of W . It is easy to see that [a, u, b, v]∗ = −ia, b.

(17)

Hence, if a, a = 1, then a, u and a, v form a canonical conjugate pair. Let m−1 m−1 SC = {a ∈ Cm | a, a = 1}. For every a ∈ SC , and α, β, γ ∈ C, we consider a quadratic form B∗ (α, β, γ ) = αa, u ∗ a, u + βa, v ∗ a, v + γ (a, u ∗ a, v + a, v ∗ a, u),

(18)

which is called a rank one quadratic form. In the following, we assume that the discriminant D = γ 2 − αβ = 1. We now write down the star exponential for the rank one quadratic form B∗ (α, β, γ ). We denote by FM (α, β, γ ) and FN (α, β, γ ) the solution of (11) for A∗ (z) = B∗ (α, β, γ ) with respect to K = 0 and [ I0m Im0 ], respectively. Then we have the following (see also [9]). m−1 Corollary 3.3. Assume that D = γ 2 − αβ = 1. Then for a ∈ SC , we have

FM (t, α, β, γ ) = gM (t, α, β, γ ) · exp QM (t, α, β, γ ),

(19)

where gM (t, α, β, γ ) = (cos t)−1 , 1 QM (t, α, β, γ ) = (tan t) · (αa, u2 + βa, v2 + 2γ a, ua, v).

(20) (21)

Similarly, we have FN (t, α, β, γ ) = gN (t, α, β, γ ) · exp QN (t, α, β, γ ).

(22)

Here gN and QN are given by gN (t, α, β, γ ) = e−i tγ · (cos 2t − iγ sin 2t)−1/2 ,

(23)

Star exponential functions as two-valued elements

QN (t, α, β, γ ) =

1 (XN (t)a, u2 + YN (t)a, v2 + 2ZN (t)a, u ◦ a, v), ⎧

α sin 2t ⎪ X (t) = ⎪ N ⎪ 2 cos 2t−iγ sin 2t , ⎨ β 2t YN (t) = 2 cos 2sin t−iγ sin 2t ,

⎪ ⎪ ⎪ 1 ⎩ZN (t) = i 1 − 2 cos 2t−iγ sin 2t

where

489

(24)

(25)

and the ◦ in the product denotes the standard ordering.

4 Polar elements are two-valued elements 4.1 Polar elements Using the formulas of the star exponential functions (19) and (22), we show how two-valued elements appear. An idea of justiﬁcations of the star exponential function of quadratic forms B∗ (α, β, γ ) is to consider the star exponential functions expressed by the standard ordering and the Weyl ordering. Looking at the formulas in Corollary 3.3 and evaluating at t = 2π , we have that FM ( 2π , α, β, γ ) diverges, while FN ( 2π , α, β, γ ) is ﬁnite. Thus we think of FN ( 2π , α, β, γ ) as a realization of the star exponential function π of 2 B∗ (α, β, γ ), which is denoted by exp∗ ( 2π B∗ (α, β, γ )). However, by the formula (22) in Corollary 3.3, we obtain the following. m−1 Theorem 4.1. Assume a ∈ SC . For any (α, β, γ ) with γ 2 − αβ = 1, we have

exp∗

π 2

√ 2i a,u◦a,v B∗ (α, β, γ ) = −1e◦ ,

(26)

which is independent of the choice of α, β, γ . √ We will show that the ambiguity of −1 cannot be eliminated for all (α, β, γ ). m−1 . Deﬁnition 4.2. Assume a ∈ SC

ε00 (a) = exp∗

π 2

B∗ (0, 0, 1)

(27)

is called the polar element. 4.2 Two-valued elements m−1 We explain below that the polar elements ε00 (a), a ∈ SC act like two-valued elements. Since (α, β, γ ) = (0, 0, 1) and (0, 0, −1) are arcwise connected in the set

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Y. Maeda, N. Miyazaki, H. Omori, and A. Yoshioka

γ 2 − αβ = 1, and thus they have to be viewed as a single element. By Theorem 4.1, we have π

e∗2

(a,v∗a,u+a,u∗a,v)

=

2i √ a,u◦a,v −1e◦ π − 2 (a,v∗a,u+a,u∗a,v)

= e∗

(28) .

However, considering the exponential law of the ∗-exponential function t

e∗2

(a,v∗a,u+a,u∗a,v)

for t ∈ C − {singular set}, we must set π

e∗2

(a,v∗a,u+a,u∗a,v)

− π (a,v∗a,u+a,u∗a,v) e∗ 2

2i

= ie◦

a,u◦a,v

,

2i a,u◦a,v

= −ie◦

(29) .

If one attempts to ﬁx the sign ambiguity, the exponential law and (28) give π

(a,v∗a,u+a,u∗a,v)

∗ e∗2

π

(a,v∗a,u+a,u∗a,v)

∗ e∗

−1 = e∗2 = e∗2

π

(a,v∗a,u+a,u∗a,v)

π − 2 (a,v∗a,u+a,u∗a,v)

(30)

= 1.

We choose a continuous path of (α, β, γ ) from (0, 0, 1) to (0, 0, −1) for the case m = 1 concretely as follows: Set a, u = u, a, v = v and ε00 = ε00 (a). By a careful calculation, we see iθ

Ad(e∗2

(u2 +v 2 )

)e∗2tuv 2 −v 2 )+cos 2θ

= e∗t (sin 2θ (u

(31) 2uv)

.

Since the discriminant of the quadratic form of the right-hand side is identically 1, the πi

(u2 +v 2 )

π

uv

right-hand side is identically ε00 for t = 2π . In particular, Ad(e∗4 )e∗ = ε00 . iθ 2 +v 2 ) (u However, consider, for each θ, the one parameter subgroup Ad(e∗2 )e∗2tuv with π respect to t, t ∈ [0, 2 ]. • ε00 2θ

This line is identically ε00 .

1• −ε00 •

?

Star exponential functions as two-valued elements πi

We easily see that Ad(e∗4

(u2 +v 2 ) πi

Ad(e∗4

491

)e∗2tuv = e∗−2tuv . In particular,

(u2 +v 2 )

)ε00 = −ε00

by the exponential law. Move 2θ from 0 to π. Note also that by (31), (23), there is a singularity at 2θ = π2 , t = 4π . We also have in the standard ordering for D = 1, π

e∗2

(αa,u2 +βa,v2 +γ (a,u∗a,v+a,v∗a,u)

=

2i √ a,u◦a,v −1e◦ .

By the exponential law, we see that ε00 (a) satisﬁes −1 (a))2 = −1, ε00 (a)2 = (ε00

−1 ε00 (a) ∗ ε00 (a) = 1.

(32)

Therefore, we must conclude that the sign ambiguity cannot be eliminated. One 2i √ a,u◦a,v with the sign ambiguity. Similar phenomena has to set ε00 (a) = −1e◦ have been discussed by Olver [6]. By the above observation, the polar element ε00 (a) should be regarded as a twovalued element. Otherwise we obtain a contradiction 1 = −1. Only in this way can one permit the identity −ε00 (a) = ε00 (a). Since such a notion does not exist in the set theory, it is impossible to deﬁne ε00 (a) as a point in a point set. In what follows, we set π

ε00 (k) = e∗2

(uk ∗vk +vk ∗uk )

=

2i √ uk ◦vk −1e◦ ,

k = 1, 2, . . . , m.

These are all regarded as two-valued elements. Although it is natural to write ε00 (k) ∗ ε00 (l) = ε00 (l) ∗ ε00 (k), we simultaneously have the equality ε00 (k) ∗ ε00 (l) = −ε00 (l) ∗ ε00 (k)

(k = l).

Hence we have ε00 (k) ∗ ε00 (k) = ±1, but we see that ε00 (k)2 = −1. This is just the same as {±1}{±1} = {±1}, but {±1}2 = 1. Hence ε00 (k)2 behaves like an ordinary number in the extended Weyl algebra. In spite of this, ε00 (k) does not behave like an ordinary number i, since it is easy to see with the bumping identity (see [9]) that Ad(ε00 (k))uk = −uk ,

Ad(ε00 (k))vk = −vk .

Using this we easily have Ad(ε00 (1))

' m i=1

* bi ui

= −b1 u1 +

m

b i ui .

(33)

i=2

m−1 Since every a, u, a ∈ SC is translated to u1 by a symplectic transformation, we have in general the reﬂection w.r.t. a:

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Ad(ε00 (a))b, u = b − 2a, ba, u, Ad(ε00 (a))b, v = b − 2a, ba, v

(34)

We introduce a notion called a blurred double covering group, which is a grouplike object formed by 2-valued elements [9]. Theorem 4.3. Ad(ε00 (a) ∗ ε00 (b)) generate SO(m, C), hence {ε00 (a) ∗ ε00 (b)} generate a blurred double covering group of SO(m, C). However, this blurred double cover has a point set picture as SO(m, C) × C× . If a, b are restricted to real vectors, then Ad(ε00 (a)∗ε00 (b)) generate SO(m), hence {ε00 (a)∗ε00 (b)} generate a blurred double covering group of SO(m), which may be viewed as Spin(m).

References [1] G. S. Agawal and E. Wolf, Calculus for functions of noncommuting operators and general phase-space method of functions, Phys. Rev. D, 2-10 (1970), 2161–2186. [2] M. Bordemann, N. Neumaier, M. Pﬂaum, and S. Waldmann, On representation of star product algebra over cotangent spaces on Hermitian line bundles, J. Functional Anal., 199 (2003), 1–47. [3] J. L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, Vol. 107, Birkhäuser, Boston, 1993. [4] N. Hitchin, Lectures on special Lagrangian submanifolds, in C. Vafa and S.-T. Yau, eds., Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, Studies in Advanced Mathematics, Vol. 23, American Mathematical Society, Providence, RI, 2001, 151–182. [5] J. M. Maillard, Star exponential functions for any ordering of the elements of the inhomogeneous symplectic Lie algebra, to appear. [6] P. J. Olver, Non-associative local Lie groups, J. Lie Theory, 6 (1996), 23–59. [7] H. Omori, One must break symmetry in order to keep associativity, Banach Center Publ., 55 (2002), 1–11. [8] H. Omori, Beyond Point Set Topology, informal preprint for the Alanfest, Erwin Schrödinger Institute, Vienna, 2003. [9] H. Omori, Y. Maeda, N. Miyazaki, and A. Yoshioka, Strange phenomena related to ordering problems in quantizations, J. Lie Theory, 13-2 (2003), 481–510.

From momentum maps and dual pairs to symplectic and Poisson groupoids Charles-Michel Marle Institut de Mathématiques Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 France [email protected] It is a great pleasure to submit a contribution for this volume in honour of Alan Weinstein. He is one of the four or ﬁve persons whose works have had the greatest inﬂuence on my own scientiﬁc interests, and I am glad to have this opportunity to express to him my admiration and my thanks.

Introduction In this survey, we will try to indicate some important ideas, due in large part to Alan Weinstein, which led from the study of momentum maps and dual pairs to the current interest in symplectic and Poisson groupoids. We hope that it will be useful for readers new to the subject; therefore, we begin by recalling the deﬁnitions and properties which will be used in what follows. More details can be found in [4, 26, 47]. 1. A Poisson manifold [27,E48] is a smooth manifold P equipped with a bivector ﬁeld (i.e., a smooth section of 2 T P ) $ which satisﬁes [$, $] = 0, the bracket on the left-hand side being the Schouten bracket [44, 40]. The bivector ﬁeld $ will be called the Poisson structure on P . It allows us to deﬁne a composition law on the space C ∞ (P , R) of smooth functions on P , called the Poisson bracket and denoted by (f, g) → {f, g}, by setting, for all f and g ∈ C ∞ (P , R) and x ∈ P , {f, g}(x) = $ df (x), dg(x) . That composition law is skew-symmetric and satisﬁes the Jacobi identity and therefore turns C ∞ (P , R) into a Lie algebra. 2. Let (P , $) be a Poisson manifold. We denote by $ : T ∗ P → T P the vector bundle map deﬁned by

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I J η, $ (ζ ) = $(ζ, η), where ζ and η are two elements in the same ﬁbre of T ∗ P . Let f : P → R be a smooth function on P . The vector ﬁeld Xf = $ (df ) is called the Hamiltonian vector ﬁeld associated to f . If g : P → R is another smooth function on P , the Poisson bracket {f, g} can be written I J I J {f, g} = dg, $ (df ) = − df, $ (dg) . 3. Every symplectic manifold (M, ω) has a Poisson structure, associated to its symplectic structure, for which the vector bundle map $ : T ∗ M → M is the inverse of the vector bundle isomorphism v → −i(v)ω. We will always consider that a symplectic manifold is equipped with that Poisson structure, unless otherwise speciﬁed. 4. Let (P1 , $1 ) and (P2 , $2 ) be two Poisson manifolds. A smooth map ϕ : P1 → P2 is called a Poisson map if for every pair (f, g) of smooth functions on P2 , {ϕ ∗ f, ϕ ∗ g}1 = ϕ ∗ {f, g}2 . 5. The product P1 × P2 of two Poisson manifolds (P1 , $1 ) and (P2 , $2 ) has a natural Poisson structure: it is the unique Poisson structure for which the bracket of functions of the form (x1 , x2 ) → f1 (x1 )f2 (x2 ) and (x1 , x2 ) → g1 (x1 )g2 (x2 ), where f1 and g1 ∈ C ∞ (P1 , R), f2 and g2 ∈ C ∞ (P2 , R), is (x1 , x2 ) → {f1 , g1 }1 (x1 ){f2 , g2 }2 (x2 ). The same property holds for the product of any ﬁnite number of Poisson manifolds. 6. Let (V , ω) be a symplectic vector space, which means a real, ﬁnite-dimensional vector space V with a skew-symmetric nondegenrate bilinear form ω. Let W be a vector subspace of V . The symplectic orthogonal of W is K orth W = v ∈ V ; ω(v, w) = 0 for all w ∈ W }. It is a vector subspace of V , which satisﬁes dim W + dim(orth W ) = dim V ,

orth(orth W ) = W.

The vector subspace W is said to be isotropic if W ⊂ orth W , coisotropic if orth W ⊂ W , and Lagrangian if W = orth W . In any symplectic vector space, there are many Lagrangian subspaces; therefore, the dimension of a symplectic vector space is always even; if dim V = 2n, the dimension of an isotropic (resp., coisotropic, resp., Lagrangian) vector subspace is ≤ n (resp., ≥ n, resp., = n). 7. A submanifold N of a Poisson manifold (P , $) is said to be coisotropic if the bracket of two smooth functions, deﬁned on an open subset of P and which vanish on N, vanishes on N too. A submanifold N of a symplectic manifold (M, ω) is coisotropic if and only if for each point x ∈ N , the vector subspace Tx N of the symplectic vector space Tx M, ω(x) is coisotropic. Therefore, the dimension of a coisotropic submanifold in a 2n-dimensional symplectic manifold is ≥ n; when it is equal to n, the submanifold N is said to be Lagrangian.

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8. A dual pair [48, 4] is a pair (ϕ1 : M → P1 , ϕ2 : M → P2 ) of smooth Poisson maps, deﬁned on the same symplectic manifold (M, ω), with values in the Poisson manifolds (P1 , $1 ) and (P2 , $2 ), such that for each x ∈ M, the two equivalent equalities hold: ker(Tx ϕ1 ) = orth ker(Tx ϕ2 ) , ker(Tx ϕ2 ) = orth ker(Tx ϕ1 ) . That property implies that for all f1 ∈ C ∞ (P1 , R) and f2 ∈ C ∞ (P2 , R), {ϕ1∗ f1 , ϕ2∗ f2 } = 0. A dual pair (ϕ1 , ϕ2 ) is specially interesting when ϕ1 and ϕ2 are surjective submersions. 9. Let ϕ : M → P be a surjective submersion of a symplectic manifold (M, ω) onto a manifold P . The manifold P has a Poisson structure $ for which ϕ is a Poisson map if and only if orth(ker T ϕ) is integrable [23]. When that condition is satisﬁed, this Poisson structure on P is unique. If in addition there exist a smooth manifold P2 and a smooth map ϕ2 : M → P2 such that ker T ϕ2 = orth(ker T ϕ), the manifold P2 has a unique Poisson structure $2 for which ϕ2 is a Poisson map, and (ϕ, ϕ2 ) is a dual pair. 10. A symplectic realization [48] of a Poisson manifold (P , $) is a Poisson map ϕ : M → P , deﬁned on a symplectic manifold (M, ω), with values in P . A symplectic realization ϕ : M → P is specially interesting when ϕ is a surjective submersion; such a symplectic realization is said to be surjective submersive. For example, in a dual pair (ϕ1 : M → P1 , ϕ2 : M → P2 ), ϕ1 and ϕ2 are symplectic realizations of the Poisson manifolds (P1 , $1 ) and (P2 , $2 ), respectively.

1 A typical example: action of a Lie group on its cotangent bundle Let G be a Lie group, πG : T ∗ G → G its cotangent bundle. For each g and h ∈ G, we write Lg (h) = gh, Rg (h) = hg. =g and R =g the canonical lifts to T ∗ G of Lg and Rg , respectively. We We denote by L recall that, for g and h ∈ G, ξ ∈ Th∗ G, X ∈ Tgh G and Y ∈ Thg G, we have I

J =g (ξ ), X = ξ, T Lg −1 X, L

I

J =g (ξ ), Y = ξ, T Rg −1 Y . R

=g ξ , and R = : T ∗ G × G → T ∗ G, = : G × T ∗ G → T ∗ G, (g, ξ ) → L The maps L =g ξ , are actions of the Lie group G on its cotangent bundle T ∗ G, on the (ξ, g) → R left and on the right, respectively. Theorem 1.1. These two actions are Hamiltonian and have as momentum maps, respectively, the maps JL : T ∗ G → G∗ ,

=π (ξ )−1 ξ JL (ξ ) = R G

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and

JR : T ∗ G → G ∗ ,

=π (ξ )−1 ξ, JR (ξ ) = L G

where G∗ is the dual of the Lie algebra G of G. We have identiﬁed G with Te G and G∗ with Te∗ G. We have denoted by πG : T ∗ G → G the cotangent bundle projection. Moreover, • • •

= the level sets of JL are the orbits of the action R, = the level sets of JR are the orbits of the action L, for each ξ ∈ T ∗ G, each one of the tangent spaces at ξ to the orbits of that point = and L = is the symplectic orthogonal of the other one. under the actions R

The dual G∗ of the Lie algebra G has two (opposite) natural Poisson structures, called the plus and the minus KKS-Poisson structures (the letters KKS stand for Kirillov [19], Kostant [21], and Souriau [45]). The brackets of two smooth functions f and g ∈ C ∞ (G∗ , R) are given by the formulae I

J I

J {f, g}+ (η) = η, df (η), dg(η) and {f, g}− (η) = − η, df (η), dg(η) . Then we have the following. Theorem 1.2. The map JL : T ∗ G → G∗ (resp., the map JR : T ∗ G → G∗ ) is a Poisson map when T ∗ G is equipped with the Poisson structure associated to its canonical symplectic structure and G∗ with the minus KKS-Poisson structure (resp, with the plus KKS-Poisson structure). The pair of maps (JL , JR ) in the above theorem is a very simple example of a dual pair; both JL and JR are surjective submersions. The cotangent space T ∗ G, equipped with its canonical structure, is a symplectic realization of each symplectic of the Poisson manifolds G∗ , {, }+ and G∗ , {, }− .

2 Hamiltonian action of a Lie group on a symplectic manifold The example dealt with in the preceding section is very symmetrical: the roles of the = and R = could be exchanged by means of the group anti-automorphism two actions L g → g −1 of the Lie group G. In this section, we will see that a similar situation (although less symmetrical) holds for a Hamiltonian action of a Lie group on a symplectic manifold. Deﬁnitions 2.1. Let : G × M → M be an action (for example, on the left) of a Lie group G on a symplectic manifold (M, ω). That action is said to be symplectic if, for any g ∈ G, ∗g ω = ω. It is said to be Hamiltonian if it is symplectic and if, in addition, there exists a smooth map J : M → G∗ such that, for each X ∈ G, vector ﬁeld XM is I the fundamental J Hamiltonian, with the function J, X : x → J (x), X as Hamiltonian. The map J is called the momentum map of .

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2.2 Comments Let us indicate brieﬂy some important properties of Hamiltonian actions. The fundamental vector ﬁelds Under the assumptions of the above deﬁnitions, we recall that the fundamental vector ﬁeld XM associated to an element X ∈ G (the Lie algebra of G) is the vector ﬁeld on M deﬁned by d XM (x) = exp(tX), x t=0 , dt where x ∈ M. The formula i(XM )ω(x) = −dJ, X(x) expresses that XM is the Hamiltonian vector ﬁeld associated to the function J, X. The momentum map equivariance When M is connected, there exists a smooth map θ : G → G∗ such that, for all g ∈ G and x ∈ M, J (g, x) = Ad ∗g J (x) + θ (g), where Ad ∗ : G × G∗ → G∗ is the coadjoint action, deﬁned by I ∗ J I J Ad g (ξ ), X = ξ, Ad g −1 (X) , ξ ∈ G∗ , X ∈ G. The map θ satisﬁes, for all g and h ∈ G,

θ (gh) = Ad ∗g θ(h) + θ (g),

and is called a symplectic cocycle of G [45]. As a consequence, the map aθ : G × G ∗ → G ∗ ,

aθ (g, ξ ) = Ad ∗g (ξ ) + θ (g),

is an action on the left of G on G∗ , called the afﬁne action associated to θ. The momentum map J : M → G∗ is equivariant when G acts on M by the action and on G∗ by the afﬁne action aθ . The modiﬁed KKS-Poisson structures Moreover, the momentum map J is a Poisson map when M is equipped with the Poisson structure associated to its symplectic structure, and G∗ with the modiﬁed KKS-Poisson structure [45]: I

J {f, g}θ− (ξ ) = − ξ, df (ξ ), dg(ξ ) + ' df (ξ ), dg(ξ ) , where ' : G × G → R is the map deﬁned by

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I J '(X, Y ) = Te θ (X), Y . When considered as a linear map from G to its dual G∗ , the map ' is a G∗ -valued 1cocycle of the Lie algebra G, which corresponds to the 1-cocycle θ of the Lie group G. Considered as a real-valued skew-symmetric bilinear map on G×G, ' is a real-valued 2-cocycle of the Lie algebra G. Symplectic orthogonality For M, each one of the vector subspaces of the symplectic vector space each x ∈ Tx M, ω(x) , • •

the tangent space at x to the orbit (G, x), the kernel of Tx J ,

is the symplectic orthogonal of the other one. 2.3 Remarks and questions Several features of dual pairs and symplectic realizations exist in a Hamiltonian action of a Lie group G on a symplectic manifold (M, ω): the momentum map J : M → G∗ is a symplectic realization of G∗ equipped with the modiﬁed KKS-Poisson structure; the symplectic orthogonality, for all x ∈ M, of ker Tx J and Tx (G, x), looks very much like the property of a dual pair; more precisely, if we assume that the set M/G of orbits of has a smooth manifold structure for which the canonical projection π : M → M/G is a submersion, there is on M/G a unique Poisson structure for which π is a Poisson map, and (J, π) is a dual pair. However, the momentum map J may not be a submersion (it is a submersion if and only if the action is locally free), and the set M/G may not have a smooth manifold structure for which π is a submersion. In order to make the Hamiltonian action of a Lie group more like the simple example of Section 1, we are led to the following questions: 1. Does there exist an “action’’ (in a generalized sense) on the manifold M whose orbits are the level sets of the momentum map J ? 2. In the theory of momentum maps, can we replace the dual of a Lie algebra by a more general Poisson manifold? 3. Does any Poisson manifold have a symplectic realization? The notion of symplectic groupoid, introduced in the next section, will allow us to answer some of these questions.

3 Lie groupoids and symplectic groupoids We recall below the deﬁnition of a groupoid. The reader will ﬁnd examples and more information about groupoids in [4, 6, 33, 43, 52].

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Deﬁnition 3.1. A groupoid is a set equipped with the structure deﬁned by the following data: • •

a subset 0 of , called the set of units of the groupoid; two maps α : → 0 and β : → 0 , called, respectively, the target map and the source map; they satisfy α|0 = β|0 = id 0 ;

•

a composition law m : 2 → , called the product, deﬁned on the subset 2 of × , K L 2 = (x, y) ∈ × ; β(x) = α(y) , which is associative, in the sense that whenever one side of the equality m x, m(y, z) = m m(x, y), z

•

is deﬁned, the other side is deﬁned too, and the equality holds; moreover, the composition law m is such that for each x ∈ , m α(x), x = m x, β(x) = x; a map ι : → , called the inverse, such that, for every x ∈ , x, ι(x) ∈ 2 and ι(x), x ∈ 2 , and m x, ι(x) = α(x), m ι(x), x = β(x).

3.2 Properties and comments The above deﬁnitions have the following consequences. Involutivity of the inverse The inverse map ι is involutive. We have indeed, for any x ∈ , i 2 (x) = m i 2 (x), β i 2 (x) = m i 2 (x), β(x) = m i 2 (x), m i(x), x = m m i 2 (x), i(x) , x = m α(x), x = x. Unicity of the inverse Let x and y ∈ be such that m(x, y) = α(x)

and

m(y, x) = β(x).

Then we have y = m y, β(y) = m y, α(x) = m y, m x, ι(x) = m m(y, x), ι(x) = m β(x), ι(x) = m α ι(x) , ι(x) = ι(x). Therefore, for any x ∈ , the unique y ∈ such that m(y, x) = β(x) and m(x, y) = α(x) is ι(x).

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Notations A groupoid with set of units 0 and target and source maps α and β will be denoted α

by ⇒ 0 . β

When there is no risk of error, for x and y ∈ , we will write x.y, or even simply xy for m(x, y), and x −1 for ι(x). α

Deﬁnitions 3.3. A topological groupoid is a groupoid ⇒ 0 for which is a (maybe β

non-Hausdorff ) topological space, 0 a Hausdorff topological subspace of , α and β surjective continuous maps, m : 2 → a continuous map and ι : → an homeomorphism. α

A Lie groupoid is a groupoid ⇒ 0 for which is a smooth (maybe nonβ

Hausdorff ) manifold, 0 a smooth Hausdorff submanifold of , α and β smooth surjective submersions (which implies that 2 is a smooth submanifold of × ), m : 2 → a smooth map and ι : → a smooth diffeomorphism. 3.4 Properties and examples of Lie groupoids α

1. Let ⇒ 0 be a Lie groupoid. Since α and β are submersions, for any x ∈ , β α −1 α(x) and β −1 β(x) are submanifolds of , both of dimension dim −dim 0 , called the α-ﬁbre and the β-ﬁbre through x, respectively. The inverse map ι, restricted to the α-ﬁbre through x (resp., the β-ﬁbre through x) is a diffeomorphism of that ﬁbre onto the β-ﬁbre through ι(x) (resp., the α-ﬁbre through ι(x)). The dimension of the submanifold 2 of composable pairs in × is 2 dim − dim 0 . 2. A Lie group is a Lie groupoid whose set of units has only one element e. 3. A vector bundle π : E → M is a Lie groupoid, with the base M as a set of units; the source and target maps both coincide with the projection π , the product and the inverse maps are the addition (x, y) → x + y and the opposite map x → −x in the ﬁbres. α

4. Let ⇒ 0 be a Lie groupoid. Its tangent bundle T is a Lie groupoid, with T 0 β

as a set of units, T α : T → T 0 , and T β : T → T 0 as target and source maps. Let us denote by 2 the set of composable pairs in × , by m : 2 → the composition law and by ι : → the inverse. Then the set of composable pairs in T × T is simply T 2 , the composition law on T is T m : T 2 → T and the inverse is T ι : T → T . When the groupoid is a Lie group G, the Lie groupoid T G is a Lie group too. We will see below that the cotangent bundle of a Lie groupoid is a Lie groupoid, and more precisely a symplectic groupoid. Let us now introduce the important notion of symplectic groupoid, ﬁrst considered by A. Weinstein [6], M. Karasev [18] and S. Zakrzewski [58].

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α

Deﬁnition 3.5. A symplectic groupoid is a Lie groupoid ⇒ 0 with a symplectic β

form ω on such that the graph of the composition law m K L (x, y, z) ∈ × × ; (x, y) ∈ 2 and z = m(x, y) is a Lagrangian submanifold of × × with the product symplectic form, the ﬁrst two factors being endowed with the symplectic form ω, and the third factor being with the symplectic form −ω. The next theorem states important properties of symplectic groupoids. α

Theorem 3.6. Let ⇒ 0 be a symplectic groupoid, with the symplectic 2-form ω. β

We have the following properties: 1. For any point c ∈ , each one of the two vector subspaces of the symplectic vector space Tc , ω(c) , and Tc α −1 α(c) Tc β −1 β(c) is the symplectic orthogonal of the other one. 2 The submanifold of units 0 is a Lagrangian submanifold of the symplectic manifold (, ω). 3. The inverse map ι : → is an antisymplectomorphism of (, ω) (it satisﬁes ι∗ ω = −ω). Proof. For each x ∈ , we denote by Px = β −1 β(x) and Qx = α −1 α(x) the β-ﬁbre and the α-ﬁbre through x, respectively. Let dim = 2n. We have seen that dim 2 = 2 dim − dim 0 . Since the graph of the product is a Lagrangian submanifold of × × , its dimension is half that of × × , so 2(2 dim − dim 0 ) = 3 dim ; therefore, dim 0 = (1/2) dim = n. Since α and β are smooth submersions, for each point x ∈ , Px and Qx are smooth n-dimensional submanifolds of . 1. Let (a, b) ∈ 2 and c = m(a, b). The maps La : y → m(a, y) and Rb : x → m(x, b) are diffeomorphisms which map, respectively, Qb onto Qc and Pa onto Pc . Therefore, if u ∈ Ta Pa and v ∈ Tb Qb , w1 and w2 ∈ Tc , the vectors (u, 0, w1 ) and (0, v, w2 ) ∈ T(a,b,c) ( × × ) are tangent to the graph of the product m if and only if w1 = T Rb (u) and w2 = T La (v). By writing that the graph of m is Lagrangian in × × , we obtain ω(u, 0) + ω(0, v) − ω T Rb (u), T La (v) = 0, so, for all u ∈ Ta Pa and v ∈ Tb Qb ,

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ω(T Rb (u), T La (v) = 0. Since T Rb (u) can be any vector in Tc Pc and T La (v) any vector in Tc Qc , we have shown that orth(Tc Pc ) ⊃ Tc Qc . But since these two subspaces are of dimension n, orth(Tc Pc ) = Tc Qc , and orth(Tc Qc ) = Tc Pc . 2. Let p ∈ 0 . We have m(p, p) = p; therefore, for u1 and u2 ∈ Tp 0 , (u1 , u1 , u1 ) and (u2 , u2 , u2 ) are tangent to the graph of m. Since that graph is Lagrangian in × × , we have 0 = ω(u1 , u2 ) + ω(u1 , u2 ) − ω(u1 , u2 ) = ω(u1 , u2 ). This shows that 0 is an isotropic submanifold of . But since dim 0 = n, 0 is Lagrangian. 3. Let x ∈ , u1 and u2 ∈ Tx , v1 = T ι(u1 ), v2 = T ι(u2 ). Let t → s1 (t) and t →s2 (t) be smooth curves in such that s (0) = s (0) = x, ds1 (t)/dt |t=0 = u1 1 2 and ds2 (t)/dt |t=0 = u2 . For i = 1 or 2, we have m si (t), ι ◦ si (t) = α ◦ si (x). Therefore, if we set w1 = Tα(u1 ) and w2 = T α(u2 ), the vectors (u1 , v1 , w1 ) and (u2 , v2 , w2 ) are tangent at x, ι(x), α(x) to the graph of m. Since that graph is Lagrangian in × × , we have ω(u1 , u2 ) + ω T ι(u1 ), T ι(u2 ) − ω(w1 , w2 ) = 0. But since w1 and w2 are tangent at α(x) to the Lagrangian submanifold 0 , ω(w1 , w2 ) = 0. It follows that ι : → is an antisymplectomorphism.

α

Corollary 3.7. Let ⇒ 0 be a symplectic groupoid, with symplectic 2-form ω. There β

exists on 0 a unique Poisson structure $ for which α : → 0 is a Poisson map, and β : → 0 an anti-Poisson map (i.e., β is a Poisson map when 0 is equipped with the Poisson structure −$). The pair α : (, ω) → (0 , $), β : (, ω) → (0 , −$) is a dual pair, and (, ω) is a symplectic realization of both (0 , $) and (0 , −$). Proof. According to the previous theorem, the symplectic orthogonal of ker(T α) is ker(T β). By Property 9 of the Introduction, there exists on 0 a unique Poisson structure $ such that α is a Poisson map from (, ω) to (0 , $). For the same reason there exists on 0 another unique Poisson structure $ such that β is a Poisson map from (, ω) to (0 , $ ). But β = α ◦ ι, and the previous theorem shows that ι is an antisymplectomorphism. Therefore, $ = −$.

3.8 Cotangent bundle of a Lie groupoid α

Let ⇒ 0 be a Lie groupoid. We have seen above that its tangent bundle T has a β

Lie groupoid structure, determined by that of . Similarly (but much less obviously)

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the cotangent bundle T ∗ has a Lie groupoid structure determined by that of . The set of units is the conormal bundle to the submanifold 0 of , denoted by N∗ 0 . We recall that N∗ 0 is the vector subbundle of T∗0 (the restriction to 0 of the cotangent bundle T ∗ ) whose ﬁbre Np∗ 0 at a point p ∈ 0 is K L Np∗ 0 = η ∈ Tp∗ ; η, v = 0 for all v ∈ Tp 0 . To deﬁne the target and source maps of the Lie algebroid T ∗ , we introduce the notion of bisection through a point x ∈ [6, 1, 4]. A bisection through x is a submanifold A of , with x ∈ A, transverse both to the α-ﬁbres and to the β-ﬁbres, such that the maps α and β, when restricted to A, are diffeomorphisms of A onto open subsets α(A) and β(A) of 0 , respectively. For any point x ∈ M, there exist bisections through x. A bisection A allows us to deﬁne two smooth diffeomorphisms between open subsets of , denoted by LA and RA and called the left and right translations by A, respectively. They are deﬁned by LA : α −1 β(A) → α −1 α(A) , LA (y) = m β|−1 A ◦ α(y), y , and RA : β −1 α(A) → β −1 β(A) ,

RA (y) = m y, α|−1 A ◦ β(y) .

The deﬁnitions of the target and source maps for T ∗ rest on the following properties. Let x be a point in and let A be a bisection through x. The two vector subspaces, Tα(x) 0 and ker Tα(x) β, are complementary in Tα(x). Forany v ∈ Tα(x) , v −T β(v) is in ker Tα(x) β. Moreover, RA maps the ﬁbre β −1 α(x) onto the ﬁbre β −1 β(x) , and its restriction to that ﬁbre does not depend on the choice of A; its depends only on x. Therefore, T RA v − T β(v) is in ker Tx β and does not depend on the choice of A. We can deﬁne the map = α by setting, for any ξ ∈ Tx∗ and any v ∈ Tα(x) , I J I J = α (ξ ), v = ξ, T RA v − T β(v) . = by setting, for any ξ ∈ Tx∗ and any w ∈ Tβ(x) , Similarly, we deﬁne β J I J I =(ξ ), w = ξ, T LA w − T α(w) . β = are unambiguously deﬁned, smooth and take their values in the We see that = α and β ∗ submanifold N 0 of T ∗ . They satisfy π ◦ = α = α ◦ π ,

= = β ◦ π , π ◦ β

where π : T ∗ → is the cotangent bundle projection. Let us now deﬁne the composition law m = on T ∗ . Let ξ ∈ Tx∗ and η ∈ Ty∗ =(ξ ) = = be such that β α (η). That implies β(x) = α(y). Let A be a bisection through ∗ and a unique x and B a bisection through y. There exist a unique ξhα ∈ Tα(x) 0 ∗ ηhβ ∈ Tβ(y) 0 such that ∗ = ∗ ξ = (L−1 A ) β (ξ ) + αx ξhα ,

η = (RB−1 )∗ = α (ξ ) + βy∗ ηhβ .

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Then m =(ξ, η) is given by ∗ ∗ ∗ = m =(ξ, η) = αxy ξhα + βxy ηhβ + (RB−1 )∗ (L−1 A ) β (x) . =(ξ ) by = We observe that in the last term of the above expression we can replace β α (η), −1 ∗ −1 ∗ ∗ (R −1 )∗ , since these two expressions are equal, and that (RB ) (LA ) = (L−1 ) A B since RB and LA commute. Finally, the inverse= ι in T ∗ is ι∗ . = α

With its canonical symplectic form, T ∗ ⇒ N∗ 0 is a symplectic groupoid. = β

When the Lie groupoid is a Lie group G, the Lie groupoid T ∗ G is not a Lie group, contrary to what happens for T G. Its set of units can be identiﬁed with G∗ , = = JR . and we recover the typical example of Section 1, with = α = JL and β

4 Lie algebroids Let us now introduce the notion of a Lie algebroid, related to that of a Lie groupoid in the same way as the notion of a Lie algebra is related to that of a Lie group. That notion is due to J. Pradines [41]. Deﬁnition 4.1. A Lie algebroid over a smooth manifold M is a smooth vector bundle π : A → M with base M, equipped with • •

a composition law (s1 , s2 ) → {s1 , s2 } on the space ∞ (π ) of smooth sections of π, called the bracket, for which that space is a Lie algebra, a vector bundle map ρ : A → T M, over the identity map of M, called the anchor map, such that, for all s1 and s2 ∈ ∞ (π) and all f ∈ C ∞ (M, R), {s1 , f s2 } = f {s1 , s2 } + (ρ ◦ s1 ).f s2 .

4.2 Examples 1. A ﬁnite-dimensional Lie algebra is a Lie algebroid (with a base reduced to a point). 2. A tangent bundle τM : T M → M to a smooth manifold M is a Lie algebroid, with the usual bracket of vector ﬁelds on M as composition law, and the identity map as anchor map. More generally, any integrable vector subbundle F of a tangent bundle τM : T M → M is a Lie algebroid, still with the bracket of vector ﬁelds on M with values in F as composition law and the canonical injection of F into T M as anchor map. 3. Let X : M → T M be a smooth vector ﬁeld on a smooth manifold M. By setting, for any pair (f, g) of smooth functions on M, [f, g] = f (X.g) − g(X.f ),

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we deﬁne on C ∞ (M, R) a composition law (which is a special case of a Jacobi bracket) which turns that space into a Lie algebra. We may identify the space C ∞ (M, R) of smooth functions on M with the space of smooth sections of the trivial vector bundle p1 : M × R → M, where p1 is the projection on the ﬁrst factor; in that identiﬁcation, a smooth function f ∈ C ∞ (M, R) is identiﬁed with the section sf : M → M × R, sf (x) = x, f (x) . The Jacobi bracket of functions deﬁned above becomes a bracket (sf , sg ) → {sf , sg } = s[f,g] on the space of smooth sections of the trivial bundle p1 : M × R → M. It is easy to see that with the anchor map ρ : M × R → T M,

ρ(x, k) = kX(x),

that bundle becomes a Lie algebroid over M. 4. Let (P , $) be a Poisson manifold. Its cotangent bundle πP : T ∗ P → P has a Lie algebroid structure, with $ : T ∗ P → T P as anchor map. The composition law is the bracket of 1-forms, ﬁrst obtained by Fuchssteiner [14], then independently by Magri and Morosi [36]. It will be denoted by (η, ζ ) → [η, ζ ] (in order to avoid any confusion with the Poisson bracket of functions). It is given by the formula, in which η and ζ are 1-forms and X a vector ﬁeld on P , I J [η, ζ ], X = $ η, dζ, X + $ dη, X, ζ + L(X)$ (η, ζ ). We have denoted by L(X)$ the Lie derivative of the Poisson tensor $ with respect to the vector ﬁeld X. Another equivalent formula for that composition law is [ζ, η] = L($ ζ )η − L($ η)ζ − d $(ζ, η) . The bracket of 1-forms is related to the Poisson bracket of functions by [df, dg] = d{f, g} for all f and g ∈ C ∞ (P , R). 4.3 Properties of Lie algebroids Let π : A → M be a Lie algebroid with anchor map ρ : A → T M. 1. It is easy to see that for any pair (s1 , s2 ) of smooth sections of π, ρ ◦ {s1 , s2 } = [ρ ◦ s1 , ρ ◦ s2 ], which means that the map s → ρ ◦ s is a Lie algebra homomorphism from the Lie algebra of smooth sections of π into the Lie algebra of smooth vector ﬁelds on M. 2. The composition law (s1 , s2 ) → {s1 , s2 } on the space of sections of π extends into a composition law on the space of sections of exterior powers of (A, π, M), which

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will be called the generalized Schouten bracket. Its properties are the same as those of the usual Schouten bracket [44, 40, 22]. When the Lie algebroid is a tangent bundle τM : T M → M, that composition law reduces to the usual Schouten bracket . When the Lie algebroid is the cotangent bundle πP : T ∗ P → P to a Poisson manifold (P , $), the generalized Schouten bracket is the bracket of forms of all degrees on the Poisson manifold P , introduced by J.-L. Koszul [22], which extends the bracket of 1 forms used in Example 4 of 4.2. 3. Let " : A∗ → M be the dual bundle of the Lie algebroid π : A → M. There exists on the space of sections of its exterior powers a graded endomorphism dρ , of E E degree 1 (this means that if η is a section of k A∗ , dρ (η) is a section of k+1 A∗ ). That endomorphism satisﬁes dρ ◦ dρ = 0, and its properties are essentially the same as those of the exterior derivative of differential forms. When the Lie algebroid is a tangent bundle τM : T M → M, dρ is the usual exterior derivative of differential forms. We can develop on the spaces of sections of the exterior powers of a Lie algebroid and of its dual bundle a differential calculus very similar to the usual differential calculus of vector and multivector ﬁelds and differential forms on a manifold [6, 4, 34, 37]. Operators such as the interior product, the exterior derivative and the Lie derivative can still be deﬁned and have properties similar to those of the corresponding operators for vector and multivector ﬁelds and differential forms on a manifold. 4.4 The Lie algebroid of a Lie groupoid α

Let ⇒ 0 be a Lie groupoid. Let A() be the intersection of ker T α and T0 (the β

tangent bundle T restricted to the submanifold 0 ). We see that A() is the total space of a vector bundle π : A() → 0 , with base 0 , the canonical projection π being the map which associates a point u ∈ 0 to every vector in ker Tu α. We will deﬁne a composition law on the set of smooth sections of that bundle, and a vector bundle map ρ : A() → T 0 , for which π : A() → 0 is a Lie algebroid, callled α

the Lie algebroid of the Lie groupoid ⇒ 0 . β

We observe ﬁrst that for any point u ∈ 0 and any point x ∈ β −1 (u), the map −1 y) Lx : y → Lx y = m(x, is deﬁned on the α-ﬁbre α (u), and maps that ﬁbre −1 onto the α-ﬁbre α α(x) . Therefore, Tu Lx maps the vector space Au = ker Tu α onto the vector space ker Tx α, tangent at x to the α-ﬁbre α −1 α(x) . Any vector =(x) = w ∈ Au can therefore be extended into the vector ﬁeld along β −1 (u), x → w Tu Lx (w). More generally, let w : U → A() be a smooth section of the vector bundle π : A() → 0 , deﬁned on an open subset U of 0 . By using the above described construction for every point u ∈ U , we can extend the section w into a smooth vector ﬁeld w =, deﬁned on the open subset β −1 (U ) of , by setting, for all −1 u ∈ U and x ∈ β (u), w =(x) = Tu Lx w(u) .

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We have deﬁned an injective map w → w = from the space of smooth local sections of π : A() → 0 , onto a subspace of the space of smooth vector ﬁelds deﬁned on open subsets of . The image of that map is the space of smooth vector ﬁelds w =, deﬁned = of of the form U = = β −1 (U ), where U is an open subset of 0 , on open subsets U which satisfy the two properties: (i) T α ◦ w = = 0, = such that β(x) = α(y), Ty Lx w (ii) for every x and y ∈ U =(y) = w =(xy). These vector ﬁelds are called left-invariant vector ﬁelds on . One can easily see that the space of left-invariant vector ﬁelds on is closed under the bracket operation. We can therefore deﬁne a composition law (w1 , w2 ) → {w1 , w2 } on the space of smooth sections of the bundle π : A() → 0 by deﬁning {w1 , w2 } as the unique section such that {w w1 , w =2 ]. 1 , w2 } = [= Finally, we deﬁne the anchor map ρ as the map T β restricted to A(). One can easily check that with that composition law and that anchor map, the vector bundle π : A() → 0 is a Lie algebroid, called the Lie algebroid of the Lie groupoid α

⇒ 0 . β

We could exchange the roles of α and β and use right-invariant vector ﬁelds instead of left-invariant vector ﬁelds. The Lie algebroid obtained remains the same, up to an isomorphism. α

When the Lie groupoid ⇒ is a Lie group, its Lie algebroid is simply its Lie β

algebra. 4.5 The Lie algebroid of a symplectic groupoid α

Let ⇒ 0 be a symplectic groupoid, with symplectic form ω. As we have seen β

above, its Lie algebroid π : A → 0 is the vector bundle whose ﬁbre, over each 0 point u ∈ 0 , is ker Tu α. We deﬁne a linear map ωu : ker Tu α → Tu∗ 0 by setting, for each w ∈ ker Tu α and v ∈ Tu 0 , I 0 J ωu (w), v = ωu (v, w). Since Tu 0 is Lagrangian and ker Tu α complementary to Tu 0 in the symplectic 0 vector space Tu , ω(u) , the map ωu is an isomorphism from ker Tu α onto Tu∗ 0 . By using that isomorphism for each u ∈ 0 , we obtain a vector bundle isomorphism of the Lie algebroid π : A → 0 onto the cotangent bundle π0 : T ∗ 0 → 0 . As seen in Corollary 3.7, the submanifold of units 0 has a unique Poisson structure $ for which α : → 0 is a Poisson map. Therefore, as seen in Example 4 of 4.2, the cotangent bundle π0 : T ∗ 0 → 0 to the Poisson manifold (0 , $) has a Lie

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algebroid structure, with the bracket of 1-forms as composition law. That structure is the same as the structure obtained as a direct image of the Lie algebroid structure of π : A() → 0 , by the above deﬁned vector bundle isomorphism of π : A → 0 onto the cotangent bundle π0 : T ∗ 0 → 0 . The Lie algebroid of the symplectic α

groupoid ⇒ 0 can therefore be identiﬁed with the Lie algebroid π0 : T ∗ 0 → 0 , β

with its Lie algebroid structure of cotangent bundle to the Poisson manifold (0 , $). 4.6 The dual bundle of a Lie algebroid There exist some very close relationships between Lie algebroids and Poisson manifolds, discussed, for example, in [11]. We have already seen one such relationship: the cotangent bundle of a Poisson manifold has a Lie algebroid structure. We now describe another one. Let π : A → M be a Lie algebroid over a manifold M, with ρ : A → T M as anchor map. Let " : A∗ → M be the dual bundle of π : A → M. We observe that a smooth section s of π can be considered as a smooth function onI A∗ , whose J restriction to each ﬁbre " −1 (x) (x ∈ M) is the linear function ζ → ζ, s(x) . By using that property, we see that the total space A∗ has a unique Poisson structure, whose bracket extends, for smooth functions, the bracket of sections of π . When the Lie algebroid is a ﬁnite-dimensional Lie algebra G, the Poisson structure on its dual space G∗ obtained by that means is the KKS-Poisson structure, already discussed in Section 1.

5 Integration of Lie algebroids and Poisson manifolds According to Lie’s third theorem, for any given ﬁnite-dimensional Lie algebra, there exists a Lie group whose Lie algebra is isomorphic to that Lie algebra. The same property is not true for Lie algebroids and Lie groupoids. The problem of ﬁnding necessary and sufﬁcient conditions under which a given Lie algebroid is isomorphic to the Lie algebroid of a Lie groupoid remained open for more than 30 years. Partial results were obtained by J. Pradines [42], K. Mackenzie [33], P. Dazord [9], P. Dazord ans G. Hector [10]. A complete solution of that problem was obtained by M. Crainic and R. L. Fernandes [7]. Let us describe their very important work. 5.1 The Weinstein groupoid of a Lie algebroid Admissible paths in a Lie algebroid Let π : A → M be a Lie algebroid and ρ : A → T M its anchor map. Let a : I = [0, 1] → A be a smooth path in A. We will say that a is admissible if, for all t ∈ I , ρ ◦ a(t) =

d (π ◦ a)(t). dt

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We observe that when the Lie algebroid is the tangent bundle τM : T M → M, with the identity of T M as anchor map, a smooth path a : I → T M is admissible if and only if it is the canonical lift to T M of the smooth path τM ◦ a : I → M in M. For a general Lie algebroid, a smooth path a : I → A is admissible if and only if the map = a : T I = I × R → A,

= a (t, λ) = λa(t),

is a Lie algebroid homomorphism of the Lie algebroid p1 : I × R → I into the Lie algebroid π : A → M, over the map π ◦ a : I → M between the bases of these two Lie algebroids. Associated paths in a Lie groupoid and in its Lie algebroid Let us assume that π : A → M is the Lie algebroid of a Lie groupoid over M, α

⇒ M. Let γ : I = [0, 1] → be a smooth path in , starting from a unit β

x = γ (0) ∈ M ⊂ , and contained in the α-ﬁbre α −1 (x), i.e., such that, for all t ∈ I , α ◦ γ (t) = x. dγ (t) is a vector tangent to the ﬁbre α −1 (x), at the point γ (t). dt Applying to that vector a left translation by γ (t)−1 , we obtain a vector, dγ (t) aγ (t) = T Lγ (t)−1 ∈ ker Tβ◦γ (t) α. dt For each t ∈ I ,

But we have seen (4.4) that for each y ∈ M, the vector subspace ker Ty α of Ty is the α

ﬁbre Ay over y of the Lie algebroid of the Lie groupoid ⇒ M. The map t → aγ (t) β

is therefore a smooth path in the Lie algebroid π : A → M. That path will be said to α

be associated to the path γ in the Lie groupoid ⇒ M. Since we have β

β ◦ Lγ (t)−1 = β, and since the anchor map of π : A → M is the restriction of T β, we see that the path aγ is admissible. Conversely, we see by integration that every smooth admissible path a in the Lie algebroid A is associated to a unique smooth path γ in the Lie groupoid , starting from the unit γ (0) = π ◦ a(0) ∈ M and contained in the α-ﬁbre α −1 γ (0) . A Lie groupoid with connected and simply connected α-ﬁbres We use the same assumptions as in the previous subsection. For smooth paths in starting from a unit in M and contained in an α-ﬁbre, homotopy with ﬁxed endpoints

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is an equivalence relation. The one-to-one correspondence γ → aγ , which associates, to each smooth path γ in starting from a unit and contained in an α-ﬁbre, a smooth admissible path aγ in the Lie algebroid A, allows us to obtain an equivalence relation for smooth admissible paths in A. That equivalence relation will still be called homotopy. Let γ : I → and γ : I → be two smooth paths in such that, for all t ∈ I , γ (0) = α ◦ γ (t) = x ∈ M,

γ (0) = α ◦ γ (t) = y = β ◦ γ (1) ∈ M.

We can deﬁne a path γ .γ : I → by setting + γ (t) for 0 ≤ t ≤ 1/2, γ .γ (t) = γ (1).γ (2t − 1) for 1/2 ≤ t ≤ 1. That path starts from the unit x and is contained in the α-ﬁbre α −1 (x). By replacing γ and γ by homotopic paths whose derivatives vanish on a neighbourhood of the endpoints 1 and 0, respectively, we may arrange things so that γ .γ is smooth (otherwise it is only piecewise smooth). By this means we obtain a composition law on the space of equivalence classes (for homotopy with ﬁxed endpoints) of smooth paths in , starting from a unit and contained in an α-ﬁbre. Crainic and Fernandes have shown [7] that this space, endowed with that composition law, is a Lie groupoid with connected and simply connected α ﬁbres, and that the Lie algebroid of that Lie α

groupoid is π : A → M, the same as that of the Lie groupoid ⇒ M. Moreover, they β

show that that Lie groupoid is, up to an isomorphism, the unique Lie groupoid with connected and simply connected α-ﬁbres whose Lie algebroid is π : A → M. That unicity property is the analogue of Lie’s ﬁrt theorem for Lie algebras and Lie groups. It was also proved by Moerdijk and Mrc˘ un [39] and by Mackenzie for transitive Lie algebroids [33]. Weinstein groupoid of a general Lie algebroid We no longer assume now that the Lie algebroid π : A → M is the Lie algebroid of a Lie groupoid. We cannot use any more paths in the groupoid , since may not exist, but we still can use admissible paths in the Lie algebroid A. As seen above, α

when π : A → M is the Lie algebroid of the Lie groupoid ⇒ M, homotopy with β

ﬁxed endpoints for paths in starting from a unit and contained in an α-ﬁbre induces an equivalence relation, still called homotopy, on the space of admissible paths in A. That equivalence relation can be expressed in terms involving the Lie algebroid π : A → M only, and still makes sense when that Lie algebroid is no longer the Lie algebroid of a Lie groupoid. That key observation allows us to consider the quotient of the space of smooth admissible paths in A by the homotopy equivalence relation. That quotient will be denoted by G(A). Crainic and Fernandes have shown that G(A) is endowed with a natural topological groupoid structure, and have called it the Weinstein

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groupoid of the Lie algebroid π : A → M (the idea of that construction being due to A. Weinstein). When that Lie algebroid is the Lie algebroid of a Lie groupoid, its Weinstein groupoid G(A) is a Lie groupoid, and is in fact the unique Lie groupoid with connected and simply connected α-ﬁbres whose Lie algebroid is π : A → M. Conversely, Crainic and Fernandes have shown that when the topological groupoid G(A) is in fact a Lie groupoid, its Lie algebroid is isomorphic to π : A → M. The problem of ﬁnding necessary and sufﬁcient conditions under which π : A → M is the Lie algebroid of a Lie groupoid amounts therefore to ﬁnding necessary and sufﬁcient conditions under which the Weinstein groupoid G(A) is smooth, i.e., is a Lie groupoid. Crainic and Fernandes have solved that problem by introducing monodromy groups. 5.2 Monodromy groups Let π : A → M be a Lie algebroid, ρ : A → T M its anchor map and G(A) its Weinstein groupoid. For each x ∈ M, let Ax = π −1 (x), ρx = ρ|Ax and gx = ker ρx . Let u and v be two elements in gx , σu and σv two smooth sections of π : A → M such that σu (x) = u, σv (x) = v. The Leibniz rule shows that {σu , σv }(x) depends only on u and v, not on the sections σu and σv . This allows us to deﬁne the bracket [u, v] by setting [u, v] = {σu , σv }(x). With that bracket, gx is a Lie algebra. Let Nx (A) be the set of elements u in the center Z(gx ) of the Lie algebra gx such that the constant path t → u, t ∈ I = [0, 1], is equivalent (for the homotopy equivalence relation on the space of admissible smooth paths in A deﬁned above) to the trivial path t → 0x , where 0x is the origin of the ﬁbre Ax . Crainic and Fernandes have shown that Nx is a group, with the usual addition as composition law, which is at most countable, called the monodromy group at x. Let us take a Riemannian metric on the vector bundle π : A → M, and let d be the associated distance in the ﬁbres of A. We set + d 0x , Nx (A) − {0x } if Nx (A) − {0x } = ∅, r(x) = +∞ if Nx (A) − {0x } = ∅. We see that r(x) > 0 if and only if the monodromy group Nx (A) is discrete. We may now state the main theorem. Theorem 5.3 (Crainic and Fernandes [7]). The Weinstein groupoid G(A) of the Lie algebroid π : A → M is a Lie groupoid if and only if the following two conditions are satisﬁed: (i) for all x ∈ M, the monodromy group Nx (A) is discrete, or in other words r(x) > 0; (ii) for all x ∈ M, we have lim inf r(y) > 0. y→x

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When these conditions are satisﬁed, the Lie algebroid π : A → M is said to be integrable. The reader will ﬁnd in [7] several examples of Lie algebroids for which the monodromy groups are fully determined. There are Lie algebroids which are locally integrable (it means that any point in the base space of the Lie algebroid has an open neighbourhood such that when restricted to that neighbourhood, the Lie algebroid is integrable), but not globally integrable; in other words, integrability of Lie algebroids is not a local property. There are also Lie algebroids which are not locally integrable. 5.4 Applications and comments Integration of a Lie algebroid by a local Lie groupoid One may think of a local Lie groupoid as the structure deﬁned on a smooth manifold by the following data: a smooth submanifold 0 ⊂ , smooth maps α, β, a composition law m and an inverse map ι, with the same properties as those of the corresponding maps of a Lie groupoid, but with the restriction that these maps are deﬁned, and have these properties, only for elements in close enough to the space 0 of units. The Lie algebroid of a local Lie groupoid can be deﬁned in the same way as that of a true Lie groupoid. J. Pradines [42] has shown that every Lie algebroid is the Lie algebroid of a local Lie groupoid. K. Mackenzie and P. Xu [35] have shown that when that Lie algebroid is the cotangent bundle to a Poisson manifold, the corresponding local Lie groupoid is a local symplectic groupoid. Crainic and Fernandes [7, 8] have shown that these results are now easy consequences of their general Theorem 5.3. Some integrable Lie algebroids By using Theorem 5.3, Crainic and Fernandes have proved the integrability of several types of Lie algebroids. Let us indicate some of their results A Lie algebroid with zero anchor map (i.e., a sheaf of Lie algebras) is always integrable, and the corresponding Lie groupoid is a sheaf of Lie groups (that result was already found by Douady and Lazard [12]). A regular Lie algebroid (i.e., a Lie algebroid whose anchor map is of constant rank) is locally integrable (but may be not globally integrable). A Lie algebroid whose anchor map is injective, or more generally injective on a dense open set, is integrable. 5.5 Symplectic realizations of a Poisson manifold The existence of local symplectic realizations of regular Poisson manifolds (i.e., with a Poisson structure of constant rank) was proved by Sophus Lie as early as 1890. The problem of ﬁnding a local symplectic realization of a Poisson manifold near a nonregular point is much more difﬁcult. It was solved by Alan Weinstein in [48],

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where it is shown that if (P , $) is a Poisson manifold of dimension n and a ∈ P a point where the rank of $ is 2k, there exists a symplectic realization ϕ : M → U of an open neighbourhood U of a by a symplectic manifold M of dimension 2(n − k). Moreover, that realization is universal, in the sense that if ϕ : M → U is another symplectic realization of U , then dim M ≥ dim M and there exist (maybe after restriction of U and M ) a surjective Poisson submersion χ : M → M, whose ﬁbres are symplectic submanifolds of M , such that ϕ = ϕ ◦ χ . By patching together symplectic realizations of open subsets of a Poisson manifold, A. Weinstein proved [49, 6] that any Poisson manifold has a global, surjective submersive symplectic realization. The same result was obtained independently by M. Karasev [18]. Another proof of that result was given by C. Albert and P. Dazord [1]. Weinstein and Karasev observed that such a realization automatically admits a local symplectic groupoid structure (this means a structure with a composition law, an inverse, source and target maps which look like those of a neighbourhood of the submanifold of units in a symplectic groupoid), the Poisson manifold being identiﬁed with the submanifold of units and the realization map either with the source or with the target map. The Lie algebroid of the (local) Lie groupoid obtained in that way can be identiﬁed with the cotangent bundle to the initially given Poisson manifold. The problem of ﬁnding a symplectic realization of a Poisson manifold (P , $) appears therefore as closely linked to the problem of ﬁnding a Lie groupoid whose Lie algebroid is the cotangent bundle T ∗ P , with the Lie algebroid structure described in Example 4 of 4.2. Integrable Poisson manifolds APoisson manifold (P , $) is said to be integrable if there exists a symplectic groupoid α

⇒ P , with P as submanifolds of units, the Poisson structure $ being that for which β

α is a Poisson map and β an anti-Poisson map. By using Theorem 5.3, Crainic and Fernandes have reinterpreted all known results about integrability of Poisson manifolds [8]. They have shown that a Poisson manifold (P , $) is integrable if and only if its cotangent bundle T ∗ P , with the Lie algebroid structure described in Example 4 of 4.2, is an integrable Lie algebroid. The symplectic groupoid which integrates (P , $) can be obtained as a symplectic quotient, as shown by Cattaneo and Felder [5]. They have shown that a Poisson manifold admits a complete realization if and only if it is integrable. 5.6 Examples Let us indicate the symplectic groupoids which can be associated to some examples of Poisson manifolds. 1. Let G∗ be the dual of a ﬁnite-dimensional Lie algebra G, with its KKS-Poisson structure. Let G be a Lie group with Lie algebra G. We can take as a symplectic groupoid associated to G∗ , the cotangent bundle T ∗ G with its natural Lie algebroid

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structure (described in 3.8). By choosing G connected and simply connected, we obtain a Lie groupoid structure on T ∗ G with connected and simply connected α- and β-ﬁbres. 2. Let (M, ω) be a connected symplectic manifold. We can take as a symplectic groupoid associated to M (considered as a Poisson manifold) the product manifold M ×M, with the symplectic form p1∗ ω −p2∗ ω. We deﬁne on K M ×M the pair L groupoid structure, for which the set of units is the diagonal = (x, x); x ∈ M , identiﬁed with M. The target and source maps are α(x, y) = (x, x),

β(x, y) = (y, y),

x and y ∈ M.

But there is a better choice for the symplectic groupoid associated to M: the fundamental groupoid of M, denoted by $(M). It is the set of homotopy classes (with ﬁxed ends) of smooth paths ϕ : [0, 1] → M. The set of units is now the set of homotopy classes of constant paths, which can be identiﬁed with M. The target and source maps associate, to the homotopy class = ϕ of a smooth path ϕ : [0, 1] → M, the homotopy classes of the constant paths equal to ϕ(0) and to ϕ(1), respectively. We have natural projections p1 : $(M) → M and p2 : $(M) → M, deﬁned by p1 (= ϕ ) = ϕ(0) ϕ ) = ϕ(1). We take as a symplectic form on $(M) p1∗ ω − p2∗ ω. The fundaand p2 (= mental groupoid $(M) has, over the pair groupoid M × M, the advantage of having connected and simply connected α- and β-ﬁbres. In the next section we describe a third example: the symplectic groupoid associated to a Poisson–Lie group.

6 Double symplectic groupoid of a Poisson–Lie group The notion of a Poisson–Lie group is due to Drinfel’d [13]. Let us recall its deﬁnition and some properties. The reader will ﬁnd more details in [32, 50]. Deﬁnition 6.1. A Poisson–Lie group is a Lie group G, equipped with a Poisson structure $ such that the product m : G × G → G,

(x, y) → m(x, y) = xy

is a Poisson map (when G×G is equipped with the natural product Poisson structure). 6.2 Properties and examples 1. Any Lie group can be considered as a Poisson–Lie group, with the zero Poisson structure. 2. The Poisson tensor $ of a Poisson–Lie group G vanishes at the unit element e of G. The linearized Poisson structure at that point is a skew-symmetric composition law on Te∗ G which satisﬁes the Jacobi identity. Therefore, the dual G∗ of the Lie algebra G of G, which can be identiﬁed with Te∗ G, has a Lie algebra structure, determined by

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the Poisson structure on G. Therefore, we have a pair of vector spaces (G, G∗ ), each of these spaces being the dual of the other, both equipped with a Lie algebra structure. The pair (G, G∗ ) is called a Lie bialgebra. 3. The connected and simply connected Lie group G∗ with Lie algebra G∗ is called the dual Lie group of the Poisson–Lie group G. It also has a structure of Poisson–Lie group $∗ . When the Lie group G itself is connected and simply connected, the roles of G and G∗ are the same, they can be exchanged (in other words, the dual Lie group of the Poisson–Lie group (G∗ , $∗ ) is the Poisson–Lie group (G, $)). 4. On the vector space D = G ⊕ G∗ , there is a natural Lie algebra structure, of which G and G∗ are Lie subalgebras, whose bracket is [X1 + α1 , X2 + α2 ] = [X1 , X2 ] − ad ∗α2 X1 + ad ∗α1 X2 + [α1 , α2 ] + ad ∗X1 α2 − ad ∗X2 α1 , where X1 and X2 ∈ G, α1 and α2 ∈ G∗ . With that structure, D is called the double Lie algebra of G. 5. The connected and simply connected Lie group D with Lie algebra D is called the double Lie group of the Poisson–Lie group G. When G is assumed to be connected and simply connected, there exist natural injective Lie group homomorphisms G → D and G∗ → D, which will be denoted by g → g and u → u, respectively (with g ∈ G, u ∈ G∗ , g and u ∈ D). 6.3 Lu and Weinstein’s construction Let (G, $) be a connected and simply connected Poisson–Lie group (the assumption of simple connexity is made for simplicity, and can easily be removed). J.-H. Lu and A. Weinstein [31] have obtained a very nice description of a symplectic groupoid whose set of units is (G, $). We describe now that construction, with the above deﬁned notations. Let K L = (g, u, v, h) ∈ G × G∗ × G∗ × G; gu = vh . Two different structures of Lie groupoid exist on , the ﬁrst one with G as a set of units and the second one with G∗ as a set of units. We will denote by α1 , β1 , m1 and ι1 the target map, the source map, the composition law and the inverse map for the ﬁrst structure, and by α2 , β2 , m2 and ι2 the corresponding maps for the second structure. They are given by the following formulae: α1 : (g, u, v, h) → (g, e, e, g), β1 : (g, u, v, h) → (h, e, e, h), m1 : (g1 , u1 , v1 , h1 = g2 ), (g2 = h1 , u2 , v2 , h2 ) → (g1 , u1 u2 , v1 v2 , h2 ), ι1 : (g, u, v, h) → (h, u−1 , v −1 , g), α2 : (g, u, v, h) → (e, v, v, e), β2 : (g, u, v, h) → (e, u, u, e), m2 : (g1 , u1 = v2 , v1 , h1 ), (g2 , u2 , v2 = u1 , h2 ) → (g1 g2 , u2 , v1 , h1 h2 ),

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ι2 : (g, u, v, h) → (g −1 , v, u, h−1 ). Moreover, has a symplectic structure ω compatible with each of its two Lie groupoid structures; in other words, with that symplectic structure, it is a symplectic groupoid in two different ways, with either G or G∗ as a set of units. Therefore, (, ω) is a symplectic realization of each of the two Poisson–Lie groups (G, $) and (G∗ , $∗ ).

7 Lie groupoid actions on a smooth manifold We have seen that not every pair of elements in a Lie groupoid can be composed: a pair (x, y) can be composed if and only if the image β(x), by the source map β, of the left element x, is equal to the image α(y), by the target map α, of the right element y. Similarly, to extend to a Lie groupoid action the well-known notion of a Lie group action on a smooth manifold, we must observe that not every pair made of an element in the groupoid and an element in the smooth manifold will be composable. For that reason, we need an extra ingredient, a map from the smooth manifold into the set of units of the goupoid, as shown in the following deﬁnition. α

Deﬁnition 7.1. Let ⇒ 0 be a Lie groupoid, M a smooth manifold and µ : M → 0 β

a smooth map. Let K L ×µ M = (x, m) ∈ × M; β(x) = µ(m) . We assume that ×µ M is a smooth submanifold of ×M. (This is true, for example, α

when µ is a submersion.) An action on the left of the Lie groupoid ⇒ 0 on the β

manifold M with moment map µ is a smooth map : ×µ M → M which satisﬁes the following properties: (i) µ (x, m) = α(x) for all (x, m) ∈ ×µ M, (ii) x, (y, m) = (xy, m) for all x and y ∈ , m ∈ M such that (x, y) ∈ 2 and (y, m) ∈ ×µ M, (iii) µ(m), m = m for all m ∈ M. A similar deﬁnition holds for an action on the right of a Lie groupoid on a smooth manifold. 7.2 Properties and examples α

1. A Lie groupoid ⇒ 0 acts on itself on the left with moment map α, and on on β

the right with moment map β. α

2. A Lie groupoid ⇒ 0 acts on both sides on the submanifold of units 0 , with β

moment map id 0 . These left and right actions are, respectively,

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x, u = β(x) → α(xu) = α(x)

and

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u = α(x), x → β(ux) = β(x).

3. Property (ii) of the above deﬁnition implies that the moment map µ of a left action α

of a Lie groupoid ⇒ 0 on a smooth manifold M satisﬁes, for all x ∈ and β

m ∈ M such that β(x) = µ(m), µ (x, m) = α xµ(m) = α(x). In other words, the moment map µ is equivariant with respect to the left actions of α

the Lie groupoid ⇒ 0 on M and the left action of that groupoid on its submanifold β

of units 0 . A similar property holds for right actions. The following deﬁnition generalizes the notion of a symplectic (or Poisson) action of a Lie group on a symplectic (or Poisson) manifold. α

Deﬁnition 7.3. Let ⇒ 0 be a symplectic groupoid with symplectic form ω, (M, $) β

a Poisson manifold and µ : M → 0 a smooth map. Let : ×µ M → M be a α

smooth action of the Lie groupoid ⇒ 0 on the manifold M, with moment map µ. β

That action is called a Poisson action if its graph K L (x, m, p) ∈ × M × M; (x, m) ∈ ×µ M, p = (x, m) is a coisotropic submanifold of the product Poisson manifold × M × M, where is equipped with the Poisson structure associated to its symplectic structure, M with the Poisson structure $ and where M means the manifold M equipped with the Poisson structure −$. K. Mikami and A. Weinstein [38] have developed a theory of reduction for Poisson actions of symplectic groupoids very similar to the classical reduction theory for symplectic actions. The following proposition indicates an important property of the moment map for Poisson actions. Proposition 7.4. Let : ×µ M → M be a Poisson action of the symplectic α

groupoid ⇒ 0 on the Poisson manifold (M, $). The manifold of units 0 will be β

equipped with the unique Poisson structure for which the target map α : → 0 is a Poisson map. Then the moment map µ : M → 0 of that action is a Poisson map. Proof. Let s be a point in 0 , θ1 and θ2 ∈ Ts∗ 0 and n ∈ µ−1 (s). Let x ∈ , m ∈ M be such that β(x) = µ(m) and that (x, m) = n. Let ξ1 and ξ2 ∈ Tx∗ , η1 and η2 ∈ Tm∗ M and ζ1 and ζ2 ∈ Tn∗ M be such that (ξ1 , η1 , ζ1 ) and (ξ2 , η2 , ζ2 ) belong to the annihilator of the tangent space at (x, m, n)

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to the graph of . Since that graph is a coisotropic submanifold of × M × M, we have $ (ξ1 , ξ2 ) + $(η1 , η2 ) − $(ζ1 , ζ2 ) = 0. But by using the fact that µ ◦ = α ◦ p1 , where p1 : ×µ M → is the projection on the ﬁrst factor, we see that (αx∗ θ1 , 0, −µ∗n θ1 ) and (αx∗ θ2 , 0, −µ∗n θ2 ) belong to the annihilator of the tangent space at (x, m, n) to the graph of . Therefore, $ (αx∗ θ1 , αx∗ θ2 ) − $(µ∗n θ1 , µ∗n θ2 ) = 0. But since α : → 0 is a Poisson map, $0 (θ1 , θ2 ) = $ (αx∗ θ1 , αx∗ θ2 ) = $(µ∗n θ1 , µ∗n θ2 ). That equality, which holds for all s ∈ 0 , θ1 and θ2 ∈ Ts∗ 0 and x ∈ µ−1 (s), proves that µ is a Poisson map. A shorter proof could be obtained by using the coisotropic calculus presented by A. Weinstein in [51], which extends to Poisson manifolds the calculus of symplectic relations initiated by W. M. Tulczyjew [46].

Conversely, P. Dazord [9] and P. Xu [55] have proved the following theorem. α

Theorem 7.5. Let ⇒ 0 be a local symplectic groupoid and (M, $) be a Poisson β

manifold. Every Poisson map µ : M → 0 is the momentum map for a unique α

α

β

β

local Poisson action of ⇒ 0 on (M, $). If ⇒ 0 is global (as a groupoid ), αconnected and α-simply connected, that action is global if and only if µ is a complete Poisson map. Let us explain some terms used in the above statement. The symplectic groupoid α

⇒ 0 is said to be α-connected and α-simply connected if the α-ﬁbres are connected β

and simply connected. The moment map µ is said to be a complete Poisson map if for every smooth function f ∈ C ∞ (0 , R) whose Hamiltonian vector ﬁeld on 0 is complete, the Hamiltonian vector ﬁeld on M associated to the function µ∗ f = f ◦ µ is complete. 7.6 Example Let : G × M → M be a Hamiltonian action (on the left) of a Lie group G on a Poisson manifold (M, $), with an Ad ∗ -equivariant momentum map J : M → G∗ . α

Let T ∗ G ⇒ G∗ be the cotangent space of G, equipped with its canonical symplectic β

groupoid structure (3.8). Let K L T ∗ G ×J M = (ξ, m) ∈ T ∗ G × M; β(ξ ) = J (m) .

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= : T ∗ G ×J M → M of that symplectic groupoid on the Poisson We deﬁne an action manifold (M, $), with moment map J , by setting, for all (ξ, m) ∈ T ∗ G ×J M, =(ξ, m) = πG (ξ ), m , where πG : T ∗ M → M is the canonical projection. That action is a Poisson action in the sense of Deﬁnition 7.3. This example shows that Hamiltonian actions of Lie groups, with Ad ∗ -equivariant momentum maps, appear as special cases of Poisson actions of symplectic groupoids. By a modiﬁcation of the symplectic structure on T ∗ G, that result can be extended to Hamiltonian actions whose momentum map is not Ad ∗ -equivariant, but rather equivariant with respect to an afﬁne action of G on G∗ . 7.7 Poisson actions of Poisson–Lie groups Let (G, $G ) be a Poisson–Lie group and (P , $P ) a Poisson manifold. An action on the left : G × P → P of the Lie group G on the manifold P is called a Poisson action if its graph K L (g, m, p) ∈ G × M × M; p = (g, m) is a coisotropic submanifold of the product Poisson manifold G × M × M, where M is the manifold M equipped with the Poisson structure −$. J.-H. Lu [30] has deﬁned the notion of a momentum map for such an action. It is a map J : P → G∗ , with values in the dual Poisson–Lie group G∗ , such that, for each X in the Lie algebra G of G, XP = $P (J ∗ X r ),

where X r is the right-invariant 1-form on G∗ whose value at the unit element is X ∈ G (the vector space G being identiﬁed with the dual of Te G∗ , itself identiﬁed with G∗ ). A momentum map in the sense of Lu is a Poisson map. Along lines similar to those followed in 7.6, a Poisson–Lie group action on a Poisson manifold, with a momentum map in the sense of Lu, can be lifted to a Poisson action of the symplectic groupoid of G∗ .

8 Poisson groupoids Poisson groupoids were introduced by A. Weinstein [51] as a generalization of both symplectic groupoids and Poisson–Lie groups. α

Deﬁnition 8.1. A Poisson groupoid is a Lie groupoid ⇒ 0 , with a Poisson structure β

$ on , such that the graph of the product K L (x, y, z) ∈ × × ; (x, y) ∈ 2 , z = xy is a coisotropic submanifold of the product Poisson manifold × × , where is the manifold with the Poisson structure −$.

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8.2 Properties and examples 1. A symplectic groupoid is a Poisson groupoid whose Poisson structure is nondegenerate (i.e., symplectic). 2. A Poisson–Lie group is a Poisson groupoid whose groupoid structure is that of a Lie group (i.e., whose submanifold of units is reduced to one point). 3. As for a symplectic groupoid, the submanifold of units 0 of a Poisson groupoid α

⇒ 0 has a unique Poisson structure for which the maps α and β are, respectively, β

a Poisson and a anti-Poisson map. 4. The Lie algebroid π : A() → 0 of a Poisson groupoid has an additional structure: its dual bundle " : A()∗ → 0 also has a Lie algebroid structure, compatible in a certain sense (indicated below) with that of π : A() → 0 (K. Mackenzie and P. Xu [34], Y. Kosmann-Schwarzbach [20], Z.-J. Liu and P. Xu [29]). The compatibility condition between the two Lie algebroid structures on the two vector bundles in duality π : A → M and " : A∗ → M can be written as follows: d∗ [X, Y ] = L(X)d∗ Y − L(Y )d∗ X, where X and Y are two sections of π, or, using the generalized Schouten bracket (Property 2 of 4.3) of sections of exterior powers of the Lie algebroid π : A → M, d∗ [X, Y ] = [d∗ X, Y ] + [X, d∗ Y ]. In these formulae d∗ is the generalized exterior derivative, which acts on the space of sections of exterior powers of the bundle π : A → M, considered as the dual bundle of the Lie algebroid " : A∗ → M, deﬁned in Property 3 of 4.3. These conditions are equivalent to the similar conditions obtained by exchange of the roles of A and A∗ . α When the Poisson groupoid ⇒ 0 is a symplectic groupoid, we have seen (4.5) β

that its Lie algebroid is the cotangent bundle π0 : T ∗ 0 → 0 to the Poisson manifold 0 (equipped with the Poisson structure for which α is a Poisson map). The dual bundle is the tangent bundle τ0 : T 0 → 0 , with its natural Lie algebroid structure deﬁned in Property 2 of 4.2. 5. Conversely, K. Mackenzie and P. Xu [35] have shown that if the Lie algebroid of a Lie groupoid is a Lie bialgebroid (that means, if there exists on the dual vector bundle of that Lie algebroid a compatible structure of Lie algebroid, in the above deﬁned sense), that Lie groupoid has a Poisson structure for which it is a Poisson groupoid. More information about Poisson groupoids can be found in [57].

9 Other developments Poisson geometry, symplectic and Poisson groupoids and their generalizations, are currently very active ﬁelds of research, and several aspects could not be discussed here.

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Let us quote the works of S. Bates and A. Weinstein [3] on geometric quantization, of J. Huebschmann [15] and P. Xu [56] on Poisson cohomology, of J.-C. Marrero and D. Iglesias [16, 17] on Lie groupoids associated to Jacobi structures [28], of M. Bangoura and Y. Kosmann-Schwarzbach [2, 20] on the dynamical Yang–Baxter equation and Gerstenhaber algebras, of P. Libermann [24, 25] on contact groupoids and Lie algebroids. We refer to the excellent review papers of Alan Weinstein [53, 54] for a much more extensive survey and other references. Acknowledgments The author thanks the referee, who suggested important improvements upon the ﬁrst version of that paper, and Ann Kostant for her patient help.

References [1] C. Albert and P. Dazord, Théorie des groupoïdes symplectiques: Chapitre II: Groupoïdes symplectiques, Publ. Dépt. Math. Univ. Claude Bernard Lyon I (N.S.), (1990), 27–99. [2] M. Bangoura and Y. Kosmann-Schwarzbach, Équation de Yang-Baxter dynamique classique et algébroïde de Lie, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 541–546. [3] S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, Vol. 8, American Mathematical Society, Providence, RI, 1997. [4] A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, Vol. 10, American Mathematical Society, Providence, RI, 1999. [5] A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, Progress in Mathematics, Vol. 198, Birkhäuser, Boston, 2001, 41–73. [6] A. Coste, P. Dazord, and A. Weinstein, Groupoïdes symplectiques, Publ. Dépt. Math. Univ. Claude Bernard Lyon I (N.S.), 2/A (1987), 1–64. [7] M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. Math., 157 (2003), 575–620. [8] M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom., 66-1, 71–137. November 2001. [9] P. Dazord, Groupoïdes symplectiques et troisième théorème de Lie “non linéaire,’’ in C. Albert, ed., Géombetrie Symplectique et Mécanique: Colloque International, La Grande Motte, France, 23–28 Mai, 1988, Lecture Notes in Mathematics, Vol. 1416, Springer-Verlag, Berlin, New York, Heidelberg, 1990, 39–74. [10] P. Dazord and G. Hector, Intégration symplectique des variétés de Poisson totalement asphériques, in P. Dazord and A. Weinstein eds., Symplectic Geometry, Groupoids, and Integrable Systems, Mathematical Sciences Research Institute Publications, Vol. 20, SpringerVerlag, New York, 1991, 37–72. [11] P. Dazord and D. Sondaz, Variétés de Poisson, Algébroïdes de Lie, Publ. Dépt. Math. Univ. Claude Bernard Lyon I (N.S.), 1/B (1988), 1–68. [12] A. Douady and M. Lazard, Espaces ﬁbrés en algèbres de Lie et en groupes, Invent. Math., 1 (1966), 133–151. [13] V. G. Drinfel’d, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation, Soviet Math. Dokl., 27 (1983), 68–71. [14] B. Fuchssteiner, The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progr. Theoret. Phys., 68 (1982), 1082–1104.

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[15] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math., 408 (1990), 57–113. [16] D. Iglesias Ponte, Lie Groups and Groupoids and Jacobi Structures, Thesis memory, Department of Fundamental Mathematics, Section of Geometry and Topology, Universidad de la Laguna, Spain, 2003. [17] D. Iglesias and J. C. Marrero, Generalized Lie bialgebroids and Jacobi structures, J. Geom. Phys., 40 (2001), 176–199. [18] M. Karasev, Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. USSR Izvest., 28 (1987), 497–527. [19] A. Kirillov, Local Lie algebras, Russian Math. Surveys, 31 (1976), 55–75. [20] Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math., 41 (1995), 153–165. [21] B. Kostant, Quantization and unitary representations, in Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, Vol. 170, Springer-Verlag, Berlin, 1970, 87–207. [22] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque, hors série (1985), 257–271. [23] P. Libermann, Problèmes d’équivalence et géométrie symplectique, Astérisque, 107–108 (1983) (IIIe rencontre de géométrie du Schnepfenried, 10–15 Mai 1982, Vol. I), 43–68. [24] P. Libermann, On symplectic and contact groupoids, in O. Kowalski and D. Krupka, eds., Differential Geometry and Its Applications: Proceedings of the 5th International Conference, Opava, August 24–28, 1992, Silesian University, Opava, Czech Republic, 1993, 29–45. [25] P. Libermann, Lie algebroids and mechanics, Arch. Math., 32 (1996), 1147–162. [26] P. Libermann and Ch.-M. Marle, Symplectic Geometry and Analytical Mechanics, Kluwer, Dordrecht, the Netherlands, 1987. [27] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom., 12 (1977), 253–300. [28] A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl., 57 (1978), 453–488. [29] Z.-J. Liu and P. Xu, Exact Lie algebroids and Poisson groupoids, Geom. Functional Anal., 6-1 (1996), 138–145. [30] J.-H. Lu, Momentum mappings and reduction of Poisson actions, in P. Dazord and A. Weinstein, eds., Symplectic Geometry, Groupoids, and Integrable Systems, Mathematical Sciences Research Institute Publications, Vol. 20, Springer-Verlag, New York, 1991, 209– 226. [31] J.-H. Lu and A. Weinstein, Groupoïdes symplectiques doubles des groupes de LiePoisson, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 951–954. [32] J.-H. Lu and A. Weinstein, Poisson Lie groups, Dressing transformations and Bruhat decomposition, J. Differential Geom., 31 (1990), 501–526. [33] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathmatical Society Lecture Notes, Vol. 124, Cambridge University Press, Cambridge, UK, 1987. [34] K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415–452. [35] K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445–467. [36] F. Magri and C. Morosi, A Geometrical Characterization of Integrable Hamiltonian Systems through the Theory of Poisson-Nijenhuis Manifolds, Quaderno S 19, University of Milan, Milan, 1984.

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[37] C.-M. Marle, Differential calculus on a Lie algebroid and Poisson manifolds, in The J. A. Pereira da Silva Birthday Schrift, Textos de Matemática 32, Departamento de Matematica, Universidade de Coimbra, Coimbra, Portugal, 2002, 83-149. [38] K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoid actions, Publ. RIMS Kyoto Univ., 24 (1988), 121–140. [39] I. Moerdijk and J. Mr˘cun, On integrability of inﬁnitesimal actions, Amer. J. Math., 124 (2002), 567–593. [40] A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor ﬁelds, Indag. Math., 17 (1955), 390–403. [41] J. Pradines, Théorie de Lie pour les groupoïdes différentiables: Calcul différentiel dans la catégorie des groupoïdes inﬁnitésimaux, C. R. Acad. Sci. Paris Sér. A, 264 (1967), 245–248. [42] J. Pradines, Troisième théorème de Lie pour les groupoïdes différentiables, C. R. Acad. Sci. Paris Sér. A, 267 (1968), 21–23. [43] J. Renault, A Groupoid Approach to C ∗ -Algebras, Lecture Notes in Mathematics, Vol. 793, Springer-Verlag, Berlin, New York, Heidelberg, 1980. [44] J. A. Schouten, On the differential operators of ﬁrst order in tensor calculus, in Convengo di Geometria Differenziale, Cremonese, Roma, 1953, 1–7. [45] J.-M. Souriau, Structure des Systèmes Dynamiques, Dunod, Paris, 1970. [46] W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Napoli, 1989. [47] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, Basel, Boston, Berlin, 1994. [48] A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523–557. [49] A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1987), 101–103. [50] A. Weinstein, Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 35 (1988), 163–167. [51] A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705–727. [52] A. Weinstein, Groupoids: Unifying internal and external symmetry: A tour through some examples, Not. Amer. Math. Soc., 43 (1996), 744–752. [53] A. Weinstein, Poisson geometry, Differential Geom. Appl., 9 (1998), 213–238. [54] A. Weinstein, The Geometry of Momentum, 2002, arXiv: math.SG/0208108 v1. [55] P. Xu, Morita equivalence of Poisson manifolds, Comm. Math. Phys., 142 (1991), 493– 509. [56] P. Xu, Poisson cohomology of regular Poisson manifolds, Ann. Inst. Fourier Grenoble, 42-4 (1992), 967–988. [57] P. Xu, On Poisson groupoids, Internat. J. Math., 6-1 (1995), 101–124. [58] S. Zakrzewski, Quantum and classical pseudogroups I, II, Comm. Math. Phys., 134 (1990), 347–370, 371–395.

Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds∗ Yong-Geun Oh Department of Mathematics University of Wisconsin Madison, WI 53706 USA and Korea Institute for Advanced Study 207-43 Cheongnyangni 2-dong Dongdaemun-gu, Seoul 130-722 Korea [email protected] Dedicated to Alan Weinstein in honor of his 60th birthday. Abstract. In this paper, we develop a mini-max theory of the action functional over the semiinﬁnite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian paths on arbitrary, especially on nonexact and nonrational, compact symplectic manifold (M, ω). To each given time dependent Hamiltonian function H and quantum cohomology class 0 = a ∈ QH ∗ (M), we associate an invariant ρ(H ; a) which varies continuously over H in the C 0 -topology. This is obtained as the mini-max value over the semiinﬁnite cycles whose homology class is “dual’’ to the given quantum cohomology class a on the covering 0 (M) of the contractible loop space 0 (M). We call them the Novikov Floer cycles. space We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper. We assume that (M, ω) is strongly semipositive here, to be removed in a sequel to this paper.

1 Introduction and main results The group Ham(M, ω) of (compactly supported) Hamiltonian diffeomorphisms of the symplectic manifold (M, ω) carries a remarkable invariant norm deﬁned by φ = inf H H →φ

H =

1

(1.1) (max Ht − min Ht )dt

0 ∗ This research was partially supported by the NSF grant DMS-9971446, by NSF grant DMS-

9729992 at the Institute for Advanced Study, by a Vilas Associate Award at the University of Wisconsin, and by a grant of the Korean Young Scientist Prize.

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which was introduced by Hofer [Ho]. Here H → φ means that φ is the time-one map 1 of the Hamilton equation x˙ = X (x) of the Hamiltonian H : [0, 1] × M → R, φH H where the Hamiltonian vector ﬁeld is deﬁned by ω(XH , ·) = dH.

(1.2)

This norm can be easily deﬁned on arbitrary symplectic manifolds although proving nondegeneracy is a nontrivial matter (See [Ho, Po1, LM] for its proof of increasing generality. See also [Ch] for a Floer-theoretic proof and [Oh3] for a simple proof of the nondegeneracy in tame symplectic manifolds). On the other hand Viterbo [V] deﬁned another invariant norm on R2n . This was deﬁned by considering the graph of the Hamiltonian diffeomorphism φ : R2n → R2n and compactifying the graph in the diagonal direction in R4n = R2n ×R2n into T ∗ S 2n . He then applied the critical point theory of generating functions of the Lagrangian submanifold, graph φ ⊂ T ∗ S 2n , which he developed on the cotangent bundle T ∗ N of the arbitrary compact manifold N . To each cohomology class a ∈ H ∗ (N ), Viterbo associated certain homologically essential critical values of generating functions of any Lagrangian submanifold L Hamiltonian isotopic to the zero section of T ∗ N and proved that they depend only on the Lagrangian submanifold but not on the generating functions, at least up to normalization. The present author [Oh1, Oh2] and Milinkovic´ [MO1, MO2, Mi] developed a Floer-theoretic approach to the construction of Viterbo’s invariants using the canonically deﬁned action functional on the space of paths, utilizing the observation made by Weinstein [W] that the action functional is a generating function of the given Lagrangian submanifold deﬁned on the path space. This approach is canonical including normalization and provides a direct link between Hofer’s geometry and Viterbo’s invariants in a transparent way. One of the key points in our construction in [Oh2] is the emphasis on the usage of the existing group structure on the space of Hamiltonians deﬁned by t −1 (H, K) → H #K := H + K ◦ (φH ) (1.3) in relation to the pants product and the triangle inequality. However, we failed to fully exploit this structure and fell short of proving the triangle inequality at the time of writing [Oh1, Oh2]. This construction can be carried out for the Hamiltonian diffeomorphisms as long as the action functional is single valued, e.g., on weakly-exact symplectic manifolds. Schwartz [Sc] carried out this construction in the case of symplectically aspherical (M, ω), i.e., for (M, ω) with c1 |π2 (M) = ω|π2 (M) = 0. Among other things he proved the triangle inequality for the invariants constructed using the notion of Hamiltonian ﬁbration and (ﬂat) symplectic connection on it. It turns out that the proof of this triangle inequality [Sc] is closely related to the notion of the K-area of the Hamiltonian ﬁbration [Po2] with connections [GLS, Po2], especially to the one with ﬁxed monodromy studied by Entov [En1]. In this context, the choice of the triple (H, K; H #K) we made in [Oh2] can be interpreted as the one which makes inﬁnity the K-area of the corresponding Hamiltonian ﬁbration over the Riemann surface of genus zero with three punctures equipped with the given monodromy around the

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punctures. Entov [En1] develops a general framework of Hamiltonian connections with ﬁxed boundary monodromy and relates the K-area with various quantities of the given monodromy which are of the Hofer length type. This framework turns out to be particularly useful for our construction of spectral invariants in this paper. On nonexact symplectic manifolds, the action functional is not single valued and the Floer homology theory has been developed as a circle-valued Morse theory or a 0 (M) of the space 0 (M) of contractible (free) Morse theory on a covering space loops on M in the literature related to Arnold’s conjecture which was initiated by Floer himself [Fl]. The Floer theory now involves quantum effects and uses the Novikov ring in an essential way [HoS]. The presence of quantum effects and denseness of the action spectrum in R (as in nonrational symplectic manifolds), had been the most serious obstacle that has plagued the study of the family of Hamiltonian diffeomorphisms, until the author [Oh4] developed a general framework of the mini-max theory over 0 (M) which we call the Novikov natural semiinﬁnite cycles on the covering space Floer cycles. In this paper, we will exploit the “ﬁniteness’’ condition in the deﬁnitions of the Novikov ring and the Novikov Floer cycles in a crucial way for the proofs of various existence results of pseudoholomorphic curves that are needed in the proofs of the axioms of spectral invariants and nondegeneracy of the norm that we construct [Oh8]. Although the Novikov ring is essential in the deﬁnition of the Floer homology and the quantum cohomology in the literature, as far as we know it is the ﬁrst time for the ﬁniteness condition to be explicitly used beyond the purpose of giving the deﬁnition of the quantum cohomology and the Floer homology. A brief description of the setting of the Floer theory [HoS] is in order, partly to ﬁx our convention: Let (γ , w) be a pair of γ ∈ 0 (M) and w a disc bounding γ . We say that (γ , w) is -equivalent to (γ , w ) iff ω([w #w]) = 0 and

c1 ([w #w]) = 0,

(1.4)

where w is the map with opposite orientation on the domain and w #w is the obvious glued sphere. Here stands for the group =

π2 (M) . ker(ω|π2 (M) ) ∩ ker(c1 |π2 (M) )

0 (M) the set of We denote by [γ , w] the -equivalence class of (γ , w) and by 0 (M) → 0 (M) be the canonical projection. We -equivalence classes. Let π : 0 (M) the -covering space of 0 (M). The action functional A0 : 0 (M) → R call is deﬁned by A0 ([γ , w]) = −

w∗ ω.

(1.5)

Two -equivalent pairs (γ , w) and (γ , w ) have the same action and so the action is 0 (M). When a one-periodic Hamiltonian H : (R/Z) × M → R is well deﬁned on (M) → R by given, we consider the functional AH : ∗ (1.6) AH ([γ , w]) = − w ω − H (t, γ (t))dt.

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Our convention is chosen to be consistent with the classical mechanics Lagrangian on the cotangent bundle with the symplectic form ω0 = −dθ, θ = pi dq i i

when (1.2) is adopted as the deﬁnition of Hamiltonian vector ﬁeld. See the remark in the end of this introduction on other conventions in the symplectic geometry. The conventions in this paper coincide with our previous papers [Oh1, Oh2, Oh4] and Entov’s [En1, En2] but different from many other literature on the Floer homology in some ways. (There was a sign error in [Oh1, Oh2] when we compare the Floer complex and the Morse complex for a small Morse function, which was rectiﬁed in [Oh4]. In our convention, the positive gradient ﬂow of f corresponds to the negative gradient ﬂow of Af .) The mini-max theory of this action functional on the -covering space has been implicitly used in the proof of Arnold’s conjecture. Recently, the author has further developed this mini-max theory via the Floer homology and applied it to the study of Hofer’s geometry of Hamiltonian diffeomorphism groups [Oh4]. We also outlined construction of spectral invariants of Hamiltonian diffeomorphisms of the type [V, Oh2, Sc] on arbitrary nonexact symplectic manifolds for the classical cohomological classes. The main purpose of this paper is to further develop the chain level Floer theory introduced in [Oh4] and to carry out construction of spectral invariants for arbitrary quantum cohomology classes. The organization of the paper is now in order. In Section 2, we brieﬂy review various facts related to the action functional and its action spectrum. Some of these may be known to the experts, but precise details for 0 (M) of general (M, ω) ﬁrst appeared the action functional on the covering space in our paper [Oh5] especially concerning the normalization and the loop effect on the action spectrum: We deﬁne the action spectrum of H by 0 (M), dAH ([z, w]) = 0}, Spec(H ) := {AH ([z, w]) ∈ R | [z, w] ∈ 0 (M) → R. In [Oh5], we have shown that i.e., the set of critical values of AH : once we normalize the Hamiltonian H on compact M by Ht dµ = 0 M

with dµ the Liouville measure, Spec(H ) depends only on the equivalence class = [φ, H ] (see Section 2 for the deﬁnition) and so Spec(φ ) ⊂ R is a well-deﬁned φ subset of R for each φ ∈ Ham(M, ω). Here π : Ham(M, ω) → Ham(M, ω) is the universal covering space of Ham(M, ω). This kind of normalization of the action spectrum is a crucial point for systematic study of the spectral invariants of the Viterbo type in general. Schwarz [Sc] previously proved that in the aspherical case

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529

where the action functional is single valued already on 0 (M), this normalization can be made on Ham(M, ω), not just on Ham(M, ω). In Section 3, we review the quantum cohomology and its Morse theory realization of the corresponding complex. We emphasize the role of the Novikov ring in relating the quantum cohomology and the Floer homology and the reversal of upward and downward Novikov rings in this relation. In Section 4, we review the standard operators in the Floer homology theory and explain the ﬁltration naturally present in the Floer complex and how it changes under the Floer chain map. In Section 5, we give the deﬁnition of our spectral invariants for the Hamiltonian functions H , and prove ﬁniteness of the mini-max values ρ(H ; a). In Section 6, we prove all the basic properties of the spectral invariants. We summarize these into the following theorem. We ∞ ([0, 1] × M) the set of normalized continuous functions on [0, 1] × M. denote by Cm ∞ ([0, 1] × M) and Noting that there is a one–one correspondence between the set Cm the set of Hamiltonian paths t λ = φH : t ∈ [0, 1] → φH ∈ Ham(M, ω),

one may equally consider ρ(H ; a) as an invariant attached to the Hamiltonian path φH . Theorem I. Let (M, ω) be arbitrary closed symplectic manifold. For any given quantum cohomology class 0 = a ∈ QH ∗ (M), we have a continuous function denoted by ∞ ρ = ρ(H ; a) : Cm ([0, 1] × M) × QH ∗ (M) → R ∞ ([0, 1] × M) be smooth such that they satisfy the following axioms: Let H, F ∈ Cm ∗ Hamiltonian functions and a = 0 ∈ QH (M). Then ρ satisﬁes the following axioms:

(1) Projective invariance: ρ(H ; λa) = ρ(H ; a) for any 0 = λ ∈ Q. (2) Normalization: For a = A∈ aA q −A , we have ρ(0; a) = v(a), where 0 is the zero function and v(a) := min{ω(−A) | aA = 0} = − max{ω(A) | aA = 0}

(1.7)

is the (upward ) valuation of a. (3) Symplectic invariance: ρ(η∗ H ; a) = ρ(H ; a) for any symplectic diffeomorphism η. (4) Triangle inequality: ρ(H #F ; a · b) ≤ ρ(H ; a) + ρ(F ; b). (5) C 0 -continuity: |ρ(H ; a) − ρ(F ; a)| ≤ H #F = H − F , where · is the ∞ ([0, 1] × M). In particular, the function ρ : H → Hofer’s pseudonorm on Cm a 0 ρ(H ; a) is C -continuous. We will call the set spec(H ) := {ρ(H ; a) | a ∈ QH ∗ (M)}

(1.8)

the essential spectrum of H . Most of the properties stated in this theorem are direct analogues to the ones in [Oh1, Oh2, Sc]. Except for the proof of ﬁniteness of ρ(H ; a), proofs of all of the

530

Y.-G. Oh

properties are reﬁnements of the arguments used in [Oh2, Oh4, Sc]. In addition, the proof of the triangle inequality uses the concept of Hamiltonian ﬁbration with ﬁxed monodromy and the K-area [Po2, En1], which is an enhancement of the arguments used in [Oh2, Sc]. In the classical mini-max theory for the indeﬁnite functionals [Ra, BnR], there was implicitly used the notion of “semiinﬁnite cycles’’ to carry out the mini-max procedure. There are two essential ingredients needed to prove existence of actual critical values out of the mini-max values: one is the ﬁniteness of the mini-max value, or the linking property of the (semiinﬁnite) cycles associated to the class a and the other is to prove that the corresponding mini-max value is indeed a critical value of the action functional. When the global gradient ﬂow of the action functional exists as in the classical critical point theory [BnR], this point is closely related to the well-known Palais–Smale condition and the deformation lemma which are essential ingredients needed to prove the criticality of the mini-max value. Partly because we do not have the global ﬂow, we need to geometrize all these classical mini-max procedures. It turns out that the Floer homology theory in the chain level is the right framework for this purpose. In Section 7, we will restrict to the rational case and prove the following additional property of spectral invariants, the spectrality axiom. We will study the nonrational cases elsewhere for which we expect the same property holds, at least for the nondegenerate Hamiltonian functions, but its proof seems to be much more nontrivial. We now recall the deﬁnition of rational symplectic manifolds: Denote ω := {ω(A) | A ∈ π2 (M)} = ω() ⊂ R and Spec(H ) = ∪z∈Per(H ) Spec(H ; z). Recall that ω is either a discrete or a countable dense subset of R. Deﬁnition 1.1. A symplectic manifold (M, ω) is called rational if ω is discrete. Theorem II (spectrality axiom). Suppose that (M, ω) be rational. Then ρ satisﬁes the following additional properties: (1) For any smooth one-periodic Hamiltonian function H : S 1 × M → R, we have ρ(H ; a) ∈ Spec(H ) for each given quantum cohomology class 0 = a ∈ QH ∗ (M). (2) For two smooth functions H ∼ K, we have ρ(H ; a) = ρ(K; a)

(1.9)

for all a ∈ QH ∗ (M). In particular, ρ can be pushed down to the “universal covering space’’ Ham(M, ω) of Ham(M, ω) by putting ρ(φ ; a) to be the this common value for φ = [H ]. We call ) ⊂ Spec(φ ) deﬁned by the subset spec(φ

Spectral invariants

531

) = {ρ(φ ; a) | a ∈ QH ∗ (M)} spec(φ . Then we have the following reﬁned the (homologically) essential spectrum of φ version of Theorem II for the rational cases. Theorem III. Let (M, ω) be rational and deﬁne the map ρ : Ham(M, ω) × QH ∗ (M) → R ; a) := ρ(H ; a). Let φ , ψ ∈ Ham(M, by ρ(φ ω) and a = 0 ∈ QH ∗ (M). Then ρ satisﬁes the following axioms: ; a) ∈ Spec(φ ). (1) Spectrality: For each a ∈ QH ∗ (M), ρ(φ (2) Projective invariance: ρ( φ ; λa) = ρ(φ ; a) for any 0 = λ ∈ Q. (3) Normalization: For a = A∈ aA q −A , we have ρ(0; a) = v(a), where 0 is the identity in Ham(M, ω) and v(a) := min{ω(−A) | aA = 0} = − max{ω(A) | aA = 0}

(1.10)

is the (upward ) valuation of a. ; a) for any symplectic diffeomorη−1 ; a) = ρ(φ (4) Symplectic invariance: ρ(ηφ phism η. · ψ ; a · b) ≤ ρ(φ ; a) + ρ(ψ ; b). (5) Triangle inequality: ρ(φ ; a) − ρ(ψ ; a)| ≤ φ ◦ ψ −1 , where · is the Hofer’s (6) C 0 -continuity: |ρ(φ → ρ(φ ; a) is pseudonorm on Ham(M, ω). In particular, the function ρa : φ 0 C -continuous. act on Ham(M, (7) Monodromy shift: Let [h, h] ∈ π0 (G) ω)×QH ∗ (M) by the map , a) → (h · φ , (φ h∗ a), where h∗ a is the image of the (adjoint) Seidel’s action [Se] by [h, h] on the quantum cohomology QH ∗ (M). Then we have , a)) = ρ(φ ; a) + Iω ([h, ρ([h, h] · (φ h]).

(1.11)

It would be an interesting question to ask whether these axioms characterize the spectral invariants ρ. It is related to the question whether the graph of the sections → ρ(φ ; a); ρa : φ

Ham(M, ω) → Spec(M, ω)

can be split into other “branch’’ in a way that the other branch can also satisfy all the above axioms or not. Here the action spectrum bundle Spec(M, ω) is deﬁned by A ) ⊂ Ham(M, Spec(M, ω) := Spec(φ ω) × R. ∈H φ am(M,ω)

We will investigate this question elsewhere.

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To get the main stream of ideas in this paper without getting bogged down with technicalities related with transversality question of various moduli spaces, we assume in this paper that (M, ω) is strongly semipositive in the sense of [Se, En1]: A closed symplectic manifold is called strongly semipositive if there is no spherical homology class A ∈ π2 (M) such that ω(A) > 0,

2 − n ≤ c1 (A) < 0.

Under this condition, the transversality problem concerning various moduli spaces of pseudoholomorphic curves is standard. We will not mention this generic transversality question at all in the main body of the paper unless it is absolutely necessary. In Section 8, we will brieﬂy explain how this general framework can be incorporated in our proofs in the context of Kuranishi structure [FOn] all at once. In the appendix, we introduce the notion of continuous quantum cohomology and explain how to extend our deﬁnition of spectral invariants to the continuous quantum cohomology classes. This work originated from a part of our paper entitled “Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group’’ [Oh6] that has been circulated since July, 2002. We isolate and streamline the construction part of spectral invariants from [Oh6] in this paper with some minor corrections and addition of more details. In particular, we considerably simplify the deﬁnition of ρ(H ; a) from [Oh6] here. We leave the application part of [Oh6] to a separate paper [Oh8] in which we construct the homological norm of Hamiltonian diffeomorphism and apply them to the study of geometry of Hamiltonian diffeomorphisms on general compact symplectic manifolds. Another application of the spectral invariants to the study of length minimizing property of Hamiltonian paths is given by the author [Oh7, Oh8]. See also [En2, EnP] for other interesting applications of spectral invariants. In another sequel to this paper, we will provide a description of spectral invariants in terms of the Hamiltonian ﬁbration. Convention (1) The Hamiltonian vector ﬁeld Xf associated to a function f on (M, ω) is deﬁned by df = ω(Xf , ·). (2) The addition F #K and the inverse K on the set of time periodic Hamiltonians C ∞ (M × S 1 ) are deﬁned by F #G(x, t) = F (x, t) + G((φFt )−1 (x), t), t G(x, t) = −G(φG (x), t).

There is another set of conventions which are used in the literature (e.g., in [Po3]): (1) Xf is deﬁned by ω(Xf , ·) = −df (2) The action functional has the form AH ([z, w]) = −

w∗ ω +

H (t, z(t))dt.

(1.12)

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533

Because our Xf is the negative of Xf in this convention, the action functional is the one for the Hamiltonian −H in our convention. While our convention makes the positive Morse gradient ﬂow correspond to the negative Cauchy–Riemann ﬂow, the other convention keeps the same direction. The reason we keep our convention is that we would like to keep the deﬁnition of the action functional the same as the classical Hamilton functional pdq − H dt (1.13) on the phase space and to make the negative gradient ﬂow of the action functional for the zero Hamiltonian become the pseudoholomorphic equation. It appears that the origin of the two different conventions is the choice of the convention as to how one deﬁnes the canonical symplectic form on the cotangent bundle T ∗ N or in the classical phase space: If we set the canonical Liouville form pi dq i θ= i

for the canonical coordinates q 1 , . . . , q n , p1 , . . . , pn of T ∗ N , we take the standard symplectic form to be ω0 = −dθ = dq i ∧ dpi , while the people using the other convention (see, e.g., [Po3]) take ω0 = dθ = dpi ∧ dq i . As a consequence, the action functional (1.12) in the other convention is the negative of the classical Hamilton functional (1.13). It seems that there is not a single convention that makes everybody happy and hence one has to live with some nuisance in this matter one way or the other.

2 The action functional and the action spectrum Let (M, ω) be any compact symplectic manifold and let 0 (M) be the set of con0 (M) be its the covering space mentioned before. We will always tractible loops and consider normalized functions f : M → R by f dµ = 0, (2.1) M

where dµ is the Liouville measure of (M, ω). When a periodic normalized Hamiltonian H : M × (R/Z) → R is given, we (M) → R by consider the action functional AH : AH ([γ , w]) = − w∗ ω − H (γ (t), t)dt. We denote by Per(H ) the set of periodic orbits of XH .

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Deﬁnition 2.1. We deﬁne the action spectrum of H , denoted as Spec(H ) ⊂ R, by 0 (M), z ∈ Per(H )}, Spec(H ) := {AH (z, w) ∈ R | [z, w] ∈ (M) → R. For each given z ∈ Per(H ), we i.e., the set of critical values of AH : denote Spec(H ; z) = {AH (z, w) ∈ R | (z, w) ∈ π −1 (z)}. Note that Spec(H ; z) is a principal homogeneous space modeled by the period group of (M, ω) ω := {ω(A) | A ∈ π2 (M)} = ω() ⊂ R and Spec(H ) = ∪z∈Per(H ) Spec(H ; z). The following was proved in [Oh4]. Lemma 2.2. For any closed symplectic manifold (M, ω) and for any smooth Hamiltonian H , Spec(H ) is a measure zero subset of R for any H . For given φ ∈ Ham(M, ω), we denote F → φ if φF1 = φ, and denote H(φ) = {F | F → φ}. We say that two Hamiltonians F and K are equivalent and denote F ∼ K if they are connected by one parameter family of Hamiltonians {F s }0≤s≤1 such that F s → φ for all s ∈ [0, 1]. We write [F ] for the equivalence class of F . Then the universal (étale) covering space Ham(M, ω) of Ham(M, ω) is realized by the set of such equivalence classes. Note that the group G := (Ham(M, ω), id) of based loops naturally acts on the loop space (M) by (h · γ )(t) = h(t)(γ (t)), where h ∈ (Ham(M, ω)) and γ ∈ (M). An interesting consequence of Arnold’s conjecture is that this action maps 0 (M) to itself (see, e.g., [Se, Lemma 2.2]). Seidel 0 (M). The set of lifts (h, [Se, Lemma 2.4] proves that this action can be lifted to h) forms a covering group G → G ⊂ G × Homeo( 0 (M)) G whose ﬁber is isomorphic to . Seidel relates the lifting (h, h) of h : S 1 → Ham(M, ω) to a section of the Hamiltonian bundle associated to the loop h (see [Se, Section 2]). When a Hamiltonian H generating the loop h is given, the assignment z → h · z provides a natural one–one correspondence

Spectral invariants

h : Per(F ) → Per(H #F ),

535

(2.2)

t )−1 . Let F, K be normalized Hamiltonians with where H #F = H + F ◦ (φH F, K → φ and let H be the Hamiltonian such that K = H #F , and ft , gt , and ht the corresponding Hamiltonian paths as above. In particular, the path h = {ht }0≤t≤1 deﬁnes a loop. We also denote the corresponding action of h on 0 (M) by h. Let h 0 (M)). Then a straightforward calculation shows (see be any lift of h to Homeo( [Oh5]) that h∗ (dAF ) = dAK (2.3)

0 (M). In particular, since 0 (M) is connected, we have as a one-form on h∗ (AF ) − AK = C(F, K, h),

(2.4)

where C = C(F, K, h) is a constant a priori depending on F, K, h. Theorem 2.3 ([Oh5, Theorem II]). Let h be the loop as above and h be a lift. Then the constant C(F, K, h) in (2.4) depends only on the homotopy class [h, h] ∈ π0 (G). In particular, if F ∼ K, we have AF ◦ h = AK and hence Spec F = Spec K ∈ Ham(M, as a subset of R. For any φ ω), we deﬁne ) := Spec F Spec(φ . for a (and so any) normalized Hamiltonian F with [φ, F ] = φ Deﬁnition 2.4 (action spectrum bundle). We deﬁne the action spectrum bundle of (M, ω) by A Spec(M, ω) = Specφ(M, ω) ⊂ Ham(M, ω) × R, ∈H φ am(M,ω)

where = [F ]} ⊂ R Specφ(M, ω) = {AF ([z, w]) | dAF ([z, w]) = 0, φ and denote by π : Spec(M, ω) → Ham(M, ω) the natural projection.

3 Quantum cohomology in the chain level We ﬁrst recall the deﬁnition of the quantum cohomology ring QH ∗ (M). As a module, it is deﬁned as QH ∗ (M) = H ∗ (M, Q) ⊗ ↑ω , ↑

where ω is the (upward) Novikov ring

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Y.-G. Oh

+ ↑ω

=

aA q

−A

| aA ∈ Q, # A | ai = 0,

A∈

−A

3 ω < λ < ∞ ∀λ ∈ R .

Due to the ﬁniteness assumption on the Novikov ring, we have the natural (upward) valuation v : QH ∗ (M) → R deﬁned by ⎛ ⎞ v⎝ aA q −A ⎠ = min{ω(−A) : aA = 0} (3.1) A∈ω

which for any a, b ∈ QH ∗ (M) satisﬁes that v(a + b) ≥ min{v(a), v(b)}. Deﬁnition 3.1. For each homogeneous element a = A∈ aA q −A ∈ QH k (M),

aA ∈ H ∗ (M, Q),

(3.2)

of degree k, we also call v(a) the level of a and the corresponding term in the sum the leading order term of a and denote by σ (a). Note that the leading order term σ (a) of a homogeneous element a is unique among the summands in the sum by the deﬁnition (1.4) of . The product on QH ∗ (M) is deﬁned by the usual quantum cup product, which we denote by “·’’ and which preserves the grading, i.e., satisﬁes QH k (M) × QH (M) → QH k+ (M). Often the homological version of the quantum cohomology is also useful, sometimes called the quantum homology, which is deﬁned by QH∗ (M) = H∗ (M) ⊗ ↓ω , ↓

where ω is the (downward) Novikov ring ⎧ ⎫ + 3 ⎨ ⎬ ↓ω = bj q Bj | bj ∈ Q, # Bj | bj = 0, ω > λ < ∞ ∀λ ∈ R . ⎩ ⎭ Bj Bj ∈

We deﬁne the corresponding (downward) valuation by ' * B v = max{ω(B) : aB = 0} aB q B∈

which for f, g ∈ QH∗ (M) satisﬁes that v(f + g) ≤ max{v(f ), v(g)}.

(3.3)

Spectral invariants

537

↓

We like to point out that the summand in ω is written as bB q B while the one in ↑ ω as aA q −A with the minus sign. This is because we want to clearly show which one we use. Obviously −v in (3.1) and v in (3.3) satisfy the axiom of non-Archimedean norm which induce a topology on QH ∗ (M) and QH∗ (M), respectively. In each case the ﬁniteness assumption in the deﬁnition of the Novikov ring allows us to numerate the nonzero summands in each given Novikov chain (3.2) so that λ1 > λ2 > · · · > λj > · · · → −∞ with λj = ω(Bj ) or ω(Aj ). Since the downward Novikov ring appears mostly in this paper, we will just use ω ↓ or for ω , unless absolutely necessary to emphasize the direction of the Novikov ring. We deﬁne the level and the leading order term of b ∈ QH∗ (M) similarly as in Deﬁnition 3.1 by changing the role of upward and downward Novikov rings. We have a canonical isomorphism 0 : QH ∗ (M) → QH∗ (M); ai q −Ai → P D(ai )q Ai and its inverse : QH∗ (M) → QH ∗ (M);

bj q Bj →

P D(bj )q −Bj .

We denote by a 0 and b# the images under these maps. There exists the canonical nondegenerate pairing ·, · : QH ∗ (M) ⊗ QH∗ (M) → Q deﬁned by

M

ai q −Ai ,

N (ai , bj )δAi Bj , bj q Bj =

(3.4)

where δAi Bj is the delta function and (ai , bj ) is the canonical pairing between H ∗ (M, Q) and H∗ (M, Q). Note that this sum is always ﬁnite by the ﬁniteness condition in the deﬁnitions of QH ∗ (M) and QH∗ (M) and so is well deﬁned. This is equivalent to the Frobenius pairing in the quantum cohomology ring. However, we would like to emphasize that the dual vector space (QH∗ (M))∗ of QH∗ (M) is not isomorphic to QH ∗ (M) even as a Q-vector space. Rather the above pairing induces an injection QH ∗ (M) → (QH∗ (M))∗ whose images lie in the set of continuous linear functionals on QH∗ (M) with respect to the topology induced by the valuation v. (3.3) on QH∗ (M). We refer to [Br] for a good introduction to non-Archimedean analytic geometry. In fact, the description of the standard quantum cohomology in the literature is not really a “cohomology’’ but a “homology’’ in that it never uses linear functionals in its deﬁnition. To keep our exposition consistent with the standard literature in the Gromov–Witten invariants and quantum cohomology, we prefer to call them the quantum cohomology rather

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than quantum homology as some authors did (e.g., [Se]) in the symplectic geometry community. In the appendix, we will introduce a genuinely cohomological version of quantum cohomology which we call continuous quantum cohomology using the continuous linear functionals on the quantum chain complex below with respect to the topology induced by the valuation v. Let (C∗ , ∂) be any chain complex on M whose homology is the singular homology H∗ (M). One may take for C∗ the usual singular chain complex or the Morse chain complex of a Morse function f : M → R, (C∗ (−f ), ∂−f ) for some sufﬁciently small > 0. However, since we need to take a nondegenerate pairing in the chain level, we should use a model which is ﬁnitely generated. We will always prefer to use the Morse homology complex because it is ﬁnitely generated and avoids some technical issue related to singular degeneration problem of the type studied in [FOh1, FOh2]. The negative sign in (C∗ (−f ), ∂−f ) is used to make the correspondence between the Morse homology and the Floer homology consistent with our conventions of the Hamiltonian vector ﬁeld (1.2) and the action functional (1.6). In our conventions, solutions of the negative gradient of −f correspond to those for the negative gradient ﬂow of the action functional Af . We denote by (C ∗ (−f ), δ−f ) the corresponding cochain complex, i.e., C k := Hom(Ck , Q),

∗ δ−f = ∂−f .

Now we extend the complex (C∗ (−f ), ∂−f ) to the quantum chain complex, denoted by (CQ∗ (−f ), ∂Q ) CQ∗ (−f ) := C∗ (−f ) ⊗ ω ,

∂Q := ∂−f ⊗ ω .

(3.5)

This coincides with the Floer complex (CF∗ (f ), ∂) as a chain complex if is sufﬁciently small. Similarly, we deﬁne the quantum cochain complex (CQ∗ (−f ), δ Q ) by changing the downward Novikov ring to the upward one. In other words, we deﬁne CQ∗ (−f ) := CM2n−∗ (−f ) ⊗ ↑ ,

δ Q := ∂f ⊗ ↑ω .

Again we would like to emphasize that CQ∗ (−f ) is not isomorphic to the dual space of CQ∗ (−f ) as a Q-vector space. We refer to the appendix for some further discussion on this issue. It is well known that the corresponding homology of this complex is independent of the choice f and isomorphic to the above quantum cohomology (respectively, the quantum homology) as a ring (see [PSS, LT2, Lu] for its proof). This isomorphism, however, plays no signiﬁcant role in the current paper, except as a bookkeeping device for the family of invariants ρ(H ; a) that we associate to each quantum cohomology class a ∈ QH ∗ (M) later. (See Section 5.1 for more explanation about this point.) To emphasize the role of the Morse function in the level of complex, we denote the corresponding homology by H Q∗ (−f ) ∼ = QH ∗ (M).

Spectral invariants

539

With these deﬁnitions, we have the obvious nondegenerate pairing CQ∗ (−f ) ⊗ CQ∗ (−f ) → Q

(3.6)

in the chain level which induces the pairing (3.4) above in homology. We now choose a generic Morse function f . Then for any given homotopy H = {H s }s∈[0,1] with H 0 = f and H 1 = H , we denote by hH : CQ∗ (−f ) = CF∗−n (f ) → CF∗−n (H )

(3.7)

the standard Floer chain map from f to H via the homotopy H. This induces a homomorphism hH : H Q∗ (−f ) ∼ = H F∗−n (f ) → H F∗−n (H ).

(3.8)

Although (3.7) depends on the choice H, (3.8) is canonical, i.e., does not depend on the homotopy H. One confusing point in this isomorphism is the issue of grading. See the next section for a review of the construction of this chain map and the issue of grading of H F∗ (H ).

4 Filtered Floer homology For each given generic nondegenerate H : S 1 × M → R, we consider the free Q vector space over 0 (M) | z ∈ Per(H )}. Crit AH = {[z, w] ∈

(4.1)

To be able to deﬁne the Floer boundary operator correctly, we need to complete this vector space downward with respect to the real ﬁltration provided by the action AH ([z, w]) of the element [z, w] of (4.1). More precisely, we have the following. Deﬁnition 4.1. We call the formal sum β= a[z,w] [z, w],

a[z,w] ∈ Q

(4.2)

[z,w]∈Crit AH

a Novikov chain if there are only ﬁnitely many nonzero terms in the expression (4.2) above any given level of the action. We denote by CF∗ (H ) the set of Novikov chains. We call those [z, w] with a[z,w] = 0 generators of the chain β and just denote as [z, w] ∈ β in that case. Note that CF∗ (H ) is a graded Q-vector space which is inﬁnite dimensional in general, unless π2 (M) = 0. As in [Oh4], we introduce the following notion which is a crucial concept for the mini-max argument we carry out later.

540

Y.-G. Oh

Deﬁnition 4.2. Let β be a Novikov chain in CF∗ (H ). We deﬁne the level of the cycle β and denote by λH (β) = max{AH ([z, w]) | a[z,w] = 0 in (4.2)} [z,w]

if β = 0, and we put λH (0) = −∞ as usual. We call the unique critical point [z, w] that realizes the maximum value λH (β) the peak of the cycle β, and denote it by pk(β). We brieﬂy review construction of basic operators in the Floer homology theory [Fl]. Let J = {Jt }0≤t≤1 be a periodic family of compatible almost complex structure on (M, ω). For each given pair (J, H ), we deﬁne the boundary operator ∂ : CF∗ (H ) → CF∗ (H ) considering the perturbed Cauchy–Riemann equation + ∂u ∂u ∂τ + J ∂t − XH (u) = 0, limτ →−∞ u(τ ) = z− , limτ →∞ u(τ ) = z+ .

(4.3)

0 (M), deﬁnes nothing but the negative gradient ﬂow This equation, when lifted to 0 (M) induced by the family of metrics on M of AH with respect to the L2 -metric on gJt = (·, ·)Jt := ω(·, Jt ·): This L2 -metric is deﬁned by

1

ξ, ηJ := 0

(ξ, η)Jt dt.

We will also denote v2J0 = (v, v)J0 = ω(v, J0 v)

(4.4)

for v ∈ T M. For each given [z− , w− ] and [z+ , w+ ], we deﬁne the moduli space M(H, J ; [z− , w− ], [z+ , w+ ]) of solutions u of (4.3) with ﬁnite energy ' 2 * ∂u 2 ∂u 1 + dtdτ < ∞ EJ (u) = ∂τ ∂t − XH (u) 1 2 R×S

and satisfying

Jt

w − #u ∼ w+ .

∂ has degree −1 and satisﬁes ∂ ◦ ∂ = 0.

Jt

(4.5)

Spectral invariants

541

When we are given a family (j, H) with H = {H s }0≤s≤1 and j = {J s }0≤s≤1 , the chain homomorphism h(j,H) : CF∗ (H 0 ) → CF∗ (H 1 ) is deﬁned by the nonautonomous equation + ∂u ρ1 (τ ) ∂u − X H ρ2 (τ ) (u) = 0, ∂τ + J ∂t limτ →−∞ u(τ ) = z− , limτ →∞ u(τ ) = z+

(4.6)

also with the condition (4.5). Here ρi , i = 1, 2 is the cutoff functions of the type ρ : R → [0, 1], + 0 for τ ≤ −R, ρ(τ ) = 1 for τ ≥ R, ρ (τ ) ≥ 0 for some R > 0. h(j,H) has degree 0 and satisﬁes ∂(J 1 ,H 1 ) ◦ h(j,H) = h(j,H) ◦ ∂(J 0 ,H 0 ) . Two such chain maps for different homotopies (j 1 , H1 ) and (j 2 , H2 ) connecting the same endpoints are also known to be chain homotopic [Fl]. Finally, when we are given a homotopy (j , H) of homotopies with j = {jκ }, H = {Hκ }, consideration of the parameterized version of (4.6) for 0 ≤ κ ≤ 1 deﬁnes the chain homotopy map HH : CF∗ (H 0 ) → CF∗ (H 1 ),

(4.7)

which has degree +1 and satisﬁes h(j1 ,H1 ) − h(j0 ,H0 ) = ∂(J 1 ,H 1 ) ◦ HH + HH ◦ ∂(J 0 ,H 0 ) .

(4.8)

By now, construction of these maps using these moduli spaces has been completed with rational coefﬁcients (see [FOn, LT1, Ru]) using the techniques of virtual moduli cycles. We will suppress this advanced machinery from our presentation throughout the paper. The main stream of the proof is independent of this machinery except that it is implicitly needed to prove that various moduli spaces we use are nonempty. Therefore, we do not explicitly mention these technicalities in the main body of the paper until Section 8, unless it is absolutely necessary. In Section 8, we will provide justiﬁcation of this in the general case all at once. The following upper estimate of the action change can be proved by the same argument as used in the proof of [Ch, Oh1, Oh4]. We would like to emphasize that in general there does not exist a lower estimate of this type. The upper estimate is just one manifestation of the “positivity’’ phenomenon in symplectic topology through the existence of pseudoholomorphic curves ﬁrst discovered by Gromov [Gr]. On the other

542

Y.-G. Oh

hand, the existence of lower estimate is closely tied to some nontrivial homological property of (Floer) cycles, and best formulated in terms of Floer cycles instead of individual critical points [z, w] for the nondegenerate Hamiltonians. However, we would like to point out that the equations (4.3) and (4.6) or the numerical estimate of the action changes for solutions u with ﬁnite energy can be studied for any H or (H, j ) which are not necessarily nondegenerate or generic, although the Floer complex or the operators may not be deﬁned for such choices. Proposition 4.3. Let H, F be any Hamiltonian not necessarily nondegenerate and j = {J s }s∈[0,1] be any given homotopy and let Hlin = {H s }0≤s≤1 be the linear homotopy H s = (1 − s)H + sF . Suppose that (4.6) has a solution satisfying (4.5). Then we have the identity AF ([z+ , w+ ]) − AH ([z− , w− ]) 2 ∞ ∂u = − − ρ (τ )(F (t, u(τ, t)) − H (t, u(τ, t)))dtdτ (4.9) ∂τ J ρ1 (τ ) −∞ 2 1 ∂u ≤ − + − min (Ft − Ht )dt (4.10) x∈M ∂τ J ρ1 (τ ) 0 1 ≤ − min (Ft − Ht )dt. (4.11) 0

x∈M

By considering the case F = H , we immediately have the following. Corollary 4.4. For a ﬁxed H and for a given one parameter family j = {J s }s∈[0,1] , let u be as in Proposition 4.3. Then we have AH ([z+ , w+ ]) − AH ([z− , w− ]) = −

2 ∂u ∂τ ρ (τ ) ≤ 0. J 1

(4.12)

Remark 4.5. We would like to remark that similar calculation proves that there is also an uniform upper bound C(j, H) for the chain map over general homotopy (j, H) or for the chain homotopy maps (4.7). In this case, the identity (4.9) is replaced by AF ([z+ , w+ ]) − AH ([z− , w− ]) 2 ∞ ∂u ∂H s =− − ρ (τ ) (t, u(τ, t)) dtdτ ∂τ J ρ1 (τ ) ∂s s=ρ(τ ) −∞ 2 1 ∂u ∂H s ≤ − + − min dt x∈M ∂τ J ρ1 (τ ) ∂s s=ρ(τ ) 0 1 ∂Hts ≤ − min dt. (4.13) x∈M ∂s 0 This upper estimate is also crucial for the construction of these maps. This upper estimate depends on the choice of homotopy (j, H) and is related to the curvature estimates of the relevant Hamiltonian ﬁbration (see [Po2, En1]).

Spectral invariants

543

Now we recall that CF∗ (H ) is also a -module: each A ∈ acts on Crit AH , and so on CF∗ (H ) by “gluing a sphere’’ A : [z, w] → [z, w#A]. Then ∂ is -linear and induces the standard Floer homology H F∗ (H ; ) with as its coefﬁcients (see [HoS] for a detailed discussion on the Novikov ring and on the Floer complex as a -module). However, the action does not preserve the ﬁltration we deﬁned above. Whenever we talk about ﬁltration, we will always presume that the relevant coefﬁcient ring is Q. For a given nondegenerate H and an λ ∈ R \ Spec(H ), we deﬁne the relative chain group CFkλ (H ) := {β ∈ CFk (H ) | λH (β) < λ}. Corollary 4.4 implies that between the two chain complexes (CFk (H ), ∂(H,J ) ) and (CFk (H ), ∂(H,J ) , there is a canonical ﬁltration-preserving chain isomorphism h(j,H ) : (CFk (H ), ∂(H,J ) ) → (CFk (H ), ∂(H,J ) ), where j is any homotopy from J and J , and H ≡ H is the constant homotopy of H . Therefore, from now on we suppress J -dependence on the Floer homology in our exposition unless it is absolutely necessary. For each given pair of real numbers λ, µ ∈ R \ Spec(H ) with λ < µ, we deﬁne (λ,µ]

CF∗

(H ) := CF µ (H )/CF λ (H ).

Then for each triple λ < µ < ν, where λ = −∞ or ν = ∞ are allowed, we have the short exact sequence of the complex of graded Q vector spaces (λ,µ]

0 → CFk

(λ,ν]

(H ) → CFk

(µ,ν]

(H ) → CFk

(H ) → 0

for each k ∈ Z. This then induces the long exact sequence of graded modules (λ,µ]

· · · → H Fk

(λ,ν]

(H ) → H Fk

(µ,ν]

(H ) → H Fk

(λ,µ]

(H ) → H Fk−1 (H ) → · · ·

whenever the relevant Floer homology groups are deﬁned. We close this section by ﬁxing our grading convention for H F∗ (H ). This convention is the analogue to the one used in [Oh1, Oh2] in the context of Lagrangian submanifolds. We ﬁrst recall that solutions of the negative gradient ﬂow equation of −f , (i.e., of the positive gradient ﬂow of f χ˙ − grad f (χ ) = 0 corresponds to the negative gradient ﬂow of the action functional Af ). This gives rise to the relation between the Morse indices µMorse −f (p) and the Conley–Zehnder =]; f ) (see [SZ, Lemma 7.2] but with some care about the different indices µCZ ([p, p convention of the Hamiltonian vector ﬁeld. Their deﬁnition of XH is −XH in our convention):

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Y.-G. Oh

µCZ ([p, p =]; f ) = µMorse −f (p) − n

(4.14)

in our convention. On the other hand, obviously we have Morse (p)) − n = n − µMorse (p). µMorse −f (p) − n = (2n − µf f

We will always grade H F∗ (H ) by the Conley–Zehnder index k = µH ([z, w]) := µCZ ([z, w]; H ).

(4.15)

This grading convention makes the degree k of [q, = q ] in CFk (f ) coincides with the Morse index of q of f for each q ∈ Crit f . Recalling that we chose the Morse complex CM∗ (−f ) ⊗ ↓ for the quantum chain complex CQ∗ (−f ), it also coincides with the standard grading of the quantum cohomology via the map 0 : QH k (M) → QH2n−k (M). Form now on, we will denote by µH ([z, w]) the Conley–Zehnder index of [z, w] for the Hamiltonian H . Under this grading, we have the following grading-preserving isomorphism QH n−k (M) → QHn+k (M) ∼ = H Qn+k (−f ) → H Fk (f ) → H Fk (H ). (4.16) We will also show in Section 6 that this grading convention makes the pants product, denoted by ∗, having the degree −n: ∗ : H Fk (H ) ⊗ H F (K) → H F(k+)−n (H #K)

(4.17)

which will be compatible with the degree-preserving quantum product · : QH a (M) ⊗ QH b (M) → QH a+b (M).

5 Construction of the spectral invariants of Hamiltonian functions In this section, we associate some homologically essential critical values of the action functional AH to each Hamiltonian functions H and quantum cohomology class a, and call them the spectral invariants of H . We denote this assignment by ∞ ([0, 1] × M) × QH ∗ (M) → R ρ : Cm

as described in the introduction of this paper. Before launching our construction, some overview of our construction of spectral invariants is necessary. 5.1 Overview of the construction We recall the canonical isomorphism

Spectral invariants

545

hαβ : H F∗ (Hα ) → H F∗ (Hβ ) which satisﬁes the composition law hαγ = hαβ ◦ hβγ . We denote by H F∗ (M) the corresponding model Q-vector space. We also note that H F∗ (H ) is induced by the ﬁltered chain complex (CF∗λ (H ), ∂), where CF∗λ (H ) = spanQ {α ∈ CF∗ (H ) | λH (α) ≤ λ}, i.e., the subcomplex generated by the critical points [z, w] ∈ Crit AH with AH ([z, w]) ≤ λ. Then there exists a canonical inclusion iλ : CF∗λ (H ) → CF∗∞ (H ) := CF∗ (H ) which induces a natural homomorphism iλ : H F∗λ (H ) → H F∗ (H ). For each given element ∈ F H∗ (M) and Hamiltonian H , we represent the class by a Novikov cycle α of H and measure its level λH (α), and deﬁne ρ(H ; ) := inf {λ ∈ R | ∈ Im iλ } or, equivalently, ρ(H ; ) :=

inf

α;iλ [α]=

λH (α).

The crucial task then is to prove that for each (homogeneous) element = 0, the value ρ(H ; ) is ﬁnite, i.e., “the cycle α is linked and cannot be pushed to inﬁnity by the negative gradient ﬂow of the action functional.’’ In the classical critical point theory (see [BnR], for example), this property of semiinﬁnite cycles is called the linking property. We like to point out that there is no manifest way to see the linking property or the critical nature of the mini-max value ρ(H ; ) from the deﬁnition itself. We will prove this ﬁniteness ﬁrst for the Hamiltonian f , where f is a Morse function and is sufﬁciently small. This ﬁniteness strongly relies on the facts that the Floer boundary operator ∂f in this case has the form Morse ⊗ ω , ∂f = ∂−f

(5.1)

i.e., “there is no quantum contribution on the Floer boundary operator,’’ and the Morse cannot push down the level of a nontrivial classical Morse theory proves that ∂−f cycle more than − max f (see [Oh4]). Once we prove the ﬁniteness for f , then we consider the general nondegenerate Hamiltonian H . We compare the cycles in CF∗ (H ) and the transferred cycles in CF∗ (f ) by the chain map h−1 H : CF∗ (H ) → CF∗ (f ), where H is a homotopy

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Y.-G. Oh

connecting f and H . The change of the level can then be measured by judicious use of (4.7) and Remark 4.5 which will prove the ﬁniteness for any H . After we prove ﬁniteness of ρ(H ; a) for general H , we study the continuity property of ρ(H ; a) under the change of H . This will be done, via the equation (4.6), considering the level change between arbitrary pair (H, K). Finally, we prove that the limit lim ρ(f ; )

→0

exists and is independent of the choice of Morse function f . If the Floer homology class is identiﬁed with a 0 for a quantum cohomology class a ∈ QH ∗ (M) under the PSS-isomorphism [PSS], then this limit is nothing but the valuation v(a). In this procedure, we can avoid considering the “singular limit’’ of the “chains.’’ (See [Oh8, Section 8] for some illustration of the difﬁculty in studying such limits.) We only need to consider the limit of the values ρ(H ; ) as H → 0 which is a much simpler task than considering the limit of chains which involves highly nontrivial analytical work. (We refer to the forthcoming work [FOh2] for the consideration of this limit in the chain level.) 5.2 Finiteness; the linking property of semiinﬁnite cycles With this overview, we now start with our construction. We ﬁrst recall the natural pairing ·, · : CQ∗ (−f ) ⊗ CQ∗ (−f ) → Q, where we have CQk (−f ) := (CMk (−f ), ∂−f ) ⊗ ↓ , CQk (−f ) := (CM2n−k (f ), ∂f ) ⊗ ↑ .

Remark 5.1. We would like to emphasize that in our deﬁnition CQk (−f ) is not isomorphic to HomQ (CQk (−f ), Q) in general. However, there is a natural homomorphism CQk (−f ) → HomQ (CQk (−f ), Q); a → a, · (5.2) whose image lies in the subset of continuous linear functionals Homcont (CQk (−f ), Q) := CQkcont (−f ) ⊂ HomQ (CQk (−f ), Q). See the appendix for more discussions on this aspect. We would like to emphasize that (5.2) is well deﬁned because of the choice of directions of the Novikov rings ↑ and ↓ . In general, the map (5.2) is injective but not an isomorphism. Polterovich [Po4, EnP] observed that this point is closely related to a certain failure of “Poincaré duality’’ of the Floer homology with Novikov rings as its coefﬁcients.

Spectral invariants

547

Now we are ready to give the deﬁnition of our spectral invariants. Previously in [Oh4], the author outlined this construction for the classical cohomology class in H ∗ (M) ⊂ QH ∗ (M). Deﬁnition 5.2. Let H be a generic nondegenerate Hamiltonian. For each given a ∈ QH k (M) ∼ = H Qk (−f ), we deﬁne ρ(H, a) = inf {λH (α) | [α] = a 0 , α ∈ CFk (H )}. α

(5.3)

Theorem 5.3. Let 0 = a ∈ QH ∗ (M). (1) Let H be a generic nondegenerate Hamiltonian. Then ρ(H, a) is ﬁnite. (2) For any pair of generic nondegenerate Hamiltonians H, K, we have the inequality

1

1

− max(K − H )dt ≤ ρ(K, a) − ρ(H, a) ≤

0

− min(K − H )dt. (5.4)

0

0 ([0, 1]×M). In particular, the function H → ρ(H ; a) continuously extends to Cm

Proof. We will prove the ﬁniteness in two steps: ﬁrst we prove the ﬁniteness for f for sufﬁciently small > 0 for any given Morse function f , and then we prove it for general H using this ﬁniteness for f . After this we will prove the inequality (5.4). Step 1: The ﬁniteness of for f . Let f be any ﬁxed Morse function and ﬁx > 0 so small that there is no quantum contribution for the Floer boundary operator ∂(f,J0 ) for a time independent family Jt ≡ J0 for any compatible almost complex structure J0 , i.e., we have Morse ∂(f,J0 ) ! ∂−f ⊗ ↓ω . (5.5) It is well known [Fl, FOn, LT1] that this is possible. Fixing such and J0 , we denote ∂f = ∂(f,J0 ) . Then by considering the Morse homology of −f with respect to the Riemannian metric gJ0 = ω(·, J0 ·), we have the identity Morse Morse QH ∗ (M) ∼ ⊗ ↑ / Im ∂f ⊗ ↑ = H M∗ (f ) ⊗ ↑ , = ker ∂f Morse Morse QH∗ (M) ∼ ⊗ ↓ / Im ∂−f ⊗ ↓ = H M∗ (−f ) ⊗ ↓ . = ker ∂−f

Recalling

CFk (f ) ∼ = CQn+k (−f ),

from (5.5), we represent a 0 ∈ QHn+k (M) by a Novikov cycle of f , where α= ap⊗q A p ⊗ q A A

with ap ∈ Q and p ∈ Crit ∗ (−f ) and

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Y.-G. Oh

n + k = µf (p ⊗ q A ),

(5.6)

where µf (p ⊗ q A ) is the Conley–Zehnder index of the element p ⊗ q A = [p, p =#A]. We recall the general index formula µH ([z, w ⊗ A]) = µH ([z, w]) + 2c1 (A) in our convention (see [Oh9] for the proof of this index formula). Applying this to H = f , we have obtained µf ([p, p =#A]) = µf ([p, p =]) + 2c1 (A). Combining this with µMorse =]) + n, −f (p) = µf ([p, p we derive that (5.6) is equivalent to µMorse −f (p) = n + k − 2c1 (A). Next, we see that α has the level λf (α) = max{−f (p) − ω(A) | ap⊗q A = 0}

(5.7)

because Af ([p, p =#A]) = −f (p) − ω(A). Now the most crucial point in our construction is to prove the ﬁniteness inf λf (α) > −∞.

(5.8)

[α]=a 0

The following lemma proves this linking property. We ﬁrst like to point out that the quantum cohomology class a= aA q −A A

uniquely determines the set (a) := {A ∈ | aA = 0}. By the ﬁniteness condition in the formal power series, we can enumerate (a) so that λ1 > λ2 > λ3 > · · ·

(5.9)

without loss of generality. In particular, we have v(a) = −ω(A1 ) = λ1 .

(5.10)

Lemma 5.4. Let a = QH n−k (M) and a 0 ∈ QHn+k (M) its dual. Suppose that a0 = aj q Aj j

Spectral invariants

549

with 0 = aj ∈ Hn+k−2c1 (Aj ) (M), where λj = −ω(Aj ) are arranged as in (5.9). Denote by γ a Novikov cycle of f with [γ ] = a 0 ∈ H Fk (f ) ∼ = QHn+k (M) and deﬁne the gap c(a) := λ1 − λ2 . Then we have 1 1 v(a) − c(a) ≤ inf {λf (γ ) | [γ ] = a 0 } ≤ v(a) + c(a) γ 2 2

(5.11)

for any sufﬁciently small > 0 and, in particular, (5.8) holds. We also have lim inf {λf (γ ) | [γ ] = a 0 } = v(a)

→0 γ

(5.12)

and so the limit is independent of the choice of Morse functions f . Proof. We represent a 0 by a Novikov cycle γA q A , γj ∈ CM∗ (−f ) γ = A

of f . It follows from (5.3) that if A ∈ (a), all the coefﬁcient Morse chains in this sum must be cycles, and if A ∈ / (a), the corresponding coefﬁcient cycle must be a boundary. Therefore, we can decompose γ as γ = γ(a) + γ(a) C,

(5.13)

where γ(a) :=

γA q A ,

A∈(a)

γ(a) C :=

γB q B ,

B ∈(a) /

and we have γ(a) C = ∂f (ν) for some Floer chain ν of f . Since the summands in γ(a) C cannot cancel those in γ(a) , we have ⎛ ⎞ λf (γ ) ≥ λf (γ(a) ) = λf ⎝ γA q A ⎠ . A∈(a)

Therefore, by removing the exact term ∂f (ν) when we take the inﬁmum over the cycles γ with [γ ] = a 0 for the deﬁnition of ρ(f ; a), we may always assume that γ has the form γ = γj q Aj j

550

Y.-G. Oh

with Aj ∈ (a). Then again by (5.5), we have [γj ] = aj ∈ H∗ (M). Furthermore, we note that we have −ω(Aj ) − max(f ) ≤ λf (γj q Aj ) ≤ −ω(Aj ) − min(f ). Therefore, if we choose > 0 so small that (max f − min f ) ≤ c(a) = λ1 − λ2 , then we have λf (γ1 q A1 ) ≥ λf (γj q Aj ) for all j = 1, 2, . . . , and so λf (γ ) = λf (γ1 q A1 ). Combining them, we derive −ω(A1 ) − max f ≤ λf (γ ) ≤ −ω(A1 ) + max f. (5.11) follows from (5.14) if we choose so that (max f − min f ) < also immediately follows from (5.14).

(5.14) c(a) 2 .

(5.12)

Step 2: The ﬁniteness for general H . Now we consider generic nondegenerate H s. We ﬁx f be any Morse function and and > 0 as in Lemma 5.4. Let α ∈ CF∗ (H ) be a Floer cycle of H with [α] = a 0 , and H = Hlin the linear homotopy Hlin : s → (1 − s)(f ) + sH. Applying (4.12) to the “inverse’’ linear homotopy −1 Hlin : s → (1 − s)H + s(f ),

we obtain the inequality λf (hH−1 (α)) ≤ λH (α) + lin

1

− min(f − H )dt.

(5.15)

0

More precisely, it follows from the deﬁnition of the chain map hH−1 that for any lin generator [z , w ] of the cycle hH−1 (α) of f , there is a generator [z, w] of the cycle lin α such that the equation (4.6) has a solution. Then we derive, from (4.11), 1 Af ([z , w ]) ≤ AH ([z, w]) + − min (f − Ht )dt 0

1

≤ λH (α) + 0

x∈M

− min (f − Ht )dt. x∈M

Spectral invariants

551

−1 Since this holds for any generator [z , w ] of Hlin (α), we obtain

−1 λf (Hlin (α)) ≤ λH (α) +

1 0

− min (f − Ht )dt. x∈M

(5.16)

−1 (α)] = a 0 that we have On the other hand, it follows from [Hlin −1 ρ(a; f ) ≤ λf (Hlin (α)).

Combining this with (5.16), we derive '

1

λH (α) ≥ ρ(a; f ) −

* − min (f − Ht )dt . x∈M

0

(5.17)

Since this holds for any cycle α of H with [α] = a 0 , by taking the inﬁmum over all such α in (5.17), we have ﬁnally obtained ' * 1

ρ(H ; a) ≥ ρ(a; f ) − 0

− min (f − Ht )dt . x∈M

(5.18)

Since Lemma 5.4 shows that ρ(a; f ) > −∞, this implies in particular that ρ(H ; a) > −∞ and so ρ(H ; a) is ﬁnite. Step 3: Proof of (5.4). Finally, we prove the inequality (5.4). For this purpose, we consider general generic nondegenerate pairs H, K. Let δ > 0 be any given number. We choose a cycle α of H , respectively, so that [α] = a 0 and λH (α) ≤ ρ(H ; a) + δ.

(5.19)

We would like to emphasize that this is possible, because we have already shown that ρ(H ; a) > −∞. By considering the linear homotopy hlin H K from H to K, we derive λK (hlin − min(Kt − Ht )dt. (5.20) H K (α)) ≤ λH (α) + x

On the other hand (5.19) implies that λH (α) + − min(Kt − Ht )dt x ≤ ρ(H ; a) + δ + − min(Kt − Ht )dt.

(5.21)

x

0 Since [hlin H K (α)] = a , we have

λK (hlin H K (α)) ≥ ρ(K; a) by the deﬁnition of ρ(K; a). Combining (5.20)–(5.22), we have derived

(5.22)

552

Y.-G. Oh

1

ρ(K; a) − ρ(H ; a) ≤ δ +

− min(Kt − Ht )dt. x

0

Since this holds for arbitrary δ, we have derived 1 ρ(K; a) − ρ(H ; a) ≤ − min(Kt − Ht )dt. 0

x

By changing the role of H and K, we also derive 1 ρ(H ; a) − ρ(K; a) ≤ − min(Ht − Kt )dt = 0

x

1 0

Hence we have the inequality 1 − max(Kt − Ht )dt ≤ ρ(K; a) − ρ(H ; a) ≤ 0

x

1 0

max(Kt − Ht )dt. x

− min(Kt − Ht )dt x

which is precisely (5.4). Obviously the inequality (5.4) enables us to extend the deﬁnition of ρ by continuity to arbitrary C 0 -Hamiltonians. This ﬁnishes the proof of Theorem 5.3.

6 Basic properties of the spectral invariants In this section, we will prove all the remaining properties stated in Theorem I in the introduction. We ﬁrst restate the main axioms of the spectral invariants. Theorem 6.1. Let H, F be arbitrary smooth Hamiltonian functions, and a = 0 ∈ QH ∗ (M) and let ∞ ρ : Cm ([0, 1] × M) × QH ∗ (M) → R be as deﬁned in Section 5. Then ρ satisﬁes the following properties: (1) Projective invariance: ρ(H ; λa) = ρ(H ; a) for any 0 = λ ∈ Q. (2) Normalization: For a = A∈ aA ⊗ q A , ρ(0; a) = v(a), the valuation of a. ; a) for any symplectic diffeomor(3) Symplectic invariance: ρ(η∗ H ; a) = ρ(H phism η. (4) Triangle inequality: ρ(H #F ; a · b) ≤ ρ(H ; a) + ρ(F ; b). (5) C 0 -continuity: |ρ(H ; a) − ρ(F ; a)| ≤ H ◦ F = H − F and, in particular, ρ(·, a) is continuous with respect to the C 0 -topology of Hamiltonian functions. We have already proved the properties of normalization and C 0 -continuity in the course of proving the linking property of the Novikov Floer cycles in Section 5. The remaining parts of the proofs deal with the symplectic invariance and the triangle inequality. 6.1 Proof of symplectic invariance We consider the symplectic conjugation

Spectral invariants

φ → η−1 φη;

553

Ham(M, ω) → Ham(M, ω)

for any symplectic diffeomorphism η : (M, ω) → (M, ω). Recall that the pullback function η∗ H given by η∗ H (t, x) = H (t, η(x)) (6.1) generates the conjugation η−1 φη when H → φ. We summarize the basic facts on this conjugation relevant to the ﬁltered Floer homology here: (1) (2) (3) (4)

When H → φ, η∗ H → ηφη−1 . If H is nondegenerate, η∗ H is also nondegenerate. If (J, H ) is regular in the Floer-theoretic sense, then so is (η∗ J, η∗ H ). There exists natural bijection η∗ : 0 (M) → 0 (M) deﬁned by η∗ ([z, w]) = ([η ◦ z, η ◦ w]) under which we have the identity AH ([z, w]) = Aη∗ H (η∗ [z, w]).

(6.2)

(5) The L2 -gradients of the corresponding action functionals satisfy η∗ (grad J AH )([z, w]) = grad η∗ J (Aη∗ H )(η∗ ([z, w])).

(6.3)

(6) If u : R × S 1 → M is a solution of the perturbed Cauchy–Riemann equation for (J, H ), then η∗ u = η ◦ u is a solution for the pair (η∗ J, η∗ H ). In addition, all the Fredholm properties of (J, H, u) and (η∗ J, η∗ H, η∗ u) are the same. These facts imply that the conjugation by η induces the canonical ﬁltration-preserving chain isomorphism η∗ : (CF∗λ (H ), ∂(H,J ) ) → (CF∗λ (η∗ H ), ∂(η∗ H,η∗ J ) ) for any λ ∈ R \ Spec(H ) = R \ Spec(η∗ H ). In particular, it induces a ﬁltrationpreserving isomorphism η∗ : H F∗λ (H, J ) → H F∗λ (η∗ H, η∗ J ) in homology. The symplectic invariance is then an immediate consequence of our construction of ρ(H ; a). 6.2 Proof of the triangle inequality To start with the proof of the triangle inequality, we need to recall the deﬁnition of the “pants product’’ H F∗ (H ) ⊗ H F∗ (F ) → H F∗ (H #F ).

(6.4)

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Y.-G. Oh

We also need to straighten out the grading problem of the pants product. For the purpose of studying the effect on the ﬁltration under the product, we need to deﬁne this product in the chain level in an optimal way as in [Oh2, Sc, En1]. For this purpose, we will mostly follow the description provided by Entov [En1] with few notational changes and a different convention on the grading. As pointed out before, our grading convention satisﬁes the relation (4.17) under the pants product. Except for the grading convention, the conventions in [En1, En2] for the deﬁnition of Hamiltonian vector ﬁeld and the action functional coincide with our conventions in [Oh1, Oh2, Oh3, Oh5] and in this paper. Let be the compact Riemann surface of genus 0 with three punctures. We ﬁx a holomorphic identiﬁcation of a neighborhood of each puncture with either [0, ∞)×S 1 or (−∞, 0]×S 1 with the standard complex structure on the cylinder. We call punctures of the ﬁrst type negative and the second type positive. In terms of the “pair of pants’’ \ ∪i Di , the positive puncture corresponds to the outgoing ends and the negative corresponds to the incoming ends. We denote the neighborhoods of the three punctures by Di , i = 1, 2, 3, and the identiﬁcation by ϕi+ : Di → (−∞, 0] × S 1 for positive punctures and ϕ3− : D3 → [0, ∞) × S 1 for negative punctures. We denote by (τ, t) the standard cylindrical coordinates on the cylinders. We ﬁx a cutoff function ρ + : (−∞, 0] → [0, 1] deﬁned by + 1, τ ≤ −2, ρ= 0, τ ≥ −1 and ρ − : [0, ∞) → [0, 1] by ρ − (τ ) = ρ + (−τ ). We will just denote by ρ these cutoff functions for both cases when there is no danger of confusion. We now consider the (topologically) trivial bundle P → with ﬁber isomorphic to (M, ω) and ﬁx a trivialization i : Pi := P |Di → Di × M on each Di . On each Pi , we consider the closed two-form of the type ωPi := ∗i (ω + d(ρHt dt))

(6.5)

for a time periodic Hamiltonian H : [0, 1] × M → R. The following is an important lemma whose proof we omit (see [En1]). Lemma 6.2. Consider three normalized Hamiltonians Hi , i = 1, 2, 3. Then there exists a closed two-form ωP such that (1) ωP |Pi = ωPi ;

Spectral invariants

555

(2) ωP restricts to ω in each ﬁber; (3) ωPn+1 = 0. Such ωP induces a canonical symplectic connection ∇ = ∇ωP [GLS, En1]. In addition it also ﬁxes a natural deformation class of symplectic forms on P obtained by those

P ,λ := ωP + λω , where ω is an area form and λ > 0 is a sufﬁciently large constant. We will always normalize ω so that ω = 1. Next, let J be an almost complex structure on P such that (1) J is ωP -compatible on each ﬁber and so preserves the vertical tangent space; (2) the projection π : P → is pseudoholomorphic, i.e., dπ ◦ J = j ◦ dπ. When we are given three t-periodic Hamiltonian H = (H1 , H2 , H3 ), we say that J is (H, J )-compatible if J additionally satisﬁes (3) For each i, (i )∗ J = j ⊕ JHi , where t ∗ JHi (τ, t, x) = (φH ) J i

for some t-periodic family of almost complex structure J = {Jt }0≤t≤1 on M over a disc Di ⊂ Di in terms of the cylindrical coordinates. Here Di = ϕi−1 ((−∞, −Ki ] × S 1 ), i = 1, 2, and ϕ3−1 ([K3 , ∞) × S 1 ) for some Ki > 0. Later we will particularly consider the case where J is in the special form adapted to the Hamiltonian H . See (6.23). Condition (3) implies that the J-holomorphic sections v over Di are precisely the solutions of the equation ∂u ∂u + Jt − XHi (u) = 0 (6.6) ∂τ ∂t if we write v(τ, t) = (τ, t, u(τ, t)) in the trivialization with respect to the cylindrical coordinates (τ, t) on Di induced by φi± above. Now we are ready to deﬁne the moduli space which will be relevant to the deﬁnition of the pants product that we need to use. To simplify the notation, we denote = z = [z, w] in general and = z = (= z1 ,= z2 ,= z3 ), where = zi = [zi , wi ] ∈ Crit AHi for i = 1, 2, 3. Deﬁnition 6.3. Consider the Hamiltonians H = {Hi }1≤i≤3 with H3 = H1 #H2 , and let J be a H -compatible almost complex structure. We denote by M(H, J;= z) the space of all J-holomorphic sections u : → P that satisfy the following: (1) The maps ui := u ◦ (φi−1 ) : (−∞, Ki ] × S 1 → M, which are solutions of (6.6), satisfy lim ui (τ, ·) = zi , i = 1, 2, τ →−∞

and similarly for i = 3 changing −∞ to +∞.

556

Y.-G. Oh

(2) The closed surface obtained by capping off prM ◦ u() with the discs wi taken with the same orientation for i = 1, 2, and the opposite one for i = 3 represents zero in π2 (M). Note that M(H, J;= z) depends only on the equivalence class of = zs: we say that = z ∼ = z if they satisfy zi = zi , wi = wi #Ai for Ai ∈ π2 (M) and 3i=1 Ai represents zero (mod) Z-torsion elements. The (virtual) dimension of M(H, J;= z) is given by dim M(H, J;= z) = 2n − (−µH1 (z1 ) + n) − (−µH2 (z2 ) + n) − (µH3 (z3 ) + n) = −n + (−µH3 (z3 ) + µH1 (z1 ) + µH2 (z2 )). (6.7) Note that when dim M(H, J;= z) = 0, we have n = −µH3 (= z3 ) + µH1 (= z1 ) + µH2 (= z2 ) which is equivalent to µH3 (= z3 ) = (µH1 (= z1 ) + µH2 (= z2 )) − n,

(6.8)

which provides the degree of the pants product (4.17) in our convention of the grading of the Floer complex we adopt in this paper. Now the pair-of-pants product ∗ for the chains is deﬁned by = z1 ∗= z2 = #(M(H, J;= z))= z3 (6.9) = z3

for the generators = zi and then by linearly extending over the chains in CF∗ (H1 ) ⊗ CF∗ (H2 ). Our grading convention makes this product of degree −n. Now with this preparation, we are ready to prove the triangle inequality. Proof of the triangle inequality. Let α ∈ CF∗ (H ) and β ∈ CF∗ (F ) be Floer cycles with [α] = [β] = a 0 and consider their pants product cycle α ∗β := γ ∈ CF∗ (H #F ). Then we have [α ∗ β] = (a · b)0 and so ρ(H #F ; a · b) ≤ λH #F (α ∗ β).

(6.10)

Let δ > 0 be any given number and choose α ∈ CF∗ (H ) and β ∈ CF∗ (F ) so that δ 2 δ λH (β) ≤ ρ(F ; b) + . 2 λH (α) ≤ ρ(H ; a) +

Then we have the expressions

(6.11)

Spectral invariants

α=

ai [zi , wi ] with AH ([zi , wi ]) ≤ ρ(H ; a) +

i

557

δ 2

and β=

j

δ aj [zj , wj ] with AH ([zj , wj ]) ≤ ρ(H ; b) + . 2

Now using the pants product (6.9), we would like to estimate the level of the chain α#β ∈ CF∗ (H #F ). The following is a crucial lemma whose proof we omit, but we refer to [Sc, Section 4.1] or [En1, Section 5]. Lemma 6.4. Suppose that M(H, J;= z) is nonempty. Then we have the identity v ∗ ωP = −AH1 #H2 ([z3 , w3 ]) + AH1 ([z1 , w1 ]) + AH2 ([z2 , w2 ]) (6.12) for any ∈ M(H, J;= z). Now since v is J-holomorphic and J is P ,λ -compatible, we have ∗ ∗ ∗ 0 ≤ v P ,λ = v ωP + λ v ω = v ∗ ωP + λ. Lemma 6.5 ([En1, Theorems 3.6.1 and 3.7.4]). Let Hi s be as in Lemma 6.2. Then for any given δ > 0, we can choose a closed two-form ωP so that P ,λ = ωP + λω becomes a symplectic form for all λ ≥ δ. In other words, the size size(H ) (see [En1, Deﬁnition 3.1]) is ∞. We recall that from the deﬁnition of ∗ that for any [z3 , w3 ] ∈ α ∗ β there exist [z1 , w1 ] ∈ α and [z2 , w2 ] ∈ β such that M(J, H ;= z) is nonempty with the asymptotic condition = z = ([z1 , w1 ], [z2 , w2 ]; [z3 , w3 ]). Applying this and the above two lemmas to H and F for λ arbitrarily close to 0, and also applying (6.10) and (6.11), we immediately derive AH #F ([z3 , w3 ]) ≤ AH ([z1 , w1 ]) + AF ([z2 , w2 ]) + δ ≤ λH (α) + λF (β) + δ ≤ ρ(H ; a) + ρ(F ; b) + 2δ

(6.13)

for any [z3 , w3 ] ∈ α ∗ β. Combining (6.10), (6.11), and (6.13), we derive ρ(H #F ; a · b) ≤ ρ(H ; a) + ρ(F ; b) + 2δ. Since this holds for any δ, we have proved ρ(H #F ; a · b) ≤ ρ(H ; a) + ρ(F ; b). The triangle inequality mentioned in Theorem 6.1 immediately follows from the ; a) = ρ(H ; a) in Theorem 5.5. deﬁnition ρ(φ

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Y.-G. Oh

7 The rational case; proof of the spectrality In this section, we will prove the spectrality for the rational symplectic manifolds: we recall that a symplectic manifold (M, ω) is rational if the period group ω is discrete. We will further study the spectrality property on general symplectic manifolds elsewhere, which turns out to be much more nontrivial to prove. Theorem 7.1. Suppose that (M, ω) is rational. Then for any smooth one-periodic Hamiltonian function H : S 1 × M → R, we have ρ(H ; a) ∈ Spec(H ) for each given quantum cohomology class 0 = a ∈ QH ∗ (M). Proof. We need to show that the mini-max value ρ(H ; a) is a critical value, i.e., that 0 (M) such that there exists [z, w] ∈ AH ([z, w]) = ρ(H ; a), dAH ([z, w]) = 0, i.e.,

z˙ = XH (z).

(7.1)

We have already shown the ﬁniteness of the value ρ(H ; a) in Section 5. If H is nondegenerate, we use the ﬁxed Hamiltonian. If H is not nondegenerate, we approximate H by a sequence of nondegenerate Hamiltonians Hj in the C 2 topology. Let [zj , wj ] ∈ Crit AHj be the peak of a Floer cycle αj ∈ CF∗ (Hj ) such that lim AHj ([zj , wj ]) = ρ(H ; a).

j →∞

(7.2)

Such a sequence can be chosen from the deﬁnition of ρ(·; a) and the ﬁniteness thereof. Since M is compact and Hj → H in the C 2 topology, and z˙ j = XHj (zj ) for all j , it follows from the standard bootstrap argument that zj has a subsequence, which we still denote by zj , converging to z∞ which solves z˙ = XH (z). Now we show 0 (M). Since we ﬁx the quantum cohomology class that [zj , wj ] are precompact on 0 = a ∈ QH ∗ (M) (or more speciﬁcally since we ﬁx its degree) and the Floer cycle satisﬁes [αj ] = a, we have µHj ([zj , wj ]) = µHi ([zi , wi ]).

(7.3)

0 (M) is a closed subset of R Lemma 7.2. When (M, ω) is rational, Crit(AK ) ⊂ for any smooth Hamiltonian K, and is locally compact in the subspace topology of the covering space 0 (M) → 0 (M). π : Proof. First note that when (M, ω) is rational, the covering group ω of π above is discrete. Together with the fact that the set of solutions of z˙ = XK (z) is compact (on compact M), it follows that 0 (M) | z˙ = XK (z)} Crit(AK ) = {[z, w] ∈ is a closed subset which is also locally compact.

Spectral invariants

559

Now consider the bounding discs of z∞ wj = wj #ucan j for all sufﬁciently large j , where ucan is the homotopically unique thin cylinder between zj and z∞ : more precisely, ucan j is given by the formula ξj (t) = (expzj (t) )−1 (z∞ (t)),

ucan j (s, t) = expzj (t) (sξj (t)),

(7.4)

where exp is the exponential map with respect to a ﬁxed metric gJref = ω(·, Jref ·) for a ﬁxed compatible almost complex structure. We note that as j → ∞ the geometric area of ucan j converges to 0. We compute the action of the critical points [z∞ , wj ] ∈ Crit AH , AH ([z∞ , wj ])

=−

wj

ω−

=−

1

ω−

wj

' = −

0

ucan j

0

H (t, z∞ (t))dt *

Hj (t, zj (t))dt 0

1

−

1

ω−

1

ω−

wj

'

(7.5)

H (t, z∞ (t))dt

0

H (t, z∞ (t)) − '

= AHj ([zj , wj ]) −

1 0

1

*

Hj (t, zj (t)) −

0

H (t, z∞ (t)) −

1

ω ucan j

*

Hj (t, zj (t)) −

0

From the explicit expression (7.4), it follows that ω=0 lim j →∞ ucan j

ω. ucan j

(7.6)

since the geometric area of ucan j converges to zero, and we have can Area(uj ) ≥ ω . ucan j

Since zj converges to z∞ uniformly and Hj → H , we have ' * 1

− 0

H (t, z∞ (t)) −

1

H (t, zj (t)) → 0.

0

Therefore, combining (7.2), (7.6), and (7.7), we derive lim AH ([z∞ , wj ]) = ρ(H ; a).

j →∞

(7.7)

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Y.-G. Oh

In particular, AH ([z∞ , wj ]) is a Cauchy sequence, which implies that ω− w w j

i

ω = AH ([z∞ , wj ]) − AH ([z∞ , wi ]) → 0;

i.e.,

Since ω is discrete and

wj #w i

wj #w i

ω → 0.

ω ∈ ω , this indeed implies that wj #w i

ω=0

(7.8)

for all sufﬁciently large i, j . Since the set { w ω}j ∈Z+ ⊂ R is bounded, these imply j that the sequence w ω eventually stabilizes. Going back to (7.5), we have proved j

that the actions

AH ([z∞ , wj ])

stabilize, and so we have AH ([z∞ , wN ]) = lim AH ([z∞ , wj ]) = ρ(H ; a) j →∞

for a ﬁxed sufﬁciently large N ∈ Z+ . This proves that ρ(H ; a) is indeed a critical ]. This ﬁnishes the proof. value of AH at the critical point [z∞ , wN

We now state the following theorem. ∞ (M × [0, 1], R) the set of normalized Theorem 7.3. Let (M, ω) be rational and Cm ∞ ∞ (M × [0, 1], R) → R the extended C -Hamiltonians on M. We denote by ρa : Cm continuous function deﬁned by ρa (H ) = ρ(H ; a).

= [φ, H ]. Hence ρa (1) The image of ρa depends only on the homotopy class φ pushes down to a well-deﬁned function ρ : Ham(M, ω) × QH ∗ (M) → R;

; a) := ρ(H ; a) ρ(φ

(7.9)

= [φ, H ]. for any H with φ (2) We have the formula ρ(H ; a) = inf {λ | a 0 ∈ Im(iλ : H F∗λ (H ) → H F∗ (H ))}. λ

(7.10)

Proof. We have shown in Theorem 7.1 that ρ(H ; a) is indeed a critical value of AH , i.e., lies in Spec(H ). With this fact, the well-deﬁnedness of the deﬁnition (7.9), i.e., = [φ, H ] is an immediate consequence of the combination independence of H with φ of the following results:

Spectral invariants

561

(1) H → ρ(H ; a) is continuous. (2) Spec(H ) is a measure zero subset of R (Lemma 2.2). ) depends only on its homotopy class [H ] = φ and so ﬁxed (3) Spec(H ) = Spec(φ as long as [H ] = φ (Theorem 2.3). (4) The only real-valued continuous function from a connect space (e.g., the unit interval [0, 1]) whose image has measure zero in R is a constant function. (7.10) is just a rephrasing of the deﬁnition of ρ(H ; a). This ﬁnishes the proof of Theorem 7.3.

One more important property concerns the effect of ρ under the action of π0 (G). ∗ acts on Ham(M, ω) × QH (M) following (and adapting We ﬁrst explain how π0 (G) into cohomological version) Seidel’s description of the action on QH∗ (M).According acts on QH∗ (M) by the quantum product of an to [Se], each element [h, h] ∈ π0 (G) even element #([h, h]) on QH∗ (M). We take the adjoint action of it on a ∈ QH ∗ (M) and denote it by h∗ a. More precisely, h∗ a is deﬁned by the identity h∗ a, β = a, #([h, h]) · β

(7.11)

with respect to the nondegenerate pairing ·, · between QH ∗ (M) and QH∗ (M). act on Ham(M, Theorem 7.4 (monodromy shift). Let π0 (G) ω) × QH ∗ (M) as above, i.e., , a) = (h · φ , [h, h] · (φ h∗ a). (7.12) Then we have

; a)) = ρ(φ ; a) + Iω ([h, h]). ρ([h, h] · (φ

Proof. This is immediate from the construction of #([h, h]) in [Se]. Indeed, the map [h, h]∗ : CF∗ (F ) → CF∗ (H #F ) is induced by the map and we have

(7.13)

[z, w] → h([z, w]),

AH #F ( h([z, w])) = AF ([z, w]) + Iω ([h, h])

by (2.5). Furthermore, the map (7.12) is a chain isomorphism whose inverse is given by ([h, h]−1 )∗ . This immediately implies the theorem from the construction of ρ. Remark 7.5. Strictly speaking, h∗ a may not lie in the standard quantum cohomology ∗ QH (M) because it is deﬁned as the linear functional on the complex CQ∗ (M) that is dual to the Seidel element #([h, h]) ∈ CQ∗ (M) under the canonical pairing between CQ∗ (M) and CQ∗ (M). A priori, the bounded linear functional h∗ a = #([h, h]), · may not lie in the image of : QH∗ (M) → QH ∗ (M), mentioned in Section 3, in general. In that case, one should consider h∗ a as a continuous quantum cohomology

562

Y.-G. Oh

class in the sense of the appendix. We refer readers to the appendix for the explanation on how to extend the deﬁnition of our spectral invariants to the continuous quantum cohomology classes. Now we can deﬁne ρ : Ham(M, ω) × QH ∗ (M) by putting

; a) := ρ(H ; a) ρ(φ

with [H ] = φ when φ is nondegenerate, and then extending to for any H → φ arbitrary φ by continuity. ; a) for each a ∈ QH ∗ (M), we have constructed Then by the spectrality of ρ(φ continuous “sections’’ of the action spectrum bundle Spec(M, ω) → Ham(M, ω). by We deﬁne the essential spectrum of φ ) := {ρ(φ ; a) | 0 = a ∈ QH ∗ (M)}, spec(φ ) := {ρ(φ ; a) | 0 = a ∈ QH k (M)} speck (φ and the bundle of essential spectra by spec(M, ω) =

A

); spec(φ

∈H φ am(M,ω)

similarly for speck (M, ω).

8 Remarks on transversality Our construction of various maps in the Floer homology works as in the previous section for the strongly semipositive case [Se, En1] by the standard transversality argument. On the other hand, in the general case where constructions of operations in the Floer homology theory require the machinery of virtual fundamental chains through multivalued abstract perturbation [FOn, LT1, Ru], we need to explain how this general machinery can be incorporated into our construction. The full details will be provided elsewhere. We will use the terminology “Kuranishi structure’’ adopted by Fukaya and Ono [FOn] for the rest of the discussion. One essential point in our proofs is that various numerical estimates concerning the critical values of the action functional and the levels of relevant Novikov cycles do not require transversality of the solutions of the relevant pseudoholomorphic sections, but depends only on the nonemptiness of the moduli space M(H, J;= z)

Spectral invariants

563

which can be studied for any, not necessarily generic, Hamiltonian H . Since we always have suitable a priori energy bound which requires some necessary homotopy assumption on the pseudoholomorphic sections, we can compactify the corresponding moduli space into a compact Hausdorff space, using a variation of the notion of stable maps in the case of nondegenerate Hamiltonians H . We denote this compactiﬁcation again by M(H, J;= z). This space could be pathological in general. But because we assume that the Hamiltonians H are nondegenerate, i.e., all the periodic orbits are nondegenerate, the moduli space is not completely pathological but at least carries a Kuranishi structure in the sense of Fukaya–Ono [FOn] for any H -compatible J. This enables us to apply the abstract multivalued perturbation theory and to perturb the compactiﬁed moduli space by a Kuranishi map + so that the perturbed moduli space M(H, J;= z, +) is transversal in that the linearized equation of the perturbed equation ∂ J(v) + +(v) = 0 is surjective and so its solution set carries a smooth (orbifold) structure. Furthermore, the perturbation + can be chosen so that as + → 0, the perturbed moduli space M(H, J;= z, +) converges to M(H, J;= z) in a suitable sense (see [FOn] for the precise description of this convergence). Now the crucial point is that nonemptiness of the perturbed moduli space will be guaranteed as long as certain topological conditions are met. For example, the followings are the prototypes that we have used in this paper: (1) hH1 : CF0 (f ) → CF0 (H ) is an isomorphism in homology and so [hH1 (10 )] = 0. This is immediately translated as an existence result of solutions of the perturbed Cauchy–Riemann equation. (2) The deﬁnition of the pants product ∗ and the identity [α ∗ β] = (a · b)0 in homology guarantee nonemptiness of the relevant perturbed moduli space M(H, J;= z, +) for α ∈ CF∗ (H1 ), β ∈ CF∗ (H2 ) with [α] = a 0 , and [β] = b0 , respectively. Once we prove the nonemptiness of M(H, J;= z, +) and an a priori energy bound for the nonempty perturbed moduli space and if the asymptotic conditions= z are ﬁxed, we can study the convergence of a sequence vj ∈ M(H, J;= z, +j ) as +j → 0 by the Gromov–Floer compactness theorem. However, a priori there are inﬁnite possibilities of asymptotic conditions for the pseudoholomorphic sections that we are studying, because we typically impose that the asymptotic limit lie in certain Novikov cycles like = z1 ∈ α,

= z2 ∈ β,

= z3 ∈ α ∗ β.

564

Y.-G. Oh

Because the Novikov Floer cycles are generated by an inﬁnite number of critical points [z, w] in general, one needs to control the asymptotic behavior to carry out the compactness argument. For this purpose, we need to establish a lower bound for the actions which will enable us to consider only ﬁnite possibilities for the asymptotic conditions because of the ﬁniteness condition in the deﬁnition of Novikov chains. We would like to emphasize that obtaining a lower bound is the heart of the matter in the classical mini-max theory of the indeﬁnite action functional which requires a linking property of semiinﬁnite cycles. On the other hand, obtaining an upper bound is usually an immediate consequence of the identity as in (4.10). With such a lower bound for the actions, we may then assume, by taking a subsequence if necessary, that the asymptotic conditions are ﬁxed when we take the limit and so we can safely apply the Gromov–Floer compactness theorem to produce a (cusp)-limit lying in the compactiﬁed moduli space M(H, J;= z). This will then justify all the statements and proofs in the previous sections for the complete generality.

Appendix: Continuous quantum cohomology In this appendix, we deﬁne the genuinely cohomological version of the quantum cohomology and explain how we can extend the deﬁnition of the spectral invariants to the classes in this cohomological version. We call this continuous quantum cohomology and denote it by ∗ QHcont (M).

In this respect, we call the usual quantum cohomology ring QH ∗ (M) = H ∗ (M) ⊗ ∗ (M) and QH ∗ (M) ↑ the ﬁnite quantum cohomology. We call elements in QHcont continuous (respectively, ﬁnite) quantum cohomology classes. ∗ (M). Let f be a Morse We ﬁrst deﬁne the chain complex associated to QHcont function and consider the complex of Novikov chains CQ2n−k (−f ) = CM2n−k (−f ) ⊗ ↓ (= CFk (f )).

(A.1)

On nonexact symplectic manifolds, this is typically inﬁnite dimensional as a Q-vector space. Therefore, it is natural to put some topology on it rather than to consider it only as an algebraic vector space. For this purpose, we recall the deﬁnition of the level λ(α) = λf (α) of an element: α= αA q A , αA ∈ CM∗ (−f ): A

λ(α) = max{Af (αA q A ) | αA = 0} = max{λMorse −f (αA ) − ω(A)}. As we saw before, this level function satisﬁes the inequality λ(α + β) ≤ max{λ(α), λ(β)}

(A.2)

Spectral invariants

565

and provides a natural ﬁltration on CQ2n−k (−f ), which deﬁnes a non-Archimedean topology. We refer to [Br] for a nice exposition to the non-Archimedean topology and geometry. Deﬁnition and Proposition A.1. For each degree ∗, consider the collection A B= {U (α, R) ⊂ CQ∗ (−f )} α∈CQ∗ (−f ),R∈R

of the subsets U (α, R) deﬁned by U (α, R) = {β ∈ CQ∗ (−f ) | λ(β − α) < R}. Then B satisﬁes the properties of a basis of topology. We equip CQ∗ (−f ) with the topology generated by the basis B. Proof. We need to prove that for any given U (α1 , R1 ) and U (α2 , R2 ) with U (α1 , R1 )∩ U (α2 , R2 ) = ∅ and for any β ∈ U (α1 , R1 ) ∩ U (α2 , R2 ), there exists R3 such that U (α, R3 ) ⊂ U (α1 , R1 ) ∩ U (α2 , R2 ).

(A.3)

Let β ∈ U (α1 , R1 ) ∩ U (α2 , R2 ). Then β satisﬁes λ(β − αi ) < Ri ,

i = 1, 2.

(A.4)

Suppose γ ∈ U (β, R), where R is to be determined. Then we derive from (A.2) λ(γ − αi ) ≤ max{λ(γ − β), λ(β − αi )} = max{R, Ri }.

(A.5)

Therefore, if we choose R ≤ min{R1 , R2 }, then we will have U (β, R) ⊂ U (α1 , R1 ) ∩ U (α2 , R2 ), which ﬁnishes the proof of the fact that B really deﬁnes a basis of topology.

By the non-Archimedean triangle inequality (A.2), it follows that the basis element U (α, R) is nothing but the afﬁne subspace R U (α, R) = CQR ∗ (−f ) + α = CF2n−∗ (f ) + α,

where CF∗R is deﬁned as in Section 4. The following is an easy consequence of the deﬁnition of the boundary operator. Lemma A.2. The boundary operator Morse ∂f = ∂−f ⊗ : CQ2n−k (−f ) → CQ2n−k−1 (−f )

is continuous with respect to this topology. Proof. Let U (α, R) be a basis element and consider the preimage

566

Y.-G. Oh

(∂f )−1 (U (α, R)). Suppose β ∈ (∂f )−1 (U (α, R)), i.e., ∂f (β) ∈ U (α, R) and so λ(∂f (β) − α) < R.

(A.6)

λ(∂f (δ)) ≤ λ(δ)

(A.7)

Recall that for any Novikov Floer chain δ. Now we consider the basis element U (β, R). Then if γ ∈ U (β, R), we have λ(∂f (γ ) − α) ≤ max{λ(∂f (γ − β)), λ(∂f (β) − α)} ≤ max{λ(γ − β), λ(∂f (β) − α)}

(A.8)

< max{R, R} = R, where the second inequality comes from (A.7). This ﬁnishes the proof of ∂f (U (β, R)) ⊂ U (α, R), i.e., U (β, R) ⊂ (∂f )−1 (U (α, R) for any β ∈ U (α, R). Hence the proof.

Now we deﬁne the following. Deﬁnition A.3. A linear functional µ : CQ2n−k (−f ) → Q is called continuous (or bounded) if it is continuous with respect to the topology induced by the above ﬁltration. We denote by CQcont (−f ) the set of continuous linear functionals on CQ2n−k (−f ). The following is easy to see from the deﬁnition of Novikov chains. Lemma A.4. A linear functional µ is continuous if and only if there exists λµ ∈ R such that µ(αA q A ) = 0 (A.9) for all A with −ω(A) ≤ λµ . Proof. The sufﬁciency part of the proof is easy and so we will focus on the necessary condition. We will prove this by contradiction. Suppose that µ : CQ2n−k (−f ) → Q is a continuous linear functional, but there exists a sequence of Aj with −ω(Aj ) → −∞,

i.e.,

ω(Aj ) → +∞

(A.10)

and αj ∈ CM∗ (−f ) such that µ(αj q Aj ) = 0. Now consider the sequence of Novikov chains

(A.11)

Spectral invariants

βN =

N

αj q Aj .

567

(A.12)

j =1

It is easy to check from (A.10) that βN converges to the Novikov chain β=

∞

αj q Aj

j =1

in the given non-Archimedean topology on CQ∗ (−f ). In fact, this convergence holds for the sequence N βc,N = (cj αj )q Aj (A.13) j =1

for any given sequence c = {cj ∈ Q}1≤j <∞ . We choose cj s so that cj =

1 , µ(αj q Aj )

which is well deﬁned by (A.11). However, we then have µ(βc,N +1 ) − µ(βc,N ) = µ(cN +1 αN +1 q AN+1 ) = cN +1 µ(αN +1 q AN+1 ) = 1 for all N. This proves that µ cannot be continuous, a contradiction. This ﬁnishes the proof.

It then follows that ∗ ∗ ∂Q = ∂−f : (CQ (−f ))∗ → (CQ+1 (−f ))∗

maps continuous linear functionals to continuous ones and so deﬁnes the canonical complex ∗ (CQ∗cont (−f ), ∂Q ) and hence deﬁnes the homology ∗ QHcont (M) := H (CQ∗cont (−f ), ∂Q )).

We recall the canonical embedding σ : CQ (−f ) = CM2n− f ⊗ ↑ → CQcont (−f ); a → a, ·

(A.14)

mentioned in Remark 5.1. We have the following proposition which is straightforward to prove. We refer to the proof of [Oh2, Proposition 2.2] for the details. Proposition A.5. The map σ in (A.14) is a chain map from (CQ (−f ), δ Q ) to ∗ ). In particular, we have a natural degree-preserving homomor(CQcont (−f ), ∂Q phism ∗ σ : QH ∗ (M) ∼ (M). = H Q∗ (−f ) → H Q∗cont (−f ) ∼ = QHcont

(A.15)

568

Y.-G. Oh

∗ (H ) for Now we can deﬁne the notion of continuous Floer cohomology H Fcont any given Hamiltonian in a similar way. Then the cochain map

(hH )∗ : CF k (H ) → CF k (f ) restricts to the cochain map k k (hH )∗ : CFcont (H ) → CFcont (f ).

Once we have deﬁned the continuous quantum cohomology and the continuous Floer cohomology, it is straightforward to deﬁne the spectral invariants for the continuous cohomology class in the following way. (M). Then we deﬁne Deﬁnition A.6. Let µ ∈ QHcont

ρ(H ; µ) := inf {λ ∈ R | µ ∈ Im iλ∗ }.

(A.16)

Now it is straightforward to generalize all the axioms in Theorem I to the continuous quantum cohomology class. The only nonobvious axiom is the triangle inequality. But the proof will be a verbatim modiﬁcation of [Oh2, Theorem II (5)] incorporating the argument in this paper that uses the Hamiltonian ﬁbration and pseudoholomorphic sections. We leave the details to the interested readers. We hope to investigate further properties of the continuous quantum cohomology and its applications elsewhere. Acknowledgments We would like to thank the Institute for Advanced Study in Princeton for the excellent environment and hospitality during our participation of the year 2001–2002 program “Symplectic Geometry and Holomorphic Curves.’’ Much of this work was ﬁnished during our stay in IAS. We thank D. McDuff for some useful communications in IAS. The ﬁnal writing was carried out while we visited the Korea Institute for Advanced Study in Seoul. We thank KIAS for providing uninterrupted quiet time for writing and excellent atmosphere of research. We thank M. Entov and L. Polterovich for enlightening discussions on spectral invariants and for explaining their applications [En2, EnP] of the spectral invariants to the study of Hamiltonian diffeomorphism group, and Y. Ostrover for explaining his work from [Os] to us during our visit to Tel-Aviv University. We also thank P. Biran and L. Polterovich for their invitation to Tel-Aviv University and hospitality.

References [BnR] [Br]

[Ch] [En1]

Benci, V., and Rabinowitz, P., Critical point theorems for indeﬁnite functionals, Invent. Math., 52 (1979), 241–273. Berkovich, V., Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, Vol. 33, American Mathematical Society, Providence, RI, 1990. Chekanov, Y., Lagrangian intersections, symplectic energy and areas of holomorphic curves, Duke J. Math., 95 (1998), 213–226. Entov, M., K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math., 146 (2001), 93–141.

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569

Entov, M., Commutator length of symplectomorphisms, Comment. Math. Helv., 79-1 (2004), 58–104. [EnP] Entov, M., and Polterovich, L., Calabi quasimorphism and quantum homology, Internat. Math. Res. Notices, 30 (2003), 1635–1676. [Fl] Floer, A., Symplectic ﬁxed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575–611. [FOh1] Fukaya, K., and Oh, Y.-G., Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math., 1 (1997), 96–180. [FOh2] Fukaya, K., and Oh, Y.-G., in preparation. [FOOO] Fukaya, K., Oh, Y.-G., Ohta, H., and Ono, K., Lagrangian Intersection Floer Theory: Anomaly and Obstruction, preprint, Kyoto University, Kyoto, 2000. [FOn] Fukaya, K., and Ono, K.,Arnold conjecture and Gromov-Witten invariants, Topology, 38 (1999), 933–1048. [Gr] Gromov, M., Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307–347. [GLS] Guillemin, V., Lerman, E., and Sternberg, S., Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press, Cambridge, UK, 1996. [HaL] Harvey, F., and Lawson, B., Finite volume ﬂows and Morse theory, Ann. Math., 153 (2001), 1–25. [Ho] Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh, 115 (1990), 25–38. [HoS] Hofer, H., and Salamon, D., Floer homology and Novikov rings, in Hofer, H., Taubes, C., Weinstein, A., and Zehnder, E., eds., The Floer Memorial Volume, Progress in Mathematics, Vol. 133, Birkhäuser, Basel, 1995, 483–524. [LM] Lalonde, F., and McDuff, D., The geometry of symplectic energy, Ann. Math., 141 (1995), 349–371. [LT1] Liu, G., and Tian, G., Floer homology and Arnold’s conjecture, J. Differential Geom., 49 (1998), 1–74. [LT2] Liu, G., and Tian, G., On the equivalence of multiplicative structures in Floer homology and quantum homology, Acta Math. Sinica (English Ser.), 15-1 (1999), 53–80. [Lu] Lu, G., Arnold conjecture and PSS isomorphism between Floer homology and quantum homology, preprint, 2000. [Mi] Milinkovi´c, D., On equivalence of two constructions of invariants of Lagrangian submanifolds, Paciﬁc J. Math., 195 (2000), 371–415. [MO1] Milinkovi´c, D., and Oh, Y.-G., Floer homology and stable Morse homology, J. Korean Math. Soc., 34 (1997), 1065–1087. [MO2] Milinkovi´c, D., and Oh, Y.-G., Generating functions versus the action functional: Stable Morse theory versus Floer theory, in Lalonde, F., ed., Geometry, Topology, and Dynamics: Proceedings of the Workshop on Geometry, Topology, and Dynamics Held at the CRM, Université de Montréal, June 26–30, 1995, CRM Proceedings and Lecture Notes, Vol. 15, American Mathematical Society, Providence, RI, 1998, 107–125. [Oh1] Oh, Y.-G., Symplectic topology as the geometry of action functional I, J. Differential Geom., 46 (1997), 499–577. [Oh2] Oh, Y.-G., Symplectic topology as the geometry of action functional II, Comm. Anal. Geom., 7 (1999), 1–55. [Oh3] Oh, Y.-G., Gromov-Floer theory and disjunction energy of compact Lagrangian embeddings, Math. Rec. Lett., 4 (1997), 895–905. [Oh4] Oh, Y.-G., Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group, Asian J. Math., 6 (2002), 579–624; erratum, 7 (2003), 447–448.

570 [Oh5] [Oh6] [Oh7] [Oh8] [Oh9] [Os] [PSS]

[Po1] [Po2] [Po3] [Po4] [Ra] [Ru] [Sc] [Se] [V] [W]

Y.-G. Oh Oh, Y.-G., Normalization of the Hamiltonian and the action spectrum, J. Korean Math. Soc., to appear; math.SG/0206090. Oh, Y.-G., Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group, preprint, 2002; math.SG/0206092. Oh, Y.-G., Spectral invariants and length minimizing property of Hamiltonian paths, Asian J. Math., to appear; math.SG/0212337. Oh, Y.-G., Spectral invariants, analysis of the Floer moduli spaces and geometry of Hamiltonian diffeomorphisms, submitted. Oh, Y.-G., Length minimizing property, Conley-Zehnder index and C 1 -perturbation of Hamiltonian functions, submitted; math.SG/0402149. Ostrover, Y., A comparison of Hofer’s metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, Comm. Contemp. Math., 5-5 (2003), 803–812. Piunikhin, S., Salamon, D., and Schwarz, M., Symplectic Floer-Donaldson theory and quantum cohomology, in Thomas, C. B., ed., Contact and Symplectic Geometry, Publications of the Newton Institute, Vol. 8, Cambridge University Press, Cambridge, UK, 1996, 171–200. Polterovich, L., Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems, 13 (1993), 357–367. Polterovich, L., Gromov’s K-area and symplectic rigidity, Geom. Functional Anal., 6 (1996), 726–739. Polterovich, L., The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2001. Polterovich, L., private communication. Rabinowitz, P., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157–184. Ruan, Y., Virtual neighborhood and pseudo-holomorphic curves, Turkish J. Math., 23 (1999), 161–231. Schwarz, M., On the action spectrum for closed symplectically aspherical manifolds, Paciﬁc J. Math., 193 (2000), 419–461. Seidel, P., π1 of symplectic diffeomorphism groups and invertibles in quantum homology rings, Geom. Functional Anal., 7 (1997), 1046–1095. Viterbo, C., Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685–710. Weinstein, A., graduate course, University of California at Berkeley, Berkeley, CA, 1987.

The universal covering and covered spaces of a symplectic Lie algebra action Juan-Pablo Ortega1 and Tudor S. Ratiu2 1 Centre National de la Recherche Scientiﬁque

Département de Mathématiques de Besançon Université de Franche-Comté UFR des Sciences et Techniques 16, route de Gray F-25030 Besançon Cedex France [email protected] 2 Centre Bernoulli École Polytechnique Fédérale de Lausanne CH-1015 Lausanne Switzerland [email protected] Dedicated to Alan Weinstein on the occasion of his 60th birthday. Abstract. We show that the category of Hamiltonian covering spaces of a given connected and paracompact symplectic manifold (M, ω) acted canonically upon by a Lie algebra admits a universal covering and covered space.

1 Introduction Let (M, ω) be a connected symplectic manifold and g a Lie algebra acting symplectically on it. A Lie algebra action of g on M is a Lie algebra antihomomorphism ξ ∈ g → ξM ∈ X(M) such that the map (m, ξ ) ∈ M × g∗ → ξM (m) ∈ T M is smooth. The action is symplectic when £ξM ω = 0 for any ξ ∈ g and where £ξM is the Lie derivative operator deﬁned by the vector ﬁeld ξM . Deﬁnition 1.1. Let (M, ω) be a connected symplectic manifold and let g be a Lie algebra acting symplectically on it. We say that the map pN : N → M is a Hamiltonian covering map of (M, ω) when it satisﬁes the following conditions: (i) pN is a smooth covering map. (ii) (N, ωN ) is a connected symplectic manifold. (iii) pN is a symplectic map.

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(iv) g acts symplectically on (N, ωN ) and admits a momentum map KN : N → g∗ . (v) pN is g-equivariant, that is, ξM (pN (n)) = Tn pN (ξN (n)) for any n ∈ N and any ξ ∈ g. The connectedness hypothesis on N that we assumed in the previous deﬁnition implies that the momentum map KN : N → g∗ is determined up to a constant element in g∗ . We will denote by [KN ] the equivalence class consisting of all the maps N → g∗ that differ from KN by a constant map. Deﬁnition 1.2. Let (M, ω) be a connected symplectic manifold and g be a Lie algebra acting symplectically on it. Let H be the category whose objects Ob(H) are the fourtuples (pN : N → M, ωN , g, [KN ]) with pN : N → M a Hamiltonian covering map of (M, ω) and whose morphisms Mor(H) are the smooth maps q : (N1 , ω1 ) → (N2 , ω2 ) that satisfy the following properties: (i) q is a symplectic covering map. (ii) q is g-equivariant. (iii) The diagram g∗

KN1

@ I @

@

KN2

@ @ - N2

q N1 @ pN1

@

@

@ R @

pN2

M commutes for some KN1 ∈ [KN1 ] and KN2 ∈ [KN2 ]. We will refer to H as the category of Hamiltonian covering maps. The main goal of this paper is to show that the category H admits universal covering and covered spaces. More explicitly, we will show that there exist two = → M, ωM = (universal Hamiltonian covering space) and objects (= p : M = , g, [K]) ( p : M → M, ωM , g, [K]) (universal Hamiltonian covered space) in H such that for any other object (pN : N → M, ωN , g, [KN ]) in H, there exist morphisms (not = → N and in Mor(H). Even though the objects necessarily unique) = q:M q:N →M that satisfy these properties are not necessarily unique, they are all isomorphic to = → M, ωM → M, ωM = and ( in H, respectively, which (= p:M p:M = , g, [K]) , g, [K]) justiﬁes the adjective “universal’’ when we will refer to the Hamiltonian covering and covered spaces.

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The universal Hamiltonian covering space will be easily obtained in Section 2 from the standard simply connected universal covering manifold. The universal Hamiltonian covered space is constructed in Section 3 using a connection introduced by Condevaux, Dazord, and Molino [CDM88, Section I.3.1] and used by them in the deﬁnition of the so called “reduced momentum map.’’ The universality of the Hamiltonian covered space is presented in Section 4.

2 The standard universal covering as a Hamiltonian covering Let (M, ω) be a connected symplectic manifold and g be a Lie algebra acting sym= → M be a simply connected universal covering of M. plectically on it. Let p =: M This can be made into a Hamiltonian covering map in a straightforward manner. First, since p = is a local diffeomorphism, the two-form ωM =∗ ω is a symplectic form on = := p = Thus properties (i), (ii), and (iii) of Deﬁnition 1.1 hold. Second, we can use the M. = ωM g-action on M to deﬁne a symplectic g-action on (M, = ) by ξM =)−1 ξM (= p (z)) = (z) := (T(z) p

for any ξ ∈ g and

= z ∈ M.

(2.1)

This is a good deﬁnition since p = is a covering map and hence a local diffeomorphism. = × g → ξM = is clearly smooth. Note that by Moreover, the map (z, ξ ) ∈ M = (z) ∈ T M deﬁnition, the vector ﬁelds ξM =-related for all ξ ∈ g. This immediately = and ξM are p shows that [ξ, η]M =∗ ω = = = −[ξM = , ηM = ] for any ξ, η ∈ g and that £ξM = = £ξM = ωM =p ∗ p = £ξM ω = 0 for any ξ ∈ g. Thus expression (2.1) deﬁnes a symplectic action of g = ωM on (M, = is equivariant by construction. Finally, the g-action = ) relative to which p = = → g∗ because M = is simply connected and = :M on M admits a momentum map K 1 = therefore H (M, R) = 0; we recall from [We77] that the canonical action of a Lie algebra h on the symplectic manifold (S, ω) admits an associated momentum map if and only if the linear map [ξ ] ∈ h/[h, h] → [iξS ω] ∈ H 1 (S, R) is identically zero. = → M, ωM = into Thus conditions (iv) and (v) also hold, which makes (= p:M = , g, [K]) an object of H. Proposition 2.1. Let (M, ω) be a connected symplectic manifold and g be a Lie = → M, ωM = be the object in H algebra acting symplectically on it. Let (= p:M = , g, [K]) constructed above using a simply connected universal covering of M. Then for any =→N other object (pN : N → M, ωN , g, [KN ]) of H, there exists a morphism q : M in Mor(H). Any other object in H that satisﬁes the same universality property is = → M, ωM = isomorphic to (= p:M = , g, [K]). = is the universal covering space of M, there exists a smooth covering Proof. Since M = → M (in general not unique) such that pN ◦ q = p map q : M =. We shall prove that = are symplectic maps, we have this is a morphism in H. Indeed, since pN and p ∗ ωM =∗ ω = (pN ◦ q)∗ ω = q ∗ pN ω = q ∗ ωN , = =p

so condition (i) in Deﬁnition 1.2 is satisﬁed. Additionally, since p = and pN are g= and ξ ∈ g, equivariant, we have, for any z ∈ M

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Tq(z) pN (Tz q(ξM =(ξM p (z)) = ξM (pN (q(z))) = (z))) = Tz p = (z)) = ξM (= = Tq(z) pN (ξN (q(z))). Since Tq(z) pN is an isomorphism it follows that Tz q(ξM = (z)) = ξN (q(z)) and so (ii) is satisﬁed. To verify (iii) it sufﬁces to note that KN ◦ q is a momentum map for the = g-action on M. = → M, ωM = ]) In order to prove the last sentence in the statement let (= p : M = , g, [K = be another object in H satisfying the same universality property as (= p : M → = → M = and q : M = → M = be the corresponding = Let q : M M, ωM = , g, [K]). morphisms. Since both q and q are symplectic covering maps their composition =→M = is also a symplectic covering map (see [Sp66, Theorems 3, 5, and 6 q ◦ q : M in Section 2.2 and Theorem 10 in Section 2.4]). Thus q ◦ q is a local diffeomorphism. = is simply connected, this map is also injective [Sp66, Theorem 9, p. 73]. Since M Consequently, ϕ := q ◦ q is a bijective local diffeomorphism, hence a diffeomorphism. Finally, this proves that both q and q are isomorphisms in H with inverses ϕ −1 ◦ q and q ◦ ϕ −1 , respectively.

= → Remark 2.2. It should be noticed that the universality property for (= p : M = M, ωM = , g, [K]) stated in the previous proposition does not imply that this is an initial object in H due to the nonuniqueness of the morphism q. This is in agreement with the situation encountered for general manifolds.

3 The universal Hamiltonian covered space in H In this section we will construct an object in the category H deﬁned in the introduction using a principal connection introduced by Condevaux, Dazord, and Molino in [CDM88, Section I.3.1]. In the next section we will prove that this object has the universality property stated in the introduction to deﬁne the universal Hamiltonian covered space. The setup is identical to the one in the introduction, but from now on we will assume that M is also paracompact. The connection α Let π : M × g∗ → M be the projection onto M. Consider π as the bundle map of the trivial principal ﬁber bundle (M × g∗ , M, π, g∗ ) that has (g∗ , +) as Abelian structure group. The group (g∗ , +) acts on M × g∗ by ν · (m, µ) := (m, µ − ν), with m ∈ M and µ, ν ∈ g∗ . Let α ∈ 1 (M × g∗ , g∗ ) be the connection one-form deﬁned by α(m, µ)(vm , ν), ξ := (iξM ω)(m)(vm ) − ν, ξ ,

(3.1)

where (m, µ) ∈ M × g∗ , (vm , ν) ∈ Tm M × g∗ , ξ ∈ g, and ·, · denotes the natural pairing between g∗ and g. We brieﬂy check that α is indeed a connection one-form on M × g∗ . Notice that the inﬁnitesimal generator νM×g∗ associated to an element ν ∈ g∗ is given by

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575

νM×g∗ (m, µ) = (0, −ν). Consequently, for any ξ ∈ g, α(m, µ)(νM×g∗ (m, µ)), ξ = α(m, µ)(0, −ν), ξ = ν, ξ , that is, α(m, µ)(νM×g∗ (m, µ)) = ν. Also, it is obvious that for any ρ ∈ g∗ we have α(m, µ − ρ)(vm , ν), ξ = (iξM ω)(m)(vm ) − ν, ξ = α(m, µ)(vm , ν), ξ , hence α is a well deﬁned connection one-form on M × g∗ . Horizontal and vertical bundles of α By deﬁnition, the horizontal subspace H (m, µ) at the point (m, µ) determined by α is given by H (m, µ) = {(vm , ν) ∈ T(m,µ) (M × g∗ ) | (iξM ω)(m)(vm ) − ν, ξ = 0 ∀ξ ∈ g}. (3.2) Consequently, given any vector (vm , ν) ∈ T(m,µ) (M × g∗ ), its horizontal (vm , ν)H and vertical (vm , ν)V parts are such that (vm , ν)H = (vm , ρ)

and

(vm , ν)V = (0, ρ ),

where ρ, ρ ∈ g∗ are uniquely determined by the relations ρ, ξ = (iξM ω)(m)(vm )

and ρ = ν − ρ

for any ξ ∈ g.

α is a ﬂat connection We compute the curvature form associated to α. Let (m, µ) ∈ M × g∗ , vm , um ∈ Tm M, ξ ∈ g, and ν, ρ ∈ g∗ arbitrary. By deﬁnition, (m, µ)((vm , ν), (um , ρ)), ξ = dα(m, µ)((vm , ν)H , (um , ρ)H ), ξ .

(3.3)

Now let (X1 , Y1 ) and (X2 , Y2 ) be vector ﬁelds on M ×g∗ such that (X1 (m), Y1 (µ)) = (vm , ν) and (X2 (m), Y2 (µ)) = (um , ρ). Using these vector ﬁelds, the right-hand side of (3.3) can be rewritten as (X1 , Y1 )[α(X2 , Y2 )](m, µ), ξ − (X2 , Y2 )[α(X1 , Y1 )](m, µ), ξ − α([X1 , X2 ], 0)(m, µ), ξ .

(3.4)

Let (m1t , µ1t ) and (m2t , µ2t ) be the ﬂows of (X1 , Y1 ) and (X2 , Y2 ), respectively. We choose Y1 and Y2 such that their ﬂows are given by µ1t (µ) = µ + tν and µ2t (µ) = µ + tρ. We can use these ﬂows to compute (X1 , Y1 )[α(X2 , Y2 )](m, µ), ξ d = α(m1t (m), µ1t (µ))(X2 (m1t (m)), Y2 (µ1t (µ))), ξ dt t=0 d = ((iξM ω)(m1t )(X2 (m1t (m)) − Y2 (µ1t (µ)), ξ ) dt t=0 d d = X1 [iξM ω(X2 )](m) − µ + tν + sρ, ξ dt t=0 ds s=0

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= X1 [iξM ω(X2 )](m). Analogously, we have (X2 , Y2 )[α(X1 , Y1 )](m, µ), ξ = X2 [iξM ω(X1 )](m). Consequently, the expression (3.4) equals X1 [iξM ω(X2 )](m) − X2 [iξM ω(X1 )](m) − iξM ω(m)([X1 , X2 ](m)) = d(iξM ω)(m)(X1 (m), X2 (m)) = (£ξM ω)(m)(X1 (m), X2 (m)) − (iξM dω)(m)(X1 (m), X2 (m)) = 0, which guarantees the ﬂatness of α. Holonomy bundles of α The ﬂatness of α implies that the associated horizontal distribution is integrable and → that its maximal integral leaves coincide with the holonomy bundles ( p : M (the M, H) of α, where H is the holonomy group of α based at any point of M paracompactness of M is used at this point in the proof [KN63, Theorem 7.1, p. 83] → M, H) is a reduction of the bundle (π : M × g∗ → M, g∗ )). Notice that ( p:M 1 and M 2 are isomorphic that since (g∗ , +) is Abelian, any two holonomy bundles M 2 → M 1 as principal bundles with the same structure group H, via the map Rτ : M 2 . deﬁned by Rτ (m, µ) := (m, µ + τ ), for some ﬁxed τ ∈ g∗ and for any (m, µ) ∈ M The group H will be referred to as the holonomy group of the g-action. It is easy to prove that the g-action on (M, ω) admits a standard momentum map if and only if the holonomy group of the action H is trivial. are initial A fact that will be important later on is that the holonomy bundles M ∗ submanifolds of M × g , that is, they satisfy the following universality property: the → M × g∗ is a smooth immersion such that for any manifold Z, a inclusion i : M is smooth if and only if i ◦ f : Z → M × g∗ is smooth. given mapping f : Z → M The initial submanifold property is satisﬁed by the maximal integral leaves of any smooth integrable distribution, such as the horizontal distribution in our case. The holonomy bundles of α are Hamiltonian coverings of (M, ω, g) We now prove the following proposition. Proposition 3.1. Let (M, ω) be a connected paracompact symplectic manifold and let g be a Lie algebra acting symplectically on it. Let α be the connection on the trivial → M, H) be one of bundle (π : M × g∗ → M, g∗ ) introduced in (3.1) and ( p:M ∗ ωM its holonomy bundles. If we deﬁne ωM ω, then the pair (M, := p ) is a symplectic manifold acted symplectically upon by the Lie algebra g via the expression ξM (m, µ) := (ξM (m), −#(m)(ξ, ·)) for any ξ ∈ g and (m, µ) ∈ M.

(3.5)

Universal covering and covered spaces of a symplectic Lie algebra action

577

The symbol # denotes the Chu map [Ch75] # : M → Z 2 (g), deﬁned by #(m)(ξ, η) := ω(m)(ξM (m), ηM (m)) for any ξ, η ∈ g, m ∈ M. Finally, the projec → g∗ of M onto g∗ is a momentum map for this action. Moreover, the :M tion K is an object in the category H introduced in four tuple ( p : M → M, ωM , g, [K]) Deﬁnition 1.2. → M is a smooth covering Proof. We start by noticing that the projection p : M projection as a consequence of the ﬂatness of α. Indeed, since the connection is ﬂat, the Ambrose–Singer theorem (Theorem [AS53]) implies that the Lie algebra Lie(H) of the holonomy group is trivial and hence H is a discrete (possibly not closed) subgroup → M, H) is a locally trivial bundle, any point m ∈ M has of (g∗ , +). As ( p :M an open neighborhood U such that p −1 (U ) is diffeomorphic to U × H. Since H is discrete, each subset U × {µ}, µ ∈ H, is an open subset diffeomorphic to U . Hence p is a covering map. Now, as p is a local diffeomorphism, the equality ωM ∗ ω deﬁnes a symplectic := p with respect to which p form on M is a symplectomorphism. We have hence shown → M satisﬁes properties (i), (ii), and (iii) in Deﬁnition 1.1. that p : M by We now deﬁne a g-action on M ξM )−1 ξM (m) for any ξ ∈ g and (m, µ) := (T(m,µ) p

(m, µ) ∈ M.

(3.6)

This is a good deﬁnition since p is a covering map and hence a local diffeomorphism. × g → ξM is clearly smooth. Moreover, the map ((m, µ), ξ ) ∈ M (m, µ) ∈ T M Note that, by deﬁnition, the vector ﬁelds ξM and ξ are p -related for all ξ ∈ g. This M = −[ξ , η ] for any ξ, η ∈ g and that £ξM ωM immediately shows that [ξ, η]M M = M £ξM p ∗ ω = p ∗ £ξM ω = 0 for any ξ ∈ g. Thus expression (3.6) deﬁnes a symplectic ωM action of g on (M, ). We now show that (3.6) can be rewritten as (3.5), that is, ξM (m, µ) := (ξM (m), −#(m)(ξ, ·))

for any ξ ∈ g and

(m, µ) ∈ M.

(3.7)

We start by checking that the right-hand side of this expression is a horizontal vector which means that α(m, µ)(ξM with respect to α and thereby tangent to M, (m, µ)), η = 0 for any η ∈ g. By the deﬁnition of α, we have that α(m, µ)(ξM (m, µ)), η = (iηM ω)(m)(ξM (m)) + #(m)(ξ, ·), η = ω(m)(ηM (m), ξM (m)) + ω(m)(ξM (m), ηM (m)) = 0. Consequently, (ξM (m), −#(m)(ξ, ·)) is horizontal, and therefore it sufﬁces to notice that p is the projection onto M to prove the equivalence between (3.6) and (3.7). The → M. same remark proves the g-equivariance of p : M → g∗ of M onto g∗ is : M We conclude by showing that the projection K a momentum map for the g-action on M deﬁned in (3.5). Let ξ ∈ g be arbi ξ := K, ξ . On one hand we have that dK ξ (m, µ)(vm , ν) = ν, ξ trary and K for any (m, µ) ∈ M and any (vm , ν) ∈ T(m,µ) M = H (m, µ). On the other hand, iξM ωM p ∗ ω)(m, µ)(vm , ν) = ( p ∗ ω)(m, µ)(ξM (m, µ)(vm , ν) = iξM (m, µ), ( ∗ (vm , ν)) = ( p ω)(m, µ)((ξM (m), −#(m)(ξ, ·)), (vm , ν)) = ω(m)(ξM (m), vm ) = ν, ξ , which proves the claim.

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is not equivariant in general. Indeed, its inﬁnitesRemark 3.2. The momentum map K imal nonequivariance cocycle is given by [ξ,η] (m, µ) − {K ξ , K η }(m, µ) (ξ, η) := K = µ, [ξ, η] − ( p∗ ω)(m, µ)(ξM (m, µ), ηM (m, µ)) = µ, [ξ, η] − ω(m)(ξM (m), ηM (m))

(3.8)

= µ, [ξ, η] − #(m)(ξ, η) used to for any ξ, η ∈ g. The value of does not depend on the point (m, µ) ∈ M the function f (m, µ) := µ, [ξ, η] − deﬁne it because for any (vm , ν) ∈ T(m,µ) M #(m)(ξ, η) is such that df (m, µ)(vm , µ) = ν, [ξ, η] − Tm #(vm )(ξ, η) = ν, [ξ, η] − ω(m)([ξ, η]M (m), vm ) = 0, where we used the horizontality of (vm , ν) in the last equality. The connectedness of concludes the argument. M

4 The universality theorem In this section we state and prove the main result of the paper. Theorem 4.1. Let (M, ω) be a connected paracompact symplectic manifold and g → a Lie algebra acting symplectically on it. The Hamiltonian covering ( p : M M, ωM , g, [K]) constructed in Proposition 3.1 is a universal Hamiltonian covered space in the category H of Hamiltonian covering maps, that is, given any other object (pN : N → M, ωN , g, [KN ]) in H, there exists a (not necessarily unique) morphism in Mor(H). Any other object of H that satisﬁes this universality property q:N →M → M, ωM is isomorphic to ( p:M , g, [K]). Proof. Let (pN : N → M, ωN , g, [KN ]) ∈ H and n0 ∈ N. Deﬁne m 0 := (pN (n0 ), KN (n0 )) ∈ M × g∗ . Since M × g∗ is foliated by the holonomy bundles . Let τ ∈ g∗ of the connection α in (3.1), the point m 0 lies in one of them, say M be such that M = Rτ (M) and deﬁne K N := KN − τ . The map K N : N → g∗ is also a momentum map for the g-action on N , [K N ] = [KN ], and, moreover, Hence we can assume without loss of generality that (pN (n0 ), K N (n0 )) ∈ M. Using this (pN : N → M, ωN , g, [KN ]) is such that (pN (n0 ), KN (n0 )) ∈ M. ∗ choice we deﬁne the map g : N → M × g by n −→ (pN (n), KN (n)), n ∈ N . We start by proving that Tn g(vn ) ∈ We will now show that g(N) ⊂ M. H (pN (n), KN (n)) for all n ∈ N and vn ∈ Tn N . Indeed, since Tn g(vn ) = (Tn pN (vn ), Tn KN (vn )), we have for any ξ ∈ g α(g(n))(Tn g(vn )), ξ = ω(pN (n))(ξM (pN (n)), Tn pN (vn )) − Tn KN (vn ), ξ ξ

= ω(pN (n))(Tn pN (ξN (n)), Tn pN (vn )) − dKN (n)(vn )

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579

ξ

= ωN (n)(ξN (n), vn ) − dKN (n)(vn ) = 0, where we used the g-equivariance and the symplectic character of pN . Let now n ∈ N be arbitrary. As N is connected, there exists a smooth curve c : [0, 1] → N such that c(0) = n0 and c(1) = n. Since the derivative T g of g maps into the horizontal bundle of α, the chain rule implies that g(c(t)) is a horizontal curve starting at g(c(0)) = Hence by the deﬁnition of the holonomy bundle, g(c(1)) = g(n) ∈ M. g(n0 ) ∈ M. This argument and the arbitrary character of n ∈ N show that g(N) ⊂ M. be the map obtained from g by restriction of the range. We will Let q : N → M show that q is the morphism needed to prove the statement of the theorem. First, the is an initial submanifold of M ×g∗ . Second, map q is smooth since g is smooth and M we verify that q satisﬁes the three conditions in Deﬁnition 1.2 that characterize an element in Mor(H). → M are (i) q is a symplectic covering projection: Since pN : N → M and p : M is a covering covering projections and p ◦ q = pN it follows that q : N → M are symplectic so is q. projection [Sp66, Lemma 1, p. 79]. Since pN and p (ii) q is g-equivariant: Let ξ ∈ g, n ∈ N be arbitrary. On one hand, ξM (q(n)) = ξM (pN (n), KN (n)) = (ξM (pN (n)), −#(pN (n))(ξ, ·)). On the other hand, Tn q(ξN (n)) = (Tn pN (ξN (n)), Tn KN (ξN (n))) = (ξM (pN (n)), Tn KN (ξN (n))). Consequently, the map q is g-equivariant if and only if Tn K(ξN (n)) = −#(pN (n))(ξ, ·). This identity holds because for any η ∈ g, we have η

Tn KN (ξN (n)), η = dKN (n)(ξN (n)) = ωN (n)(ηN (n), ξN (n)) ∗ = (pN ω)(n)(ηN (n), ξN (n))

= ω(pN (n))(Tn pN (ηN (n)), Tn pN (ξN (n))) = ω(pN (n))(ηM (pN (n)), ξM (pN (n))) = −#(pN (n))(ξ, η). ◦ q = KN by (iii) The diagram in Deﬁnition 1.2 commutes since p ◦ q = pN and K the deﬁnition of q. → M, ωM ]) that We conclude by showing that any other object ( p : M , g, [K satisﬁes the just proved universality property of ( p : M → M, ωM , g, [K]) is nec → essarily isomorphic to it. Indeed, the universality property satisﬁed by ( p : M M, ωM p : M → M, ωM , g, [K]) and ( , g, [K ]) implies the existence of two mor and q : M → M →M in Mor(H) and of an element τ ∈ g∗ such phisms q : M that the following diagram commutes:

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*

g∗ YH 6 H H H H K HHK H

+ τ K HH q q - M - M M H HH H H HH p p p H HH j ? M →M (see [Sp66, Since q and q are covering maps so is the composition q ◦ q : M Theorems 3, 5, and 6 in Section 2.2 and Theorem 10 in Section 2.4]). Thus q ◦ q is a local surjective diffeomorphism. We now show that it is also injective. If (m, µ) ∈ M, the deﬁnition of q and the commutativity of the diagram above yield (q (m, µ))) (q ◦ q )(m, µ) = ( p (q (m, µ)), K = ( p (m, µ), K(m, µ) + τ ) = (m, µ + τ ).

(4.1)

satisfy (q ◦q )(m, µ) = (q ◦q )(m , µ ), then (4.1) imHence if (m, µ), (m , µ ) ∈ M plies that (m, µ) = (m , µ ). Consequently, ϕ := q ◦ q is a bijective local diffeomorphism and hence a diffeomorphism. This proves that both q and q are isomorphisms is H.

→ Remark 4.2. It should be noticed that the universality property for ( p : M M, ωM , g, [ K]) stated in the theorem does not imply that it is a ﬁnal object in H due to the nonuniqueness of the morphism q. Remark 4.3. One could consider larger categories than H in which case the universality result in Theorem 4.1 would be weaker. For example, if we drop the condition that pN : N → M is a covering map in the deﬁnition of the objects of H then the is not necessarily a covering map. morphism q : N → M Example 4.4. We shall illustrate the difference between the universal Hamiltonian covering and covered spaces by considering the following elementary example. Let T2 = {(eiθ1 , eiθ2 )} be the two-torus considered as a symplectic manifold with its area form ω := dθ1 ∧ dθ2 and the circle S 1 = {eiφ } acting canonically on it by eiφ · (eiθ1 , eiθ2 ) := (ei(θ1 +φ) , eiθ2 ). Proposition 2.1 guarantees that the universal covering space R2 of T2 can be endowed with the necessary structure to make it the universal Hamiltonian covering space of (T2 , ω). On the other hand, a straightforward computation shows that, in this case, the horizontal vectors in T (T2 ×R) with respect to the connection α deﬁned in (3.1) are of

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G2 τ := {((eiθ1 , eiθ2 ), τ + θ2 ) ∈ the form ((a, b), b), with a, b ∈ R. Since any surface T 2 T × R | θ1 , θ2 ∈ R} integrates the horizontal distribution, it is immediately clear that the universal Hamiltonian covered space is given in this example by any of the G2 τ . cylinders T Acknowledgments We thank James Montaldi and Alan Weinstein for several illuminating discussions. This research was partially supported by the European Commission and the Swiss Federal Government through funding for the Research Training Network Mechanics and Symmetry in Europe (MASIE). Partial support of the Swiss National Science Foundation is also acknowledged.

References [AS53]

W. Ambrose and I. M. Singer [1953], A theorem on holonomy, Trans. Amer. Math. Soc., 75, 428–443. [Ch75] R. Y. Chu [1975], Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197, 145–159. [CDM88] M. Condevaux, P. Dazord, and P. Molino [1988], Géométrie du moment: Travaux du séminaire sud-Rhodanien de géométrie I, Publ. Dépt. Math. Univ. Claude Bernard Lyon I (N.S. B), 88-1, 131–160. [KN63] S. Kobayashi and K. Nomizu [1963], Foundations of Differential Geometry, Vol. I, Interscience Tracts in Pure and Applied Mathematics, Vol. 15, Wiley, New York. [Sp66] E. H. Spanier [1966], Algebraic Topology, McGraw–Hill, New York; reprinted by Springer-Verlag. [We77] A. Weinstein [1977], Lectures on Symplectic Manifolds, CBMS Conference Series, Vol. 29, American Mathematical Society, Providence, RI.

Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraic topics related to Alan’s interests∗ Jim Stasheff Department of Mathematics University of North Carolina at Chapel Hill Chapel Hill, NC 27599 USA [email protected] Dedicated to Alan Weinstein on his 60th birthday. Abstract. Homotopy algebra is playing an increasing role in mathematical physics. Especially in the Hamiltonian and Lagrangian settings, it is intimately related to some of Alan’s interests, e.g., Courant and Lie algebroids. There is a comparatively long history of such structure in cohomological physics in terms of equations that hold mod exact terms (typically, divergences) or only “on shell,’’ meaning modulo the Euler–Lagrange equations of “motion’’; more recently, higher homotopies have come into prominence. Higher homotopies were developed ﬁrst within algebraic topology and may not yet be commonly available tools for symplectic geometers and mathematical physicists. This is an expanded version of my talk at Alanfest, planned as a gentle introduction to the basic point of view with a variety of applications to substantiate its relevance. Most technical details are supplied by references to the original work or to [MSS02].

1 Introduction Cohomological physics is a phrase I introduced some time ago in the context of anomalies in gauge theory, but it all began with Gauss’s invention of the (elctromagnetic or asteroidal) linking number in 1833 (or even earlier—see Kirchoff’s laws—as Nash’s “Topology and physics: A historical survey’’ [Nas98] reminded me). The cohomology referred to in Gauss was that of differential forms, div, grad, curl and especially Stokes’s theorem (the de Rham complex).

2 Basic concepts: DG space of states, maps, and homotopies Whatever “states’’ are physically, mathematically it is crucial that they form a vector space, in fact, usually a Hilbert or pre-Hilbert space. In cohomological physics, the ∗ This research was supported in part by NSF Focussed Research Grant DMS 0139799.

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physical space of states H is the (co)homology of a dg (differential graded) vector space, V = ⊕V n , d : V n → V n+1 with d 2 = 0. (As is more common in physics, we have adopted the cohomological conventions with the grading as a superscript and the differential of degree 1. Of course there is a corresponding theory with differentials of degree −1 for which we would indicate the grading with a subscript. These conventions are equivalent just by raising/lowering: V n ⇔ V−n .) The space H is often considered as a subspace of the dg vector space by some (implicit) choice of representatives. In physical language, this might be referred to as gauge ﬁxing. Although much of physics is phrased in terms of manifolds and even analysis, my point of view is almost entirely (differential graded) algebraic, e.g., think of an algebra of observables without considering them as functions. Maps (morphisms) of dg vector spaces f : (V , dV ) → (W, dW ) are linear maps of degree 0 which respect the differentials: dW f = f dV . For cochains/differential forms on topological spaces/manfolds, maps of spaces induce morphisms of the dg vector spaces. Assuming the differentials are of degree 1, a homotopy between two such maps is a linear map h : (V , dV ) → (W, dW ) of degree −1 such that f − g = dW h + hdV . Homotopies in the topological or smooth sense induce such dg homotopies. Notice, when applied to cocycles (representatives of physical states), f = g mod exact terms. Higher homotopies refer to homotopies of homotopies, and so on. For example, for given f and g with two such homotopies h and k as above, a second level homotopy is a linear map h2 : (V , dV ) → (W, dW ) of degree −2 such that h − k = d W h2 − h 2 dV . In full generality, higher homotopies refer to a family of linear maps hn : (V , dV ) → (W, dW ) of degree −n such that dW hn − (−1)n hn dV satisﬁes some relation among the hi for i < n. It is time to look at examples.

3 A∞ -structure The examples currently most relevant to physics and Poisson structure are those of A∞ - and L∞ -structures [MSS02]. The topological version of A∞ -structure came ﬁrst and is the easiest to visualize. It also has an obvious relevance to open string ﬁeld theory (OSFT) [Kaj03] as L∞ -structure has to closed string ﬁeld theory (CSFT) [Zwi93]. Consider the space of based loops X on a space X with base point ∗. That is, a based loop is a map λ of the unit interval I into X such that λ(0) = ∗ = λ(1). Because we deﬁne the “product’’ of two loops by reparameterizing the result of following one loop by another, this product is only homotopy associative. There are ﬁve ways of parenthesizing the product of four loops, which results in a pentagon of loops, where the sides represent a single application of a speciﬁc

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associating homotopy h(a, b, c) from a(bc) to (ab)c for any three based loops: a, b, c. For example, the bottom edge from left to right of Figure 1 is given by h(a, bc, d). By looking at the parameterizations in more detail, it can be seen that the pentagon can be ﬁlled in by a two-parameter family of loops. (ab)(cd) •Z

Z

Z

Z

Z

a(b(cd)) •

B

B

B

B

a((bc)d)

B

B B•

•

Z• ((ab)c)d

(a(bc))d

Fig. 1. The pentagon K4 .

Now there are 14 ways of parenthesizing the product of ﬁve loops, and so on. The combinatorics, in general for n-loops, can be realized in terms of a polyhedron, called an associahedron and denoted by Kn , described as a convex polytope with one vertex for each way of associating n ordered variables, that is, ways of inserting parentheses in a meaningful way in a word of n letters. For n = 5, a portrait due to Masahico Saito is in Figure 2: a rotatable image is available at http://www.labmath.uqam.ca/˜chapoton/stasheff.html. An A∞ -space Y is a topological space Y together with a family of maps mn : Kn × Y n → Y (where Y n is the n-fold topological product of Y with itself) which ﬁt together in a way suggested by the pentagon. (In technical language, these maps make Y an algebra over the non- operad K = {Kn }n≥1 .) The main result is the following theorem. Theorem 3.1. A connected space Y (of the homtopy type of a CW-complex with a nondegenerate base point) has the homotopy type of a based loops space X for some space X if and only if Y admits the structure of an A∞ -space. Now consider a chain complex functor from topological spaces to dg modules (over a commutative ground ring) which respects products. Applied to an A∞ -space, such a functor reveals an algebraic structure generalizing that of a dg associative algebra. Deﬁnition 3.1. An A∞ -algebra (or strongly homotopy associative algebra) consists of a graded module V with maps

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•X XX XX X• • A B A B B A B A B• • A L A • • L H • C H @ L C HH @ H C H @ • • L C @ L C @ L C• a @• !L• aa ! aa !! aa !!! a•! Fig. 2. Saito’s portrait of K5 .

mn : V ⊗n → V of degree 2 − n, satisfying suitable compatibility conditions (An )n≥1 . In particular, (A1 ) m1 = d is a differential; (A2 ) m = m2 : V ⊗ V → V is a chain map, that is, d is a derivation with respect to m = m2 ; (A3 ) m3 : V ⊗3 → V is a chain homotopy for associativity of the multiplication m, i.e., m3 d ⊗3 + dm3 = m(m ⊗ 1) − m(1 ⊗ m), where d ⊗3 denotes d ⊗ 1 ⊗ 1 + 1 ⊗ d ⊗ 1 + 1 ⊗ 1 ⊗ d; (A4 ) m4 is a “higher homotopy’’such that m4 d ⊗4 −dm4 has ﬁve terms, corresponding to the edges of the pentagon K4 : m4 d ⊗4 −dm4 = m3 (m2 ⊗1⊗1−1⊗m2 ⊗1+1⊗1⊗m2 )−m2 (m3 ⊗1+1⊗m3 ). An alternate formulation generalizes the bar construction on an associative differential graded algebra. Deﬁne the suspension sA of a graded vector space A by shifting the grading down: (sA)n = An+1 . Alternate Deﬁnition 3.2. An A∞ -algebra structure on a positively graded vector space A is equivalent to a coderivation differential δ of degree 1 with respect to the total grading on the tensor coalgebra Tc (sA) on the suspension of the graded vector space A. As a coderivation, δ is determined by the formula δ = δ1 + δ2 + · · · , where δn (sa1 ⊗ · · · ⊗ san ) := · smn (a1 ⊗ · · · ⊗ an ) for a1 , . . . , an ∈ A and is an appropriate sign.

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4 L∞ -structure Although there was no topological predecessor, the notion of an L∞ -algebra (also known as a strong homotopy Lie or sh-Lie algebra) follows a similar pattern algebraically. 4.1 A Lie algebra is… A grad student in mathematics is likely to encounter the following. Deﬁnition 4.1. A Lie algebra is a vector space g with a skew-bilinear operation [ , ] : g ⊗ g → g satisfying the Jacobi identity. Physicists are more likely to assume a basis {Xi } for g and write [Xi , Xj ] = Cijk Xk with the structure constants Cijk being skew-symmetric and satisfying the Jacobi identity. There is another alternative that plays a key role in the homological study of Lie algebras. Consider the familiar exterior (=Grassmann) algebra Eg except think of it as a coalgebra. That is, deﬁne : Eg → Eg ⊗ Eg by (X1 ∧ · · · ∧ Xn ) =

(σ )Xi1 ∧ · · · ∧ Xip ⊗ Xip+1 ∧ · · · ∧ Xin ,

where the summation is over all 0 ≤ p ≤ n and all permutations σ = (i1 , . . . , in ), (σ ) being the sign of the permutation. Grade Eg by wedge degree and deﬁne a coderivation δ : Eg → Eg by δ(X ∧ Y ) = [X, Y ] extended as a coderivation. The Jacobi identity is equivalent to the condition δ 2 = 0. This dg coalgebra is the Chevalley–Eilenberg chain complex for Lie algebra homology. Many will be more familiar with the dual algebra and dual derivation, which is the standard Chevalley–Eilenberg coboundary and a special example of a BRST operator. Now I want to up the ante and consider dg (differential graded) Lie algebras. For a dg Lie algebra, the classical Chevalley–Eilenberg chain complex needs a slight adjustment to account for the grading. Deﬁnition 4.2. A dg Lie algebra is a graded vector space g = ⊕gn with a differential d : gn → gn+1 with d 2 = 0 and a graded-skew bilinear operation [ , ] : gp ⊗ gq → gp+q , which is a chain map satisfying the graded Jacobi identity.

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Following our example of having associativity satisﬁed only up to homotopy, we can do the same for the Jacobi identity and then consider higher homotopies. That is, the Jacobiator is not zero but is exact, i.e., a boundary. One can give the full deﬁnition in terms of a family of maps ln = [−, . . . , −] : g⊗n → g, which are suitably “skew symmetric’’ and compatible, but it is deﬁnitely a lot simpler to give the deﬁnition in terms of the graded symmetric coalgebra sg. The notation c for the graded symmetric coalgebra is well established in rational homotopy theory; the alternative S c is used elsewhere. Deﬁnition 4.3. An L∞ -algebra structure on a graded vector space g is equivalent to a coderivation differential δ of degree 1 with respect to the total grading on the graded symmetric coalgebra c (sg) on the suspension of the graded vector space g. Note that if we write d := −sl1 and B := δ − d, then the condition that δ is a differential, i.e., δ 2 = 0, can be written in (generalized) Maurer–Cartan form dB + Bd + 1/2[B, B] = 0, where [ , ] denotes the commutator of coderivations. If you would like some hands-on examples, consider very small ﬁnite-dimensional L∞ -algebras. There are two versions of the classiﬁcation, depending on whether we consider L∞ -algebras in the original Z-graded sense or the super, i.e., = Z/2Zgraded, sense. There are classiﬁcations here by, respectively, Daily [Dai02] or Fialowski–Penkava [FP03] for very small-dimensional examples. In the Z-graded situation, particularly important are the cases considered below with g−1 → g0 and g0 → g1 . The ﬁrst of these, but with d of degree −1, i.e., g1 → g0 , are considered extensively and categorically as Lie 2-algebras by Baez in a recent arXiv preprint [BC03]. Now what does this have to do with physics or symplectic geometry or Alan’s interests? The answers include moment maps, symplectic reduction, and Courant algebroids. 4.2 Courant algebroids The most straightforward connection with Alan’s interests appears in his paper with Roytenberg [RW98] on Courant algebroids. There is no point in repeating the very clear exposition in their original paper, so I will mention only the salient facts in re: higher homotopies. Deﬁnition 4.4. A Courant algebroid is a vector bundle E → M equipped with a nondegenerate bilinear form ·, · on the bundle, a skew-symmetric bracket [·, ·] on the space of sections (E) and a bundle map a : E → T M satisfying ﬁve properties. Kosmann-Schwarzbach has simpliﬁed the deﬁnition to two properties, but in the non-skew-symmetric form [KS03].

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The skew bracket does not in general satisfy the Jacobi identity, but property number 5 addresses the defect in terms of a map D : C ∞ (M) → (E) related to the deRham differential on M. Roytenberg and Weinstein consider the following small complex: X1 = C ∞ (M) → X0 = (E) (1) with the differential l1 given by D. They deﬁne a very speciﬁc L∞ -structure as follows: • • •

l2 is given by the bracket on X0 ⊗ X0 and by ·, D· on X0 ⊗ X1 and 0 in higher degrees, l3 is given by one third of the sum of the cyclic permutations of [·, ·], · on X0 ⊗ X0 ⊗ X0 and 0 otherwise, while ln for n > 3 is 0.

(An alternate nomenclature would refer to such an algebra as an L3 -algebra.) Of course, many of these 0s follow just from the fact that Xn = 0 for n ≥ 1, in contrast to the L∞ -structures that appear in the work of Fulp, Lada, and myself concerning higher spin algebras [FLS02b, FLS02a] where the complex is of the form X0 → X 1 (see Section 6.2 below).

5 Homological reduction of constrained Poisson algebras Cohomological physics had a major breakthrough with the “ghosts’’ introduced by Fade’ev and Popov [FP67]. These were incorporated into what came to be known as BRST cohomology (Becchi–Rouet–Stora [BRS75] and Tytutin [Tyu75]) and which was applied to a variety of problems in mathematical physics. There the ghosts were reinterpreted by Stora [Sto77] and others in terms of the Maurer–Cartan forms in the case of a ﬁnite-dimensional Lie group and more generally as generators of the Chevalley–Eilenberg cochain complex [CE48] for Lie algebra cohomology. If, as geometers, you feel more comfortable with manifolds, one can make the following algebra seem more palatable as functions on “supermanifolds,’’ but most (all?) of the work is just algebraic (homological). Warning! The term “BRST cohomology’’ has a variety of meanings in the existing literature. From time to time, it threatens to be used for any cohomology in physics, at least if the coboundary operator is called “Q.’’At other times, it refers (only implicitly) to the case in which the Lie algebra is the Virasoro algebra. I prefer to reserve the term for situations in which the coboundary operator has at least some part corresponding to that of Chevalley–Eilenberg. Such is the case for the ghost technology for the cohomological reduction of constrained Poisson algebras, introduced by Batalin, Fradkin and Vilkovisky [BF83, FF78, FV75], which extended the complex of BRST by adjoining odd generators, called ghosts and antighosts, thus reinventing the Koszul–Tate [Tat57] resolution of the ideal of constraints and producing a synergistic combination of both Chevalley– Eilenberg and resolution cohomology. Here it was that I saw the essential features of a strong homotopy Lie algebra (L∞ -algebra).

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5.1 Moment maps, momaps, and symplectic reduction The setting is one of Alan’s favorites [Wei02], that of a moment map, though generalized in an important way. Alan considers a phase space with a symmetry group consists of a manifold P equipped with a symplectic structure ω and a hamiltonian action of a Lie group G. By the latter, we mean a symplectic action of G on P together with an equivariant moment map J from P to the dual g∗ of the Lie algebra of G such that, for each v ∈ g, the one-parameter group of transformations of P generated by v is the ﬂow of the hamiltonian vector ﬁeld with hamilitonian x →< J (x), v >. The map J is called the momentum map (or, by many authors, moment map) of the hamiltonian action. If one is simply given a symplectic action of G on P , any map J satisfying the condition in italics above, even if it is not equivariant, is called a momentum map for the action. By contrast, Batalin–Fradkin–Vilkovisky consider constraints on the symplectic manifold P to be paramount. A Hamiltonian system with constraints means we have functions φα : P → R, 1 ≤ α ≤ r, the constraints. Solutions of the system are constrained to lie in a subspace V ⊂ P given as the zero set of a smooth momap φ : P → W = R r with components φα . In contrast to the more restrictive case in which W = R r has the structure of the dual of a Lie algebra g and φ is assumed to be equivariant with respect to the action of the corresponding Lie group on P , here we do not assume any Lie group G action. To emphaisze this, I refer to φ as a momap. (This also avoids the moment versus momentum controversy revealed to me last night.) The algebra C ∞ (V ) is given by C ∞ (P )/I , where I is the ideal generated by the φα . Dirac calls the constraints ﬁrst class if I is closed under the Poisson bracket. In terms of the constraints, the condition is then γ

{φα , φβ } = fαβ φγ , γ

where we have structure functions fαβ on P , not structure constants. In other words, we have an analogue of a Lie algebroid with anchor map a : C ∞ (P ) → (T P ) given by the Hamiltonian vector ﬁeld associated to a function. If we let W denote the vector space spanned by the φα , physicists speak of W as an open algebra since the bracket deﬁned on W does not close in W . Compare this with Lie’s notion of function group [Lie90] as discussed by Alan [Wei02]. In this ﬁrst class case, the Hamiltonian vector ﬁelds Xφα determined by the constraints are tangent to V (where V is smooth) and give a foliation F of V . Similarly, C ∞ (P )/I is an I -module with respect to the bracket. (In symplectic geometry, the corresponding variety is called coisotropic. The passage from P to V /F is known as symplectic reduction.) The true physics of the system is the induced system on the space of leaves V /F. If that space is a smooth manifold, C ∞ (V /F) is the true algebra of observables. When C ∞ (V /F) makes no sense, the Batalin–Fradkin–Vilkovisky construction provides a replacement, as described below (see [HT92] for a comprehensive treatment).

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In this context, the classical BRST construction, at least as developed by Batalin– Fradkin–Vilkovisky in the case of regular constraints, is a homological model for C ∞ (V /F) or rather for the full de Rham complex (V , F) consisting of forms on vertical vector ﬁelds, those tangent to the leaves. The model is constructed as follows: First, consider the most common case of an equivariant moment map φ : P → W = g∗ with respect to a Lie group action of G on P , where g is the Lie algebra of G. Let A denote C ∞ (P ) considered as a Poisson algebra. Extend A as a graded commutative algebra to BFV = A ⊗ Eg∗ ⊗ Eg

(2)

and extend the Poisson bracket {·, ·} (still of degree 0) as determined by the fundamental pairing g∗ ⊗g → R. Note: Elements of g are called antighosts and have degree −1, while elements of g∗ are called ghosts and have degree 1. Now make BFV a dg Poisson algebra by deﬁning dBFV = dK + δ ∗ , (3) where δ ∗ is the Chevalley–Eilenberg coboundary and dK is the Koszul differential on A ⊗ Eg regarded as a resolution of the ideal of constraints. In terms of a basis {eα } for g so that φα = eα ◦ φ, this means that dK is the graded derivation determined by dK (eα ) = φα . If we denote eα as Pα and deﬁne ηα in terms of a dual basis, then dBFV = {Q0 + Q1 , ·} for γ Q0 = ηα Pα and Q1 = 1/2ηα ηβ Cαβ Pγ , the formula that often appears in the physics literature. Because we have a strict Lie group action and, hence, structure constants, it is 2 = 0, but this is not the case for our momaps. The straightforward to verify dBFV deﬁnition of the algebra is no problem: BFV = A ⊗ EW ⊗ EW ∗

(4)

and dK + δ ∗ is deﬁned as before but fails to square to 0, essentially because we now have structure functions. In the regular case, the brilliance of Batalin–Fradkin– Vilkovisky was to deﬁne dBFV by adding terms of higher order to dK + δ ∗ so that (dBFV )2 = 0. With hindsight, the existence of such terms of higher order was due to the fact that A ⊗ EW provided a resolution of the ideal of constraints, thus permitting the techniques of homological perturbation theory. However, the proof crucially involves keeping dBFV as an inner derivation {Q, ·} by adding terms of higher order to Q0 + Q1 . If we write Q = Q0 + Q∞ , then we see we are seeking a solution of the master equation (a.k.a. Maurer–Cartan) (see Section 6.3): {Q0 , Q∞ } + 1/2{Q∞ , Q∞ } = 0, or, equivalently, we seek to deform Q0 in the direction of Q1 (see Section 7). The point of doing this is as follows.

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Theorem 5.1. If the ﬁrst class constraints generate a regular ideal, then the cohomology of BFV is isomorphic to the cohomology of (V , F) with respect to the leaf-wise exterior differential. In particular, H 0 (BFV) is isomorphic to H 0 ( (V /F)), the algebra of “observables’’ on the reduced phase space. In the more general nonregular case, the Koszul complex can be extended to the Koszul–Tate resolution by adding the polynomial algebra generated by “antighosts of antighosts’’ (given degree 2), etc. To preserve the crucial Poisson algebra structure, one also adds “ghosts of ghosts’’ (given degree −2), etc. In general, the quotient space is not a manifold, often not even Hausdorff, then H 0 (BFV) provides a suitable candidate for the algebra of observables on the “reduced phase space.’’ Since BFV is a free graded commutative algebra over A, assuming sufﬁcient ﬁniteness, the differential derivation dBFV is graded dual to a differential coderivation on a free graded cocommutative coalgebra over A and hence is equivalent to an L∞ -algebra. This is spelled out in considerable detail by Kjeseth [Kje01a, Kje01b], subsequent to some relevant observations by Huebschmann [Hue90] and myself.

6 Lagrangians with symmetries Lagrangian physics derives “equations of motion’’ from a variational principle of least action. Here an action refers to an integral S(φ) = L((j n φ)(x)) volM M

over some manifold M where φ is a (possibly vector valued) function on M or section of a bundle E over M. The action may have symmetries, i.e., variations in φ which do not change the value of S and hence are physically irrelevant in the sense that φ and its transformed value encode the same physical information. Emmy Noether had two major theorems regarding the variational calculus. The ﬁrst, much better known and often referred to as Noether’s theorem, asserts a correspondence between symmetries and conserved quantities. Noether’s second variational theorem establishes a correspondence between symmetries, notably gauge symmetries, and differential algebraic relations among the Euler–Lagrange equations. It is this second theorem that has an important role in the Batalin–Vilkovisky construction for Lagrangians with symmetries. These symmetries create difﬁculties for quantization of such physical theories. The method of Batalin and Vilkovisky [BV84, BV83] was invented to handle these difﬁculties, but turns out to be of interest also in a classical context. The construction is quite parallel to that of Batalin–Fradkin–Vilkovisky in the constrained Hamiltonian case, but with one crucial difference: instead of a grading preserving bracket, they use an “antibracket’’ (independently due to Zinn-Justin [ZJ75, ZJ76]) which is of degree 1. Therefore, it is also known as an odd Poisson or Gerstenhaber bracket. In this Lagrangian setting, Batalin and Vilkovisky extend the BRST cohomological

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approach by introducing antiﬁelds (independently and previously due to Zinn-Justin) dual to the original ﬁelds and antighosts which (with hindsight) correspond to the Noether relations and are dual to the ghosts which generate the BRST complex for the Lie algebra of symmetries. The original version of Noether in “Invariante Variationsprobleme’’ [Noe18] was written in terms of an inﬁnite continuous group, G∞ρ , “understood to be a group whose most general transformations depend on ρ essential arbitrary functions and their derivatives.’’ Noether’s Theorem II refers to an integral I (= S in our notation) and reads as follows: If the integral I is invariant with respect to G∞ρ in which the arbitrary functions occur up to the σ th derivatives, there subsist ρ identity relationships between the Lagrange expressions and their derivatives up to the σ th order. . . . The converse holds. Later in that paper these relations are called dependencies. The relevance of Noether’s theorem is not emphasized in most of the literature using the BV approach. As with BFV, part of the differential of the Batalin– Vilkovisky complex BV is that of the Koszul–Tate resolution, in this case of the differential ideal generated by the Euler–Lagrange equations. The antiﬁelds generate the Koszul complex, which is not a resolution; the antighosts provide the next level of generators, as described by Tate [Tat57], corresponding to the relations among the Euler–Lagrange equations. It is the full acyclicity of the Koszul– Tate resolution that permits the application of homological perturbation theory [Gug82, GL89, GLS90, GS86, Hue84, HK91] and thus guarantees the existence of the terms of higher order in the full differential dBV . As in the concluding remark in Section 5, the graded dual to dBV is equivalent to an L∞ -algebra. We comment on this further in Section 6.2. Rather than carrying out this analysis in the abstract, we mention two particularly striking realizations of this structure: the Poisson sigma models of Cattaneo and Felder [CF99] and our analysis with Fulp and Lada [FLS02a, FLS02b] of Lagrangians with ﬁeld dependent symmetries as in the case of higher spin particles. 6.1 The Poisson sigma model The ﬁelds of the Cattaneo–Felder σ -model are ordered pairs (X, η) such that X is a mapping from a two-dimensional manifold into a Poisson manifold M and η is a section of the bundle Hom(T , X∗ T ∗ M) −→ . These ﬁelds are subject to boundary conditions, namely they should satisfy the conditions X(u) = 0 and η(u)(v) = 0 for arbitrary u in the boundary of and for v tangent to the boundary of at u. Observe that for each u ∈ , we can regard η(u) as a linear mapping ∗ M. In local coordinates {uµ } on and {x i } on M, we write from Tu into TX(u) ∂ dX = (dXj ) ∂x j and η( ∂u∂ µ ) = ηi,µ dx i . The Poisson structure is given by a Poisson tensor α, which is a skew-symmetric tensor on M: ∂ ∂ α = α ij ∧ , (5) ∂x i ∂x j

594

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which satisﬁes a Jacobi condition: α il ∂l α j k + α j l ∂l α ki + α kl ∂l α ij = 0. The action S of the model is deﬁned in such local coordinates by 1 S(X, η) = (ηi ∧ dX i ) + (α ij ◦ X)(ηi ∧ ηj ). 2

(6)

(7)

According to the variational principle, we obtain extrema of S as those ﬁelds (X, η) which satisfy the Euler–Lagrange equations 1 EXi := dηi + ∂i α j k (ηj ∧ ηk ) = 0 2

(8)

Eηi := −dXi − α ij ηj = 0.

(9)

and The gauge symmetries of the action are parameterized by all sections β of the bundle X∗ T ∗ M −→ which vanish on the boundary of . For each such β, deﬁne δβ acting on the ﬁelds by (δβ X)i = (α ◦ X)(dx i , β), (δβ η)(W ◦ X) = −(dβ)(W ◦ X) − ((LW α) ◦ X)(η, β),

(10) (11)

where W is a vector ﬁeld on M, and LW α is the Lie derivative of α with respect to W . In terms of components of the ﬁelds, we write 1 EXi = (∂µ ηi,ν + ∂i α j k ηj,µ ηk,ν ) µν 2

(12)

Eηi,ν = −(∂µ X i + α ij ηj,µ ) µν .

(13)

and

It follows from Noether’s theorem that α ik EXi + ∂µ Eηk,µ − ∂i α j k ηj,µ Eηi,µ = 0

(14)

are the Noether identities corresponding to the gauge symmetry δβ deﬁned above. Applied to this Poisson sigma model, the Batalin–Vilkovisky graded algebra BV is a graded commutative algebra over LocE (the algebra of local functions on E) with generators Xi+ and η+i , called “antiﬁelds,’’ γi , called “ghosts,’’ and γ +i , called “antighosts.’’ (If only the ghosts were added as generators, this would be just a BRST algebra.) These generators are bigraded. The graded commutativity is with respect to the sum (which we call the total degree) of the ghost degree and the form degree. The pairing between symmetries and identities is now expressed as the pairing between ghosts and antighosts, which plays a crucial role in the Batalin–Vilkovisky antibracket ( , ).

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Now the initial differential on BV can be expressed as (S 0 + S 1 ), where S 0 is our original action and S 1 is 1 Xi+ α ij (X)γj − η+i ∧ (dγi + ∂i α kl (X)ηk γl ) − γ +i ∂i α j k (X)γj γk . (15) 2 Corresponding to the fact that (dKT + δ)2 = 0, as in BFV, we have (S 0 + S 1 , S 0 + S 1 ) = 0. Batalin and Vilkovisky show that, in much more general situations, one can add terms S i of ghost degree i > 1 to achieve a total SBV such that (SBV , SBV ) = 0. The reason for this is that the dKT homology vanishes in appropriate degrees, so the methods of homological perturbation theory apply again. In the Cattaneo–Felder model, only one more term is needed: 1 S2 = − η+i ∧ η+j ∂i ∂j α kl (X)γk γl . (16) 4 Thus the total Batalin–Vilkovisky generator SBV is 1 ηi ∧ dX i + α ij (X)ηi ∧ ηj 2 1 + Xi+ α ij (X)γj − η+i ∧ (dγi + ∂i α kl (X)ηk γl ) − γ +i ∂i α j k (X)γj γk 2 1 +i +j kl − η ∧ η ∂i ∂j α (X)γk γl . (17) 4 6.2 Field dependent gauge symmetries Field dependent gauge symmetries appear in several ﬁeld theories, most notably in a class due to Ikeda [Ike94] and Schaller and Strobl [SS94], including the Poisson sigma model of Cattaneo and Felder [CF99] above. A signiﬁcant generalization occurs in the Berends, Burgers, and van Dam [Bur85, BBvD86, BBvD85] approach to “particles of spin ≥ 2.’’ The physics of “particles of spin ≤ 2’’ leads to representations of a Lie algebra + of gauge parameters on a vector space of ﬁelds. By a ﬁeld dependent action of + on , Berends, Burgers, and van Dam mean a polynomial (or power series) map δ(ξ )(φ) = i≥0 Ti (ξ, φ), where Ti is linear in ξ and polynomial of homogeneous degree i in φ. Berends, Burgers, and van Dam consider arbitrary ﬁeld theories, subject only to the requirement that the commutator of two gauge symmetries be another gauge symmetry whose gauge parameter is possibly ﬁeld dependent. Thus they do not require an a priori given Lie structure to induce the algebraic structure of the gauge symmetry “algebra.’’

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Let denote the vector space of ﬁelds and + the vector space of gauge parameters. Let ∗ denote the free graded cocommutative coalgebra cogenerated by . Although the space + of gauge parameters has no natural Lie structure, the space of linear maps from ∗ into + is a Lie algebra under certain mild assumptions along with a hypothesis which we refer to as the BBvD hypothesis. Under these assumptions, the gauge algebra gives rise to an L∞ -algebra on a differential graded vector space V with + in degree 0, in degree 1 and 0 in all other degrees. The vector space Coder(∗ ) of coderivations on ∗ is a Lie algebra with bracket given by the commutator with respect to composition. The vector space Hom(∗ , ) is isomorphic to Coder(∗ ) and hence inherits a Lie algebra structure; the bracket on Hom(∗ , ) is known as the Gerstenhaber bracket [Ger63, Sta93]. Suppose that we are given a linear map δ : + → Hom(∗ , ), a “ﬁeld dependent action’’ of + on . By a ﬁeld dependent action of + on , Berends, Burgers, and van Dam [Bur85, BBvD84, BBvD85] mean a polynomial (or power series) map δ(ξ )(φ) = i=0 Ti (ξ, φ) where Ti is linear in ξ and polynomial of homogeneous degree i in φ. We adopt the corresponding polarized and adjoint point of view and write δ(ξ ) = i=0 Ti (ξ ) where Ti is zero except on i . We extend δ to a map δˆ : Hom(∗ , +) → Hom(∗ , ) by

ˆ δ(π) = ev ◦ (δ ◦ π ⊗ 1) ◦

,

where ev is the evaluation map. Consider the possibility of inducing a Lie-type bracket ˆ Under certain conditions, e.g., δ is injective, on Hom(∗ , +) via the mapping δ. such a bracket may then be used to obtain a bracket on the parameter space + by restricting the bracket on Hom(∗ , +), where we think of + as being contained in the space Hom(∗ , +) by identifying ξ ∈ + with the map (also denoted ξ ) in Hom(∗ , +), which is 0 except on the scalars, where ξ(1) = ξ . In order to assure the Jacobi property of the bracket on Hom(∗ , +), we introduce a correction term. We accomplish this, following Berends, Burgers, and van Dam, by assuming there is a map C : + ∧ + → Hom(∗ , +) such that

[δ(ξ ), δ(η)] = δC(ξ, η) ∈ Hom(∗ , )

for all ξ, η ∈ +. We refer to this as the BBvD hypothesis. We then extend C to a mapping Cˆ : Hom(∗ , +) ∧ Hom(∗ , +) → Hom(∗ , +) by

ˆ 1 , π2 ) = C ◦ ((π1 ⊗ π2 ) ⊗ 1) ◦ ( C(π

⊗ 1) ◦

,

where we have identiﬁed C with its adjoint mapping, the mapping from +⊗+⊗∗ into + deﬁned by (ξ, η, φ1 ∧ · · · ∧ φn ) −→ C(ξ, η)(φ1 ∧ · · · ∧ φn ).

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Next, we redeﬁne the bracket on Hom(∗ , +) by including the correction term ˆ 1 , π2 ). [π1 , π2 ] = π1 δ(π2 ) − π2 δ(π1 ) + C(π

(18)

ˆ 1 , π2 ] = [δ(π1 ), δ(π2 )]. Theorem 6.1. The mapping δˆ preserves brackets, that is, δ[π ∗ ∗ Moreover, if δ : Hom( , +) → Hom( , ) is injective, then [π1 , π2 ] satisﬁes the Jacobi identity. This suggests that the parameter space should be enlarged to include all of Hom(∗ , +), but this is apparently unacceptable to physicists since the number of independent parameters is linked to the number of independent Noether identities. However, the polynomial equations of physical relevance deﬁne an L∞ -structure on an appropriate graded vector space. We restrict our attention to the constant maps in Hom(∗ , +) and show that the algebraic structure of Hom(∗ , +) induces an L∞ -structure on + ⊕ . We assume the hypothesis and that δˆ is injective so Theorem 6.1 holds. Deﬁne a differential graded vector space V with + in degree 0, in degree 1, and 0 in all other degrees. Take ∂ : + → , given by ∂(ξ ) = δ(ξ )(1) ∈ , as the only nontrivial differential. Deﬁne D : ∗ (sV ) → sV by D(ξ ) = ∂(ξ ), D(ξ ∧ φ1 ∧ · · · ∧ φn ) = δ(ξ )(φ1 ∧ · · · ∧ φn )

for n ≥ 1,

D(ξ1 ∧ ξ2 ∧ φ1 ∧ · · · ∧ φn ) = C(ξ1 , ξ2 )(φ1 ∧ · · · ∧ φn ), and D = 0 on elements of ∗ (sV ) with more than two entries from + or with no entry from +. Notice this is essentially not of the same form as that of Roytenberg and Weinstein in Section 4.2, although both have just two components. The crucial difference is in the grading: 0, 1 here versus −1, 0 for them. Theorem 6.2 ([FLS02a]). D : ∗ (sV ) → sV as deﬁned above gives V the structure of an L∞ -algebra. 6.3 The master equation and homological perturbation theory The name “master equation’’ derives from the physics literature, especially of the Batalin–Vilkovisky approach considered above, but equations of this form are well known in mathematics, though under various names: • • •

the deﬁning equation for a twisting cochain, the integrability equation of deformation theory, the Maurer–Cartan equation;

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the latter is perhaps the most famous. The equation makes sense as applied to elements of a dg algebra, associative or Lie or Gerstenhaber, etc., and has generalizations to higher homtopy versions. In [HS02], Huebschmann and I give an extensive comparison of these various interpretations. We show how to construct solutions using the tools of homological perturbation theory, working in characteristic zero. In particular, we endow the homology H (g) of a strict dg Lie algebra g with the structure of an L∞ -algebra such that g and H (g) are equivalent as L∞ -algebras, i.e., via L∞ maps (see Section 7). The much older analogous result for A∞ -algebras is due to Kadeishvili [HK91]. Note that H (g) is a strict dg Lie algebra with d = 0, but the higher-order operations li are often nontrivial. If g is equivalent as L∞ -algebra to H (g) with all li = 0 for i ≥ 3, then g is called formal.

7 L∞ -maps, deformation quantization, string ﬁeld theory (SFT), and more Deﬁnition 7.1. An L∞ -map f : h → g of L∞ -algebras (or dgLie algebras) is a dg coalgebra map c (sh) → c (sg). The Cattaneo and Felder Poisson sigma model was developed to provide an alternative, “path integral,’’ proof of Kontsevich’s theroem that any Poisson manifold can be deformation quantized. In both proofs, the key issue is the formality of a certain dgLie algebra g. The L∞ -equivalence of this g and H (g) implies that all the obstructions to deformation quantization vanish. For this important application, was a disk so the maps → M could be considered as world sheets as in SFT. Because BBvD give an explicit expansion Ti , Ci , the corresponding multi-brackets li are visible or at least easy to extract. In contrast, in the Cattaneo and Felder Poisson sigma model, they are hidden in the single function α and its derivatives. The relevance of A∞ - and L∞ -structure to (respectively) OSFT and CSFT has a particularly “physical’’ interpretation. The higher order operations describe multiple string interactions, not obtained from three-string interactions (multiplication, respectively, bracketing of two strings) or the equivalent correlation functions [Zwi93]. Here too there is contact with Alan and his student Tang in their recent paper [TW04]. A combined open-closed string ﬁeld theory [Zwi98] leads to an intricate combination of an L∞ -algebra acting by strong homotopy derivations of an A∞ -algebra [HK91].

8 Homological Legendre transform In their concluding remarks in [RW98], Dmitry and Alan muse: L∞ -algebras occur in physics in the framework of the Batalin–Vilkovisky procedure for quantizing gauge theories. On the other hand, the Courant bracket seems to provide a geometric framework for constrained Hamiltonian

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systems. It is known [HT92] that gauge Lagrangians lead to constrained theories in the Hamiltonian formalism. This suggests that homotopy Lie algebras arising in the Batalin–Vilkovisky formalism and those in the Courant formalism might be somehow related. In response to my paper for the Alanfestschrift, Dmitry pointed out to me the paper of Grigoriev and Damgaard which establishes an analogue of the Legendre transform in terms of the BFV and BV constructions. Here at Alanfest, we have investigated their transforms in further detail. This bears further investigation, but for now let me mention only that in either direction, Hamiltonian to Lagrangian of vice versa, the essential idea is the “oddiﬁcation’’ of all the ﬁelds, ghosts, etc., then substituting these into the respective formulas for the Hamiltonian, the Lagrangian, etc., and keeping only the parts of the appropriate total degree. Dmitry can interpret this further by looking at the algebra as that of a graded path space.

9 Coda There are still other examples of A∞ - and L∞ -structures with potential physical relevance, for example, the notion of A∞ -category. A∞ -categories have been used by Fukaya [Fuk95] for remarkable applications to Morse theory and Floer homology. More recently, they play a role in string and D-brane theory and homological mirror symmetry. But that takes us further aﬁeld from today’s topic: the L∞ -structures directly involved in some of Alan’s work and closely related to his foundational work on symplectic reduction. There are further relations to be discovered, as he has indicated. Leaving that for the future, let me conclude with best wishes for the continuation of a long happy and inspiring career to Alan on his 60th birthday.

References [BBvD84] F. A. Berends, G. J. H. Burgers, and H. van Dam, On spin three selﬁnteractions, Z. Phys. C, 24 (1984), 247–254. [BBvD85] F. A. Berends, G. J. H. Burgers, and H. van Dam, On the theoretical problems in constructing intereactions involving higher spin massless particles, Nuclear Phys. B, 260 (1985), 295–322. [BBvD86] F. A. Berends, G. J. H. Burgers, and H. van Dam, Explicit construction of conserved currents for massless ﬁelds of arbitrary spin, Nuclear Phys. B, 271 (1986), 429–441. [BF83] I. A. Batalin and E. S. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett., 122B (1983), 157–164. [BRS75] C. Becchi, A. Rouet, and R. Stora, Renormalization of the abelian Higgs-Kibble model, Comm. Math. Phys. 42 (1975), 127–162. [Bur85] G. J. H. Burgers, On the Construction of Field Theories for Higher Spin Massless Particles, Ph.D. thesis, Rijksuniversiteit te Leiden, Leiden, the Netherlands, 1985. [BV83] I. A. Batalin and G. S. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D, 28 (1983), 2567–2582.

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I. A. Batalin and G. S. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feyneman rules, Nuclear Phys. B, 234 (1984), 106–124. [BC03] J. C. Baez andA. S. Crans, Higher-dimensional algebra VI: Lie 2-algebras, preprint, 2003; math.QA/0307263. [CE48] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85–124. [CF99] A. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys., 212-3 (2000), 591–611 [Dai02] M. Daily, L-inﬁnity structures on spaces with 3 one-dimensional components, preprint, 2002; math.QA/0212030. [FF78] E. S. Fradkin and T. E. Fradkina, Quantization of relativistic systems with boson and fermion ﬁrst and second class constraints, Phys. Lett., 72B (1978), 343–348. [FLS02a] R. Fulp, T. Lada, and J. Stasheff, Noether’s variational theorem II and the BV formalism, Suppl. Rend. Circ. Mat. Palermo Ser. II, 71 (2002), 115–126. [FLS02b] R. Fulp, T. Lada, and J. Stasheff, Sh-Lie algebras induced by gauge transformations, Comm. Math. Phys., 231 (2002), 25–43. [FP67] L. D. Fade’ev and V. N. Popov, Feynman diagrams for the Yang-Mills ﬁeld, Phys. Lett., 25B (1967), 29–30. [FP03] A. Fialowski and M. Penkava, Versal deformations of three dimensional lie algebras as L-inﬁnity algebras, preprint, 2003; math.RT/0303346. [Fuk95] K. Fukaya, Floer homology, A∞ -categories and topological ﬁeld theory, in J. Andersen, J. Dupont, H. Pertersen, and A. Swan, eds., Geometry and Physics, Lecture Notes in Pure and Applied Mathematics, Vol. 184, Marcel Dekker, New York, 1995, 9–32 (notes by P. Seidel). [FV75] E. S. Fradkin and G. S. Vilkovisky, Quantization of relativistic systems with constraints, Phys. Lett., 55B (1975), 224–226. [Ger63] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267–288. [GL89] V. K. A. M. Gugenheim and L. Lambe, Applications of perturbation theory to differential homological algebra I, Illinois J. Math., 33 (1989), 556–582. [GLS90] V. K. A. M. Gugenheim, L. Lambe, and J. Stasheff, Algebraic aspects of Chen’s twisting cochain, Illinois J. Math., 34 (1990), 485–502. [GS86] V. K. A. M. Gugenheim and J. D. Stasheff, On perturbations and A∞ -structures, Bull. Soc. Math. Belgium, 38 (1986), 237. [Gug82] V. K. A. M. Gugenheim, On a perturbation theory for the homology of a loop space, J. Pure Appl. Algebra, 25 (1982), 197–207. [HK91] J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Math. Z., 207 (1991), 245–280. [HS02] J. Huebschmann and J. Stasheff, Formal solution of the master equation via HPT and deformation theory, Forum Math., 14 (2002), 847–868. [HT92] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, Princeton, NJ, 1992. [Hue84] J. Huebschmann, Perturbation Theory and Small Models for the Chains of Certain Induced Fibre Spaces, Habilitationsschrift, 1984. [Hue90] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math., 408 (1990), 57–113. [Ike94] N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Ann. Phys., 235 (1994), 435–464. [Kaj03] H. Kajiura, Noncommutative Homotpy Algebras Associated with Open Strings, Dissertation, University of Tokyo, Tokyo, 2003.

Poisson homotopy algebra [Kje01a] [Kje01b] [KS03]

[Lie90] [MSS02]

[Nas98] [Noe18]

[RW98] [SS94] [Sta93]

[Sto77] [Tat57] [TW04] [Tyu75] [Wei02] [ZJ75]

[ZJ76]

[Zwi93] [Zwi98]

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L. Kjeseth, A homotopy Lie-Rinehart resolution and classical BRST cohomology, Homology Homotopy Appl., 3-1 (2001), 165–192 (electronic). L. Kjeseth, Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs, Homology Homotopy Appl., 3-1 (2001), 139–163 (electronic). Y. Kosmann-Schwarzbach, Quasi, twisted, and all that… in Poisson geometry and Lie algebroid theory, in J. E. Marsden and T. S. Ratiu, eds., The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein (this volume), Progress in Mathematics, Vol. 232, Birkhäuser, Boston, 2005, 363–390. S. Lie, Theorie der Tranformationensgruppen, Teubner, Leipzig, 1890. M. Markl, S. Shnider, and J. Stasheff, Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, Vol. 96, American Mathematical Society, Providence, RI, 2002. C. Nash, Topology and physics: A historical essay, in I. M. James, ed., A History of Topology, Elsevier–North Holland, Amsterdam, 1998, 359–415. E. Noether, Invariante variationsprobleme, Nachr. König. Gesell. Wissen. Göttingen Math.-Phys. Kl., (1918), 235–257 (in German); Transport Theory Stat. Phys., 1 (1971), 186–207 (in English). D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys., 46-1 (1998), 81–93. P. Schaller and T. Strobl, Poisson structure induced (topological) ﬁeld theories, Mod. Phys. Lett. A, 9 (1994), 3129–3136. J. D. Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra, 89 (1993) (festschrift in honor of Alex Heller), 231–235. R. Stora, Continuum gauge theories, in M. Levy and P. Mitter, eds., New Developments in Quantum Field Theory, Plenum, New York, 1977, 201–224. J. Tate, Homology of Noetherian rings and local rings, Illinois J. Math., 1 (1957), 14–27. X. Tang and A. Weinstein, Quantization and morita equivalence for constant dirac structures on tori, Ann. Inst. Fourier, 54 (2004), to appear; math.QA/0305413. I. V. Tyutin, Gauge invariance in ﬁeld theory and statistical physics in operator formulation, technical report, Lebedev Physics Institute, Moscow, 1975 (in Russian). A. Weinstein, The Geometry of Momentum, 2002, math.SG/0208108. J. Zinn-Justin, Renormalization of gauge theories, in H. Rollnick and K. Dietz, eds., Trends in Elementary Particle Theory, Lecture Notes in Physics, Vol. 37, Springer-Verlag, 1975, 2–39. J. Zinn-Justin and R. Balian, eds., Méthodes en théorie des champs = Methods in Field Theory, North-Holland, Amsterdam, 1976 (École d’Été de Physique Théorique, Session XXVIII, tenue à Les Houches, 28 Juillet–6 Septembre, 1975). B. Zwiebach, Closed string ﬁeld theory: Quantum action and the BatalinVilkovisky master equation, Nuclear Phys. B, 390 (1993), 33–152. B. Zwiebach, Oriented open-closed string theory revisited, Ann. Phys., 267 (1998), 193–248.

Dirac submanifolds of Jacobi manifolds Izu Vaisman Department of Mathematics University of Haifa 31905 Haifa Israel [email protected] Dedicated to Alan Weinstein on the occasion of his sixtieth birthday. Abstract. The notion of a Dirac submanifold of a Poisson manifold studied by Xu is interpreted in terms of a general notion of tensor ﬁelds soldered to a normalized submanifold. This interpretation is used to deﬁne the notion of a Dirac submanifold of a Nambu–Poisson and Jacobi manifold. Several properties and examples are discussed.

1 Normalized submanifolds In this section we make some general considerations on submanifolds of a differentiable1 manifold. These considerations were inspired by the theory of Dirac submanifolds of a Poisson manifold developed in [10]. The notion of a normalized submanifold was used in afﬁne and projective differential geometry half a century ago. Deﬁnition 1.1. Let N n be a submanifold of M m (indices denote dimensions) and ι : N ⊆ M the corresponding embedding. A normalization of N by a normal bundle νN is a splitting T M|N = T N ⊕ νN (1) (T denotes tangent bundles). A submanifold endowed with a normalization is called a normalized submanifold. Then if X ∈ T M ( denotes spaces of global cross-sections) is a vector ﬁeld on M such that X|N ∈ T N , respectively X|N ∈ νN , X is said to be tangent, respectively, normal, to N. Let (N n , νN) be a normalized submanifold of M m . Let σ : W → N be a tubular neighborhood of N such that ∀x ∈ N , Tx (Wx ) = νx N (Wx is the ﬁber of W and νx N is the ﬁber of νN at x). Then W is said to be a compatible tubular neighborhood of N ; obviously, such neighborhoods exist. Furthermore, each 1 In this paper, everything is of class C ∞ and all submanifolds are embedded.

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point x ∈ N has a σ -trivializing neighborhood U endowed with coordinates (x a ) (a, b, c, · · · = 1, . . . , m − n) on the ﬁbers of σ and such that x a |N ∩U = 0, and coordinates (y u ) (u, v, w, · · · = m − n + 1, . . . , m) on N ∩ U . We will say that (x a , y u ) are adapted local coordinates. With respect to adapted coordinates, N has the local equations x a = 0, and ∂ ∂ T N|U ∩N = span , νN | . (2) = span U ∩N ∂y u x a =0 ∂x a x a =0 Accordingly, the transition functions between two systems of adapted local coordinates must be of the form x˜ a = x˜ a (x b , y v ),

y˜ u = y˜ u (y v ),

(3)

and satisfy the conditions ∂ x˜ a = 0, ∂y v x b =0

∂ y˜ u ≡ 0. ∂x b

(4)

The splitting (1) induces a similar relation for the dual bundles T ∗ M|N = T ∗ N ⊕ ν ∗ N,

(5)

and locally, with respect to adapted coordinates, one has T ∗ N = ann(νN ) = span{dy u |x a =0 }, ν ∗ N = ann(T N ) = span{dx a |x a =0 }

(6)

(ann denotes annihilator spaces). Two normal bundles νN and ν˜ N of the same submanifold N are connected as follows. Let pν , pT be the projections deﬁned by the splitting (1) and pν˜ , p˜ T the similar projections of the second normalization. The mapping (v ∈ νN ) → pν˜ v is an isomorphism ϕ : νN → ν˜ N , and with respect to adapted coordinates, we have ∂ c ν˜ N = span ϕ | = X (7) a x =0 , ∂x a x c =0 where2 Xa =

∂ ∂ − θau u ∂x a ∂y

(8)

are vector ﬁelds on U , and (θau |x c =0 ) is the local matrix of the homomorphism ψ : νN → T N deﬁned by ψ(v) = p˜ T (v) (v ∈ νN ). Notice that ϕ + ψ is the inclusion of νN in T M|N , and ν˜ N is uniquely determined by any of the mappings ϕ, ψ. Some geometric objects of M may have a strong relationship with the normalized submanifold (N, νN ). 2 In this paper, we use the Einstein summation convention.

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Deﬁnition 1.2. A differential k-form κ ∈ k (M) ( denotes spaces of differential forms) is soldered to N if ι∗ [LX κ] = 0 for any vector ﬁeld X ∈ T M normal to N . Since ∀f ∈ C ∞ (M) and any vector ﬁelds X, Y1 , . . . , Yk , one has (Lf X κ)(Y1 , . . . , Yk ) = f (LX κ)(Y1 , . . . , Yk ) −

k

(9)

(−1)j (Yj f )[i(X)κ](Y1 , . . . , Yj −1 , Yj +1 , . . . , Yk ),

j =1

it follows that κ is soldered iff for any vector ﬁeld X ∈ T M normal to N , one has ι∗ [i(X)κ] = 0,

ι∗ [LX κ] = 0,

(10)

ι∗ [i(X)κ] = 0,

ι∗ [i(X)dκ] = 0.

(11)

or, equivalently,

With respect to adapted local coordinates, κ has the expression κ=

s+t=k

1 κa ...a u ...u dx a1 ∧ · · · ∧ dx as ∧ dy u1 ∧ · · · ∧ dy ut , s!t! 1 s 1 t

and κ is soldered to N iff κau1 ...uk−1 |x b =0 = 0,

∂κu1 ...uk = 0. ∂x a x b =0

(12)

(13)

In particular, the space of soldered functions is C ∞ (M, N, νN) = {f ∈ C ∞ (M) /(∂f/∂x a )x a =0 = 0}.

(14)

We will denote by k (M, N, νN) the space of soldered k-forms (for k = 0 we have the space (14)). Obviously, the soldering conditions (11) are compatible with the exterior product and the exterior differential. Therefore, we get a cohomology k (M, N, νN), which will be called the soldered de Rham cohomolalgebra ⊕k HsdeR ogy algebra, deﬁned by the cochain spaces k (M, N, νN) and the operator d. The inclusion in the usual de Rham complex induces homomorphisms k k ιk : HsdeR (M, N, νN) −→ HdeR (M).

(15)

k (W, N, νN), where W is a compatible tubular neighIn principle, the spaces HsdeR borhood, should provide interesting information about the normalized submanifold (N, νN).

Deﬁnition 1.3. A k-vector ﬁeld Q ∈ V k (M) (V denotes spaces of multivector ﬁelds) is soldered to the normalized submanifold (N, νN ) if

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(i) for any (k − 1)-form λ ∈ ∧k−1 [ann(νN )], the vector ﬁeld deﬁned along N by i(λ)Q|N is tangent to N ; (ii) for any vector ﬁeld X on M normal to N , (LX Q)|ann(νN ) = 0. We will denote by V k (M, N, νN) the space of soldered k-vector ﬁelds. Using adapted local coordinates, we see that V 0 (M, N, νN) is, again, (14), and for k = 1 one has ∂ηu 1 1 u ∂ V (M, N, νN) = Y ∈ V (M) / Y |N = η , =0 . (16) ∂y u N ∂x a x c =0 Generally, we have an expression of the form Q=

s+t=k

1 a1 ...as u1 ...ut ∂ ∂ ∂ ∂ Q ∧ ··· ∧ a ∧ u ∧ ··· ∧ u , s!t! ∂x a1 ∂x s ∂y 1 ∂y t

and Q ∈ V k (M, N, νN) iff Qau1 ...uk−1 |x c =0 = 0,

∂Qu1 ...uk−1 c = 0. ∂x a x =0

(17)

(18)

The spaces of soldered forms and multivector ﬁelds are components of important algebraic structures; namely, we have the following. Proposition 1.1. 1. The space V 1 (M, N, νN) is a Herz–Reinhart Lie algebra over (R, C ∞ (M, N, νN )). 2. The complex of N-soldered differential forms is a complex over the Lie algebra V 1 (M, N, νN). 3. The triple (⊕k V k (M, N, νN), ∧, [ , ]), where [ , ] is the Schouten–Nijenhuis bracket is a Gerstenhaber algebra. Proof. For the deﬁnition of the algebraic structures above, see, for instance, [2] and [3]. The use of adapted local coordinates shows that if f ∈ C ∞ (M, N, νN) and Y, Z ∈ V 1 (M, N, νN), then f Y and [Y, Z] belong to V 1 (M, N, νN), which proves 1. Furthermore, for the same Y and any κ ∈ k (M, N, νN), i(Y )κ ∈ k−1 (M, N, νN), and we get 2. Finally, the exterior product of soldered multivector ﬁelds is obviously soldered, and in order to get 3, it remains to prove that the Schouten– Nijenhuis bracket3 of P ∈ V p (M, N, νN) and Q ∈ V q (M, N, νN) belongs to V p+q−1 (M, N, νN). To see that soldering condition (i) is satisﬁed, we look at the known formula (e.g., [8]) i([P , Q])ω = (−1)(p+1)(q+1) i(P )d[i(Q)ω] − i(Q)d[i(P )ω] + (−1)p i(P ∧ Q)dω,

(19)

3 We take this bracket with the sign convention of the axioms of graded Lie algebras, e.g., [8,

Proposition 4.21].

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where ω is an arbitrary (p + q − 1)-form on M, and use this formula for ω = dx a ∧ dy u1 ∧ · · · ∧ dy up+q−2 . Then soldering condition (ii) follows by using (19) to evaluate the terms of the equality LX [P , Q] = [LX P , Q] + [P , LX Q], where X is normal to N , on dy u1 ∧ · · · ∧ dy up+q−1 .

The following proposition extends a result proven for Poisson bivector ﬁelds in [10]. Proposition 1.2. If the involutive diffeomorphism ϕ : M → M (ϕ 2 = Id) preserves a k-form κ, respectively, a k-vector ﬁeld Q, then κ, respectively, Q, is soldered to the ﬁxed point locus N of ϕ. Proof. It is well known that N is a submanifold of M, the tangent bundle T N consists of the (+1)-eigenspaces of ϕ∗ along N , and N has a normalization with the normal bundle νN deﬁned by the (−1)-eigenspaces of ϕ∗ along N [10]. This also implies that ann(νN) = T ∗ N consists of the (+1)-eigenspaces, and ann(T N ) = ν ∗ N consists of the (−1)-eigenspaces, of ϕ ∗ along N . If ϕ ∗ κ = κ, if X is a normal vector ﬁeld of N on M and Y1 , . . . , Yk−1 are tangent to N , then κ(X, Y1 , . . . , Yk−1 )|N = (ϕ ∗ κ)(X, Y1 , . . . , Yk−1 )|N = −κ((X, Y1 , . . . , Yk−1 )|N ; hence ι∗ [i(X)κ] = 0. The same holds for dκ; therefore, κ is soldered to N . The proof for a k-vector ﬁeld Q is similar, and uses the fact that ϕ∗ (LX Q) = Lϕ∗ X (ϕ∗ Q) ◦ ϕ for any diffeomorphism ϕ : M → N .

The N-soldered differential forms and multivector ﬁelds have a nice interpretation by means of the geometry of the tangent bundle T M, and by looking at the normal bundle νN as a submanifold of the former. (See [10] for the case of a Poisson bivector ﬁeld.) The tensor ﬁelds of the manifold M may be lifted to T M by various processes and, in particular, there exists a complete lift, which comes from the lift of the ﬂow of a vector ﬁeld [11]. In the case of differential forms and multivector ﬁelds, the complete lift has the coordinate expression 1 κi ...i d z˙ i1 ∧ dzi2 ∧ · · · ∧ dzik (k − 1)! 1 k 1 ∂κi1 ...ik i1 + z˙ j dz ∧ · · · ∧ dzik , k! ∂zj 1 ∂ ∂ ∂ Qi1 ...ik i ∧ i ∧ · · · ∧ i QC = 1 2 (k − 1)! ∂z ∂ z˙ ∂ z˙ k ...i i 1 ∂Q 1 k ∂ ∂ + z˙ j ∧ ··· ∧ i . j i 1 k! ∂z ∂ z˙ ∂ z˙ k κC =

(20)

(21)

In formulas (20) and (21) κ and Q are given by the expressions (12) and (17), respectively, while (zi ) = (x a , y u ) (i = 1, . . . , m), and (˙zi ) are the corresponding natural vector coordinates.

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Proposition 1.3. (i) The differential form κ is soldered to the normalized submanifold (N, νN ) of M iff ∀Z ∈ T (νN), the form i(Z)κ C belongs to the ideal generated by [ann(T (νN ))]. (ii) The k-vector ﬁeld Q is soldered to (N, νN ) iff ∀α ∈ [ann(T (νN ))], i(α)QC belongs to the ideal generated by T (νN). Proof. On M, we use N-adapted local coordinates and on T M the corresponding natural vector coordinates as described above. Then the submanifold νN has the local equations x a = 0, y˙ u = 0, and the results are immediate consequences of formulas (20) and (21).

We remark that it is possible to deﬁne soldered symmetric tensor ﬁelds, similarly. The following proposition provides a nice example. Proposition 1.4. Let M be a Riemannian manifold with the metric tensor g. Then g is soldered to a submanifold N normalized by the normal bundle νN ⊥ T N iff N is a totally geodesic submanifold. Proof. If ∇ is the Levi-Civita connection of g, i.e., ∇g = 0 and ∇ has no torsion, one has (LX g)(Y, Z) = g(∇Y X, Z) + g(Y, ∇Z X). Then if X is normal and Y, Z are tangent to N, the restriction of the previous formula to N yields (LX g)(Y, Z) = 2g(b(Y, Z), X), (22) where b is the second fundamental form of the submanifold N . Thus the soldering condition of g is equivalent to b = 0.

The notion of soldering has other interesting extensions, too. First, for a multivector ﬁeld Q ∈ V k (M), condition (ii) of Deﬁnition 1.3 is itself a geometric condition, since it is easy to check that if it holds for X normal to N , it also holds for f X ∀f ∈ C ∞ (M). If Q satisﬁes only this condition, we will call it quasi-soldered to (N, νN). In fact, this notion extends to any contravariant tensor ﬁeld. The notion of a soldered differential form also extends to any covariant tensor ﬁeld, but it implies algebraic conditions like the ﬁrst condition in (10) as well. Then we may look at objects that only satisfy the algebraic condition of soldering (e.g., the ﬁrst condition (10), condition (i) of Deﬁnition 1.3, etc.), and call them algebraically compatible with the normalized submanifold . If the algebraic condition holds, the Lie derivative in the normal directions yields an important object for the submanifold. For instance, a Riemannian metric is algebraically compatible with any submanifold N , if the normal bundle is the g-orthogonal bundle of T N, and formula (22) shows that (LX g)|T N is equivalent to the second fundamental form of the submanifold. Finally, we may add to the algebraic condition a condition that is weaker than soldering. For instance, a tensor ﬁeld that is algebraically compatible with a normalized submanifold will be called weakly soldered if, along the submanifold, the

Dirac submanifolds of Jacobi manifolds

609

Lie derivative in the normal directions are proportional to the pullback of the tensor ﬁeld to the submanifold. For instance, the Riemannian metric g is weakly soldered to a submanifold N with the g-normal bundle if, for some 1-form α, one has (LX g)|T N = α(X)g|T N , for any normal vector ﬁeld X, and this happens iff N is a totally umbilical submanifold.

2 Dirac submanifolds of Poisson and Nambu–Poisson manifolds In this section, we recall the deﬁnition and characteristic properties of the Dirac submanifolds of a Poisson manifold studied by Xu [10], and give a few additional facts. In particular, we extend Xu’s deﬁnition to Nambu–Poisson manifolds. We refer to [8] for generalities of Poisson geometry and to [9] for the Nambu–Poisson geometry. A Dirac submanifold of a Poisson manifold inherits an induced Poisson structure, and the cotangent Lie algebroid of the latter may be seen as a Lie subalgebroid of the cotangent Lie algebroid of the original manifold. Deﬁnition 2.1. A submanifold N n of a Poisson manifold M m , with the Poisson bivector ﬁeld $, is a Dirac submanifold if N has a normalization (1) with the following properties: (i) $ (ann(νN)) ⊆ T N ($ is the morphism T ∗ M → T M deﬁned by $); if this condition holds, we will say that νN is algebraically $-compatible; (ii) ∀x ∈ N, there exists an open neighborhood U of x in M such that ∀f, g ∈ C ∞ (U ) which satisfy the conditions df |νN = 0, dg|νN = 0 one has d{f, g}|νN = 0. ({f, g} is the Poisson bracket deﬁned by $.) Proposition 2.1 ([10]). With the notation of Deﬁnition 2.1, the submanifold N is a Dirac submanifold iff there exists a normalization (1) such that the Poisson bivector ﬁeld $ is soldered to (N, νN ). Proof. $ is soldered to (N, νN ) iff with the notation and the adapted coordinates as deﬁned in Section 1, one has $=

∂ ∂ 1 ∂ ∂ ∂ 1 ab ∂ $ ∧ b + $au a ∧ u + $uv u ∧ v , a 2 ∂x ∂x ∂x ∂y 2 ∂y ∂y

(23)

where $au |x c =0 = 0, ∂$uv = 0. ∂x a c

(24) (25)

x =0

On the other hand, if $ is given by (23), condition (i) of Deﬁnition 2.1 is equivalent to $ (dy u ) ∈ span{∂/∂y v } along N , which means (24), and condition ii) is equivalent

to the fact that d{y u , y v }|x c =0 = 0, which is (25). Furthermore, from the deﬁnition of soldered multivector ﬁelds, we also get the following.

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Proposition 2.2 ([10]). The submanifold N of (M, $) is a Dirac submanifold iff N has a $-compatible normal bundle νN such that for any normal to N vector ﬁeld X of M, one has (LX $)|ann(νN ) = 0. (26) Corollary 2.1. Let {Xα } be a family of Poisson inﬁnitesimal automorphisms of (M, $). Then any submanifold N such that span{Xα |N } is a $-compatible normal bundle of N is a Dirac submanifold. Proof. The hypotheses of Corollary 2.1 imply the characteristic conditions stated by Proposition 2.2.

The $-compatibility hypothesis of Corollary 2.1 also has the following meaning. A family {Xα } of Poisson inﬁnitesimal automorphisms has an associated generalized distribution D(Xα ) spanned by the $-Hamiltonian vector ﬁelds of the functions f ∈ C ∞ (M) that are constant along the orbits of the vector ﬁelds Xα , and this distribution is involutive. span{Xα |N } is a $-compatible normal bundle of N iff D(Xα ) ⊆ T N. If the family {Xα } reduces to one Hamiltonian vector ﬁeld Xh$ , we have no submanifolds N as in Corollary 2.1 since N should be both tangent and normal to Xh$ . However, we may have the required type of submanifolds if the family consists of a single inﬁnitesimal automorphism X that is not a Hamiltonian vector ﬁeld. For instance, if D(X) is a regular distribution, it must be a foliation and the leaves of this foliation are Dirac hypersurfaces of M. Before going on with the discussion of Dirac submanifolds, let us also consider some of the situations mentioned at the end of Section 1. Deﬁnition 2.2. A normalized submanifold (N, νN ) of the Poisson manifold (M, $) will be called an algebraically Poisson-compatible (a.P.c.) submanifold, respectively, a quasi-Dirac submanifold, if the Poisson bivector ﬁeld $ is algebraically compatible, respectively, quasi-soldered, to the submanifold. Thus the a.P.c. property is characterized by condition (i) of Deﬁnition 2.1, respectively, by the local condition (24), and the quasi-Dirac property is characterized by the local condition (25). The a.P.c. and Dirac properties of a submanifold may hold for more than one normal bundle [10]. A second normal bundle ν˜ N may be deﬁned by (7), and the corresponding local expression of the Poisson bivector ﬁeld is obtained by switching to the bases (Xa , ∂/∂y u ) in (23). Accordingly, for ν˜ N , the a.P.c. condition is equivalent to ($ab θbu )|N = 0 ⇔ ψ ◦ ($ |ann(T N ) ) = 0, (27) and condition (ii) of Deﬁnition 2.1 is equivalent to Xc [$ab θau θbv − $au θav + $av θau + $uv ]|x c =0 = 0. In view of (24), (25), and (27), (28) becomes

(28)

Dirac submanifolds of Jacobi manifolds

∂$ab u v ∂$av u ∂$ua v ∂$uv w θ θ + θ − θ − θ ∂x c a b ∂x c b ∂x c a ∂y w c

611

x c =0

= 0.

(29)

Condition (27) shows that if $ |ann(T N ) is a surjection, therefore, an isomorphism onto νN (equivalently, det ($ab ) = 0), then νN = $ (ann(νN )) provides the only normalization which makes N an a.P.c. submanifold of M. The submanifolds N such that $ (ann(νN )) is a complement of T N in T M|N are called cosymplectic submanifolds, and it is known that they are Dirac submanifolds [10]. Indeed, the a.P.c. property follows from the skew symmetry of $, and (25) follows from the following component of the Poisson condition [$, $] = 0 in the local adapted coordinates of (23) [8]: ∂$uv [$, $]auv |x c =0 = 2 $ab = 0. (30) ∂x b x c =0 The following result is, obviously, important. Proposition 2.3. If N is either an a.P.c. or a quasi-Dirac submanifold, the bivector ﬁeld $ = pT ($|N ) is a Poisson bivector ﬁeld on N . Moreover, in the a.P.c. case, $ does not depend on the choice of the normal bundle among those which satisfy Deﬁnition 2.2. Proof. From (23), it follows that 1 ∂ uv ∂ ∧ . $ = $ 2 ∂y u ∂y v x c =0

(31)

Then from [$, $] = 0 and either (24) or (25), we get ⎞ ⎛ u2 u3 ∂$ ⎠ [$, $]u1 u2 u3 |x c =0 = 2 ⎝ $u1 w ∂y w Cycl(u1 ,u2 ,u3 )

= [$ , $ ]u1 u2 u3 = 0;

(32)

x c =0

i.e., $ is a Poisson bivector ﬁeld on N. Finally, (24) shows that ι∗ ($ (dy u )) = $ (dy u ) (ι : N ⊆ M), and this proves the last assertion.

The Poisson structure $ is said to be induced by the Poisson structure $, and was deﬁned and studied for Dirac submanifolds in [10]. In the a.P.c. case, the submanifold has a second fundamental form (LX $)|ann(νN ) , which vanishes iff N is a Dirac submanifold. Remark 2.1. Several authors have studied a generalization of Poisson structures that consists of a pair ($, ), where $ is a bivector ﬁeld and is a closed, differential 3-form on the manifold M, such that [$, $] = $ .

(33)

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Such a pair is called a twisted Poisson structure. For details, see [7] and the references of that paper. The notions of an algebraically compatible, a quasi-Dirac and a Dirac submanifold may be deﬁned for twisted Poisson manifolds by asking the bivector ﬁeld $ to satisfy the same conditions as in the Poisson case. Then the proof of Proposition 2.3 shows that if ι : N → M is a submanifold of one of these three categories, ($ , ι∗ ) is an induced twisted Poisson structure. If σ : W → N is a compatible tubular neighborhood of (N, νN ), the induced Poisson structure is characterized by {f, g}$ = {f ◦ σ, g ◦ σ }$ ◦ ι

(34)

∀f, g ∈ C ∞ (N ) and ι : N ⊆ M. With the same tubular neighborhood, the a.P.c. property is equivalent to the fact that ∀f ∈ C ∞ (N ), the $-Hamiltonian vector ﬁeld of f ◦ σ is tangent to N or that one has ι∗ ◦ $ = $ ◦ σ ∗ .

(35)

Furthermore, N is a Dirac submanifold if, besides the above, it also has the property that ∀x ∈ N, ∀X ∈ Tx (Wx ), ∀f, g ∈ C ∞ (N ), one has X{f ◦ σ, g ◦ σ }$ = 0.

(36)

We summarize the above remarks in the following. Proposition 2.4. A submanifold N of the Poisson manifold (M, $) is an a.P.c. submanifold iff N is endowed with a Poisson structure $ and has a tubular neighborhood σ : W → N such that conditions (34) and (35) hold. Furthermore, N is a Dirac submanifold iff it is a.P.c. and condition (36) holds, too. Remark 2.2. In [10] the author uses the independence of the induced Poisson structure on the choice of the normal bundle to deﬁne local Dirac submanifolds [10] as submanifolds N of the Poisson manifold (M, $) such that ∀x ∈ N , there exists an open neighborhood U in M where N ∩ U is a Dirac submanifold. Proposition 2.3 shows that a local Dirac submanifold also inherits a well-deﬁned, global, induced Poisson structure. In fact, Proposition 2.3 shows that local a.P.c. submanifolds may be deﬁned similarly. Remark 2.3. Proposition 2.4 suggests considering submanifolds N of (M, $) which come endowed with a Poisson structure $ such that for some tubular neighborhood σ : W → N, σ is a Poisson mapping or, equivalently, the brackets {f ◦ σ, g ◦ σ }|$ , which are deﬁned on W , are constant along the ﬁbers of σ . Such submanifolds deserve the name of strong Dirac submanifolds. Obviously, they are Dirac submanifolds, and, with respect to adapted coordinates, one must have ∂$uv /∂x a ≡ 0. The complete lift of the Poisson bivector ﬁeld $ is a Poisson structure $C called the tangent Poisson structure of $. The tangent structure is exactly the one induced by the Lie algebroid structure of T ∗ M deﬁned by $ on the total space of its dual vector bundle T M. From Proposition 1.3, we get the following characteristic property of Dirac submanifolds.

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Proposition 2.5 ([10]). The submanifold N of (M, $) is a Dirac submanifold iff there exists a normal bundle νN which is a coisotropic submanifold of (T M, $C ). The following result is signiﬁcant for the next section. Recall that a Poisson structure $ is homogeneous if there exists a vector ﬁeld Z ∈ T M such that LZ $ = −$.

(37)

Proposition 2.6. Let N be an a.P.c. submanifold of the homogeneous Poisson manifold (M, $, Z) such that Z|N ∈ T N . Then (N, $ , Z|N ), where $ is the induced Poison structure, also is a homogeneous Poisson manifold. Proof. If the homogeneity condition (37) is evaluated on (dϕ, dψ) (ϕ, ψ ∈ C ∞ (M)), one gets the equivalent condition $ {ϕ, ψ}$ = Z{ϕ, ψ}$ − [Z, Xϕ$ ]ψ + [Z, Xψ ]ϕ,

(38)

where X denotes Hamiltonian vector ﬁelds. Now let σ : W → N be a tubular neighborhood of N where the conditions of Proposition 2.4 hold. Then (38), written for ϕ = f ◦ σ , ψ = g ◦ σ (f, g ∈ C ∞ (M)) and composed by ι, provides the similar condition (38) for $ .

Now we point out the existence of a speciﬁc cohomology related with a Dirac submanifold (N, νN ) of the Poisson manifold (M, $). Namely, since the Poisson bivector ﬁeld $ is soldered to N , V(M, N, νN) = (V k (M, N, νN), ∂$ = −[$, .])

(39)

is a subcomplex of the Lichnerowicz–Poisson cochain complex of (M, $) (e.g., [8]); k (M, N, νN) that will be called soldered therefore, it deﬁnes cohomology spaces HsP Poisson cohomology spaces. Proposition 2.7. The homomorphism $ : T ∗ M → T M induces homomorphisms k k k$ : HsdeR (M, N, νN) −→ HsP (M, N, νN).

(40)

If the Poisson structure $ is deﬁned by a symplectic form ω, the mappings (40) are isomorphisms. Proof. Using adapted local coordinates, it is easy to check that ∀κ ∈ k (M, N, νN), $ κ ∈ V k (M, N, νN) and, as for the general Poisson cohomology, ∂$ ◦$ = −$ ◦d. This justiﬁes the existence of the homomorphisms (40). In the symplectic case, N must be a symplectic (2n)-dimensional submanifold of (M 2m , ω), and νN must be the ω-orthogonal bundle of T N [10]. Indeed, each ∗ ∗ point x ∈ N has an open neighborhood with coordinates (x a , x a , y u , y u ) (a = 1, . . . , m − n, u = 2m − 2n + 1, . . . , 2m − n, a ∗ = a + m − n, u∗ = u + n) such that ∗ ∗ ω= dx a ∧ dx a + dy u ∧ dy u , (41) a

u

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I. Vaisman ∗

and N has the local equations x a = 0, x a = 0 [6]. Obviously, these coordinates also are adapted coordinates with respect to the ω-orthogonal bundle νN of T N. The uniqueness of the normal bundle follows from (27). Then $ has the inverse −0ω , and the latter induces the inverses of the homomorphisms (40).

We end this section by a brief discussion of the same subject in Nambu–Poisson geometry. A Nambu–Poisson manifold of order k is a manifold M m endowed with a Nambu–Poisson tensor ﬁeld , i.e., a k-vector ﬁeld Q ∈ V k (M) (k ≥ 3) which, when evaluated on differentials, deﬁnes a skew-symmetric bracket of k functions {f1 , . . . , fk } = Q(df1 , . . . , dfk ) that satisﬁes the generalized Jacobi identity {f1 , . . . , fk−1 , {g1 , . . . , gk }} =

k

(42)

{g1 , . . . , gh−1 , {f1 , . . . , fk−1 , gh }, gh+1 , . . . , gk }.

h=1

It is known that Q is a Nambu–Poisson tensor ﬁeld iff it is decomposable (i.e., a wedge product of k vectors) at each point, and its components in every local chart satisfy the equality ( ) m k i1 ...ik−1 h j1 ...jk j1 ...jl−1 hjl+1 ...jn i1 ...ik−1 jl Q . (43) ∂h Q − Q ∂h Q h=1

l=1

Another basic result tells that around each point x ∈ M where Qx = 0, there are local coordinates such that Q=

∂ ∂ ∧ ··· ∧ k ∂x ∂x 1

(44)

(e.g., see [9] for details). Accordingly, it is natural to deﬁne the following. Deﬁnition 2.3. Let (M m , Q) be a Nambu–Poisson manifold of order k and (N n , νN) (n ≥ k) a normalized submanifold. Then N is an algebraically compatible, a quasiDirac, or a Dirac submanifold if Q is algebraically compatible, quasi-soldered or soldered, respectively, to N. Obviously, the projection Q = pT (Q|N ) is a decomposable k-vector ﬁeld on N. Then a straightforward examination of condition (43), where the components of Q are as in formula (17), and an argument similar to that at the end of the proof of Proposition 2.3 show that we have the following. Proposition 2.8. If (N, νN ) is either an algebraically compatible or a quasi-Dirac (Dirac) submanifold of the Nambu–Poisson manifold (M, Q) of order k, Q is an induced Nambu–Poisson k-vector ﬁeld on N . Moreover, if algebraic compatibility holds, Q does not depend on the choice of νN .

Dirac submanifolds of Jacobi manifolds

615

Example 2.1. Consider the Nambu–Poisson manifold (Rm , Q), where Q is globally deﬁned by (44) with the canonical coordinates of Rm . Let N n (n ≥ k) be the ndimensional plane deﬁned by the equations x n+s =

n

cun+s x u + p n+s ,

s = 1, . . . , m − n,

(45)

u=1

where cun+s , pn+s = const. Then the tangent spaces of N are spanned by the vector ﬁelds m−n ∂ ∂ Xu = u + cun+s n+s , u = 1, . . . , n, ∂x ∂x s=1

and we may deﬁne a normalization where the normal bundle νN is spanned by (∂/∂x n+s )m−n s=1 . It follows easily that with this normalization, N is a quasi-Dirac submanifold with the induced Nambu–Poison structure Q = X1 ∧ · · · ∧ Xk . Furthermore, (N, νN ) above is a Dirac submanifold iff cun+s = 0 for u = 1, . . . , k; s = 1, . . . , m − n. In the Nambu–Poisson case, we may also distinguish a situation similar to that of the cosymplectic submanifolds of a Poisson manifold. Indeed, the interior product by a (k − 1)-form deﬁnes a vector bundle morphism Q : ∧k−1 T ∗ M → T M and the submanifold N of M will be called cosymplectic if im(Q |ann(T N )⊗∧k−2 T ∗ M ) is a normal bundle νN of the submanifold N . Then we get the following. Proposition 2.9. Any cosymplectic submanifold N n (n ≥ k) of a Nambu–Poisson manifold (M, Q) of order k is a Dirac submanifold. Proof. Of course, the normalization we have in mind is that produced by the deﬁnition of a cosymplectic submanifold, i.e., νN is spanned by the vector ﬁelds i(dx a ∧ dy u1 ∧ · · · ∧ dy uk−2 )Q = (−1)k−1 Qbau1 ...uk−2 + Qau1 ...uk−2 v

∂ ∂x b

(46)

∂ , ∂y v

where the local coordinates and components are those of the formula (17). The normality of these vectors implies that Q satisﬁes the condition of algebraic compatibility Qau1 ...uk−2 v = 0. The condition of cosymplecticity means that there are m−n choices of the group of indices (a, u1 , . . . , uk−2 ) with linearly independent corresponding vector ﬁelds (46). Then (43) written for these choices of the indices implies the fact that Q is quasi-soldered to (N, νN ).

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3 Dirac submanifolds of Jacobi manifolds For a detailed study of Jacobi manifolds we refer the reader to [1] and its references. Jacobi manifolds are a natural generalization of Poisson manifolds namely, a Jacobi structure on a manifold M m is a Lie algebra bracket {f, g} on C ∞ (M), which is given by bidifferential operators. It follows that one must have {f, g} = (df, dg) + f Eg − gEf,

(47)

where E is a vector ﬁeld and is a bivector ﬁeld on M such that4 [, ] = −2E ∧ ,

LE = 0.

(48)

Thus if E = 0, we have a Poisson structure. The Jacobi structure (M, , E) is equivalent to the homogeneous Poisson structure ∂ ∂ ∧ E), Z= $ = e−τ ( + (49) ∂τ ∂τ on M × R; M will then be identiﬁed with M × {0}. For instance [5], let $ be a linear Poisson structure on Rn \{0}, and consider the diffeomorphism S n−1 × R ≈ Rn \{0} deﬁned by n x i = e−τ ui , (ui )2 = 1, τ ∈ R. i=1

Then it is easy to check that $ must be of the form (49), which provides a Jacobi structure on S n−1 . We call this structure a Lichnerowicz–Jacobi structure of the sphere. On a Jacobi manifold, one may deﬁne Hamiltonian vector ﬁelds Xf = df + f E

(f ∈ C ∞ (M)),

(50)

and they span a generalized foliation S such that the leaves of S are either contact or locally conformal symplectic manifolds. For instance, in the case of a Lichnerowicz– Jacobi structure the leaves are the orbits of the quotient coadjoint representation of a connected Lie group G with the Lie algebra G of structure constants deﬁned by the corresponding linear Poisson structure $ (see above), i.e., the action deﬁned on S n−1 by the coadjoint action of G on G ∗ \{0} ≈ Rn \{0} if S n−1 is seen as a quotient space of G ∗ \{0} [5]. Another important fact we want to recall is that ∀ϕ ∈ C ∞ (M), the bracket {f, g}ϕ = e−ϕ {eϕ f, eϕ g}

(51)

is a Jacobi bracket said to have been obtained by a conformal change of the original bracket. The tensor ﬁelds of the new bracket are ϕ = eϕ ,

E ϕ = eϕ (E + i(dϕ)).

(52)

Now we begin our considerations on submanifolds. 4 The minus sign comes from our sign convention for the Schouten–Nijenhuis bracket in

Section 1.

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Deﬁnition 3.1. Let (N, νN ) be a normalized submanifold of the Jacobi manifold (M, , E). Then (1) N is an almost Dirac submanifold if is an N -soldered bivector ﬁeld; (2) N is an algebraically Jacobi-compatible (a.J.c.) submanifold if and E are algebraically compatible with the normalization of N ; (3) N is a (quasi-)Dirac submanifold if and E are (quasi-)soldered to (N, νN ). Equivalently, N is an almost Dirac submanifold if each point x ∈ N has a neighborhood with adapted coordinates as in Section 1 such that =

1 ab ∂ ∂ ∂ ∂ ∂ 1 ∂ ∧ + au a ∧ u + uv u ∧ v , 2 ∂x a ∂x b ∂x ∂y 2 ∂y ∂y

(53)

where au |x c =0 = 0, ∂uv = 0. ∂x a c

(54) (55)

x =0

Then N is a Dirac submanifold if, furthermore, the vector ﬁeld E is tangent to N and has the local expression ∂ ∂ E = a a + u u , (56) ∂x ∂y where ∂ u = 0. (57) a |x c =0 = 0, ∂x a x c =0 For the quasi-Dirac case, we only have the condition (55) and the second equality (57). Finally, N is an a.J.c. submanifold if (54) and the ﬁrst condition (57) hold. Proposition 3.1. Let (N, νN ) be either an almost Dirac or an a.J.c. or a quasiDirac submanifold of the Jacobi manifold (M, , E). Then [ = pT (|N ), E = pT (E|N )] is a Jacobi structure on N . Furthermore, in the almost Dirac and the a.J.c. case does not depend on the choice of the normalization. Proof. With local adapted coordinates, we have 1 ∂ ∂ = uv u ∧ v 2 ∂y ∂y x c =0

(58)

and ⎛ [, ]u1 u2 u3 |x c =0 = 2 ⎝

u1 a

Cycl(u1 ,u2 ,u3 )

+

Cycl(u1 ,u2 ,u3 )

∂u2 u3 ∂x a

(59)

⎞ u2 u3 ∂ ⎠ u1 w ∂y w

x c =0

= [ , ]u1 u2 u3 .

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Hence in all the cases of the proposition, we have [ , ] = pT ([, ]|N ) = −2E ∧ . Then an examination of the coordinate expression of LE , where E and are given by (56) and (53), respectively, shows that the conditions (54) and (55), as well as either (54) and the ﬁrst condition (57) or (55) and the second condition (57), imply that LE = pT (LE |N ) = 0. Finally, where asserted, the independence of of the normalization follows from

ι∗ ( (dy u ) = (dy u ), ι : N ⊆ M. We notice that formula (59) also implies the following. Proposition 3.2. Let (N, νN ) be a normalized submanifold of the Jacobi manifold (M, , E) such that is algebraically compatible with the normalization and E is a normal ﬁeld of N. Then is a Poisson structure on N , and it is independent on the choice of νN among all possible choices that contain E|N . The Jacobi or Poisson structures deﬁned on N by ( , E ) are said to be induced by (, E). In the cases where only algebraic compatibility holds, invariants of the second fundamental form type (LX )|ann(νN ) , [X, E]|ann(νN ) , where X is normal to N, appear. Proposition 1.2 allows us to give some simple examples. Consider the Jacobi manifold M = R3n+1 with ⎞ ⎛ n n ∂ ∂ ∂ ∂ ∂ ⎠, E=t (60) ui i ∧ + t pj = ∧⎝ ∂q ∂pi ∂t ∂pj ∂t i=1

j =1

(the variables of (60) are the natural coordinates of M). Then the hyperplane t = 0 is the ﬁxed point locus of the involution (ui , q i , pi , t) → (ui , q i , pi , −t). This involution preserves the tensor ﬁelds (60); hence the hyperplane t = 0 is a Dirac submanifold with an induced Poisson structure. For the same M, the involution (ui , q i , pi , t) → (−ui , −q i , pi , t) also preserves (60); hence its ﬁxed point locus, which is the (n + 1)-plane ui = 0, q i = 0, is a Dirac submanifold with an induced Jacobi structure. Finally, if we restrict M to the domain pi > 0, t > 0 and consider the involution that sends t to 1/t and preserves the other coordinates, only of (60) is preserved hence, the ﬁxed point locus, which is the hyperplane t = 1, is an almost Dirac submanifold. Moreover, the last involution sends E to −E. Therefore, E is normal to the submanifold, and the induced structure is a Poisson structure. Another interesting fact is the following. Proposition 3.3. Let (M, , E) be a Jacobi manifold, and (M × R, $), with $ deﬁned by (49), the corresponding homogeneous Poisson manifold. Then N is an a.J.c. or a Dirac submanifold of the former iff N × R is an a.P.c., respectively, Dirac, submanifold of the latter.

Dirac submanifolds of Jacobi manifolds

619

Proof. We will use the lift of νN to N ×R as a normal bundle. τ of (49) is a coordinate along N × R, and we see that if and E are of the local form (53), (56) then $ satisﬁes the conditions (24), (25), and conversely.

From Proposition 1.3 it follows that the almost Dirac and Dirac submanifolds of a Jacobi manifold may also be characterized by using the tangent bundle T M; namely, we have the following. Proposition 3.4. The normalized submanifold (N, νN ) is almost Dirac iff the submanifold νN of T M is such that C (ann(T νN )) ⊆ T νN , where C is the complete lift of . Furthermore, N is a Dirac submanifold iff besides the previous condition, one also has E C ∈ (T νN), where E C is the complete lift of the vector ﬁeld E. Remark 3.1. The tensor ﬁelds (C , E C ) do not deﬁne a Jacobi structure on T M. A tangent Jacobi structure can be obtained by considering the Poisson structure induced on the manifold T M × R by the Lie algebroid J 1 M = T ∗ M × R of (, E) [4]. Namely, with the notation of (20), (21), if we associate with each cross-section (f, αi dzi ) ∈ J 1 M the function eτ (f + αi z˙ i ) ∈ C ∞ (T M × R), the Lie algebroid bracket of J 1 M yields a Poisson bracket of the speciﬁed kind of functions, which extends to a Poisson bracket on C ∞ (T M × R). Computations show that the Poisson bivector of this structure is 4 5 ∂ ˜ = e−τ C − V − E ∧ (E C − E V ) + $ ∧ EC , (61) ∂τ where the upper index V denotes the vertical lift [11] and E is the Euler vector ﬁeld , i.e., EV = Ei

∂ , ∂ z˙ i

V =

1 ij ∂ ∂ ∧ j, i 2 ∂ z˙ ∂ z˙

E = z˙ i

∂ . ∂ z˙ i

(62)

Accordingly, (C − V − E ∧ (E C − E V ), E C ) is a Jacobi structure on T M, which deserves the name of tangent Jacobi structure. A particular class of Dirac submanifolds was studied in [1], and here we reprove the following. Proposition 3.5 ([1]). Assume that N is a submanifold of (M, , E) such that (ann(T N )) is a normal bundle νN of N. Then N is a Dirac submanifold iff the vector ﬁeld E is tangent to N . Furthermore, there always exists a conformal change (51), with ϕ|N = 0, such that N is a Dirac submanifold of (M, ϕ , E ϕ ). Proof. If N is a Dirac submanifold, E ∈ T N by deﬁnition. For the converse, we use the normal bundle of the hypothesis, and represent and E by (53) and (56), respectively. Clearly, the choice of νN is such that (dx a ) ∈ νN along N , which is equivalent to (54). If this condition holds, the auv-component of the ﬁrst equality (48) yields

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ab ∂

uv

∂x b

+E a

uv

|N = 0.

(63)

Since (ann(T N )) is normal to N iff the matrix (ab ) is nondegenerate, we see that E tangent to N implies (55). Furthermore, if E|N ∈ T N , the second equality (48) yields ∂E u ab b = 0; (64) ∂x N therefore, (57) holds. Concerning the last part of the proposition, (52) shows that the required conformal transformation exists if there exists a function ϕ ∈ C ∞ (M), which vanishes on N and is such that ∂ϕ E a |N = ab b . (65) ∂x N Since (ab ) is nondegenerate, the conditions for ϕ prescribe the 1-jet with respect to the variables (x a ) of ϕ at the points of N . Therefore, a required function ϕ exists around every point of N . Then these local solutions may be glued up by a partition of unity along N. (See also the argument of [1].)

We end by a discussion of Dirac submanifolds of the transitive Jacobi manifolds, i.e., locally conformal symplectic (l.c.s.) and contact manifolds [1]. Proposition 3.6. A submanifold N is an almost Dirac submanifold of the l.c.s. manifold M iff it is Dirac, and this happens iff N inherits from M an induced l.c.s. structure. Moreover, there is only one possible normal bundle, the symplectic orthogonal bundle of T N. Proof. Recall that the l.c.s. structure of M is a nondegenerate 2-form such that for some open covering M = ∪α Uα , ∀α |Uα = e−σα α , where σα are functions, α are 2-forms and d α = 0. Equivalently, d = ω ∧ , where ω is the closed 1-form deﬁned by gluing up the local forms dσα (ω is called the Lee form). It is known that M is a Jacobi manifold with the structure deﬁned by the bivector ﬁeld , where = 0−1

, and the vector ﬁeld E = ω [1]. Assume that N is an almost Dirac submanifold with the normal bundle νN . Then |ann(T N ) is an isomorphism onto νN, which, as in (27), ensures the uniqueness of νN , and |ann(νN ) is an isomorphism on T N, which is equivalent to the fact that ι∗ (ι : N ⊆ M) is nondegenerate and provides an l.c.s. structure on N . Accordingly, we may use Marle’s theorem [6], and ﬁnd local coordinates of N on some neighborhood Uα such that with the notation of (41), one has ' * ∗ ∗ (66)

|Uα = e−σα dx a ∧ dx a + dy u ∧ dy u . a

u

Obviously, this expression of implies that νN is -orthogonal to T N and that the coordinates used in (66) are adapted coordinates.

Dirac submanifolds of Jacobi manifolds

621

Now condition (55) applied to (66) becomes (∂σα /∂x a )x c =0 = 0, i.e., ωa |x c =0 = 0. Furthermore, one of the conditions that express dω = 0 is ∂ωa /∂y u = ∂ωu /∂x a , whence we also get (∂ωu /∂x a )x c =0 = 0. Therefore, the Lee form ω, and the vector ﬁeld E too, are soldered to N , and N must be a Dirac submanifold of M. The converse part of the proposition follows from (66).

Proposition 3.7. Let M 2m+1 be a contact manifold with the contact 1-form θ. Then a submanifold ι : N ⊆ M is a Dirac submanifold iff ι∗ θ is a contact form on N. Furthermore, the normal bundle of N is unique, and it is the dθ -orthogonal bundle of T N. Proof. Recall that θ is a contact form iff θ ∧ (dθ )m vanishes nowhere. A contact form produces a Jacobi structure [1], which consists of the vector ﬁeld E deﬁned by i(E)θ = 1,

i(E)dθ = 0,

(67)

and the bivector ﬁeld (df, dg) = dθ(Xfθ , Xgθ )

(f, g ∈ C ∞ (M)),

(68)

where the Hamiltonian vector ﬁeld Xfθ is deﬁned by i(Xfθ )θ = f,

i(Xfθ )dθ = −df + (Ef )θ.

(69)

From (69) we get Xfθ = df + f E,

(df, dg) = dθ ( df, dg).

(70)

If M is the contact manifold above, M × R has the Poisson bivector $ given by (49), and it also has the symplectic form

= eτ (dθ + dτ ∧ θ ).

(71)

An easy computation shows that all the functions of the form eτ f ∈ C ∞ (M × R) (τ ∈ R, f ∈ C ∞ (M)) have the same Hamiltonian vector ﬁelds with respect to $ and . Therefore, $ ◦ 0 = − Id . Now Proposition 3.3 states that N is a Dirac submanifold of M iff N ×R is a Dirac submanifold of M × R. In the present case, this means that N × R is a symplectic submanifold of (M × R, ), and it follows that (N, ι∗ θ ) must be a contact manifold, and that the normal bundle must be the one indicated by the proposition.

References [1] P. Dazord, A. Lichnerowicz, and Ch.-M. Marle, Structure locale des variétés de Jacobi, J. Math. Pure Appl., 70 (1991), 101–152. [2] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, New York, 1993.

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[3] J. Huebschmann, Lie-Reinhart algebras, Gerstenhaber algebras and Batalin-Vilkoviski algebras, Ann. Inst. Fourier (Grenoble), 48 (1998), 425–440. [4] Y. Kerbrat and Z. Souici-Benhammadi, Variétés de Jacobi et groupoïdes de contact, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 81–86. [5] A. Lichnerowicz, Représentation coadjointe quotient et espaces homogènes de contact ou localement conformément symplectiques, J. Math. Pures Appl., 65 (1986), 193–224. [6] Ch.-M. Marle, Sous-variétés de rang constant d’une variété symplectique, Astérisque, 107–108 (1983), 69–86. [7] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background, Progr. Theorer. Phys. Suppl., 144 (2001), 145–154. [8] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, Vol. 118, Birkhäuser, Basel, 1994. [9] I. Vaisman, A survey on Nambu-Poisson brackets, Acta Math. Univ. Comenianae, 68 (1999), 213–241. [10] P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. Ec. Norm. Sup., 36 (2003), 403–430. [11] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, New York, 1973.

Quantum maps and automorphisms∗ Steve Zelditch Department of Mathematics Johns Hopkins University Baltimore, MD 21218 USA [email protected] Abstract. There are several inequivalent deﬁnitions of what it means to quantize a symplectic map on a symplectic manifold (M, ω). One deﬁnition is that the quantization is an automorphism of a ∗-algebra associated to (M, ω). Another is that it is unitary operator Uχ on a Hilbert space associated to (M, ω), such that A → Uχ∗ AUχ deﬁnes an automorphism of the algebra of observables. A yet stronger one, common in partial differential equations, is that Uχ should be a Fourier integral operator associated to the graph of χ. We compare the deﬁnitions in the case where (M, ω) is a compact Kähler manifold. The main result is a Toeplitz analogue of the Duistermaat–Singer theorem on automorphisms of pseudodifferential algebras, and an extension which does not assume H 1 (M, C) = {0}. We illustrate with examples from quantum maps.

1 Introduction Much attention has been focussed recently on ∗ products on Poisson manifolds (M, {, }) (see, among others, [Kontsevich (1997)], [Cattaneo–Felder (2000)], [Karabegov–Schlichenmaier (2001)], [Tamarkin (1998)], [Reshetikhin–Takhtajan (1999)], [Etingof–Kazhdan (1996)], [Weinstein–Xu (1998)], [Boutet de Monvel 5 (2002)]). Such ∗ products are viewed as quantizing functions on M to an algebra of observables, which we will refer to as a ∗ product algebra, or more simply as a ∗-algebra. This article is concerned with the related problem of quantizing symplectic maps χ on Kähler manifolds (M, ω), a special case of the problem of quantizing Poisson maps. From the ∗-algebra viewpoint, it seems most natural to quantize such a symplectic map as an automorphism of a ∗-algebra associated with (M, ω), specifically the (complete) symbol algebra T ∗ /T −∞ of Berezin–Toeplitz operators over (M, ω). These symbol algebras are basic examples of abstract ∗-algebras arising in deformation quantization of Poisson manifolds (see [Boutet de Monvel 1 (1999)], [Boutet de Monvel 3 (1999)], [Boutet de Monvel 5 (2002)], [Charles (2003)], [Cattaneo–Felder (2000)], [Guillemin (1995)], [Schlichenmaier (1999)], [Schlichenmaier ∗ This research was partially supported by NSF grant DMS-0071358 and by the Clay Math-

ematics Institute.

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(1998)] for more on this aspect). But they carry more structure than bare ∗-algebras: the Toeplitz operator algebra T ∗ of which T ∗ /T −∞ is the symbol algebra also comes with a representation as operators on a Hilbert space. In the Hilbert space setting, it is most natural to try to quantize a symplectic map χ as a unitary operator Uχ , and to induce the automorphism Uχ AUχ∗ on T ∗ . As will be explained below (see also [Zelditch (1997)]), it is not always possible to quantize a symplectic map this way. When possible, the quantization Uχ is an example of what is known as a quantum map in the literature of quantum chaos. Such quantum maps have also been the focus of much attention in recent years by a virtually disjoint group (see, e.g., [de Biévre–degli Esposti (1998)], [Keating (1991)], [degli Esposti–Grafﬁ–Isola (1995)], [Hannay–Berry (1980)], [Marklof–Rudnick (2000)], [Zelditch (1997)]). The main purpose of this article is to contrast the different notions of quantizing symplectic maps, as as they arise in Toeplitz ∗-algebras, partial differential equations and quantum chaos. Aside from its intrinsic interest, the relation between quantum maps and automorphisms of ∗-algebras has practical consequences in quantum chaos, i.e., in the relations between dynamical properties of χ and the eigenvalues/eigenfunctions of its quantization Uχ . In the physics literature of quantum chaos, quantum maps are studied through examples such as quantum kicked tops (on S 2 ), cat maps, rotors, baker’s map and standard maps (on the 2-torus T2 ). Almost always, the quantizations are given as explicit unitary matrices UN (depending on a Planck constant 1/N ) on special Hilbert spaces HN , often using some special representation theory, and no formal deﬁnition is given of the term “quantum map.’’The need for precise deﬁnitions is felt, however, as soon as one aims at quantizing maps which lie outside the range of standard examples. Even symplectic maps on surfaces of genus g ≥ 2 count as nonstandard, and only seem to have been quantized by the Toeplitz method discussed in this paper and in [Zelditch (1997)]. A further reason to study quantum maps versus automorphisms is to better understand obstructions to quantizations. It is often said that Kronecker translations Tα,β (x, ξ ) = (x + α, ξ + β),

(x, ξ ) ∈ R2n /Z2n

and afﬁne symplectic torus maps fα (x, ξ ) = (x + ξ, ξ + α) are not quantizable, for reasons explained in Proposition 2.2. Nevertheless, the paper [Marklof–Rudnick (2000)] proposes a quantization of such maps. Of course, the resolution of this paradox is that a weaker notion of quantization is assumed in [Marklof–Rudnick (2000)] than elsewhere, as will be explained below. The implicit criterion (including that in [Marklof–Rudnick (2000)]) that UN quantize a symplectic map χ is that the Egorov-type formula UN∗ OpN (a)UN ∼ OpN (a ◦ χ )

(N → ∞)

(1)

hold for all elements OpN (a) of the algebra TN of observables, where χ is a symplectic map of (M, ω). Postponing precise deﬁnitions, we see that the operative condition is

Quantum maps and automorphisms

625

that UN∗ OpN (a)UN deﬁnes an automorphism of TN , at least to leading order. Here, our notation for observables and quantum maps are in terms of sequences as the inverse Planck constant N varies. We temporarily write T ∗ for sequences {OpN (a)} of observables (with T −∞ the sequences which are rapidly decaying in N ), and U ∼ {UN } for sequences of unitary quantum maps. We will soon give more precise deﬁnitions. We now distinguish several notions of quantizing a symplectic map and make a number of assertions which will be justiﬁed in the remainder of the article. •

•

•

•

•

•

There is a geometric obstruction to quantizing a symplectic map χ as a Toeplitz quantum map Uχ ,N on HN (see Deﬁnition 1.5 and Proposition 2.2). Kronecker translations and parabolic maps of the torus are examples of nonquantizable symplectic maps in the Toeplitz sense (see Propositions 5.1 and 5.3). There is no obstruction to quantizing a symplectic map as an automorphism of the Toeplitz symbol algebra T ∗ /T −∞ (see Theorem 1.6). For instance, Kronecker maps and cat maps are quantizable as automorphisms (see Propositions 5.2–5.4). Conversely, if H 1 (M, C) = {0}, then every order-preserving automorphism of the symbol algebra T ∗ /T −∞ on M is induced by a symplectic map of (M, ω) (see Theorem 1.6 for this and for the case where H 1 (M, C) = {0}). There is an obstruction to “extending’’ an automorphism α of T ∗ /T −∞ as an automorphism of T ∗ . In particular, there is an obstruction to inducing automorphisms αN of the ﬁnite-dimensional algebras of operators TN acting on HN (see Theorem 1.6). Again, Kronecker maps are examples (see Section 5). Any sequence UN of unitaries on HN which deﬁnes an automorphism of T ∗ /T −∞ must be a Toeplitz quantum map in sense of Deﬁnition 1.5(i) (see [Boutet de Monvel 4 (1985)], [Zelditch (1997)]). Many of the key problems of quantum chaos, e.g., problems on eigenvalue level spacings or pair correlation, on ergodicity and mixing of eigenfunctions (etc.) concern only the spectral theory of the automorphism quantizing χ and not the unitary map per se (see Section 6).

1.1 The Toeplitz setup In order to state our results precisely, we need to specify the framework in which we are working. The framework of Toeplitz operators used in this paper is the same as in [Boutet de Monvel 1 (1999)], [Boutet de Monvel 2 (1998)], [Guillemin (1995)], [Bleher–Shiffman–Zelditch (2001)], [Shiffman–Zelditch], [Zelditch (1997)], [Zelditch (1998)]. We brieﬂy recall the notation and terminology. Our setting consists of a compact, connected Kähler manifold (M, ω) with 1 [ω] ∈ H 1 (M, Z). Under this integrality condition, there exists a positive Her2π mitian holomorphic line bundle (L, h) → M over M with curvature form √ −1 ¯ ∂ ∂ log eL h = ω, c1 (h) = − π where eL is a nonvanishing local holomorphic section of L, and where eL h = 1 m h(eL , eL )1/2 denotes the h-norm of eL . We give M the volume form dV = m! ω .

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The Hilbert spaces “quantizing’’ (M, ω) are then deﬁned to be the spaces H 0 (M, LN ) of holomorphic sections of LN = L ⊗ · · · ⊗ L. The metric h induces 0 N Hermitian metrics hN on LN given by s ⊗N hN = sN h . We give H (M, L ) the inner product s1 , s2 = hN (s1 , s2 )dV (s1 , s2 ∈ H 0 (M, LN )), (2) M

and we write |s| = s, s1/2 . We then deﬁne the Szegö kernels as the orthogonal projections $N : L2 (M, LN ) → H 0 (M, LN ), so that ($N s)(w) = hN s ∈ L2 (M, LN ). (3) z (s(z), $N (z, w))dVM (z), M

Instead of dealing with sequences of Hilbert spaces, observables and unitary operators, it is convenient to lift them to the circle bundle X = {λ ∈ L∗ : λh∗ = 1}, where L∗ is the dual line bundle to L, and where h∗ is the norm on L∗ dual to h. ¯ X and the volume form Associated to X is the contact form α = −i∂ρ|X = i ∂ρ| dVX =

1 α ∧ (dα)m = α ∧ π ∗ dVM . m!

(4)

Holomorphic sections then lift to elements of the Hardy space H2 (X) ⊂ L2 (X) of square-integrable CR functions on X, i.e., functions that are annihilated by the Cauchy–Riemann operator ∂¯b and are L2 with respect to the inner product 1 F1 , F2 = F1 F2 dVX , F1 , F2 ∈ L2 (X). (5) 2π X We let rθ x = eiθ x (x ∈ X) denote the S 1 action on X and denote its inﬁnitesimal genO ∂ 2 (X), erator by ∂θ . The S 1 action on X commutes with ∂¯b ; hence H2 (X) = ∞ H N =0 N 2 (X) = {F ∈ H2 (X) : F (r x) = eiN θ F (x)}. A section s of LN deterwhere HN θ N mines an equivariant function sˆN on L∗ by the rule sˆN (λ) = (λ⊗N , sN (z)),

λ ∈ L∗z ,

z ∈ M,

where λ⊗N = λ⊗· · ·⊗λ. We henceforth restrict sˆ to X and then the equivariance property takes the form sˆN (rθ x) = eiN θ sˆN (x). The map s → sˆ is a unitary equivalence 2 (X). We refer to [Boutet de Monvel–Guillemin (1981)], between H 0 (M, LN ) and HN [Boutet de Monvel–Sjöstrand (1976)], [Bleher–Shiffman–Zelditch (2001)], [Zelditch (1998)] for further background. We now deﬁne the (lifted) Szegö kernel of degree N to be the orthogonal projection 2 (X). It is deﬁned by $N : L2 (X) → HN $N F (x) = $N (x, y)F (y)dVX (y), F ∈ L2 (X). (6) X

The full Szegö kernel is the direct sum

Quantum maps and automorphisms

$=

∞ D

627

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$N .

N =1

Following Boutet de Monvel–Guillemin [Boutet de Monvel–Guillemin (1981)], we then deﬁne the following. Deﬁnition 1.1. The ∗-algebra T ∗ (M) of Toeplitz operators of (M, ω) is the algebra of operators on H2 (X) of the form $A$ =

∞ D N =1

$N AN $N , A ∈ #S∗1 (X),

(8)

where #S∗1 (X) is the algebra of pseudodifferential operators of integral order over X which commute with the S 1 action, and where 1 AN = ei N θ Ae−iN θ dθ. (9) 2π S 1 ∂ Here [A, N ] = 0, where N = 1i ∂θ is the operator generating the S 1 action, 2 whose eigenvalue in HN (X) equals N . Since the symbol of A is S 1 -invariant, a Toeplitz operator of order s possesses an expansion ∞ $A$ ∼ N s $aj $N −j , (10) j =0

where aj ∈

C ∞ (M).

We may also express it in the direct sum form $A$ =

∞

$ N aN $ N ,

(11)

N =1 s is a semiclassical symbol of some order s, i.e., admits an where aN (z, z¯ ) ∈ Sscl asymptotic expansion

aN (z, z¯ ) ∼ N s

∞

N −j aj (z, z¯ ),

aj (z, z¯ ) ∈ C ∞ (M)

(12)

j =0

in the sense of symbols. We deﬁne the order of a Toeplitz operator $A$ to be the order s of the symbol. The order deﬁnes a ﬁltration of T ∗ by spaces of operators T s of order s ∈ Z. See [Guillemin (1995)] for further background. −∞ We also deﬁne “ﬂat’’ symbols f (z, N ) ∈ Sscl , denoted f ∼ 0, as functions −m satisfying f = O(N ) for all m. We then deﬁne T −∞ to be the ﬂat (or smoothing) Toeplitz operators (possessing a ﬂat symbol). The following deﬁnition is important in distinguishing the automorphisms that concern us. Deﬁnition 1.2. The complete Toeplitz symbol algebra (or smooth Toeplitz algebra) is the quotient algebra T ∗ /T −∞ .

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O We often view ∞ N =1 $N aN $N as the sequence {$N aN $N } of operators on the sequence HN ! H 0 (M, LN ) of Hilbert spaces. The physicists’ notation for $N aN $N is OpN (aN ). Viewing symbols as sequences {aN (z, z¯ ), we deﬁne the ∗N product by $N aN $N ◦ $N bN $N = $N aN ∗N bN $N . (13) In the appendix (Section 7), we will describe the calculation of aN ∗N bN so that it will not seem abstract to the reader. We now introduce automorphisms. Deﬁnition 1.3. An order-preserving automorphism α of T ∗ /T −∞ is an automorphism which preserves the ﬁltration T s /T −∞ . We denote the algebra of such automorphisms by Aut o (T ∗ /T −∞ ). It is important to distinguish • •

order-preserving automorphisms of T ∗ that preserve T −∞ ; order-preserving automorphisms of the symbol algebra T ∗ /T −∞ .

Since elements of T ∗ and of T ∗ /T −∞ commute with the S 1 action, either kind of automorphism satisﬁes * ' ∞ ∞ D D α $ N aN $ N ∼ $ N bN $ N , (14) N =1

N =1

where bN is a semiclassical symbol of the same order as aN . In the case of automorphisms of T ∗ , we can conclude that α($N aN $N ) = $N bN $N and that α induces automorphisms αN of the ﬁnite-dimensional algebras TN for ﬁxed N . However, for automorphisms of T ∗ /T −∞ in general, α($N aN $N ) is not even deﬁned since $N aN $N ∈ T −∞ . To put it another way, we cannot uniquely represent an element of the ﬁnite-dimensional algebra as $N aN $N although we can uniquely represent elements of T ∗ /T −∞ this way. 1.1.1 Covariant and contravariant symbols Let $N a$N be a Toeplitz operator. By the contravariant symbol of $N aN $N is meant the multiplier aN . By the covariant symbol of an operator F is meant the function F zN , w $N F $N (z, z) N |z=w = , (15) fˆ(z, z¯ ) = zN , w $N (z, z) N where $N (z, w) w (16) N (z) = $N (·, z) is the L2 -normalized “coherent state’’ centered at w. When F = $N a$N , we get a(z, ˆ z¯ ) =

$N a$N (z, z) . $N (z, z)

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We use the notation IN (a) = aˆ for the linear operator (the Berezin transform) which takes the contravariant symbol to the covariant symbol (see [Reshetikhin–Takhtajan (1999)] for background).

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629

1.2 Statement of results Let us now consider the senses in which we can quantize symplectic maps in our setting. The ﬁrst sense is that of quantizations of symplectic maps as Toeplitz Fourier integral operators. The deﬁnition is as follows. Deﬁnition 1.4. Suppose that the symplectic map χ of (M, ω) lifts to (X, α) as a contact transformation χ˜ . By the Toeplitz Fourier integral operator (or quantum map) deﬁned by χ we mean an operator D U= Uχ ,N , Uχ ,N = $N Tχ σN $N , N

where Tχ : L2 (X) → L2 (X) is the translation Tχ (f ) = f ◦ χ˜ −1 and where σN is a symbol designed to make Uχ ,N unitary. (Such a symbol always exists [Zelditch (1997)].) We now distinguish several notions of quantizing a symplectic map. Deﬁnition 1.5. Let χ be a symplectic map of (M, ω). In descending strength, we say the following: (a) χ is quantizable as a Toeplitz quantum map (or Toeplitz Fourier integral operator) if it lifts to a contact transformation χ˜ of (X, α). The quantization is then that of Deﬁnition 1.4. (b) χ is quantizable as an automorphism of the full observable algebra if there exists an automorphism α of T ∗ satisfying (1). (c) χ is quantizable as an automorphism of the symbol algebra if there exists an automorphism α of T ∗ /T −∞ satisfying (1). By descending strength, we mean that quantization in the sense above implies quantization in all of the following senses. The automorphisms above are order preserving in the sense that the order of α($A$) is the same as the order of $A$. Henceforth, all automorphisms will be assumed to be order preserving. We now explain the relations between these notions of quantization. We are guided in part by the analogous relations between quantizations of symplectic maps (of cotangent bundles) and automorphisms of the symbol algebra # ∗ /# −∞ of the algebra of pseudodifferential operators, as determined by Duistermaat–Singer in [Duistermaat–Singer (1976)], [Duistermaat–Singer (1975)]. Their main result was that if H 1 (S ∗ M, C) = {0}, then every order-preserving automorphism of # ∗ /# −∞ is either conjugation by an elliptic Fourier integral operator associated to the symplectic map or a transmission (we refer to [Duistermaat–Singer (1976)] for the deﬁnition). We prove an analogous theorem for Toeplitz operators and also extend it to the case where the phase space is not simply connected. To state the results, we need some notation. We denote the universal cover of (M, ω) by M˜ and denote the group of deck transformations of the natural cover p : M˜ → M by . We lift all objects on M to M˜ under p. We denote by Tγ the

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˜ We also denote by T ∗ the algebra unitary operator of translation by γ on L2 (M). ˜ of -invariant Toeplitz operators on M. It is important to understand that T∗ is not isomorphic to the algebra of Toeplitz operators on M since there are nontrivial (smoothing) operators which act trivially on automorphic (periodic) functions. In other words, the representation of T∗ on automorphic sections has a kernel, which ˜ we denote by K , and T ∗ (X) ! T∗ (X)/K . Automorphisms which descend to the ﬁnite Toeplitz algebras are precisely those which preserve the subalgebra K . For further discussion, we refer to Section 4. Theorem 1.6. With the above notation, we have the following: (0) (Essentially known.) A symplectic map of (M, ω) lifts to a contact transformation of (X, α) and hence deﬁnes a Toeplitz quantum map if and only if it preserves holonomies of all closed curves of M. (See Proposition 2.2 of Section 2). (i) Any symplectic map of any compact Kähler manifold (M, ω) is quantizable as an automorphism of the algebra T ∗ /T −∞ of smooth Toeplitz operators over M. (ii) Suppose that H 1 (M, C) = {0}. Then any order-preserving automorphism of T ∗ /T −∞ is given by conjugation with a Toeplitz Fourier integral operator on M associated to a symplectic map χ of (M, α). (The map lifts to a contact transformation of (X, α) by (0)). (iii) Suppose H 1 (M, C) = {0}. Then to each automorphism of T ∗ /T −∞ there corresponds a symplectic map χ of (M, ω) and a Toeplitz Fourier integral operator (Deﬁnition 1.5) Uχ on the universal cover M˜ which satisfy Tγ∗ Uχ Tγ = Mγ Uχ , where Mγ is a central operator. The automorphism A → Uχ∗ AUχ is -invariant and deﬁnes an order-preserving automorphism of the algebra T∗ which induces α on the -invariant symbol algebra T ∗ /T −∞ . (iv) Let K = ker ρ , where ρ is the representation of T∗ on -automorphic func˜ If α preserves K , then it induces an order-preserving automorphism tions on M. on T ∗ (M) and hence on the ﬁnite rank observables OpN (a) on HN . We separate the proof into the cases H 1 (M, C) = {0} in Section 3 and = {0} in Section 4. The latter case is very common in the physics literature on quantum maps. The difference between order-preserving automorphisms of T ∗ and T ∗ /T −∞ is very signiﬁcant, and only the former automorphisms are quantum maps in the physics sense. For instance, as will be seen in Section 2, Kronecker maps and afﬁne symplectic maps are quantizable as automorphisms of the symbol algebra, but do not lift to contact transformations of X, do not preserve the kernel K and are therefore not automorphisms of T ∗ . Regarding (iv), it is not clear to us whether this operator condition is equivalent to the holonomy-preservation condition in (0). As a corollary, we prove a result which indicates that the physicists’ quantum maps are necessarily Toeplitz quantum maps once they are conjugated to the complex (Bargmann) picture. O Corollary 1.7. If {UN } is a sequence of unitary operators on HN and if UN deﬁnes an order-preserving automorphism α of T ∗ /T −∞ , then {UN } must be a Toeplitz H 1 (M, C)

Quantum maps and automorphisms

631

Fourier integral operator associated to a quantizable symplectic map in the sense of Deﬁnition 1.4. As a gauge of our deﬁnitions, let us reconsider the Marklof–Rudnick quantizations mentioned above of Kronecker maps, parabolic maps and other “nonquantizable’’ maps [Marklof–Rudnick (2000)]. They deﬁne a sequence {UN } of unitary operators on HN satisfying the leading order condition (1), but not to any lower order. Hence conjugation Uχ OpN (a)Uχ∗ of an observable in T ∗ by their quantum map is no longer an observable, i.e., it is not an element of T ∗ . Rather, its Toeplitz symbol only possesses a one term asymptotic expansion and is not a classical symbol. Hence it need not correspond to a quantizable symplectic map. In addition to Theorem 1.6, we discuss a related issue revolving around the quantum maps versus automorphisms distinction: From the viewpoint of quantum chaos, the main interest in the quantum maps Uχ ,N lies in their spectral theory and its relation to the dynamics of χ . This is only well deﬁned when the associated symplectic map is quantizable in the strong sense as a sequence of unitary operators on HN . As stated in the corollary, the symplectic map must then lift to a contact transformation. In the last section Section 6, we point out that even when the symplectic map is quantizable as a unitary operator, it is often the automorphism it induces which is most signiﬁcant in quantum chaos. That is, much of the spectral theory in quantum chaos concerns the spectrum of the automorphism induced by Uχ ,N rather than the spectrum of Uχ ,N itself.

2 Toeplitz quantization of symplectic maps In this section, we consider the quantization of symplectic maps as Toeplitz Fourier integral operators. In some sense, the material in this section is known, but it seems worthwhile to recall the material and to complete some of the arguments. Suppose that χ : (M, ω) → (M, ω) is a symplectic diffeomorphism. There are several equivalent ways to state the condition that χ is quantizable. The most “geometric’’ one is the following. Deﬁnition 2.1. χ is quantizable if χ lifts to a contact transformation χ˜ of (X, α), i.e., a diffeomorphism of X such that χ˜ ∗ α = α. Equivalently, χ lifts to an automorphism of each power LN of the prequantum complex line bundle. It is said to be linearizable in algebraic geometry. Let us consider the obstruction to lifting a symplectic map. We follow in part the discussion in [Guillemin–Sternberg (1977), p. 220]. The key notion is that χ preserves the holonomy map of the connection 1-form α. Recall that the horizontal subbundle H ⊂ T X of the connection is deﬁned by Hx = ker αx = {v ∈ Tx X : αx (v) = 0}. The holonomy map H : → U (1),

H (γ ) = eiθγ

from the free loop space deﬁned by horizontally lifting a loop γ : [0, 1] → M to γ˜ : [0, 1] → X and expressing γ˜ (1) = eiθγ γ˜ (0). We say that χ is holonomy preserving if

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S. Zelditch

H (χ (γ )) = H (γ )

∀γ ∈ .

If the loop is contained in the domain of a local frame s : U → X, then ∗ H (γ ) = exp 2π i s α .

(18)

(19)

γ

If γ = ∂σ , then γ s ∗ α = σ ω. It follows (see [Guillemin–Sternberg (1977)]) that the symplectic map preserves the holonomy around such homologically trivial loops. Hence it is sufﬁcient to consider the map Hχ : H 1 (M, Z) → U (1),

Hχ (γ ) = H (γ )−1 H (χ (γ )) = ei(θγ −θ χ (γ )) . (20)

Proposition 2.2. A symplectic map χ of a symplectic manifold (M, g) lifts to a contact transformation of the associated prequantum S 1 bundle (X, α) if and only if Hχ ≡ 1, the trivial representation. Proof. Suppose that χ lifts to χ˜ : X → X as a contact transformation. Let γ ∈ and let γ˜ be a horizontal lift of γ . Then χ˜ (γ˜ ) is a horizontal lift of χ (γ ). Obviously, γ˜ (1) = eiθγ γ˜ (0) implies χ˜ ◦ γ˜ (1) = eiθγ χ˜ ◦ γ˜ (0), so H = H ◦ χ . Conversely, suppose that Hχ = 1. We then deﬁne χ˜ by lifting χ along paths. We ﬁx a basepoint x0 ∈ MX and deﬁne χ˜ on the orbit S 1 · x0 by ﬁxing χ˜ (x0 ) to be a chosen basepoint on π −1 (χ (π(x0 )) and then extending by S 1 invariance. We now consider horizontal paths x(t) : [0, 1] → X from x0 . At least one horizontal path exists from x0 to any given point since the curvature is positive (Chow’s theorem). We deﬁne χ˜ (x(t)) to be the horizontal lift of χ (π x(t)) to χ˜ (x0 ). To see that this is well deﬁned, we must prove independence of the path. So let x1 (t), x2 (t) be two horizontal paths from x0 to x1 (1). Thus there is trivial holonomy of the loop deﬁned by x1 followed by x2−1 (i.e., the backwards path to x2 ). Now project each path, apply χ , and horizontally lift. This deﬁnes a horizontal lift of the loop formed by the projected curves χ ◦ π ◦ xj (t) (j = 1, 2). It has trivial holonomy if χ is holonomy preserving. It follows that the horizontal lifts must agree at t = 1.

As a corollary, we note that for any contact transformation χ of a torus, there exists a unique translation Tα,β so that χ ◦ Tα,β lifts as a contact transformation. Moreover, a contact transformation χ always lifts to a contact transformation if M is simply connected. Hence if we lift χ ﬁrst to the universal cover M˜ of M, then this further lifts to X˜ as a contact transformation. We verify this in another way, since we will use it in Section 4. ˜ Then there exists a unique (up to Proposition 2.3. Let χ be a symplectic map of M. one scalar) lift χ˜ of χ to X˜ such that π χ˜ = χ π, where π : X˜ → M˜ is the S 1 -ﬁbration. Proof. The fact that χ can be lifted to X˜ is obvious since X˜ ! M˜ × S 1 . The key point is that the map can be lifted as contact transformations. Any lift that commutes with the S 1 action has the form

Quantum maps and automorphisms

χ˜ · (z, eiθ ) = (χ (z), eiθ+ϕχ (θ,z) ).

633

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The contact form on X˜ is the connection 1-form α˜ of the Hermitian line bundle ˜ In local symplectic coordinates (x, ξ ) on M, ˜ it has the form over X. α˜ =

1 (ξ dx − xdξ ) − dθ. 2

˜ and since M˜ is simply connected, Since χ ∗ (xdξ −ξ dx)−(xdξ −ξ dx) is closed on M, ∞ ˜ there exists a function fχ ∈ C (M) such that χ˜ ∗ (xdξ − ξ dx) − (xdξ − ξ dx) = dfχ (x, ξ ). Using the product structure, we have that ϕχ (x, ξ, θ) = fχ (x, ξ ) deﬁnes a lift satisfying χ˜ ∗ (α) ˜ = α, ˜ as desired. Regarding uniqueness, the only ﬂexibility in the lift is in the choice of fχ , which is deﬁned up to a constant. The constant can be ﬁxed by requiring that fχ (0, 0) = 0.

There is a weaker condition which has come up in some recent work (see [Marklof–Rudnick (2000)]): Let us say that χ is quantizable at level N if χ lifts to an automorphism of the bundle LN . Often a map is quantizable of level N along an arithmetic progression N = kN0 , k = 1, 2, 3, . . . of powers, although it is not quantizable for all N . In geometric terms, this simply means that χ fails to lift as a contact transformation of X but does lift as a contact transformation of X/ZN , where ZN ⊂ S 1 is the group of N th roots of unity. In everything that follows, the stated results have analogous for this modiﬁed version of quantization.

3 Proof of Theorem 1.6 in the case H 1 (M, C) = {0} We ﬁrst prove that if H 1 (M, C) = {0}, then every automorphism is given by conjugation with a Toeplitz Fourier integral operator. In this case, we may identify maps on M which S 1 -invariant maps on X. We emphasize that we are not considering the most general Toeplitz operators $A$ with A ∈ # ∗ (X) but only the S 1 -invariant operators whose symbols lie in C ∞ (M). The proof is modelled on that of Duistermaat–Singer [Duistermaat–Singer (1976)], but has several new features due to the holomorphic setting. In some respects the proof is simpler, since there are no transmission automorphisms, and there are natural identiﬁcations between symbols of different orders. However, in some respects, it is more complicated, and also we must be careful about using contravariant versus covariant symbols. We begin with the following. Lemma 3.1. Suppose that H 1 (M, C) = {0} and that ι is an order-preserving automorphism of T ∞ /T −∞ . Then ι is equal to conjugation by a Toeplitz Fourier integral operator in the sense of Deﬁnition 1.4.

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Proof. Since ι is order preserving, it induces automorphisms on the quotients of the ﬁltered algebra T ∗ . We ﬁrst consider T 0 /T −1 . The map to contravariant symbols deﬁnes an identiﬁcation with C ∞ (M). Thus ι induces an automorphism of C ∞ (M), viewed as an algebra of contravariant symbols under multiplication. The maximal ideal space of C(M) equals M; hence ι induces a map χ on M such that ι(p) = p ◦χ . Precisely as in [Duistermaat–Singer (1976)], one veriﬁes that χ is a smooth diffeomorphism of M. Now consider the quotients T m /T m−1 . They are are simply N m times T 0 /T −1 , so for any m ι(p) = p ◦ χ for p ∈ T m /T m−1 . This step is simpler than in the pseudodifferential case, and as a result certain steps carried out in [Duistermaat–Singer (1976)] are unnecessary here. Now let n = 1, so that T 1 /T 0 is a Lie algebra under commutator bracket. The principal symbol is an isomorphism of the quotient algebra to the Poisson algebra (M, {, }) deﬁned by the symplectic form ω. Since ι is an automorphism of the quotient algebra, we have {a ◦ χ , b ◦ χ } = {a, b} ◦ χ ; hence χ is a symplectic map of (M, ω). This step is also simpler than in [Duistermaat– Singer (1976)], and we see that no transmissions arise as possible automorphisms. By Proposition 2.3, χ lifts to a contact transformation χ˜ of X. 3.0.1 Symbol-preserving automorphisms −1 −1 . It follows Now let A−1 N = $N aTχ $N denote any Toeplitz quantization of χ˜ ∞ −∞ that preserves principal that α(P ) = AN ι(P )A−1 N is an automorphism of T /T contravariant symbols in the sense of Section 1.1.1. We now prove that any such automorphism is given by conjugation with a Toeplitz multiplier of some order s. Thus let j be a principal contravariant symbol-preserving automorphism. Let P ∈ T m . Since j (P ) − P ∈ T m−1 , we get an induced map

βm : C ∞ (M) → C ∞ (M),

βm (a) = j ($N a$N ) − $N a$N .

(22)

Then β = βm is a derivation in two ways: (i) (ii)

β(p · q) = β(p) · q + p · β(q) β({p, q}) = {β(p), q} + {p, β(q)}.

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Now any derivation in the sense of (i) is given by differentiation along a vector ﬁeld V . Since V commutes with Poisson bracket, it must be a symplectic vector ﬁeld. Since ω(V , ·) = β is a closed 1-form, there exists a local Hamiltonian H for V . Under our assumption that H 1 (M) = {0}, the Hamiltonian is global, so V = +H for some global H . Thus we have βm (a) = i{a, Hlog bm }

(24)

for some bm ∈ C ∞ (M). Thus j may be represented by j (P ) = B −1 P B for P ∈ T m m ≡ N mS0 , with B = $N eib $N (b = bm ). Because of the natural identiﬁcation of Sscl scl we ﬁnd that bm = b is the same for all m. By composing automorphisms, we now have an automorphism j2 such that

Quantum maps and automorphisms

j2 ($N aN $N ) − $N aN $N ∈ T m−2 ,

635

m aN ∈ Sscl .

We ﬁnd as above that j2 ($N aN $N )−$N aN $N = β2 (aN ), where β2 is a derivation, hence β2 = N −1 {log b−1 , ·} for some b−1 . Therefore, $N e−iN

−1 b −1

$N j2 (P )$N eiN

−1 b −1

$N − P ∈ T m−2

∀P ∈ T m .

(25)

0 such that Proceeding in this way, we get an element bN ∈ Sscl

$N e−ibN $N j (P )$N eibN $N − P ∈ T −∞

∀P ∈ T m .

This completes the proof of the lemma.

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3.0.2 Conclusion of proof when H 1 (M, C) = {0} Lemma 3 of [Duistermaat–Singer (1976)] is an abstract result which says that automorphisms of Frechet spaces satisfying a certain density condition are always given by conjugation. The density condition is easy to prove, so we omit the proof. The result is the following: Let ι denote an automorphism of T ∗ acting on H ∞ (M). Then ι(P ) = A−1 P A, where A : H ∞ (M) → H ∞ (M) is an invertible, continuous linear map, determined uniquely up to multiplicative constant. Thus i(P ) = A−1 P A for all P ∈ T ∗ , and also, by Lemma 1, there exists an elliptic Toeplitz Fourier integral operator B such that i(P ) ≡ B −1 ◦ P ◦ B for all P ∈ T ∞ /T −∞ . Let E = A ◦ B −1 . Then [E, P ] ∈ T −∞ for all P ∈ T ∞ /T −∞ . In particular, [E, P ] ∈ T −∞ for all P = {$N a$N }, a ∈ C ∞ (M). In place of Lemma 4 of [Duistermaat–Singer (1976)], we use the following. Lemma 3.2. Let E be an operator on H2 such that [E, $a$] ∈ T −∞ for all a ∈ C ∞ (M). Then there exists a constant c such that E = c$+R, where R is a smoothing operator. Proof. It is sufﬁcient to prove the statement for all a supported in a given S 1 -invariant open set U ⊂ X. We can then use a partition of unity to prove the result for all a. We use the notation A ∼U B to mean that A, B are deﬁned on U and their difference is a smoothing operator on U . In a sufﬁciently small open set U ⊂ X, there exists a Fourier integral operator F : L2 (X) → L2 (Rn ) associated to a contact transformation ϕ such that $ ∼U F $0 F ∗ modulo smoothing operators, where $0 is the model Szegö kernel discussed in [Boutet de Monvel–Sjöstrand (1976)], [Boutet de Monvel–Guillemin (1981)], namely, the orthogonal projection onto the kernel of the annihilation operators Dj = 1i (∂/∂yj + yj |Dt |) on Rn = Rp × Rq . Furthermore, it is proved in [Boutet de Monvel–Sjöstrand (1976)], [Boutet de Monvel–Guillemin (1981)] that there exists a complex Fourier integral operator R0 : L2 (Rq ) → L2 (Rn ) such that R0∗ R0 = I , R0 R0∗ = $0 . Moreover, for any pseudodifferential operator A on X, there exists a

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pseudodifferential operator Q on Rq so that $0 A$0 ∼ R0 QR0∗ . Transporting R0 to X by F , we obtain a complex Fourier integral operator R : L2 (Rq ) → L2 (X) so that RR ∗ ∼U $, R ∗ R ∼ϕ(U ) I and so that $a$ ∼U RQR ∗ . Then [E, $a$] ∼U [E, R ∗ QR] and we may rewrite the condition on E as [E, R ∗ QR] ∈ T −∞ (X)

∀Q ∈ # 0 (Rq ).

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[R ∗ ER, Q] ∈ # −∞ (X)

∀Q ∈ # 0 (Rq ).

(28)

This is equivalent to

We then apply Beals’s characterization of pseuodifferential operators: P ∈ # k (Rm ) if and only if for all {ji , k }, ad(xj1 ) · · · ad(xjr ) ad(Dxk1 ) · · · ad(Dxks )P : H s+r (Rm ) → H s (Rm ) is bounded. Here ad(L)P denotes [L, P ]. It follows ﬁrst that R ∗ ER ∈ # 0 , and easy symbol calculus shows that the complete symbol of R ∗ ER is constant. Hence R ∗ ER = I + S, where S is a smoothing operator. Applying R on the left and R ∗ on the right concludes the proof.

Remark. It is in the step in Section 3.0.1 that the distinction between symplectic maps of M and contact transformations of X enters. Ultimately it is this step which leads to Corollary 1.7. We also note that the proof above is rather different from that in [Duistermaat–Singer (1976)].

4 H 1 (M, C) = {0} The problems with quantizing symplectic maps on M are all due to the fundamental group π1 (M) or more precisely H1 (M, C). We solve them by passing to the universal ˜ In this section, we relate Toeplitz operators on M and M. ˜ cover M. 4.1 Toeplitz operators on the universal cover Since we are comparing algebras and automorphisms on covers to those on a quotient, we begin with the abstract picture as discussed in [Gromov–Henkin–Shubin (1998)]. We then specialize it to algebras of Toeplitz operators. 4.1.1 Abstract theory Suppose that p : X˜ → X is a covering map of a compact manifold X, and denote its ˜ We would deck transformation group by . We regard as acting on the left of X. ˜ like to compare operators on X and operators on X. To gain perspective, we start ˜ which with the large von Neumann algebra B of all bounded operators on L2 (X) commute with . Later we specialize to the Toeplitz algebra which is our algebra of observables.

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The Schwartz kernel of such an operator satisﬁes B(γ x, γ y) = B(x, y). If we denote by D a fundamental domain for , then there exists an identiﬁcation ˜ ! L2 () ⊗ L2 (D) = L2 () ⊗ L2 (X). L2 (X)

(29)

Elements of L2 () ⊗ L2 (X) can be viewed as functions f (γ , x) on × D. The unitary isomorphism is deﬁned by ˜ → fϕ (γ , x) = ϕ(γ · x). ϕ ∈ L2 (X) Note that both left translation Lγ and right translation Rγ by γ act on this space, namely, Lγ f (α, x) = f (γ α, x), Rγ f (α, x) = f (αγ , x). We may regard B as bounded operators commuting with all Lγ . The isomorphism (29) induces an algebra isomorphism B ! R ⊗ B(X),

(30)

where B(X) is the algebra of bounded operators on X and where R is the algebra generated by right translations Rγ on L2 (). ˜ The corresponding algebra So far we have been considering operators on L2 (X). B is much larger than B(X). To make the connection to L2 (X) tighter, we need to ˜ of -periodic functions on X. ˜ The natural Hilbert space consider the space L2 (X) structure is to deﬁne f 2 = D |f (x)|2 dV , where dV is a -invariant volume ˜ ! L2 (X). We may regard elements form. We have the obvious isomorphism L2 (X) 2 ˜ as functions f (γ , x) as above which are constant in γ . of L (X) Elements B ∈ B with properly supported kernels, or kernels which decay fast ˜ Indeed, R acts trivially on L2 (X), ˜ so B acts enough off the diagonal, act on L2 (X). by the quotient algebra B /R . We will be working with subalgebras of Toeplitz operators where the action is clearly well deﬁned. Remark. Let us deﬁne B as the subalgebra of B of elements which commute with ˜ ! B(X). We may write the (Schwartz) both Rγ and Lγ for all γ . Then we have B (X) kernel of an element of B as B(γ , x, γ , x ). It belongs to B if B(αγ , x, αγ , x ) = B(γ , x, γ , x ), and it belongs to B if addditionally B(γ α, x, α, γ α, x ) = B(γ , x, γ , x ). We have been talking about algebras of bounded operators, but our main interest is in C ∗ -algebras of Toeplitz operators. Everything we have said restricts to these subalgebras once we have deﬁned the appropriate notions. 4.2 Toeplitz operators The positive Hermitian holomorphic line bundle (L, h) → M pulls back under π to ˜ h) ˜ → M. ˜ This induces an inner product on the space H 0 (M, ˜ L˜ N ) of entire one (L,

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˜ L˜ N ) the space of L2 holomorphic holomorphic sections of L˜ N . We denote by H2 (M, sections relative to this inner product. As in the quotient, there exists an associated S 1 bundle X˜ with a contact (connection) form α˜ such that X˜ → M˜ ↓ ↓ X→M commutes. The vertical arrows are covering maps and the horizontal ones are S 1 ˜ It is isomorphic bundles. We denote the deck transformation group of X˜ → X by . to , so when no confusion is possible we drop the˜. Since all objects are lifted from quotients, it is clear that ˜ acts by contact transformations of α. ˜ Let us the denote operator of translation by γ on M˜ by Lγ . ˜ the Hardy space of L2 CR functions on X. ˜ They are boundary We denote by H2 (X) values of holomorphic functions in the strongly pseudoconvex complex manifold D˜ ∗ = {(z, v) ∈ L˜ ∗ : hz (v) < 1},

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˜ The group ˜ acts on D˜ ∗ ⊂ L˜ ∗ with quotient the compact disc bundle which are L2 (X). ∗ ∗ D ⊂ L → M. In this setting it is known (see [Gromov–Henkin–Shubin (1998), Theorem 0.2]) that ˜ = ∞. dim H2 (X) Due to the S 1 symmetry, holomorphic functions on D˜ ∗ are easily related to CR ˜ We denote by holomorphic functions on X. ˜ → H2 (X) ˜ ˜ : L2 (X) $

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the Szegö (orthogonal) projection. Under the S 1 action, we have ˜ = H2 (X)

∞ D

2 ˜ HN (X),

˜ = $

N =1

∞ D

˜ N. $

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N =1

2 (X) ˜ ! H2 (M, ˜ L˜ N ). We refer to [Gromov–Henkin–Shubin (1998), As on X, we have HN example (2), p. 559] for the proof that L˜ → M˜ has many holomorphic sections. ˜ More important are periodic funcSo far we have discussed L2 functions on X. tions. We endow them with a Hilbert space structure by setting + ˜ = {f ∈ L2 (X) ˜ : Lγ f = f }, L2 (X) loc (34) 2 2 ˜ ˜ H (X) = {f ∈ L (X), ∂ b f = 0},

with the inner product , obtained by integrating over a fundamental domain D 2 (X) ˜ for the S 1 action for . Both are direct sums of weight spaces L2,N , resp. H,N 2 ˜ There exists a Hilbert space isomorphism L : H (X) ˜ = H2 (X), namely, by on X. ∗ lifting Lϕ = p ϕ under the covering map. The adjoint of L is given by L∗ f = p∗ (f 1D ). We thus have

Quantum maps and automorphisms

˜ → H2 (X). ˜ LL∗ = Id : H2 (X) 2 (X) ˜ H,N

639

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2 (X) ˜ HN

We observe that the spaces and are completely unrelated and have different dimensions. The former is canonically isomorphic to HN (X). We now consider Toeplitz algebras. Since X˜ is noncompact and inﬁnite volume in general, we must take care that Toeplitz operators are well deﬁned and form an algebra. As for pseudodifferential operators on inﬁnite volume spaces, we deﬁne ˜ as the space of operators of the form $A ˜ is the space of ˜ $, ˜ where A ∈ # s 1 (X) T s (X) S 1 properly supported pseudodifferential operators commuting with S . We also deﬁne ˜ $ ˜ with A having a smooth properly supported kernel. T −∞ as the space of such $A ˜ off the diagonal (see (36)), the operators $A ˜ $ ˜ Due to the exponential decay of $ have exponentially decaying kernels (relative to the -invariant volume form) and have well-deﬁned compositions. We then distinguish the automorphic Toeplitz operators ˜ = {$A ˜ : L∗γ $A ˜ $L ˜ γ = $A ˜ $ ˜ ∈ T ∗ (X) ˜ $}. ˜ T∗ (X) ˜ = 0 or, equivalently, $(γ ˜ x, γ y) = $(x, ˜ We note that [Lγ , $] y). So the operative condition is that A ∈ #S 1 , , the space of pseudodifferential operators commuting ˜ with . The associated symbols aN (z, z¯ ) are exactly the -invariant symbols on M. We note the following. ˜ on H2 preserving Proposition 4.1. There exists a representation ρ of T∗ (X) 2 each H,N . Proof. Since Lγ $A$f = $A$f whenever Lγ f = f , the only issue is whether Af is well deﬁned for f ∈ H . However, A is a polyhomogeneous sum of aj N −j ,

where aj is a periodic function, so the action is certainly deﬁned. Let us denote by K = ker ρ . We further denote by ρ,N the associated repre2 , and put K = ker ρ sentation on H,N ,N . ˜ Proposition 4.2. K ⊂ T −∞ (X). ˜ annihilates H2 (X) ˜ ! H2 (X). This means that Proof. Assume that $A$ ∈ T∗ (X) ˜ In particular it implies that the “Berezin symbol’’ Af, gD = 0 for all f, g ∈ H2 (X). ˜ N aN $ ˜ N (z, z) = 0. However, asymptotically, N −m $ ˜ N aN $ ˜ N (z, z) ∼ a0 (z) for $ a zeroth-order Toeplitz operator. One sees by induction on the terms in (10) that A ∼ 0.

It could happen that K = 0, unlike the analogous representation on H2 (X) which deﬁnes Toeplitz operators. For each N there could exist aN ∈ C ∞ (M) with 2 (X) ˜ ! H2 (X) aN L2 = 1 which is orthogonal to the ﬁnite-dimensional space H,N N 2 (X). ˜ Then T = $ ˜ N aN $ ˜ N ∈ K,N . but which is not orthogonal to HN ˜ to T ∗ (X). In preparation, we relate the Szegö kernels on We now relate T∗ (X) ˜ First, we consider a ﬁxed N . The following is proved in [Shiffman–Zelditch]. X, X.

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Proposition 4.3. The degree N Szegö kernels of X, X˜ are related by ˜ N (γ · x, y). $ $N (x, y) = γ ∈ 2 . The same formula deﬁnes the Szegö projector L2,N → H,N

A key point is to use the estimate √ ˜ N d(x,y)

˜ N (x, y)| ≤ Ce− |$

,

(36)

˜ y) is the Riemannian distance with respect to the Kähler metric ω˜ to show where d(x, ˜ N acts on that the sum converges for sufﬁciently large N . The estimates show that $ ˜ Since L2 (X) ˜ ∩ C(X) ˜ ⊂ L∞ (X), ˜ $ ˜ and we have ˜ N acts on H 2 (X), L∞ (X). ˜ N (x, y)sN (y)dV (y) = ˜ N (x, γ y)sN (y)dV (y). ˜ N sN (x) = $ $ $ X˜

D γ ∈

˜ ˜ N = sN for sN ∈ H 2 (X). To complete the proof, one has to prove that $s Now we consider the full Szegö kernel. ˜ N L for each N as operators from L2 (X) to H2 . Corollary 4.4. We have L$N = $ N ,N 2 2 ˜ ˜ : L (X) → H (X). Hence L$ = $L ˜ The identity (30) for the larger algebra of bounded operators suggests that T∗ (X) should be a larger algebra than T ∗ (X). However, this is not the case. ˜ ˜ Proposition 4.5. L induces an algebra isomorphism T ∗ (X) ! T (X)/K (X). Proof. It follows from Proposition 4.1 and Corollary 4.4 that ˜ ˜ : L2 (X) → H2 (X) ˜ ∗ a $L L$a$ = $p ˜ : L2 (X) → H2 (X). ˜ ∗ a $L ⇐⇒ $a$ = L∗ $p

(37)

Equality on the designated spaces is equivalent to equality in the algebras. Further, the equality LL∗ = I : H2 → H2 implies that the linear isomorphism is an algebra isomorphism.

˜ is so small. We recall that one obtains T ∗ (X) It may seem surprising that T (X) 2 ˜ on H (X). ˜ When dealing with all bounded operators, the by representing T (X) kernel is very large (R ) and it is also large if we ﬁx N and consider the associated Toeplitz algebra. But the kernal is trivial if we consider the full Toeplitz algebra. We also have the following. ˜ Corollary 4.6. L induces algebra isomorphisms TN (X) ! T,N (X)/K ,N . ˜ The concrete Now let us relate such automorphisms to automorphisms on X. identiﬁcation with T (X) is by (37). We have the following. Proposition 4.7. There is a natural identiﬁcation of •

˜ automorphisms α on T ∗ (X) with automorphisms α˜ on T∗ (X)/K ;

Quantum maps and automorphisms

•

641

−∞ ˜ automorphisms α on T ∗ (X)/T −∞ (X) with automorphisms α˜ on T∗ (X)/T .

Proof. The ﬁrst statement is clear since the algebras are isomorphic.An automorphism of T∗ descends to T ∗ if and only if it preserves K . Concretely, we wish to set ˜ ∗ a $)L. ˜ α($a$) = L∗ α( ˜ $p

(38)

The inner operator must be determined by the left side for this to be well deﬁned. We have ˜ ∗a$ ˜ ≡ L$a$L∗ mod K . $p (39) The same equivalence is true modulo the larger subalgebra T−∞ . Since T−∞ /K = T −∞ (X), the second statement is correct.

4.3 Completion of the proof of Theorem 1.6 We now prove statements (iii)–(iv) of the theorem. The following gives an upper bound on existence of semiclassical automorphisms. Lemma 4.8. Suppose that α is an order-preserving automorphism of T ∗ (X)/T −∞ . ˜ Let α˜ denote the corresponding automorphisms of T∗ (X)/K . Then there exists a canonical transformation χ of (M, ω) and a unitary Toeplitz quantum map U˜ χ on ˜ such that H 2 (X) α(P ˜ ) = U˜ χ∗ P U˜ χ and such that

˜ ˜ L−1 γ U χ L γ = Mγ U χ ,

˜ where Mγ commutes with T∗ (M). Proof. By Lemma 3.1, we know that α˜ is given by conjugation by a Toeplitz quantum map U˜ χ . We may deﬁne Mγ by the formula above since U˜ χ is invertible. We then determine its properties. We have −1 ˜ ˜ −1 ˜ −1 ˜ L−1 γ U χ L γ P L γ Uχ L γ = U χ P U χ

˜ ∀P ∈ T∗ (M).

Hence Mγ−1 U˜ χ−1 P U˜ χ Mγ = U˜ χ−1 P U˜ χ ⇐⇒ Mγ−1 P Mγ = P It follows that Mγ is central. This proves (iii) of Theorem 1.6.

˜ ∀P ∈ T∗ (M).

The lower bound is given as follows. Lemma 4.9. Suppose that χ is a symplectic map of (M, ω). Then it lifts to a con˜ α). tact transformation χ˜ of (X, ˜ The associated quantum map Uχ˜ on X˜ deﬁnes (by ˜ which descends to an conjugation) an order-preserving automorphism α˜ of T (X) automorphism of T ∗ (X)/T −∞ . If α˜ preserves K , then it deﬁnes an automorphism of all of T ∗ (X). Proof. χ automatically lifts to M˜ as a symplectic map commuting with the action of . By Proposition 2.3 it lifts to X˜ as a contact transformation. We then deﬁne a unitary quantum map by

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˜ N σ Tχ˜ $ ˜ N, U˜ N χ˜ = $ ˜ See where σ is a function on M˜ which makes the operator unitary on H2 (X). [Zelditch (1997)] for background. ˜ When χ A crucial issue now is the commutation relations between U˜ N χ˜ and . ˜ lifts to X, i.e., is quantizable, then χ˜ commutes with . In this case, χ is quantizable as a quantum map and there was no need to lift it to X˜ to quantize it as an automorphism. Assume, however, that χ does not lift to X and consider the commutation relations ˜ The commutator of the translation by χ˜ with left translations by elements of . −1 −1 ˜ ˜ χ˜ γ˜ χ˜ γ˜ covers the identity map of M since the lift of χ to M commutes with . ˜ It follows that Furthermore, it commutes with the S 1 action on X. χ˜ γ˜ χ˜ −1 γ˜ −1 = Teiθγ ,χ ,

(40)

where the right side is translation by the element eiθγ ,χ . The angle θγ ,χ is a priori a ˜ However, the left side is a contact transformation covering the identity function on M. and therefore dθ = dθ + dθγ ,χ , i.e., θγ ,χ is a constant. After quantizing, the same commutator identity holds for the operators. Therefore the operators Mγ are central. It follows that the automorphism α(P ˜ ) = U˜ χ∗ P U˜ χ satisﬁes

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˜ ˜ N p ∗ aN $ ˜ N ) ∈ T∗ (X). α˜ N ($

It therefore descends to T ∗ (X)/T −∞ (X) by (38), i.e., as ˜ N p ∗ aN $ ˜ N ) = L∗ U˜ χ∗ $ ˜ N p ∗ aN $ ˜ N )U˜ χ L. α˜ N ($ Unitarity of L then implies that α is also an automorphism. If the automorphism preserves K , then it also descends to T ∗ (X).

An obvious question is whether the condition that the automorphism preserve K is equivalent to the quantization condition that χ lift to X. Clearly, quantizability in the sense of Deﬁnition 1.5 implies preservation of K , since the quantum map Uχ ,N is well deﬁned on the spaces HN . The converse is not obvious, since we only know a priori that the automorphism induces automorphisms αN of the ﬁnite rank observables OpN (aN ) for ﬁxed N . Abstractly, such automorphisms must be given by conjugations by unitary operators on HN , but it is not clear that these unitary operators are Toeplitz quantum maps in the sense of Deﬁnition 1.5. We end the section with the following proof. Proof of Corollary 1.7. This follows immediately from (ii) if H 1 (M, C) = {0}. If H 1 (M, C) = {0}, then by Theorem 1.6(iii) there exists a symplectic map χ of M and a Toeplitz Fourier integral operator V˜χ on M˜ and a central operator Mγ such that Tγ∗ V˜χ Tγ = Mγ V˜χ , and such that α is induced by conjugation by V˜ . But by deﬁnition, α is also given by conjugation by U . Now the Schwarz kernel of U , hence U , lifts to

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643

˜ By assumption, U˜ AU˜ ∗ has the same complete symbol a -invariant kernel U˜ on M. ∗ ˜ ˜ as V AV for any Toeplitz operator A on M, lifted to T . By Lemma 3.2, it follows that U˜ = V˜ + R, where R is a smoothing Toeplitz operator. It follows that Mγ = 1 (hence θγ ,χ = 1 for all γ ), and therefore the symplectic map χ˜ underlying V˜ , when ˜ is invariant under the deck transformation group ˜ of X˜ → X. lifted to X,

5 Quantization of torus maps To clarify the issues involved, we consider some standard examples on the symplectic 2m-torus T2m = Cm /Z2nm . Since H1 (T2m , C) = C2m , there will exist symplectic maps which cannot be quantized in the sense of Deﬁnitions 1.4 and 1.5(a) as quantum maps, though they can and will be quantized as automorphisms. In fact, the distinction can already be illustrated with the simplest maps: • •

Kronecker translations Tθ (x) = x + θ (x, θ ∈ T2m ); symplectic automorphisms A ∈ Sp(2m, Z).

We begin by describing the line bundle on the torus and its universal cover. We follow [Zelditch (1997)], [Bleher–Shiffman–Zelditch (2001)] and refer there for further discussion. The quotient setting is L → T2n , where L is the bundle with curvature j dzj ∧ d z¯ j . Sections of LN are theta-functions of level N . On the universal cover, we have the pulled back bundle LH = C × Cm → Cm . Its associated principal S 1 bundle m . We recall that it is the quotient Cm ×S 1 → Cm is the reduced Heisenberg group Hred under the subgroup (0, Z) in the center of the simply connected Heisenberg group Hm = Cm × R with group law 1 (ζ, t) · (η, s) = (ζ + η, t + s + 4(ζ · η)). ¯ 2 The identity element is (0, 0) and (ζ, t)−1 = (−ζ, −t). The reduced Heisenberg m = Hm /{(0, k) : k ∈ Z} = Cm × S 1 with group law group is thus Hred 1

¯ (ζ, e2π it ) · (η, e2π is ) = (ζ + η, e2π i[t+s+ 2 4(ζ ·η)] ). m with the left-invariant connection form We now equip Hred

αL =

1 dt (ξq dxq − xq dξq ) − 2 q 2π

(ζ = x + iξ ),

(42)

whose curvature equals the symplectic form ω = q dxq ∧ dξq . The kernel of α L is the distribution of horizontal planes. To deﬁne the Szegö kernel, we further need to split the complexiﬁed horizontal spaces into their holomorphic and antiholomorphic parts. The left-invariant (CR-)holomorphic (respectively, antiholomorphic) vector m are the horizontal lifts of the vector ﬁelds ∂ , ﬁelds ZqL (respectively, Z¯ qL ) on Hred ∂zq

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S. Zelditch

respectively,

∂ ∂ z¯ q

m ) of CR with respect to α L . We then deﬁne the Hardy space H2 (Hred

m ) satisfying the left-invariant holomorphic functions to be the functions in L2 (Hred m . For N = 1, 2, . . . , we L ¯ Cauchy–Riemann equations Zq f = 0 (1 ≤ q ≤ m) on Hred m 2 2 further deﬁne HN ⊂ H (Hred ) as the (inﬁnite-dimensional) Hilbert space of squareintegrable CR functions f such that f ◦ rθ = eiN θ f as before. The representation H12 is irreducible and may be identiﬁed with the Bargmann–Fock space of entire 2 holomorphic functions on Cn which are square integrable relative to e−|z| . The H 2 Szegö kernel $N (x, y) is the orthogonal projection to HN . It is given by

$H N (x, y) =

1 m iN (t−s) N (ζ ·η− ¯ 12 |ζ |2 − 12 |η|2 ) N e e , m π

x = (ζ, t),

y = (η, s).

(43)

We note that it satisﬁes the estimates in (36). In this model example, Proposition 4.3 was proved in [Zelditch (1997)]. Finally, we describe the circle bundle X in the quotient setting. The lattice Z2m m of Hm under the homomorphism may be embedded as a subgroup HZ R ι(m, n) = (m, n, eiπ m·n ).

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m . To clarify the role of the factors of 1 , we show that We will denote the image by HZ 2 m HZ is indeed a subgroup:

(m, n, eiπm·n ) · (m , n , eiπ m ·n )

= (m + m , n + n , eiπ(m·n+m ·n +mn −m n) ) = (m + m , n + n , eiπ(m·n+m ·n +mn +m n) )

(45)

= (m + m , n + n , eiπ((m+m )·(n+n )) ). m . The leftWe then put X = HZm \H n mR , i.e., X is the left quotient of HRm by HZ L invariant contact form α descends to X as a contact form and a connection form for the principal S 1 bundle X → Cm /Z2m .

5.1 Kronecker translations We ﬁrst show that irrational Kronecker translations T(a,b) f (x, ξ ) = f (x + a, ξ + b)

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are nonquantizable as Toeplitz quantum maps. Proposition 5.1. T(a,b) fails to be quantizable for all (a, b) ∈ R2n /Z2n . T(a,b) is quantizable at level N iff N a, N b ∈ Z2n . Proof. By Proposition 2.2, the map lifts if and only if translations preserve holonomy of homologically nontrivial loops. The loops on T2m that we need to consider are

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645

given in local coordinates by γm,n (t) = (tm, tn). Horizontal lifts to X are given by γ˜m,n (t) = (tm, tn, 1). At t = 1 we obtain (m, n, 1) ∼ (0, 0, eiπ m·n ). Hence the holonomy of the path γm,n equals eiπ m·n . Now translate the loop γ(m,n) by (a, b) to obtain the loop γ1 (t) = (tm+a, tn+b). A horizontal lift to X is given by γ˜1 (t) = (tm + a, tn + b, eπ i(b·m−a·n)t ). It is the projection to X of the left translate by (a, b, 0) of the original horizontal path. At t = 1 the endpoint is (m + a, n + b, eπ i(b·m−a·n) ) = (a, b, eiπ[m·n+2(b·m−a·n)] ). Hence the holonomy changed by e2π i(b·m−a·n) . The holonomy is preserved iff b · m − a · n ∈ Z for all (m, n) ∈ Z2m iff (a, b) ∈ Z2m . Lifting to level N means changing the holonomy to e2π iN θγ . So the condition to lift becomes (a, b) ∈ N1 Z2 .

Remark. The nonquantizability of T(a,b) is due to the left–right invariance of various objects. T(a,b) only lifts to HZ \HR as right translation by (a, b, 0). But right translation by an element of HR does not preserve the left invariant contact form. Equivalently, T(a,b) only lifts to a contact transformation of HR if it lifts to left translation by (a, b, 0). But then the lift does not descend to HZ \HR . 5.1.1 Kronecker translations as automorphisms It is easy to see that Kronecker translations deﬁne automorphisms of the revelant algebras. We lift T(a,b) to HRn as the contact transformation of left multiplication T(a,b) (x, ξ, e2π it ) := (x + a, ξ + b, e2π it eπ i(aξ −bx) ).

Although the map T(a,b) does not descend to the quotient as a map, we claim the following. Proposition 5.2. Kronecker maps T(a,b) have the following properties: (i) T(a,b) deﬁnes an automorphism of T∗ . (ii) T(a,b) deﬁnes an automorphism of T ∞ (X)/T −∞ (X) by αa,b;N (OpN (a)) = OpN (a ◦ T(a,b) ). (iii) However, α(a,b) does not preserve K and does not deﬁne an automorphism of T ∞ (X). Proof. (i) Left translation by (a, b) deﬁnes an automorphism of T∗ because, by (43), the Szegö kernel commutes with left translations, i.e., H $H N (α · x, α · y) = $N (x, y) ∀α.

(47)

Indeed, it is the kernel of a convolution operator. T , where γ ∈ = Z2m . An easy computa(ii) Consider the conjugates Tγ−1 Ta,b γ tion shows that

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S. Zelditch Tγ−1 Ta,b Tγ f (x, ξ, t) = Mγ Ta,b f (x),

where Mγ f (x, ξ, t) = f (x, ξ, t + ω(γ , (a, b)). ˜ Nσ$ ˜ N with We need to show that Mγ commutes with every Toeplitz operator $ symbol lifted from Cm /Z2m . Since the symbol is invariant under the central circle, it ˜ N ] = 0. But this follows as long as $ ˜ N (z · x, z · y) = is sufﬁcient to show that [Mγ , $ it 1 ˜ $N (x, y) for any z = e ∈ S , the center of the Heisenberg group. But this follows because ˜ N (z · x, z · y) = |z|2N $ ˜ N (x, y). ˜ N (x, y) = $ $ ˜ N , the automorphism descends to the Since left translation commutes with $ quotient as

∗ ˜ ˜ N )Tα,β ($N p ∗ aN $ L αN ($N aN $N ) = L∗ Tα,β ˜ N (Ta,b ˜ N L. = L∗ $ p ∗ aN )$

This is the stated formula. (iii) A Kronecker automorphisms α(a,b) can only preserve K,N if $N a$N = 0 implies $N (a ◦ Ta,b )$N = 0. But if this were the case, the elements $N eik,x $N would be distinct eigenoperators with eigenvalues eik,(a,b) . This contradicts the ﬁnite dimensionality of the algebra for ﬁxed N .

5.2 Quantum cat maps We now show, in a similar way, that symplectic linear maps of T2 always deﬁne quantum automorphisms even though they do not always deﬁne quantum maps. We write ab g= ∈ SL(2, Z), c d and deﬁne g(x, ξ ) = (ax +bξ, cx +dξ ) on the torus. It lifts to the reduced Heisenberg group by g(x, ξ, t) = (g(x, ξ ), t). Proposition 5.3. Tg is quantizable iff a · c, b · d ∈ 2Z. Proof. We go through the same calculation as for Kronecker translations. This time, the horizontal lift of the transformed loop is (t (a · m + b · n), t (c · m + d · n), 1). At t = 1, the endpoint is ((a · m + b · n), (c · m + d · n), 1) = (0, 0, eiπ(a·m+b·n)·(c·m+d·n) ). Since the holonomy of the original path was eiπ m·n , the change in holonomy equals eiπ(m·n−(a·m+b·n)(c·m+d·n)) = 1 ⇐⇒ ac, bd ∈ 2Z. Here we use that ad + bc ≡ 1(mod 2Z).

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5.2.1 Linear maps as automorphisms It is known that quantizable linear maps (cat maps) on the quotient deﬁne quantum maps with exact Egorov theorems (see, e.g., [Zelditch (1997)]). We now show that nonquantizable maps as well-deﬁned automorphisms by the exact Egorov formula. Proposition 5.4. Tg deﬁnes an automorphism of T ∞ (X)/T −∞ (X) by αg;N (OpN (a)) = OpN (a ◦ Tg ). Proof. Consider the conjugates Tγ−1 Tg Tγ , where γ ∈ . We have Tγ−1 Tg Tγ f (x, ξ, t) = Mγ Tg f (x), where Mγ f (x, ξ, t) = f ((x, ξ ) + (I − g)γ , t + ω(γ , z) − ω(γ , g(z + γ )). Mγ is the composition of translation T(I −g)γ with a central translation. Since (I −g)γ is in the lattice, translation by this element commutes with left invariant operators. ˜ . Thus Mγ ∈ T∗ (X) Thus the automorphism descends to the quotient as ˜ N p ∗ aN $ ˜ N )Tg L αN ($N aN $N ) = L∗ Tg∗ ($ ˜ N L. ˜ N (Tg p ∗ aN )$ = L∗ $

6 Spectra of automorphisms In this article, our interest lies in the automorphisms deﬁned by symplectic maps. But most of the interest in quantizations of quantizable symplectic maps, at least in the physics literature, is in their spectral theory as unitary operators Uχ ,N on the ﬁnite-dimensional Hilbert spaces HN (X). In this section, we point out how the most important aspects of this spectral theory of U pertain only to the spectrum of the associated automorphism U AU ∗ . The main point is that the reformulation suggests generalizations to other kinds of automorphisms. We also tie together the automorphisms of Toeplitz algebras on the torus with the well-known ones on the rotation algebra. 6.1 Spectra of automorphisms of Hilbert–Schmidt algebras We let H denote a Hilbert space, and denote by HS the algebra of Hilbert–Schmidt operators on H, i.e., the operators for which the inner product A, B := Tr AB ∗ is ﬁnite. We let ∗ denote the adjoint on H; no confusion with the ∗ for the star product should arise. A ﬁnite-dimensional algebra of Hilbert–Schmidt operators is of course a full matrix algebra, and its automorphisms are given by conjugation by unitary operators. Suppose that α is an automorphism of HS.

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Deﬁnition 6.1. We say that an automorphism α of HS is • • • •

a ∗-automorphism if α(A∗ ) = α(A)∗ ; unitary if α(A), α(B) = A, B; a conjugation if there exists a unitary operator U : H → H s.t. α(A) = U AU ∗ ; tracial if Tr α(A) = Tr A for all A of trace class.

We will consider the eigenvalues and “eigenoperators’’ of a unitary ∗-automorphism on HS: α(A) = eiθ A. If α is a unitary automorphism, then (as a unitary operator) it possesses an orthornormal basis of eigenoperators {Aj }. The following is elementary from the deﬁnitions. Proposition 6.2. We have the following: (i) A tracial ∗-automorphism is unitary. (ii) The composition of any two eigenoperators is a (possibly zero) eigenoperator. (iii) If Aj is an eigenoperator, then A∗j Aj is an invariant operator, i.e., α(A∗j Aj ) = A∗j Aj . Proof. (i) This is immediate from the fact that α(A), α(B) = Tr α(A)α(B)∗ = Tr α(AB ∗ ) = Tr AB ∗ = A, B. (ii)–(iii) These statements follow from the equations α(Aj Ak ) = α(Aj )α(Ak ) = ei(θj +θk ) Aj Ak , and α(A∗j ) = [α(Aj )]∗ = e−iθj α(A∗j ).

In the case α(A) = U ∗ AU we note that the eigenoperators of the automorphism are given by ∗ ∗ α(ϕN,j ⊗ ϕN,k ) = ei(θN,j −θN,k ) (ϕN,j ⊗ ϕN,k ), (48) where {(ϕN,j , eiθN,j )} are the spectral data of U . 6.2 Spectral problems of quantum chaos The main problems on quantum maps pertain to the spacings between eigenvalues (the pair correlation problem) and the asymptotics of matrix elements relative to eigenfunctions of the operators. 6.2.1 Pair correlation problem Let us recall that the pair correlation function ρ2N (PCF) of a quantum map {UN } with Planck constant 1/N is the function on R deﬁned by ∞ ˆ f (x)dρ2N (x) = f | Tr UN |2 . N R =0

Its limit as N → ∞, when one exists, is the PCF of the quantum map. Clearly,

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knowledge of dρ2N is equivalent to knowledge of its form factor | Tr UN |2 . We observe that the form factor depends only on the automorphism below. Proposition 6.3. The form factor of a conjugation automorphism {αN } is given by = αN Aj , Aj , Tr αN j

where {Aj } is an orthonormal basis for HS N . = U ⊗ U − on HS , and we have Indeed, if α(A)N = UN∗ AUN , then αN N N N − 2 Tr HS N UN ⊗ UN = | Tr HN UN | . Hence the pair correlation problem makes sense for all unitary automorphisms αN .

Problem. Given any unitary automorphism αN , determine N → ∞, where αN (j ) = eiϑN,j j

1 dN

dN2

j =1 δ(N (ϑN,j )) as

(49)

is the eigenvalue problem for the automorphism. 6.2.2 Problems of quantum ergodicity/mixing We observe that these too can be formulated for any sequence of automorphisms. We ∗ as the rewrite the asymptotics of matrix elements AϕN,j , ϕN,k = Tr AϕN,j ⊗ ϕN,k ∗ inner products A, j k . We observe that the eigenfunctions k,k = ϕN,k ⊗ ϕN,k always have eigenvalue 1, i.e., they are invariant states of the automorphism. It is simple to check that the proof of quantum ergodicity for quantizations of ergodic quantizable symplectic maps χ (see [Zelditch (1997)]) uses only the automorphism involved. It states that if a symplectic map χ is ergodic and quantizable, then the invariant states of the corresponding automorphism of the Toeplitz algebra are asymptotic to the traces τN (A) = dim1HM Tr A|HN . It might be interesting to ﬁnd generalizations of this result to other kinds of automorphisms. 6.3 Spectral theory of model automorphisms We now point out that the automorphisms induced by model quantum maps on the torus are the same as the well-known automorphism of the ﬁnite-dimensional rotation algebras. 6.3.1 The rotation algebra modulo N We denote by GN the ﬁnite Heisenberg group of order N 2 , generated by two elements U, V satisfying U2 U1 = e2π i/N U1 U2 ,

U1N = U2N = I,

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and its group algebra by RN . GN has a unique irreducible unitary representation ρN on CN given as follows: If we regard CN = L2 (Z/NZ), then ρN (U1 )ψ(Q) = e

2π iQ N

ψ(Q),

ρN (U2 )ψ(Q) = ψ(Q + 1).

Recall that the rotation algebra or noncommutative torus Aθ is the (pre-)C ∗ -algebra generated by unitaries U1 , U2 satisfying the Weyl commutation relation U2 U1 = e2π iα U1 U2 . When α = 1/N, Aθ has a large center generated by U1N , U2N . RN is obtained from Aθ by viewing central elements as scalars. 6.3.2 Toeplitz algebra and rotation algebra We identify the rotation algebra with θ = 2π N to the algebra of Toeplitz operators $N a$N on the torus. As veriﬁed by S. Nonnenmacher [Nonnenmacher (2003)], the elements 2 2 U1 = eπ N $N eiθ1 $N , U2 = eπ N $N eiθ2 $N (50) satisfy UjN = I . Here (eiθ1 , eiθ2 ) are the standard coordinates on the torus. Hence any quantum map on the torus deﬁnes an automorphism of RN . Thus we can identify the automorphisms αg,N quantizing ab g= c d with the well-known automorphisms of RN deﬁned by U1 → U1a U2b , U2 → U1c U2d (see, e.g., [Narnhofer (1997)]). We can also see easily from this point of view that Kronecker maps Tu,v cannot in general be quantized as automorphisms. Namely, the quantization on R2 translates the symbol, so it would descend to U1 → eiu U1 , U2 → eiv U2 . To be well deﬁned, one needs eiu , eiv to be N th roots of unity, which of course they are not in the irrational case.

7 Appendix The key elements of the Toeplitz algebra and its automorphisms are the ∗ product (13) and the Egorov formula (1). The purpose of this appendix is to direct the reader’s attention to the existence of routine calculations of the complete symbols of compositions aN ∗N bN of symbols and of conjugations UN OpN (a)UN∗ of observables by Toeplitz quantum maps. The method is to use the Boutet de Monvel– Sjöstrand parametrix for the Szegö kernel [Boutet de Monvel–Sjöstrand (1976)] as in [Zelditch (1998)]. Since the original version of this paper was written, several papers [Karabegov–Schlichenmaier (2001)], [Schlichenmaier (1999)], [Schlichenmaier (1998)], [Schlichenmaier (1999b)], [Shiffman–Tate–Zelditch (2003)] have also used this method to describe Toeplitz ∗ product on symbols, so we only brieﬂy mention how to go about calculating asymptotic expansions.

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Proposition 7.1. Let (M, ω) be a compact Kähler manifold. Then the ∗ product deﬁnes an algebra structure on classical symbols. There exists an asymptotic expansion fˆ1 ∗ fˆ2 (z, z¯ ) ∼

∞

N −k Bk (f1 , f2 ),

k=0

where B0 (a, b) = f1 · f2 , B1 (f1 , f2 ) = 12 {f1 , f2 } and where Bk is a bidifferential operator of C ∞ (M) × C ∞ (M) → C ∞ (M). Proof. Using the Boutet de Monvel-Sjostrand parametrix as in [Zelditch (1997)], [Shiffman–Tate–Zelditch (2003)], one can obtain a complete asymptotic expansion of the covariant symbol $N a$N b$N (z, z¯ ). One writes out $N (z, w) as an oscillatory integral and applies complex stationary phase. For calculations of this kind, we refer to [Shiffman–Tate–Zelditch (2003)]. To obtain a ∗N b, we invert the Berezin transform IN on symbols, as described in [Reshetikhin–Takhtajan (1999)] and elsewhere. It is invertible on formal power series, and the same inverse is well deﬁned on symbol expansions. Thus a ∗N b ∼ IN−1 $N a$N b$N (z, z¯ ). This produces the symbol expansion claimed in the proposition.

Proposition 7.2. Let Uχ ,N be a Toeplitz quantum map as in Deﬁnition 1.5. Then for any observable $N aN $N = OpN (aN ), the contravariant symbol aχ of Uχ∗,N OpN (aN )Uχ ,N possesses a complete asymptotic expansion aχ (z) ∼

∞

N −k Vk (a ◦ χ ),

k=0

where V0 (a) = a, and where Vk is a differential operator of order at most 2k. Proof. The expansion is obtained from the covariant symbol Uχ∗,N OpN (aN )Uχ ,N (z, z¯ ) by inverting the Berezin transform. The asymptotics of the covariant symbol follow by applying stationary phase to the oscillatory integral formula for Uχ∗,N OpN (aN )Uχ ,N .

Acknowledgments The author would like to thank S. de Bievre, Z. Rudnick, and particularly S. Nonnenmacher for helpful discussions on this paper during the program on Semiclassical Methods at MSRI in Spring, 2003. The author would also like to thank the Clay Mathematics Institute for partial support during this period.

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