Gauge Theory and Symplectic Geometry PDF
Preview Gauge Theory and Symplectic Geometry
Gauge Theory and Symplectic Geometry NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NA TO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics London and New York C Mathematical and Physical Sciences Kluwer Academic Publishers D Behavioural and Social Sciences Dordrecht, Boston and London E Applied Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, London, H Cell Biology Paris and Tokyo I Global Environmental Change PARTNERSHIP SUB-SERIES 1. Disarmament Technologies Kluwer Academic Publishers 2. Environment Springer-Verlag I Kluwer Academic Publishers 3. High Technology Kluwer Academic Publishers 4. Science and Technology Policy Kluwer Academic Publishers 5. Computer Networking Kluwer Academic Publishers The Partnership Sub-Series incorporates activities undertaken in collaboration with NATO's Cooperation Partners, the countries of the CIS and Central and Eastern Europe, in Priority Areas of concern to those countries. NATO-PCO-DATA BASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 50000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO-DATA BASE is possible in two ways: - via online FILE 128 (NATO-PCO-DATA BASE) hosted by ESRIN, Via Galileo Galilei, 1-00044 Frascati, Italy. - via CD-ROM "NATO-PCO-DATA BASE" with user-friendly retrieval software in English, French and German (© WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered through any member of the Board of Publishers or through NATO PCO, Overijse, Belgium. 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 Lalonde Fran~ois Departement de mathematiques et d'informatique, a Universite de Quebec 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. ISBN 978-90-481-4830-1 ISBN 978-94-017-1667-3 (eBook) DOI 10.1007/978-94-017-1667-3 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 photo copying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Table of Contents Preface Vll Participants IX Contributors xvii Michele AUDIN Lectures on gauge theory and integrable systems 1 Yakov ELIASHBERG Symplectic geometry of plurisubharmonic functions 49 Nigel HITCHIN Frobenius manifolds 69 Jacques HURTUBISE Moduli spaces and particle spaces 113 Fran<;ois LALONDE J-holomorphic curves and symplectic invariants 147 Dusa McDUFF Lectures on Gromov invariants for symplectic 4-manifolds 175 Index 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 Au din, Integrable systems and moduli spaces; Yakov Eliashberg, Pseudoconvexity; Nigel Hitchin, Frobenius manifolds; Jacques Hurtubise, Stability theorems; John Jones, Morse-Floer the ory; Franl,;ois Lalonde, Pseudo-holomorphic curves and applications; Dusa McDuff, Gromov invariants; Tomasz Mrowka, Seiberg-Wittten theory; Dietmar Salamon, Seiberg-Witten the ory; 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 prob lem. 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 pseudo holomorphic 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 or ganisation 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 Augustin BANYAGA School of Mathematics Department of Mathematics Institute for Advanced Study 218 McAllister Bldg. Princeton, NJ 08540 Pennsylvania State University USA University Park, PA 16802-6401 USA Sharad AGNIHOTRI Mathematical Institute Anne BEAULIEU Oxford University Matbematiques 24-29 St. Giles Universite de Marne la Vallee Oxford, OX1 3LB 2, rue de la Butte Verte United Kingdom 93166 Noisy-Ie-Grand Cedex France Vaughn ANDERSON Department of Mathematics Mohan BHUPAL University of Britsh Columbia Mathematical Institute Vancouver, BC, V6T 1Z2 University of Warwick Canada Coventry, CV 4 7A L United Kingdom Hassan AURAG Departement de mathematiques John BLAND et de statistique Department of Mathematics Universite de Montreal University of Toronto C.P. 6128, Succ. Centre-ville Toronto, Ont., M5S 1A1 Montreal, QC, H3C 3J7 Canada Canada Steven BRADLOW David AUSTIN Department of Mathematics Department of Mathematics University of Illinois University of British Columbia Urbana, IL 61801 Vancouver, BC, V6T lZ2 USA Canada David CALDERBANK Philippe BALCER Department of Pure Mathematics UER de Mathematiques & Mathematical Statistics Universite Louis Pasteur 16 Mill Lane 7, rue Rene Descartes Cambridge, CB2 1SB 67084 Strasbourg Cedex United Kingdom France x Participants Michael CALLAHAN Arleigh CRAWFORD Hertford College Department of Mathematics & Statistics Oxford, OX1 3BW McMaster University United Kingdom Hamilton, Ont., L8S 4K1 Canada Ana CANAS DA SILVA Department of Mathematics Mihai DAMIAN Massachusetts Institute of Technology Centre de Mathematiques Cambridge, MA 02139-4307 Ecole Poly technique USA 91128 Palaiseau Cedex France Virginie CHARETTE Department of Mathematics Jean-Paul DUFOUR University of Maryland Getodim CC 051 College Park, MD 20742 Universite de Montpellier II USA PI. Eugene Bataillon 34095 Montpellier Cedex 05 Meng-Kiat CHUAH France Department of Applied Mathematics National Chiao Thng University Mikhail ENTOV Hsinchu - Taiwan Department of Mathematics Republic of China Stanford University Stanford, CA 94305-2125 Ralph COHEN USA Department of Mathematics Stanford University Emmanuel FERRAND Stanford, CA 94305-2125 Centre de Mathematiques USA Ecole Polyt echnique 91128 Palaiseau Cedex Vincent COLIN France UMPA Ecole Normale Superieure de Lyon Daniel GATIEN 46, Allee d'Italie Departement de mathematiques 69364 Lyon Cedex 07 et d'informatique France Universite du Quebec it Montreal C.P. 8888 Succ. Centre-ville Olivier COLLIN Montreal, QC, H3C 3P8 Mathematical Institute Canada Oxford University 24-29 St. Giles Benoit GERARD Oxford OX1 3LB Department of Mathematics United Kingdom Brandeis University P.O. Box 9110 Waltham, MA 02254-9110 USA Participants xi Sophie GERARDY Eugenie HUNSICKER UER de Mathematiques Department of Mathematics Universite Louis Pasteur University of Chicago 7, rue Rene Descartes Chicago, IL 60637-1538 67084 Strasbourg Cedex USA France Stuart JARVIS Ezra GETZLER Merton College Department of Mathematics Oxford, OX1 4JD Massachusetts Institute of Technology United Kingdom Cambridge, MA 02139-4307 USA John D.S. JONES Mathematical Institute Emmanuel GIROUX University of Warwick UMPA Coventry, CV 4 7 AL Ecole Normale Superieure de Lyon United Kingdom 46, Allee d'Italie 69364 Lyon Cedex 07 Mikhail KARASEV France (Moscow State Institute of Electronics & Mathematics) Pierre GOSSELIN u1.26 Bakinskih Comissarov 3-1-316 IRMA 117571 Moscow Universite Louis Pasteur Russia 7, rue Rene Descartes 67084 Strasbourg Cedex Takashi KIMURA France Department of Mathematies University of North Carolina Andrzej GRANAS Chapel Hill, NC 27599-3250 Institute of Mathematics USA Nicholas Copernicus University Chopina 12/18 Mounia KJIRI 87100 Torun Departement de mathematiques Poland et de statistique Universite de Montreal Bertrand HAAS C.P. 6128, Succ. Centre-Ville UER de Mathematiques Montreal, QC, H3C 3J7 Universite Louis Pasteur Canada 7, rue Rene Descartes 67084 Strasbourg Cedex Dieter KOTSCHICK France Department of Mathematics Harvard University Christopher HERALD One Oxford Street Max-Planck Institut fUr Mathematik Cambridge, MA 02138 Gottfried Claren Str. 26 USA 53225 Bonn Germany
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