Twisted K-Theory from Monodromies

TL;DR

The paper analyzes how RR flux configurations may be identified by twisted K-theory despite differing in cohomology via monodromies from mortal branes and, in the presence of NS5-branes, further identifications. It demonstrates via MMS-like examples that certain fluxes, such as a Romans mass $G_0$ arising after brane decay, are not captured by twisted K-theory when $dG$ has nontrivial compactly supported cohomology, and shows monodromies shift RR potentials by integers. An $E_8$ bundle formalism ties these classifications to a unified topological framework, with the M-theory picture producing a correspondence between M2-brane charge and $\pi_3(E_8)$, and a loop-group extension $LE_8$ yielding the type II connection. The results indicate twisted K-theory is an approximate, not complete, classifier of fluxes and highlight future work on S-duality covariant AHSS and NS5-monodromies to extend the framework.

Abstract

RR fluxes representing different cohomology classes may correspond to the same twisted K-theory class. We argue that such fluxes are related by monodromies, generalizing and sometimes T-dual to the familiar monodromies of a D7-brane. A generalized theta angle is also transformed, but changes by a multiple of 2pi. As an application, NS5-brane monodromies modify the twisted K-theory classification of fluxes. Furthermore, in the noncompact case K-theory does not distinguish flux configurations in which dG is nontrivial in compactly supported cohomology. Such fluxes are realized as the decay products of unstable D-branes that wrapped nontrivial cycles. This is interpreted using the E8 bundle formalism.

Figures (6)

• Figure 1: In a spacetime with 3 units of $H$ flux, stacks of 3 D3-branes are unstable. D3-brane charge is measured by Gauss' Law, that is by the integral of $dC_4$ over a linking 5-cycle $\partial U$. By Stoke's Theorem this is an integral over the region $U$, but the two different choices of $U$ drawn above yield answers which differ by $3$. This ill-definedness in D3-brane charge is a consequence of the instability of stacks of 3 D3-branes.
• Figure 2: Consider massless IIA SUGRA on a spacetime with 4 units of $H$ flux, stacks of 4 D6-branes are unstable. The decay occurs via a mortal D8-brane. As the D8-brane sweeps out space, it leaves behind it massive IIA with a Romans mass $G_0=1$. This does not correspond to a twisted K-theory class because $H\wedge G_0\neq 0$.
• Figure 3: Consider massless IIA SUGRA on a spacetime with 3 units of $H$ flux on a 3-cycle, stacks of 3 D0-branes are unstable. The decay occurs via a mortal D2-brane. As the D2-brane sweeps out the 3-cycle, it leaves behind it one unit of $G_6$ flux. The wedge product $G_6\wedge H=k$ is nontorsion and so again the final configuration does not correspond to any twisted K-theory class.
• Figure 4: An M5-brane wraps a trivial 3-cycle $S^3_D$. This leads to a nontrivial $E_8$ bundle over every linking $S^4$, characterized by a transition function $S^3_E\longrightarrow E_8$. This transition function represents a nontrivial class in $\pi_3(E_8)$. A choice of $x\in S^3_E$ in each $S^4$ is mapped to some basepoint in $E_8$ for each $S^4$. $S^4$'s are labeled by their centers $y\in S^3_D$, and thus for each $x$ there is a map $S^3_D\longrightarrow E_8$. $E_8$ approximates the classifying space $K(\mathbb Z,3)$ and therefore the homotopy class of the map determines an element of the 3rd cohomology group of the M5-brane's worldvolume. This element is the worldvolume 3-form fieldstrength $T_3$ and so the homotopy class of this map equals the M2-brane charge of the dielectric M5-brane.
• Figure 5: An NS5-brane sweeps out a 5-cycle which supports 3 units of $G_5$ flux. When it decays, it leaves 3 D-strings. Later these D-strings decay via the same process by which they were created. The initial and final states are included in the twisted K-theory classification. However the monodromy about the NS5-brane relates distinct cohomology classes which do not represent the same K-theory class. This is one way in which the K-theory classification is modified by the inclusion of processes involving NS5-branes.