D-Branes, Derived Categories, and Grothendieck Groups

TL;DR

This paper explores how Grothendieck groups and derived categories can describe type II D-branes on complex varieties, linking algebraic geometry tools to string theory concepts and T-duality transformations.

Contribution

It introduces a framework connecting Grothendieck groups and derived categories to D-brane descriptions, incorporating holomorphic connections and T-duality insights.

Findings

Grothendieck groups encode D-brane data with connections.

Derived categories provide deeper understanding of D-brane structures.

Fourier-Mukai transforms relate to T-duality actions on D-branes.

Abstract

In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II D-branes, in the case that all D-branes are wrapped on complex varieties and all connections are holomorphic. Our proposal is in the same spirit as recent discussions of K-theory and D-branes; within the restricted class mentioned, Grothendieck groups encode a choice of connection on each D-brane worldvolume, in addition to information about the smooth bundles. We also point out that derived categories can also be used to give insight into D-brane constructions, and analyze how a Z_2 subset of the T-duality group acting on D-branes on tori can be understood in terms of a Fourier-Mukai transformation.