D-Branes, Derived Categories, and Grothendieck Groups
Eric R. Sharpe
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 Z2 subset of the T-duality group acting on D-branes on tori can be understood in terms of a Fourier-Mukai transformation.
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