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Contents
1 Introduction 5
2 Overview of mathematics of sheaves and Ext groups 8
2.1 Complexes and exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Ext groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Introduction
Using sheaves as a mathematical tool to model D-branes on large-radius Calabi-Yau manifolds was first suggested many years ago by J. Harvey and G. Moore in [1]. Since then, it
has become popular to assume that such a model is a reasonable one, and furthermore to
assume that sheaves can be used to calculate physical properties such as, for example:
1. Open string spectra between D-branes, which are believed to be counted by Ext groups.
2. Boundary-boundary OPE’s, which are believed to be given by Yoneda pairings of Ext groups.
3. T-duality, which is believed to be described by a Fourier-Mukai transformation.

Another motivation comes from understanding mirror symmetry. Before D-branes
were popularized, Kontsevich [2] proposed an understanding of mirror symmetry involving
mathematical objects known as derived categories (collections of complexes of sheaves, for
the moment). At the time, the physical meaning of this proposal was far from clear.
Using sheaves as a tool to describe D-branes was progress towards understanding the
physical meaning of Kontsevich’s proposal, but only a first step.

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