# Dirichlet Branes and Mirror Symmetry

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*Paul S. Aspinwall; Tom Bridgeland; Alastair Craw; Michael R. Douglas; Mark Gross; Anton Kapustin; Gregory W. Moore; Graeme Segal; Balázs Szendrői; P.M.H. Wilson*

A co-publication of the AMS and Clay Mathematics Institute

Research in string theory over the last several decades has yielded a rich
interaction with algebraic geometry. In 1985, the introduction of
Calabi–Yau manifolds into physics as a way to compactify ten-dimensional
space-time has led to exciting cross-fertilization between physics and
mathematics, especially with the discovery of mirror symmetry in 1989. A
new string revolution in the mid-1990s brought the notion of branes to the
forefront. As foreseen by Kontsevich, these turned out to have
mathematical counterparts in the derived category of coherent sheaves on
an algebraic variety and the Fukaya category of a symplectic manifold.

This has led to exciting new work, including the
Strominger–Yau–Zaslow conjecture, which used the theory of
branes to propose a geometric basis for mirror symmetry, the theory of
stability conditions on triangulated categories, and a physical basis
for the McKay correspondence. These developments have led to a great
deal of new mathematical work.

One difficulty in understanding all aspects of this work is that it
requires being able to speak two different languages, the language of
string theory and the language of algebraic geometry. The 2002 Clay School
on Geometry and String Theory set out to bridge this gap, and this
monograph builds on the expository lectures given there to provide an
up-to-date discussion including subsequent developments. A natural sequel
to the first Clay monograph on Mirror Symmetry, it presents the new ideas
coming out of the interactions of string theory and algebraic geometry in
a coherent logical context. We hope it will allow students and researchers
who are familiar with the language of one of the two fields to gain
acquaintance with the language of the other.

The book first introduces the notion of Dirichlet brane in the
context of topological quantum field theories, and then reviews the
basics of string theory. After showing how notions of branes arose in
string theory, it turns to an introduction to the algebraic geometry,
sheaf theory, and homological algebra needed to define and work with
derived categories. The physical existence conditions for branes are
then discussed and compared in the context of mirror symmetry,
culminating in Bridgeland's definition of stability structures, and
its applications to the McKay correspondence and quantum geometry. The
book continues with detailed treatments of the
Strominger–Yau–Zaslow conjecture, Calabi–Yau metrics
and homological mirror symmetry, and discusses more recent physical
developments.

This book is suitable for graduate students and researchers with either a
physics or mathematics background, who are interested in the interface
between string theory and algebraic geometry.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in mathematical aspects of quantum field theory, in particular string theory and mirror symmetry.