Mathematical Physics Studies 25 Topological Quantum Field Theory and Four Manifolds by Jose Labastida Marcos Marino TOPOLOGICAL QUANTUM FIELD THEORY AND FOUR MANIFOLDS 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 Topological Quantum Field Theory and Four Manifolds by JOSE LABASTIDA and MARCOS MARINO A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 1-4020-3058-4 (HB) ISBN 1-4020-3177-7 (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. Table of Contents Preface ............................vii 1. Topological Aspects of Four-Manifolds ............. 1 1.1. Homology and cohomology . ............. 1 1.2. The intersection form .................. 2 1.3. Self-dual and anti-self-dual forms . ............. 4 1.4. Characteristic classes ................... 5 1.5. Examples of four-manifolds. Complex surfaces . ....... 6 1.6. Spin and Spinc-structures on four-manifolds ......... 9 2. The Theory of Donaldson Invariants . ............12 2.1. Yang–Mills theory on a four-manifold ...........12 2.2. SU(2) and SO(3) bundles .................14 2.3. ASD connections . ....................16 2.4. Reducible connections . .................18 2.5. A local model for the moduli space . ............19 2.6. Donaldson invariants . ..................22 2.7. Metric dependence . ...................27 3. The Theory ofSeiberg–Witten Invariants . ...........31 3.1. The Seiberg–Witten equations . ............31 3.2. The Seiberg–Witten invariants . ............32 3.3. Metric dependence . ...................36 4. Supersymmetry in Four Dimensions . ............39 4.1. The supersymmetry algebra . ............39 4.2. N = 1 superspace and superfields . ............40 4.3. N = 1 supersymmetric Yang–Mills theories .........45 4.4. N = 2 supersymmetric Yang–Mills theories . ........50 4.5. N = 2 supersymmetric hypermultiplets . ..........53 4.6. N = 2 supersymmetric Yang–Millstheories with matter ....55 5. Topological Quantum Field Theories in Four Dimensions .....58 5.1. Basic properties of topological quantum field theories .....58 5.2. Twist of N =2supersymmetry . .............61 5.3. Donaldson–Witten theory . ................64 5.4. Twisted N = 2 supersymmetric hypermultiplet . .......71 5.5. Extensions of Donaldson–Witten theory ...........72 5.6. Monopole equations . ..................74 6. The Mathai–Quillen Formalism . .............78 v 6.1. Equivariant cohomology..................79 6.2. The finite-dimensional case . ............82 6.3. A detailed example ....................88 6.4. Mathai–Quillen formalism: Infinite-dimensional case . ....93 6.5. The Mathai–Quillen formalism for theories with gauge symmetry 102 6.6. Donaldson–Witten theory in the Mathai–Quillen formalism . 105 6.7. Abelian monopoles in the Mathai–Quillen formalism . ... 107 7. The Seiberg–Witten Solution of N = 2 SUSYYang–Mills Theory 110 7.1. Low energyeffective action: semi-classical aspects . .... 110 7.2. Sl(2, Z) duality of the effective action . .......... 116 7.3. Elliptic curves . .................... 120 7.4. The exact solution ofSeiberg and Witten ......... 123 7.5. The Seiberg–Witten solution in terms of modular forms . 129 8.The u-plane Integral .................... 133 8.1. The basic principle (or, ‘Coulomb + Higgs=Donaldson’) .. 133 8.2. Effective topological quantum field theory on the u-plane . 134 8.3. Zero modes . ..................... 140 8.4. Final form for the u-plane integral ............ 144 8.5. Behavior under monodromy andduality . ......... 149 9. Some Applications of the u-plane Integral ........... 154 9.1. Wall crossing ..................... 154 9.2. The Seiberg–Witten contribution . ............ 157 9.3. The blow-up formula . ................. 165 10. Further Developments in Donaldson–Witten Theory ...... 170 10.1. More formulae for Donaldson invariants . ........ 170 10.2. Applications to the geography of four-manifolds ...... 177 10.3. Extensions to higher rank gauge groups . ........ 188 Appendix A.Spinors in Four Dimensions ............ 204 Appendix B. Elliptic Functions and Modular Forms ....... 209 Bibliography ......................... 213 vi Preface The emergence of topological quantum field theory has been one of the most important breakthroughs which have occurred in the context of math- ematical 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 mathe- matics. Topological quantum field theories constitute the core of these phe- nomena, although the main drivingforce 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 mathemati- cal 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 mechan- ics since the late fifties, after discovering that global effects are important in physical phenomena. Many aspects of topology have become ordinary ele- ments 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 1987. In 1982 E. Witten studied supersymmetric quantum mechanics and supersymmetric sigma models providing a framework that led to a general- ization of Morse theory. This framework was later considered byA. 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 1987. His construction consisted of a quantum field theory representation of the theory of Donaldson invariants on four-manifolds proposed in 1982. 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 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 ofa 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 gen- eral, 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 meth- ods to the same theory waited several years but led to the discovery of the relevance of Seiberg–Witten invariants as building blocks of Donaldson in- variants. This connection was possible owing to the progress achieved in 1994 byN.Seiberg and E. Witten in understanding non-perturbative properties of supersymmetric theories. From the mathematical side the emergence ofSeiberg–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. Tounderstand 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 infor- mation 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