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

Abstract

This elementary survey article was prepared for a talk at the PIMS 2016 Alberta Superschool on Derived Categories and D-branes. The goal is to outline an identification of the (bounde d) derived category of coherent sheaves on a Calabi–Yau threefold X with the D-brane category in B-model topological string theory. This was originally conjectured by Kontsevich [1]. 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. Most naïvely, a D-brane in the B-model is a holomorphic submanifold of X and a locally-free sheaf supported on it; such objects pushforward to the category of coherent sheaves on X. 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 an element 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. The material is based largely on Eric Sharpe’s lecture notes [2] as well as Paul Aspinwall’s paper [3].