D-Branes, Derived Categories, and Grothendieck Groups

TL;DR

The paper develops a holomorphic refinement of D-brane charge classification by employing Grothendieck groups $K'_0(X)$ and $K'^0(X)$ of coherent and locally free sheaves, respectively, to encode both bundle data and holomorphic connections on complex submanifolds. It then positions derived categories $D^b(X)$ as the natural language for handling brane/antibrane configurations and explains how objects in $D^b(X)$ map to Grothendieck group elements, making contact with physics via a physically meaningful interpretation. The central technical tool is the Fourier-Mukai transform, which realizes a ${\mathbb Z}_2$ subgroup of T-duality on tori as an auto-equivalence of derived categories, acting on Grothendieck groups through a well-defined map $\mathcal{T}_K$. The paper also discusses limits of the W.I.T. condition, non-W.I.T. examples, and the sign ambiguity inherent in double dualization, highlighting how these mathematical structures illuminate brane/antibrane dynamics and dualities with potential connections to broader topics like mirror symmetry and heterotic duals.

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.