Thom isomorphism and Push-forward map in twisted K-theory

TL;DR

This paper develops the Thom isomorphism and push-forward map in twisted K-theory, extending classical results to non-K-oriented maps and applying them to D-brane charge classification in string theory.

Contribution

It establishes the Thom isomorphism and constructs the push-forward map in twisted K-theory for general differentiable maps, broadening the scope of classical K-theory tools.

Findings

Thom isomorphism in twisted K-theory for any real vector bundle.

Push-forward map in twisted K-theory for non-K-oriented maps.

Application to D-brane charges satisfying Freed-Witten anomaly cancellation.

Abstract

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable proper map $f: X\to Y$ (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any $K$-oriented differentiable proper map and the Atiyah-Singer index theorem of Dirac operators on Clifford modules. For $D$-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial $B$-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.