Twisted Homology

TL;DR

This paper demonstrates that twisted homology provides a simpler and more versatile framework than twisted K-theory for classifying D-branes and M-branes in certain compactifications, with practical computational tools.

Contribution

It establishes an isomorphism between twisted K-theory and twisted homology for compact six-manifolds, enabling easier calculations and broader classification of branes.

Findings

Twisted homology can be twisted by classes of any degree.

Provides a spectral sequence for calculating twisted homology.

Shows that brane decay transitions depend on cycle triviality.

Abstract

D-branes are classified by twisted K-theory. Yet twisted K-theory is often hard to calculate. We argue that, in the case of a compactification on a simply-connected six manifold, twisted K-theory is isomorphic to a much simpler object, twisted homology. Unlike K-theory, homology can be twisted by a class of any degree and so it classifies not only D-branes but also M-branes. Twisted homology classes correspond to cycles in a certain bundle over spacetime, and branes may decay via Kachru-Pearson-Verlinde transitions only if this cycle is trivial. We provide a spectral sequence which calculates twisted homology, the kth step treats D(p-2k)-branes ending on Dp-branes.