Twisted equivariant quasi-elliptic cohomology and M-brane charge
Zhen Huan
TL;DR
This work defines twisted quasi-elliptic cohomology $QEll^{\alpha+\ast}_G(-)$ by twisting the loop-space/ orbifold K-theory framework with $\alpha\in H^3(BG;U(1))$, and develops its loop-space and Chern-character machinery. It connects to twisted Devoto elliptic cohomology and twisted Tate K-theory, yielding a computable invariant that, for representation 4-spheres under finite SU(2) subgroups, aligns with conjectured observables for M-brane charges in a Tate-elliptic setting. The paper proves structural properties (restriction, change-of-group, induction) and provides explicit calculations for $S^1$ and $S^4$ under various finite subgroups, illustrating both the tractability and physical relevance of twisted $QEll$. A key structural point is that for all finite subgroups of SU(2), the twisting class vanishes in cohomology, so twisted and untwisted theories coincide on these spaces, supporting the proposed links to M-brane charge and higher elliptic phenomena. Overall, the framework offers a concrete, computable bridge between higher elliptic cohomology, loop-groupoid models, and mathematical physics applications in string/M-theory contexts.
Abstract
In this paper we construct a twisted version of quasi-elliptic cohomology. This theory can be constructed as a K-theory of a loop space. After establishing basic properties of the theory, including restriction, change-of-group and induction maps, we construct the Chern character map. And we compute the twisted quasi-elliptic cohomology theories of representation 4-spheres acted by the finite subgroups of SU(2), which, as conjectured by Sati and Schreiber, can produce good observables on M-brane charge in a Tate-elliptic enhancement of D-brane charge in twisted equivariant K-theory.