The Derived Category of Coherent Sheaves and B-model Topological String Theory

TL;DR

This survey explains how the derived category of coherent sheaves on Calabi-Yau threefolds models the D-brane category in B-model topological string theory, linking physics and advanced algebraic geometry.

Contribution

It provides an accessible overview of the identification between derived categories and D-branes, including physical motivation and homological algebra insights.

Findings

Identification of D-branes with complexes of coherent sheaves

Physical realization of quasi-isomorphism via RG flow

Connection between stable objects and universality classes

Abstract

This elementary survey article was prepared for a talk at the 2016 Superschool on Derived Categories and D-branes. The goal is to outline an identification of the bounded derived category of coherent sheaves on a Calabi-Yau threefold with the D-brane category in B-model topological string theory. This was originally conjectured by Kontsevich. We begin by briefly introducing topological closed string theory to acquaint the reader with the basics of the non-linear sigma model. With the inclusion of open strings, we must specify boundary conditions for the endpoints; these are what we call D-branes. After briefly summarizing the necessary homological algebra and sheaf cohomology, we argue that one should think of a D-brane as a complex of coherent sheaves, and provide a physical motivation to identify complexes up to homotopy. Finally, we argue that renormalization group (RG) flow on the worldsheet provides a physical realization of quasi-isomorphism. This identifies a stable object in the derived category with a universality class of D-branes in physics. I aim for this article to be an approachable introduction to the subject for both mathematicians and physicists. As such, it is far from a complete account.