Twisted Homology

TL;DR

The paper argues that on a simply-connected six-manifold, twisted K-theory, which classifies D-branes, is isomorphic to twisted homology, a more computable and degree-flexible framework that also extends to M-branes. It introduces a geometric realization of twisted homology via cycles in a spacetime bundle $Q$ (and a simpler subbundle $P$ when available), and constructs a spectral sequence with differentials $d_{2i+1}$ that converges to twisted homology, capturing multistep brane decays such as MMS/KPV transitions. The authors provide a concrete cohomological computation using the Gysin sequence, giving an explicit formula for $H$-cohomology in terms of the total space $P$, and illustrate with examples: the SU(2) WZW model on $S^3$ and a lens space, showing how charges and anomalies align with twisted homology predictions. This framework yields a practical, bijective bridge between brane configurations, anomalies, and conserved charges, facilitating background-brane analyses in string phenomenology and extending naturally to M-theory contexts.

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.

Figures (3)

• Figure 1: The action of $\partial_H$ on a $k$-chain $\sigma$ is sketched. The twist $H$ is depicted using its dual (dim$(M)$-3)-cycle, which is the cap product of $H$ with the fundamental class of the spacetime $M$. The intersection of this cycle with the chain $\sigma$ is the $(k-3)$-cycle $H\cap\sigma$. $\partial_H\sigma$ is the sum of $H\cap\sigma$ and the boundary of $\sigma$, and is represented by everything red.
• Figure 2: A D$p$-brane wraps the cycle $C_p$. The pullback of the $H$ flux to $C_p$ is nontrivial in its worldvolume, although $H\cap C_p$ is a boundary in spacetime. Therefore the D$p$-brane worldvolume gauge theory has a worldvolume monopole on the cycle $H\cap C_p$. A D$(p-2)$-brane ends on the monopole, it wraps the chain $C_{p-2}$ which itself supports a nontrivial $H$ flux. Therefore the D$(p-2)$ also has a monopole, actually two in this picture, on which D$(p-4)$-branes end.
• Figure 3: Upon excising $B_3$, the total space of the remaining bundle over the D$3$-brane is $P-\pi^{-1}(B_3)$, with boundary $\pi^{-1}(S^2_a)$. The bundle over the 'thin' D$3$ on the tube has total space $\tilde{P}$. A composite bundle is constructed by gluing the respective preimages $\pi^{-1}(S^2_a)$ and $\partial \tilde{P}$ over $S^2_a$.