Lectures on D-branes and Sheaves

TL;DR

This paper reviews the connection between boundary states in the open string B model on Calabi-Yau manifolds and sheaves, highlighting mathematical tools for counting open string states and exploring physical-mathematical relationships.

Contribution

It provides a comprehensive overview of how boundary states relate to sheaves and derived categories, and discusses various physical scenarios affecting these models.

Findings

Boundary states correspond to sheaves on Calabi-Yau manifolds.

Mathematical interpretations facilitate counting open string states.

Physical variations like B fields and orbifolds alter the models.

Abstract

These notes are a writeup of lectures given at the twelfth Oporto meeting on ``Geometry, Topology, and Physics,'' and at the Adelaide workshop ``Strings and Mathematics 2003,'' primarily geared towards a physics audience. We review current work relating boundary states in the open string B model on Calabi-Yau manifolds to sheaves. Such relationships provide us with a mechanism for counting open string states in situations where the physical spectrum calculation is nearly intractable -- after translating to mathematics, such calculations become easy. We describe several different approaches to these models, and also describe how these models are changed by varying physical circumstances -- flat B field backgrounds, orbifolds, and nonzero Higgs vevs. We also discuss mathematical interpretations of operator products, and how such mathematical interpretations can be checked physically. One of the motivations for this work is to understand the precise physical relationship between boundary states in the open string B model and derived categories in mathematics, and we outline what is currently known of the relationship.