Chern character in twisted K-theory: equivariant and holomorphic cases
Varghese Mathai, Danny Stevenson
TL;DR
This work develops a comprehensive Chern–Weil framework for the bundle-gerbe model of twisted K-theory and extends it to equivariant and holomorphic contexts. It provides a detailed construction of the twisted Chern character in the even and odd cases, proves its compatibility with the twisted $K$-theory module structure, and establishes functorial and multiplicativity properties, even in the presence of nontrivial B-field curvature. The authors formulate and relate multiple models of twisted K-theory, including bundle-gerbe modules and equivariant variants, and they extend the theory to holomorphic bundle gerbes and their modules, with Spin$^C$ spinor-module realizations. The results connect geometric and cohomological characters across smooth, equivariant, and holomorphic settings, with explicit constructions and examples that illuminate the role of lifting bundle gerbes and their modules in physical applications such as D-brane charges and Verlinde-type structures.
Abstract
It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary vector bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. In this paper, we study in more detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced previously, and we also extend it to the equivariant and holomorphic cases. Included is a discussion of interesting examples.