GEOMETRY, TOPOLOGY AND PHYSICS - USTC

1 GRADUATE STUDENT SERIES IN PHYSICSS eries Editor:Professor Douglas F Brewer, MA, DPhilEmeritus Professor of Experimental PHYSICS , University of SussexGEOMETRY, TOPOLOGYAND PHYSICSSECOND EDITIONMIKIO NAKAHARAD epartment of PhysicsKinki University, Osaka, JapanINSTITUTE OF PHYSICS PUBLISHINGB ristol and Philadelphia c IOP Publishing Ltd 2003 All rights reserved. No part of this publication may be reproduced, storedin a retrieval system or transmitted in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without the prior permissionof the publisher. Multiple copying is permitted in accordance with the termsof licences issued by the Copyright Licensing Agency under the terms of itsagreement with Universities UK (UUK).British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British 0 7503 0606 8 Library of Congress Cataloging-in-Publication Data are availableCommissioning Editor: Tom SpicerProduction Editor: Simon LaurensonProduction Control: Sarah PlentyCover Design: Victoria Le BillonMarketing: Nicola Newey and Verity CookePublished by Institute of PHYSICS Publishing, wholly owned by The Institute ofPhysics, LondonInstitute of PHYSICS Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UKUS Office.

2 Institute of PHYSICS Publishing, The Public Ledger Building, Suite929, 150 South Independence Mall West, Philadelphia, PA 19106, USAT ypeset in LATEX2 by Text 2 Text, Torquay, DevonPrinted in the UK by MPG Books Ltd, Bodmin, Cornwall Dedicated to my family CONTENTSP reface to the First EditionPreface to the Second EditionHow to Read this BookNotation and Conventions1 Quantum space, bras and of canonical equation, Heisenberg picture and Schr integral quantization of a Bose integral time and partition product and generating integral quantization of a Fermi harmonic of Grassmann states and completeness relation Partition function of a fermionic of a scalar scalar scalar of a Dirac gauge gauge Wu Yang The (anti-)self-dual solutionProblems2 Mathematical relation and equivalence and vector maps, images and vector product and and Hausdorff and topological characteristic.

3 An exampleProblems 3 Homology group generated Abelian groups and free Abelian and simplicial complexes and groups of simplicial group, cycle group and boundary ofH0(K) homology properties of homology and homology of homology numbers and the Euler Poincar e theoremProblems4 Homotopy and properties of fundamental connectedness and fundamental invariance of fundamental of fundamental group of groups of groups and fundamental groups of betweenH1(K)and 1(|K|) homotopy properties of higher homotopy nature of higher homotopy connectedness and higher homotopy invariance of higher homotopy homotopy groups of a product covering spaces and higher homotopy of higher homotopy groups in condensed matter and in nematic liquid parameter of nematic liquid defects in nematic liquid defects in nematic liquid dimensional Textures in Line defects and non-singular vortices Shankar monopole in3He-AProblems5 calculus on and Lie group of product and Lie derivative of of differential of groups and Lie one-parameter and structure action of Lie groups on manifolds and isotropy vector adjoint representationProblems6 de Rham Cohomology Rham cohomology ofHr(M)andHr(M).

4 De Rham s e s of de Rham cohomology e K unneth of de Rham cohomology andH1(M)7 Riemannian manifolds and pseudo-Riemannian transport, connection and covariant transport and covariant derivative of tensor transformation properties of connection metric and meaning of the Riemann tensor and thetorsion Ricci tensor and the scalar fundamental Levi-Civita connection in the classical geometry normal coordinate curvature tensor with Levi-Civita and conformal vector fields and conformal Killing vector vector Killing vector s structure local Levi-Civita connection in a non-coordinate forms and Hodge volume transformations (Hodge star) products of exterior Laplacian, harmonic forms and the Hodgedecomposition forms and de Rham cohomology Aspects of general Introduction to general Einstein Hilbert Spinors in curved Bosonic string The string Symmetries of the Polyakov stringsProblems8 Complex on complex complex differential of real differential forms on complex manifolds and Hermitian differential Hermitian ahler derivatives and ahler manifolds and K ahler differential ahler holonomy group of K ahler forms and -cohomology adjoint operators and and the Hodge on a K ahler Hodge numbers of K ahler complex examples9 Fibre of fibre and bundles and dual of vector product bundle and Whitney sum product of bundlesProblems10 Connections on Fibre Connections on principal The connection The local connection form and gauge Horizontal lift and

5 Parallel Holonomy Covariant derivatives in principal Geometrical meaning of the curvature and the Ambrose Singer Local form of the The Bianchi The covariant derivative on associated vector The covariant derivative on associated A local expression for the covariant Curvature A connection which preserves the inner Holomorphic vector bundles and Hermitian Gauge U(1) gauge The Dirac magnetic The Aharonov Bohm Yang Mills Berry s Derivation of Berry s Berry s phase, Berry s connection and Berry s curvatureProblems11 Characteristic Invariant polynomials and the Chern Weil Invariant Chern Properties of Chern Splitting Universal bundles and classifying Chern Properties of the Chern Todd Pontrjagin and Euler Pontrjagin Euler HirzebruchL-polynomial and Chern Simons Definition The Chern Simons form of the Chern Cartan s homotopy operator and Stiefel Whitney Spin Cech cohomology Stiefel Whitney classes12 Index Elliptic operators and Fredholm Elliptic Fredholm Elliptic The Atiyah Singer index Statement of the The de Rham The Dolbeault The twisted Dolbeault complex and the Hirzebruch Riemann Roch The signature The Hirzebruch The signature complex and the Hirzebruch Spin Dirac Twisted spin The heat

6 Kernel and generalized The heat kernel and index Spectral The Atiyah Patodi Singer index -invariant and spectral The Atiyah Patodi Singer (APS) index Supersymmetric quantum Clifford algebra and Supersymmetric quantum mechanics in flat Supersymmetric quantum mechanics in a Supersymmetric proof of index The Path integral and index theoremProblems 13 Anomalies in Gauge Field Abelian Fujikawa s Non-Abelian The Wess Zumino consistency The Becchi Rouet Stora operator and the Faddeev Popov The BRS operator, FP ghost and moduli The Wess Zumino Descent equations and solutions of WZ Abelian anomaliesversusnon-Abelian m+2 The parity anomaly inodd-dimensional The parity The dimensional ladder: 4 3 214 Bosonic String Differential geometry on Riemann Metric and complex Vectors, forms and Covariant The Riemann Roch Quantum theory of bosonic Vacuum amplitude of Polyakov Measures of Complex tensor calculus and string Moduli spaces of Riemann One-loop Moduli spaces, CKV, Beltrami and quadratic The evaluation of determinantsReferences PREFACE TO THE FIRST EDITIONThis book is a considerable expansion of lectures I gave at the School ofMathematical and Physical Sciences, University of Sussex during the winterterm of 1986.

7 The audience included postgraduate students and faculty membersworking in particle PHYSICS , condensed matter PHYSICS and general relativity. Thelectures were quite informal and I have tried to keep this informality as much aspossible in this book. The proof of a theorem is given only when it is instructiveand not very technical; otherwise examples will make the theorem figures will help the reader to obtain concrete images of the spite of the extensive use of the concepts of TOPOLOGY , differential ge-ometry and other areas of contemporary mathematics in recent developments intheoretical PHYSICS , it is rather difficult to find a self-contained book that is easilyaccessible to postgraduate students in PHYSICS . This book is meant to fill the gapbetween highly advanced books or research papers and the many excellent intro-ductory books. As a reader, I imagined a first-year postgraduate student in theo-retical PHYSICS who has some familiarity with quantum field theory and this book, the reader will find many examples from PHYSICS , in which topo-logical and geometrical notions are very important.

8 These examples are eclecticcollections from particle PHYSICS , general relativity and condensed matter should feel free to skip examples that are out of their direct , I believe these examples should be thetheoretical minimato studentsin theoretical PHYSICS . Mathematicians who are interested in the application oftheir discipline to theoretical PHYSICS will also find this book book is largely divided into four parts. Chapters 1 and 2 deal with thepreliminary concepts in PHYSICS and mathematics, respectively. In chapter 1,a brief summary of the PHYSICS treated in this book is given. The subjectscovered are path integrals, gauge theories (including monopoles and instantons),defects in condensed matter PHYSICS , general relativity, Berry s phase in quantummechanics and strings. Most of the subjects are subsequently explained in detailfrom the topological and geometrical viewpoints.

9 Chapter 2 supplements theundergraduate mathematics that the average physicist has studied. If readers arequite familiar with sets, maps and general TOPOLOGY , they may skip this chapterand proceed to the 3 to 8 are devoted to the basics of algebraic TOPOLOGY anddifferential geometry. In chapters 3 and4, the idea of the classification of spaceswith homology groups and homotopy groups is introduced. In chapter 5, we define a manifold, which is one of the central concepts in modern theoreticalphysics. Differential forms defined there play very important roles throughout thisbook. Differential forms allow us to define the dual of the homology group calledthe de Rham cohomology group in chapter 6. Chapter 7 deals with a manifoldendowed with a the metric, we may define such geometricalconcepts as connection, covariant derivative, curvature, torsion and many chapter 8, a complex manifold is defined as a special manifold on which thereexists a natural complex 9 to 12 are devoted to the unification of TOPOLOGY and chapter 9, we define a fibre bundle andshow that this is a natural settingfor many physical phenomena.

10 The connection defined in chapter 7 is naturallygeneralized to that on fibre bundles in chapter 10. Characteristic classes definedin chapter 11 enable us to classify fibre bundles using various cohomologyclasses. Characteristic classes are particularly important in the Atiyah Singerindex theorem in chapter 12. We do not prove this, one of the most importanttheorems in contemporary mathematics, but simply write down the special formsof the theorem so that we may use them in practical applications in 13 and 14 are devoted to the most fascinating applications oftopology and geometry in contemporary PHYSICS . In chapter 13, we apply thetheory of fibre bundles, characteristic classes and index theorems to the study ofanomalies in gauge theories. In chapter 14, Polyakov s bosonic string theory isanalysed from the geometrical point of view. We give an explicit computation ofthe one-loop would like to express deep gratitude tomy teachers, friends and thanks are due to Tetsuya Asai, David Bailin, Hiroshi Khono, DavidLancaster, Shigeki Matsutani, Hiroyuki Nagashima, David Pattarini, Felix E APirani, Kenichi Tamano, David Waxman and David Wong.