Clay Mathematics Proceedings This volume is the proceedings of Volume 3 the 2002 Clay Mathematics Institute 3 School on Geometry and String Theory. This month-long program Strings and was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England, and was organized by both mathematicians and physicists: A. Corti, R. Dijkgraaf, M. Douglas, J. Gauntlett, M. Gross, C. Hull, A. Jaffe and M. Reid. The early part of the school had many lectures that introduced various G STRINGS AND concepts of algebraic geometry eometry and string theory with a focus on improving communication between GEOMETRY these two fields. During the latter Proceedings of the part of the program there were also Clay Mathematics Institute a number of research level talks. 2002 Summer School This volume contains a selection of expository and research articles Isaac Newton Institute by lecturers at the school, and D Cambridge, United Kingdom highlights some of the current ouglas, interests of researchers working March 24–April 20, 2002 at the interface between string theory and algebraic geometry. The topics covered include manifolds G Michael Douglas of special holonomy, supergravity, auntlett and Jerome Gauntlett supersymmetry, D-branes, the McKay correspondence and the Mark Gross Fourier-Mukai transform. Editors G ross, Editors CMIP/3 www.ams.org American Mathematical Society AMS www.claymath.org CMI Clay Mathematics Institute 4-color process 392 pages • 3/4” spine STRINGS AND GEOMETRY Clay Mathematics Proceedings Volume 3 STRINGS AND GEOMETRY Proceedings of the Clay Mathematics Institute 2002 Summer School on Strings and Geometry Isaac Newton Institute Cambridge, United Kingdom March 24–April 20, 2002 Michael Douglas Jerome Gauntlett Mark Gross Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 81T30, 83E30, 53C29, 32Q25, 14J32, 83E50, 14E15, 53C80, 32S45, 14D20. ISBN 0-8218-3715-X Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribu- tion, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam- bridge, MA 02138, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2004 by the Clay Mathematics Institute. All rights reserved. Published by the American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA. Printed in the United States of America. The Clay Mathematics Institute retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the Clay Mathematics Institute home page at http://www.claymath.org/ 10987654321 090807060504 Contents The Geometry of String Theory 1 Michael R. Douglas M theory, G2-manifolds and Four Dimensional Physics 31 Bobby S. Acharya Conjectures in K¨ahler geometry 71 Simon K. Donaldson Branes, Calibrations and Supergravity 79 Jerome P. Gauntlett M-theory on Manifolds with Exceptional Holonomy 127 Sergei Gukov Special holonomy and beyond 159 Nigel Hitchin Constructing compact manifolds with exceptional holonomy 177 Dominic Joyce From Fano Threefolds to Compact G2-Manifolds 193 Alexei Kovalev An introduction to motivic integration 203 Alastair Craw Representation moduli of the McKay quiver for finite Abelian subgroups of SL(3, C) 227 Akira Ishii Moduli spaces of bundles over Riemann surfaces and the Yang–Mills stratification revisited 239 Frances Kirwan On a classical correspondence between K3 surfaces II 285 Carlo Madonna and Viacheslav V. Nikulin Contractions and monodromy in homological mirror symmetry 301 Balazs´ Szendroi˝ Lectures on Supersymmetric Gauge Theory 315 Nick Dorey vii viii CONTENTS The Geometry of A-branes 337 Anton Kapustin Low Energy D-brane Actions 349 Robert C. Myers List of Participants 371 Preface The 2002 Clay School on Geometry and String Theory was held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK from 24 March - 20 April 2002. It was organized jointly by the organizers of two concurrent workshops at the Newton Institute, one on Higher Dimensional Complex Geometry organized by Alessio Corti, Mark Gross and Miles Reid, and the other on M-theory orga- nized by Robbert Dijkgraaf, Michael Douglas, Jerome Gauntlett and Chris Hull, in collaboration with Arthur Jaffe, then president of the Clay Mathematics Institute. This volume is one of two books which will provide the scientific record of the school, and focuses on the topics of manifolds of special holonomy and supergravity. Articles in algebraic geometry, Dirichlet branes and related topics are also included. It begins with an article by Michael Douglas that provides an overview of the geom- etry arising in string theory and sets the subsequent articles in context. A second book, in the form of a monograph to appear later, will more systematically cover mirror symmetry from the homological and SYZ points of view, derived categories, Dirichlet branes, topological string theory, and the McKay correspondence. On behalf of the Organizing Committee, we thank the directors of the Isaac Newton Institute, H. Keith Moffatt and John Kingman, for their firm support. We thank the Isaac Newton Institute staff, Wendy Abbott, Mustapha Amrani, Tracey Andrew, Caroline Fallon, Jackie Gleeson, Louise Grainger, Robert Hunt, Rebecca Speechley and Christine West, for their superlative job in bringing such a large project to fruition, and providing the best possible environment for the school. We thank the dining hall staff at Kings College, Magdelene College, Corpus Christi College and Emmanuel College, and especially the Kings College singers, for some memorable evenings. We thank the staff of the Clay Mathematics Institute, and especially Barbara Drauschke, for their behind-the-scenes work, which made the school possible. Finally we thank Arthur Greenspoon, Vida Salahi and Steve Worcester for their efforts in helping to produce this volume. Michael Douglas, Jerome Gauntlett and Mark Gross September 2003 ix Clay Mathematics Proceedings Volume 3, 2004 The Geometry of String Theory Michael R. Douglas Abstract. An overview of the geometry of string theory, which sets the var- ious contributions to this proceedings in this context. 1. Introduction The story of interactions between mathematics and physics is very long and very rich, too much so to summarize in a few pages. But from the beginning, a central aspect of this interaction has been the evolution of the concept of geometry, from the static conceptions of the Greeks, through the 17th century development of descriptions of paths and motions through a fixed space, to Einstein’s vision of space-time itself as dynamical, described using Riemannian geometry. String/M theory, the unified framework subsuming superstring theory and su- pergravity, is at present by far the best candidate for a unified quantum theory of all matter and interactions, including gravity. One might expect that a worthy successor to Einstein’s theory would be based on a fundamentally new concept of geometry. At present, it would be fair to say that this remains a dream, but a very live dream indeed, which is inspiring a remarkably fruitful period of interaction between physicists and mathematicians. Our school focused on the most recent trends in this area, such as compactifi- cation on special holonomy manifolds, and approaches to mirror symmetry related to Dirichlet branes. But before we discuss these, let us say a few words about how these interactions began. To a large extent, this can be traced to before the renaissance of string theory in 1984, back to informal exchanges and schools during the mid-1970’s, at which physicists and mathematicians began to realize that they had unexpected common interests. Although not universally known, one of the most important of these encounters came at a series of seminars at Stony Brook, in which C. N. Yang would invite mathematicians to speak on topics of possible mutual interest. In 1975, Jim Simons gave a lecture series on connections and curvature, and the group quickly realized that this mathematics was the geometric foundation of Yang-Mills theory, and could be used to understand the recently discovered non-Abelian instanton and monopole solutions [23, 65]. These foundations are by now so familiar that it is 2000 Mathematics Subject Classification. 81T30, 83E30. c 2004 Clay Mathematics Institute 1 2 MICHAEL R. DOUGLAS a bit surprising to realize that, for Yang-Mills theory, they date back only to this time. Among the participants in the Simons-Yang seminars was Is Singer, who carried news of these developments to Atiyah at Oxford. Before long the mathematicians were taking the lead in exploiting these solutions, culminating in the early 1980’s with Donaldson’s use of instanton moduli spaces to formulate his celebrated invari- ants [17], which revolutionized the study of four-dimensional topology. While this case study in math-physics interaction might have ended there, with the lesson being that mathematicians can find useful inspiration in physical devel- opments but then must apply them to their own problems, of course it did not. The deeper aspects of this interaction began with Witten’s 1988 reformulation of the Donaldson invariants as observables in a topological field theory [58].