TOPOLOGICAL QUANTUM FIELD THEORY AND FOUR MANIFOLDS

Transcription

1 TOPOLOGICAL QUANTUM FIELD THEORY AND FOUR MANIFOLDS

2 MATHEMATICAL PHYSICS STUDIES Editorial Board: Maxim Kontsevich, IHES, Bures-sur-Yvette, France Massimo Porrati, New York University, New York, U.S.A. Vladimir Matveev, Université Bourgogne, Dijon, France Daniel Sternheimer, Université Bourgogne, Dijon, France VOLUME 25

3 Topological Quantum Field Theory and Four Manifolds by JOSE LABASTIDA and MARCOS MARINO

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN (HB) ISBN (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Springer, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Springer, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

5 Table of Contents Preface vii 1. Topological Aspects of Four-Manifolds Homology and cohomology The intersection form Self-dual and anti-self-dual forms Characteristic classes Examples of four-manifolds. Complex surfaces Spin and Spin c -structures on four-manifolds The Theory of Donaldson Invariants Yang Mills theory on a four-manifold SU(2) and SO(3) bundles ASD connections Reducible connections A local model for the moduli space Donaldson invariants Metric dependence The Theory of Seiberg Witten Invariants The Seiberg Witten equations The Seiberg Witten invariants Metric dependence Supersymmetry in Four Dimensions The supersymmetry algebra N = 1 superspace and superfields N = 1 supersymmetric Yang Mills theories N = 2 supersymmetric Yang Mills theories N = 2 supersymmetric hypermultiplets N = 2 supersymmetric Yang Mills theories with matter Topological Quantum Field Theories in Four Dimensions Basic properties of topological quantum field theories Twist of N = 2supersymmetry Donaldson Witten theory Twisted N = 2 supersymmetric hypermultiplet Extensions of Donaldson Witten theory Monopole equations The Mathai Quillen Formalism v

6 6.1. Equivariant cohomology The finite-dimensional case A detailed example Mathai Quillen formalism: Infinite-dimensional case The Mathai Quillen formalism for theories with gauge symmetry Donaldson Witten theory in the Mathai Quillen formalism Abelian monopoles in the Mathai Quillen formalism The Seiberg Witten Solution of N = 2 SUSY Yang Mills Theory Low energy effective action: semi-classical aspects Sl(2, Z) duality of the effective action Elliptic curves The exact solution of Seiberg and Witten The Seiberg Witten solution in terms of modular forms The u-plane Integral The basic principle (or, Coulomb + Higgs=Donaldson ) Effective topological quantum field theory on the u-plane Zero modes Final form for the u-plane integral Behavior under monodromy and duality Some Applications of the u-plane Integral Wall crossing The Seiberg Witten contribution The blow-up formula Further Developments in Donaldson Witten Theory More formulae for Donaldson invariants Applications to the geography of four-manifolds Extensions to higher rank gauge groups Appendix A. Spinors in Four Dimensions Appendix B. Elliptic Functions and Modular Forms Bibliography vi

7 Preface The emergence of topological quantum field theory has been one of the most important breakthroughs which have occurred in the context of mathematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mathematics. Topological quantum field theories constitute the core of these phenomena, although the main driving force behind it has been the enormous effort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines profit from each other. Although the most striking results have appeared on the mathematical side, theoretical physics has clearly also benefitted, since the corresponding developments have helped better to understand aspects of the fundamentals of field and string theory. Topology has played an important role in the study of quantum mechanics since the late fifties, after discovering that global effects are important in physical phenomena. Many aspects of topology have become ordinary elements in studies in quantum mechanics as well as in quantum field theory and in string theory. The origin of topological quantum field theory can be traced back to 1982, although the term itself appeared for the first time in In 1982 E. Witten studied supersymmetric quantum mechanics and supersymmetric sigma models providing a framework that led to a generalization of Morse theory. This framework was later considered by A. Floer who constructed its mathematical setting and enlarged it to a more general context. This, in turn, was reconsidered by E. Witten who, influenced by M. Atiyah, proposed the first topological quantum field theory itself in His construction consisted of a quantum field theory representation of the theory of Donaldson invariants on four-manifolds proposed in After the first formulation of a topological quantum field theory by E. Witten many others have been considered. A new area of active research has developed since then. In this book we will concentrate our attention on vii

8 aspects related to that first theory, nowadays known as Donaldson Witten theory, which is the most relevant theory in four dimensions. Other important topological theories, such as Chern Simons gauge theory in three dimensions or topological string theory, fall beyond the scope of this book. We will deal with many aspects of Donaldson Witten theory, emphasizing how its formulation has allowed Donaldson invariants to be expressed in terms of a set of new simpler invariants known as Seiberg Witten invariants. Topological quantum field theory is responsible for the discovery of Seiberg Witten invariants and their relation to Donaldson invariants. In general, quantum field theories can be studied by different methods providing several pictures of the same theory. The relation between Donaldson Witten theory and Donaldson invariants was found using perturbative methods in the context of quantum field theory. The application of non-pertubative methods to the same theory waited several years but led to the discovery of the relevance of Seiberg Witten invariants as building blocks of Donaldson invariants. This connection was possible owing to the progress achieved in 1994 by N. Seiberg and E. Witten in understanding non-perturbative properties of supersymmetric theories. From the mathematical side the emergence of Seiberg Witten invariants constitutes one of the most important results obtained in the nineties in the study of four-manifolds. These invariants turn out to be much simpler than Donaldson invariants and contain all the information provided by the latter. To understand the connection between these invariants one needs to regard Donaldson Witten theory as a theory which originates after the twisting of certain supersymmetric quantum field theories. Other pictures of topological quantum field theory, such as the one in the framework of the Mathai Quillen formalism, which is also described in this book, do not provide useful information in this respect. However, it is important to become acquainted with this picture since it provides an interesting geometrical setting. The main goal of this book is to provide a unifying treatment of all the aspects of Donaldson Witten theory as a stem theory for Donaldson and Seiberg Witten invariants. An important effort has been made so that it can be read by theoretical physicists and mathematicians. The focus of the book is on the interplay of mathematical and physical aspects of the theory, and although we have included expositions of background material such as the more mathematical aspects of Donaldson theory or the physics of the viii

9 Seiberg Witten solution we have not provided all the details, and we refer the reader to more specific references for an exhaustive treatment of some of the subjects. The book starts with a chapter that collects basic mathematical results about the topology of four-manifolds which will be needed in the rest of the chapters. Chapters 2 and 3 contain reviews of the theories of Donaldson and Seiberg Witten invariants, respectively. Chapter 4 presents supersymmetry in four dimensions and describes the supersymmetric theories which will be relevant for Donaldson Witten theory. Chapter 5 deals with the twisting of supersymmetric theories and constructs all the topological quantum field theories which will be of interest in other chapters. There is shown, in particular, in sections 5.3 and 5.6, the connection between these theories and the Donaldson and Seiberg Witten invariants introduced in Chapters 2 and 3. In Chapter 6 a different framework for dealing with topological quantum field theories, the Mathai Quillen formalism, is introduced. This formalism provides an interesting geometrical framework for these theories which is worth being be considered. However, its content is not needed for the rest of the book. The chapter could be omitted in a first reading. Chapter 7 deals with non-perturbative aspects of supersymmetric theories. A detailed analysis of the resulting solution, the Seiberg Witten solution, is presented. The structure of this solution is used in Donaldson Witten theory in Chapter 8. It allows one to write the Donaldson Witten invariants as an integral on the so called u-plane introduced by Moore and Witten. The u-plane integral is the most systematic physical framework in which to understand Donaldson Witten theory, and it also leads to the connection between Donaldson invariants and Seiberg Witten invariants. Chapter 9 deals with several applications of the u-plane integral, and Chapter 10 summarizes further developments of Donaldson Witten theory. Finally, two appendices contain useful formulae about spinors in four dimensions, elliptic functions and modular forms. Acknowledgements We would like to thank our collaborators and colleagues over all the years we have devoted to the study of topological quantum field theory. We have benefitted from their knowledge and their insight has certainly influenced our work. It is not possible to list all of them here but we wish to thank specially M. Alvarez, L. Álvarez-Gaumé, J.D. Edelstein, C. Lozano, J. Mas, G. Moore, M. Pernici, A.V. Ramallo, and E. Witten. ix

10 Several colleagues agreed to read part of the manuscript before publication, providing important remarks. We would like to give special thanks in this respect to Carlos Lozano, Gregory Moore, and Vicente Muñoz. x