Strings and geometry : proceedings of the Clay Mathematics Institute 2002 Summer School on Strings and Geometry : Isaac Newton Institute, United Kingdom, March 24 - April 20, 2002

386 Pages·2004·4.69 MB·English

by  Michael Douglas

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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
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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