Clay mathematics Proceedings This volume is the proceedings of 3 Volume 3 the 2002 Clay Mathematics Institute School on Geometry and String Theory. This month-long program was held at the Isaac Newton S Institute for Mathematical Sciences t r in Cambridge, England, and was i n organized by both mathematicians g and physicists: A. Corti, R. Dijkgraaf, s M. Douglas, J. Gauntlett, M. Gross, a C. Hull, A. Jaffe and M. Reid. The n StringS and early part of the school had many d lectures that introduced various g concepts of algebraic geometry geometry e and string theory with a focus on o improving communication between m these two fields. During the latter Proceedings of the e part of the program there were also t Clay Mathematics Institute a number of research level talks. r y 2002 Summer School This volume contains a selection Isaac Newton Institute of expository and research articles by lecturers at the school, and Cambridge, United Kingdom d highlights some of the current o March 24–April 20, 2002 interests of researchers working u g at the interface between string l a theory and algebraic geometry. The s topics covered include manifolds , michael douglas g of special holonomy, supergravity, a Jerome gauntlett supersymmetry, D-branes, the u n mark gross McKay correspondence and the t Fourier-Mukai transform. le Editors t t a n d g r o s s , E d i t CMIP/3 o r s 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. Primary81T30, 83E30, 53C29, 32Q25, 14J32, 83E50,14E15, 53C80, 32S45, 14D20. ISBN0-8218-3715-X Copyingandreprinting. Materialinthisbookmaybereproducedbyanymeansforeduca- tionalandscientificpurposeswithoutfeeorpermissionwiththeexceptionofreproductionbyser- vicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledgment ofthesourceisgiven. Thisconsentdoesnotextendtootherkindsofcopyingforgeneraldistribu- tion,foradvertisingorpromotionalpurposes,orforresale. Requestsforpermissionforcommercial use of material should be addressed to the Clay Mathematics Institute, One Bow Street, Cam- bridge,MA02138,[email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:1)c 2004bytheClayMathematicsInstitute. Allrightsreserved. PublishedbytheAmericanMathematicalSociety,Providence,RI, fortheClayMathematicsInstitute,Cambridge,MA. PrintedintheUnitedStatesofAmerica. TheClayMathematicsInstituteretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 090807060504 Contents The Geometry of String Theory 1 Michael R. Douglas M theory, G -manifolds and Four Dimensional Physics 31 2 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 G -Manifolds 193 2 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 Bala´zs Szendro˝i 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- nizedbyRobbertDijkgraaf,MichaelDouglas, JeromeGauntlettandChrisHull,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,andfocusesonthetopicsofmanifoldsofspecialholonomyandsupergravity. Articlesinalgebraicgeometry,Dirichletbranesandrelatedtopicsarealsoincluded. ItbeginswithanarticlebyMichaelDouglasthatprovidesanoverviewofthegeom- 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 mirrorsymmetry fromthehomological andSYZpointsof view,derivedcategories, 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. WethankthedininghallstaffatKingsCollege,MagdeleneCollege,Corpus ChristiCollegeandEmmanuelCollege,andespeciallytheKingsCollegesingers,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 ClayMathematicsProceedings Volume3,2004 The Geometry of String Theory Michael R. Douglas Abstract. Anoverviewofthegeometryofstringtheory,whichsetsthevar- iouscontributionstothisproceedingsinthiscontext. 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 centralaspectofthisinteractionhasbeentheevolutionoftheconceptofgeometry, 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 fairtosay thatthis remains a dream, but avery 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 renaissanceofstringtheoryin1984,backtoinformalexchangesandschoolsduring the mid-1970’s, at which physicists and mathematicians began to realize that they had unexpected common interests. Althoughnotuniversallyknown,oneofthemostimportantoftheseencounters came at a series of seminars at Stony Brook, in which C. N. Yang would invite mathematicianstospeakontopicsofpossiblemutualinterest. In1975,JimSimons 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 2000MathematicsSubjectClassification. 81T30,83E30. (cid:1)c2004 Clay Mathematics Institute 1
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