LECTURES ON CHERN-WE11 THEORY

Transcription

1 LECTURES ON CHERN-WE11 THEORY AND WITTEN DEFORMATIONS

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3 Nankai Tracts in Mathematics - Vol. 4 LECTURES ON CHERN-WEN THEORY AND WITTEN DEFORMATIONS Weiping Zhang Nankai Institute of Mathematics Tionjin, f R China World Scientific ewjersey*london *Singapore *Hang Kong

4 Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Faner Road, Singapore USA ujj5ce: Suite IB, 1060 Main Street, River Edge, NJ UK ofice; 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. LECTURES ON CHERN-WEIL THEORY AND WITTEN DEFORMATIONS Copyright by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN (pbk) Printed in Singapore.

5 Dedicated to my teachers Jean-Michel Bismut and Shiing-Shen Chern

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7 Preface These lecture notes are based on the notes of a graduate course of differential geometry I taught at the Nankai Institute of Mathematics. It consists of two parts: the first geometric part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and Andre Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; while the second part, which is analytic in nature, contains analytic proofs of the Poincark-Hopf index formula as well as the Morse inequalities based on deformations introduced by Edward Witten. We hope this book can serve as a text book to cover materials not generally contained in an introductory course in differential geometry. With this reason, we have not tried hard to make this book being completely self-contained. However, we will give detailed references when (possibly) nonstandard results will be quoted. On the other hand, we have tried to make each chapter in the text to be relatively independent from the other chapters. As a result, we will list the references of each chapter at the end of that chapter. We will work in smooth (i.e. C") category throughout this book. I would like to thank Dr. Huitao Feng for taking the preliminary notes for my lectures. Part of these notes were prepared during a short visit to the Institute of Mathematics of Fudan University in May, 2000, and during my visit to the Department of Mathematics of MIT for the Spring semester of I would like to thank Professor Jiaxing Hong of F'udan University for kind hospitality. I am also grateful to Professors Richard Melrose and Gang Tian for arranging my visit to MIT, and to MIT for financial support. vii

8 viii Preface Finally, I would like to thank the Ministry of Education and the National Natural Science Foundation of China for their support during the writing of this book.

9 Contents Preface Chapter 1 Chern-Weil Theory for Characteristic Classes Review of the de Rham Cohomology Theory Connections on Vector Bundles The Curvature of a Connection Chern-Weil Theorem Characteristic Forms, Classes and Numbers SomeExamples Chern Forms and Classes Pontrjagin Classes for Real Vector Bundles A Hirzebruch s L-class and A-class K-groups and the Chern Character The Chern-Simons Transgressed Form Bott Vanishing Theorem for Foliations Foliations and the Bott Vanishing Theorem Adiabatic Limit and the Bott Connection Chern-Weil Theory in Odd Dimension References vii Chapter 2 Bott and Duistermaat-Heckman Formulas Berline-Vergne Localization Formula Bott Residue Formula Duistermaat-Heckman Formula Bott s Original Idea ix

10 X Contents 2.5 References Chapter 3 Gauss-Bonnet-Chern Theorem A Toy Model and the Berezin Integral Mathai-Quillen s Thorn Form A Transgression Formula Proof of the Gauss-Bonnet-Chern Theorem Some Remarks Chern s Original Proof References Chapter 4 PoincarB-Hopf Index Formula: an Analytic Proof Review of Hodge Theorem Poincark-Hopf Index Formula Clifford Actions and the Witten Deformation An Estimate Outside of UpEzero(V~Up Harmonic Oscillators on Euclidean Spaces A Proof of the Poincark-Hopf Index Formula Some Estimates for DT.i S, 2 5 i An Alternate Analytic Proof References Chapter 5 Morse Inequalities: an Analytic Proof Review of Morse Inequalities Witten Deformation Hodge Theorem for (R*(M). d ~ f... ) Behaviour of UTj Near the Critical Points of f Proof of Morse Inequalities Proof of Proposition Some Remarks and Comments References Chapter 6 Thom-Smale and Witten Complexes The Thorn-Smale Complex The de Rham Map for Thom-Smale Complexes Witten s Instanton Complex and the Map et The Map P,. TeT An Analytic Proof of Theorem

11 Contents xi 6.6 References Chapter 7 Atiyah Theorem on Kervaire Semi-characteristic Kervaire Semi-characteristic Atiyah s Original Proof A proof via Witten Deformation A Generic Counting Formula for k(m) Non-multiplicativity of k(m) References Index 117