Differential Topology and Quantum Field Theory PDF
Preview Differential Topology and Quantum Field Theory
Differential Topology and Quantum Field Theory Charles Nash St Patrick’s College Maynooth Ireland ACADEMIC PRESS Harcourt Brace Jovanovich, Publishers London San Diego New York Boston Sydney Tokyo Toronto ACADEMIC PRESS LIMITED 24-28 Oval Road London NW1 7DX United States Edition published by ACADEMIC PRESS INC. San Diego, CA 92101 Copyright (©) 1991 by ACADEMIC PRESS LIMITED This book is printed on acid-free paper All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers. British Library Cataloguing in Publication Data is available from the British Library ISBN 0-12-514075-4 Typeset using TEX Printed in Great Britain by St Edmundsbury Press Limited, Bury St Edmunds, Suffolk Preface The twentieth century has been witness to a great burgeoning of mathemat- ics and physics. In the early part of the century the new physical theories of relativity and quantum mechanics made extensive use of the machinery of differential geometry and Hilbert spaces. Some time later quantum field theory began to pose difficult mathematical problems. As a partial response to this the subject of axiomatic quantum field theory was born. The main thrust of this approach was to tackle the formidable problems of quantum field theory head on using the most powerful mathematical tools available; the bulk of these tools being drawn from analysis. More recently there has been considerable evidence that the way forward in these problems is consid- erably illuminated if, in addition to analysis, one uses differential topology. There have also been advances in the differential topology of dimensions three and four which have drawn extensively on physical sources, the prin- cipal source being Yang-Mills theories. Thus there has been a genuinely two sided interaction between the worlds of mathematics and mathematical physics. This book is intended as an informal introduction to some of these mathematical and physical ideas. It should be of use to graduate students and other research workers taking an interest in this material for the first time. September 1990 Charles Nash For Jeanine Niamh Marie—-Therésé Nash 27" September 1977—15t* February 1980 Contents Preface CHAPTER I A Topological Preliminary §1 From homeomorphism to diffeomorphism § 2 Some algebraic topology: homotopy §3 Homotopy groups § 4 Cohomology and homology groups §5 Fibre bundles and fibrations 14 §6 Differentiable structures for manifolds 19 CHAPTER II Elliptic Operators 27 §1 The meaning of ellipticity 27 § 2 Ellipticity and hypo-ellipticity 32 §3 Ellipticity and vector bundles 36 § 4 Pseudo-differential operators 41 §5 Pseudo-differential operators and Sobolev spaces 47 viii Differential Topology and Quantum Field Theory CHAPTER III Cohomology of Sheaves and Bundles 56 §1 Sheaves 56 § 2 Sheaf cohomology 59 §3 K-theory 65 §4 Bott periodicity 75 §5 Some characteristic classes 78 §6 Fredholm operators and K(X) 88 CHAPTER IV Index Theory for Elliptic Operators 89 §1 The index of an elliptic operator 89 §2 Some examples /'/ 97 §3 Twisted complezes | 116 §4 The index theorem for famil;es of operators 119 §5 The index for real families 121 §6 Index theory and fized points 123 §7 Index theory for manifolds with boundary 127 CHAPTER V Some Algebraic Geometry 137 §1 Algebraic varieties 137 §2 Riemann surfaces and divisors 139 §3 Serre duality, line bundles and Kahler manifolds 144 Contents §4 The Teichmiiller space Ty § 5 The moduli space M, §6 The dimension of the moduli space §7 Weierstrass gaps and Weterstrass points CHAPTER VI Infinite Dimensional Groups §1 Some infinite dimensional groups § 2 Group extensions §3 Representations CHAPTER VII Morse Theory §1 The topology of critical points §2 Critical sub-manifolds §3 Equivariant Morse theory §4 Supersymmetric quantum mechanics and Morse theory CHAPTER VIII Instantons and Monopoles §1 The topology of gauge fields § 2 Secondary characteristic classes §3 Instantons and their modul: §4 Monopoles and symmetries of instantons §5 Monopole moduli and monopole scattering Differential Topology and Quantum Field Theory §6 Critical point theory and gauge theories 256 CHAPTER IX The Elliptic Geometry of Strings 259 §1 The Bosonic string 259 § 2 The space of metrics _ 262 §3 The Weil-Petersson metric 265 CHAPTER X Anomalies 269 §1 Introduction 269 §2 Anomalies and Yang-Mills theories 270 §3 Grauitational anomalies 279 §4 The critical dimension for strings 281 §5 Global anomalies . 283 §6 Anomalies from a Hamiltonian perspective 291 CHAPTER XI Conformal Quantum Field Theories 301 §1 Conformal invariance and quantum field theory 301 § 2 Conformal field theories in two dimensions 302 §3 Relation to the Virasoro algebra 307 §4 Statistical mechanics 311 §5 Operator products, fusion rules and aztomatics 313 Contents xi CHAPTER XII Topological Quantum Field Theories 322 §1 Introduction 322 §2 Floer theory and the Chern-Simons function 322 §3 Donaldson’s polynomial invariants 332 § 4 Knots and knot invariants 339 §5 Chern—Simons theory and knots 342 §6 CPem—Simon@ theory and the Jones polynomial 350 §7 'AS/'urgery and the Jones polynomial 355 References 361 Index 375
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