Gauge Theory and Symplectic Geometry

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Series C: Mathematical and Physical Sciences - Vol. 488

Gauge Theory and Symplectic Geometry edited by

Jacques Hurtubise Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada

and

Fran~ois Lalonde Departement de mathematiques et d'informatique, Universite de Quebec a Montreal, Montreal, Quebec, Canada

Technical Editor

Gert Sabidussi Departement de mathematiques et statistique, Universite de Montreal, Montreal, Quebec, Canada

Springer-Science+Business Media, B.V.

Proceedings of the NATD Advanced Study Institute and Seminaire de mathematiques superieures on Gauge Theory and Symplectic Geometry Montreal, Canada July 3-14,1995

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

AII Rights Reserved © 1997 Springer Science+Business Media Oordrecht Originally published by Kluwer Academic Publishers in 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, includ ing photo­copying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

ISBN 978-90-481-4830-1 ISBN 978-94-017-1667-3 (eBook)DOI 10.1007/978-94-017-1667-3

Table of Contents

Preface

Participants

Contributors

Michele AUDIN

Lectures on gauge theory and integrable systems

Yakov ELIASHBERG

Symplectic geometry of plurisubharmonic functions

Nigel HITCHIN

Frobenius manifolds

Jacques HURTUBISE

Moduli spaces and particle spaces

Fran<;ois LALONDE

J-holomorphic curves and symplectic invariants

Dusa McDUFF

Lectures on Gromov invariants for symplectic 4-manifolds

Index

Vll

IX

xvii

1

49

69

113

147

175

211

Preface

The two areas of gauge-theoretical four-dimensional topology and symplectic topology have many points in common, and for several years their developments have followed parallel paths. For example, in both areas, a main technique has been to first add an extra structure (a suitable metric), then to consider spaces of solutions to non-linear p.d.e. that the structure allows us to define (the Yang-Mills equations in gauge theory, pseudo-holomorphic curves in symplectic topology), and finally to extract the information that persists as one varies the structure. There is also a variational content to the equations that are considered, and the behaviour of the action functional in both cases is strikingly similar; to cite but one instance, in both cases the solution spaces exhibit similar non-compactness, in the form of "bubbling" , and this is both an important source of technical difficulties and an essential geometric feature.

It is not surprising then that the two areas have become very closely linked, and the 1995 Seminaire de Mathematiques Superieures at the Universite de Montreal was planned so as to encourage and stimulate this interaction. It came as an additional bonus that in the year and a half preceeding the SMS, the two subjects were both revolutionised and made even more inextricably linked by the ground-breaking discovery in gauge theory of the Seiberg-Witten invariants, and their application to symplectic topology, in particular by Taubes. Several of the principal protagonists of this new point of view were invited speakers at the 8MS, and the school turned out to be most timely.

The main lecturers of the 1995 SMS and the topics of their lectures were Michele Au­din, Integrable systems and moduli spaces; Yakov Eliashberg, Pseudoconvexity; Nigel Hitchin, Frobenius manifolds; Jacques Hurtubise, Stability theorems; John Jones, Morse-Floer the­ory; Franl,;ois Lalonde, Pseudo-holomorphic curves and applications; Dusa McDuff, Gromov invariants; Tomasz Mrowka, Seiberg- Wittten theory; Dietmar Salamon, Seiberg- Witten the­ory; and Jean-Claude Sikorav, Theory of generating functions. (Ofthese, Audin, Eliashberg, Hitchin, Hurtubise, Lalonde and McDuff have written lecture notes for this volume.) At the end of the SMS, Cliff Taubes gave a two-hour summary of his work linking the Seiberg-Witten invariants and the Gromov invariants. Additional lectures were given by Ezra Getzler, Dieter Kotschick, Kaoru Ono and Lisa Traynor.

The first chapter of this book consists then of notes by Michele Audin on integrable systems. One very useful tool in understanding the symplectic geometry of a space is the presence of such a system, and we are particularly fortunate here in being presented with two such structures on the moduli space of vector bundles over a Riemann surface, as well as results on these systems due to Goldman, Jeffrey, Weitsman, Fock and Roslyi.

The study of complex manifolds has, in some sense two extremes: the theory of compact manifolds and the theory of Stein manifolds. Yasha Eliashberg's contribution gives us an overview of some aspects of the theory of Stein manifolds and the closely linked concepts of J-convexity and pluri-subharmonic functions, from the view-point of symplectic geometry.

The Frobenius manifolds of Dubrovin occur in a number of quite different problems, in particular in the theory of Gromov-Witten invariants and quantum cohomology. Nigel Hitchin's notes provide us with an introduction to the theory, along with some of the contexts in which they appear: orthogonal coordinates in Rn, Hamiltonian flows on orbits in the Lie algebra of SO(n), moduli spaces of flat connections on a punctured sphere, isomonodromic deformations and the Painleve equations, and Hamiltonian equations of hydrodynamic type. The emphasis is on the underlying geometry of the objects involved.

vii

viii Preface

Both gauge theory and the theory of holomorphic curves have a variational aspect, in that the relevant moduli spaces appear as critical or extremal sets for a variational prob­lem. Jacques Hurtubise's notes consider the relationship between the moduli spaces and the function spaces in which they sit, in particular explaining the various topological stability theorems which one can obtain.

The last two contributions to this volume are concerned with the theory of pseudo­holomorphic curves and their applications to symplectic topology. Fran,,;ois Lalonde gives an introduction to the theory, covering both local properties and the Gromov compactness theorem, and then explains two applications: the first is to non-squeezing results and the second is to the definition of symplectic invariants of diffeomorphisms. In her notes, Dusa McDuff introduces the Gromov-Witten invariants, explains a basic structure theorem due to Taubes, and concludes with some examples such as elliptic surfaces and fiber sums. She also explains some of the difficulties involved in counting pseudo-holomorphic curves.

We would like to take this opportunity to thank all of the people associated with the or­ganisation of the SMS, in particular Aubert Daigneault, Ghislaine David and Gert Sabidussi, for their help in assuring that the event was a success. We also owe a debt of gratitude to NATO, which provides the major part of the funding for the event through its Advanced Study Institutes programme, as well as to NSERC and the Universite de Montreal for their additional support.

Jacques Hurtubise and Fran,,;ois Lalonde

Participants

Miguel ABREU School of Mathematics Institute for Advanced Study Princeton, NJ 08540 USA

Sharad AGNIHOTRI Mathematical Institute Oxford University 24-29 St. Giles Oxford, OX1 3LB United Kingdom

Vaughn ANDERSON Department of Mathematics University of Britsh Columbia Vancouver, BC, V6T 1Z2 Canada

Hassan AURAG Departement de mathematiques

et de statistique Universite de Montreal C.P. 6128, Succ. Centre-ville Montreal, QC, H3C 3J7 Canada

David AUSTIN Department of Mathematics University of British Columbia Vancouver, BC, V6T lZ2 Canada

Philippe BALCER UER de Mathematiques Universite Louis Pasteur 7, rue Rene Descartes 67084 Strasbourg Cedex France

Augustin BANYAGA Department of Mathematics 218 McAllister Bldg. Pennsylvania State University University Park, PA 16802-6401 USA

Anne BEAULIEU Matbematiques Universite de Marne la Vallee 2, rue de la Butte Verte 93166 Noisy-Ie-Grand Cedex France

Mohan BHUPAL Mathematical Institute University of Warwick Coventry, CV 4 7 AL United Kingdom

John BLAND Department of Mathematics University of Toronto Toronto, Ont., M5S 1A1 Canada

Steven BRADLOW Department of Mathematics University of Illinois Urbana, IL 61801 USA

David CALDERBANK Department of Pure Mathematics

& Mathematical Statistics 16 Mill Lane Cambridge, CB2 1SB United Kingdom

x

Michael CALLAHAN Hertford College Oxford, OX1 3BW United Kingdom

Ana CANAS DA SILVA Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 USA

Virginie CHARETTE Department of Mathematics University of Maryland College Park, MD 20742 USA

Meng-Kiat CHUAH Department of Applied Mathematics National Chiao Thng University Hsinchu - Taiwan Republic of China

Ralph COHEN Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

Vincent COLIN UMPA Ecole Normale Superieure de Lyon 46, Allee d'Italie 69364 Lyon Cedex 07 France

Olivier COLLIN Mathematical Institute Oxford University 24-29 St. Giles Oxford OX1 3LB United Kingdom

Participants

Arleigh CRAWFORD Department of Mathematics & Statistics McMaster University Hamilton, Ont., L8S 4K1

Canada

Mihai DAMIAN Centre de Mathematiques Ecole Poly technique 91128 Palaiseau Cedex France

Jean-Paul DUFOUR Getodim CC 051 Universite de Montpellier II PI. Eugene Bataillon 34095 Montpellier Cedex 05 France

Mikhail ENTOV Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

Emmanuel FERRAND Centre de Mathematiques Ecole Poly technique

91128 Palaiseau Cedex France

Daniel GATIEN Departement de mathematiques

et d'informatique Universite du Quebec it Montreal C.P. 8888 Succ. Centre-ville Montreal, QC, H3C 3P8 Canada

Benoit GERARD Department of Mathematics Brandeis University P.O. Box 9110 Waltham, MA 02254-9110 USA

Participants

Sophie GERARDY UER de Mathematiques Universite Louis Pasteur 7, rue Rene Descartes 67084 Strasbourg Cedex France

Ezra GETZLER Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 USA

Emmanuel GIROUX UMPA Ecole Normale Superieure de Lyon 46, Allee d'Italie 69364 Lyon Cedex 07 France

Pierre GOSSELIN IRMA Universite Louis Pasteur 7, rue Rene Descartes 67084 Strasbourg Cedex France

Andrzej GRANAS Institute of Mathematics Nicholas Copernicus University Chopina 12/18 87100 Torun Poland

Bertrand HAAS UER de Mathematiques Universite Louis Pasteur 7, rue Rene Descartes 67084 Strasbourg Cedex France

Christopher HERALD Max-Planck Institut fUr Mathematik Gottfried Claren Str. 26 53225 Bonn Germany

Eugenie HUNSICKER Department of Mathematics University of Chicago Chicago, IL 60637-1538 USA

Stuart JARVIS Merton College Oxford, OX1 4JD United Kingdom

John D.S. JONES Mathematical Institute University of Warwick Coventry, CV 4 7 AL United Kingdom

Mikhail KARASEV (Moscow State Institute of Electronics

& Mathematics) u1.26 Bakinskih Comissarov 3-1-316 117571 Moscow Russia

Takashi KIMURA Department of Mathematies University of North Carolina Chapel Hill, NC 27599-3250 USA

Mounia KJIRI Departement de mathematiques

et de statistique Universite de Montreal C.P. 6128, Succ. Centre-Ville Montreal, QC, H3C 3J7 Canada

Dieter KOTSCHICK Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138 USA

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xii

Alexei KOVALEV Department of Mathematics & Statistics University of Edinburgh JCMB - King's Bldgs. Edinburgh EH9 3JZ Scotland, UK

Yi-Jen LEE Department of Physics Harvard University Cambridge, MA 02138 USA

Yue LEI Department of Mathematics 253-37 California Institute of Technology Pasadena, CA 91125 USA

Veronique LIZAN UMPA Ecole Normale Superieure de Lyon 46, Allee d'Italie 69364 Lyon Cedex 07 France

Wladyslav LOREK Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794-3651 USA

Antony MACIOCIA Department of Mathematics & Statistics University of Edinburgh JCMB - King's Bldgs. Edinburgh EH9 3JZ Scotland, UK

Samuel J. MALTBY Box 59, Site 2, SS3 Calgary, AB, T3C 3N9 Canada

Participants

Shaun MARTIN Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 USA

Alan McRAE Department of Mathematics Rawles Hall Indiana University Bloomington, IN 47405-5701 USA

Christian MERCAT IRMA, UFR de Mathematiques Universite Louis Pasteur 7, rue Rene Descartes 67084 Strasbourg Cedex France

Darko MILINDOVIC Department of Mathematics Van Vleck Hall University of Wisconsin-Madison Madison, WI 53706 USA

Maung MIN-OO Department of Mathematics & Statistics McMaster University Hamilton, Ont., L8S 4K1 Canada

Tomasz MROWKA Department of Mathematics California Institute of Technology Pasadena, CA 91125 USA

Michael M. MURRAY Department of Pure Mathematics University of Adelaide 5005 Adelaide, SA Australia

Participa.nts

Gregory NABER Departement of Mathematics California State University Chico, CA 95929-0525 USA

Emile NASATYR Department of Mathematics University of Aberdeen Dunbar St. Aberdeen, AB92TY Scotland, UK

Thomas NEVINS Department of Mathematics University of Chicago Chicago, IL 60637-1538 USA

Tien Dung NGUYEN SISSA-ISAS Via Beirut 2-4 34013 Trieste Italy

Paul NORBURY Mathematical Institute University of Warwick Coventry, CV 4 7 AL United Kingdom

Kaoru ONO Department of Mathematics Ochanomizu University 2-1-1 Otsuka 112 Tokyo Japan

Keith ORPEN Department of Mathematics University of British Columbia Vancouver, BC, V6T 1Z2 Canada

Peter PANG Department of Mathematics Brown University Providence, RI 02912 USA

Elisa PRATO DMI Ecole Normale Superieure 45, rue d'Ulm 75230 Paris Cooex 05 France

Alexander REZNIKOV Institute of Mathematics Hebrew University 91904 Giv'at Ram Israel

Simon RICHARD Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794-3651 USA

Tristan RIVIERE CCMLA Ecole Normale Superieure de Cachan 61, ave President Wilson 95235 Cachan Cedex France

Lorenzo SADUN Department of Mathematics University of Texas Austin, TX 78712 USA

Dietmar SALAMON Mathematical Institute University of Warwick Coventry, CV 4 7 AL United Kingdom

Xlll

xiv

John SARDIS Mathematical Institute University of Warwick Coventry, CV 4 7 AL United Kingdom

Jean-Claude SIKORAV URA - CNRS 1408 Universite Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex France

Lesley SIBNER Polytechnic University Six Metrotech Ctr. Brooklyn, NY 11201 USA

Robert SIBNER Department of Mathematics CUNY, Brooklyn College 2900 Bedford Ave. Brooklyn, NY 11210-2889 USA

Karl F. SIBURG Mathematik (HG G36.1) ETH-Zentrum 8092 Ziirich Switzerland

Roman G. SMIRNOV Department of Mathematics & Statistics Jeffery Hall Queen's University Kingston, Ont., K7L 3N6 Canada

Margaret SYMINGTON Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

Clifford H. TAUBES Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138 USA

Key Yong TEE New College Oxford, OX1 3BN United Kingdom

Mark TEMPLE-RASTON Department of Mathematics Bishop's University Lennoxville, QC, HIM lZ7 Canada

David THERET UFR de Mathematiques Universite de Paris VII 2, Place Jussieu 75251 Paris Cedex 05 France

Lisa TRAYNOR Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010 USA

llya USTILOVSKY School of Mathematics Tel Aviv University 69978 Ramat Aviv Israel

Jianmei WANG Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138 USA

Participants

Participants

Qing YANG Mathematics 253-37 California Institute of Technology Pasadena, CA 91125 USA

Carmen YOUNG Department of Mathematics & Statistics McMaster University Hamilton, Ont., L8S 4Kl Canada

Miguel A. ZARATE REYES Mathematical Institute University of Warwick Coventry, CV 4 7 AL United Kingdom

xv

Contributors

Michele A UDIN IRMA, UFR de Mathematique

et d'Informatique Universite Louis-Pasteur 7, rue Rene Descartes 67084 Strasbourg Cedex France

Yakov ELIASHBERG Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

Nigel J. HITCHIN Department of Pure Mathematics

& Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge, CB2 1SB United Kingdom

Jacques HURTUBISE Department of Mathematics & Statistics Burnside Hall McGill University 805 Sherbrooke St. W. Montreal, QC, H3A 2K6

Canada

Franc;ois LALONDE Departement de mathematiques

et d'informatique Universite du Quebec a Montreal C.P. 8888, Succ. Centre-ville Montreal, QC, H3C 3P8 Canada

Dusa McDUFF Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794-3651 USA

xvii