Gauge potentials on the M5 brane in twisted equivariant cohomotopy
Pinak Banerjee
TL;DR
The work confronts the challenge of globally defining C-field flux and M5-brane worldvolume flux in M-theory on curved and orbifold backgrounds by positing that $G_{4}$ flux quantization is governed by 4-cohomotopy $\pi^{4}(X)$ and the worldvolume $H_{3}$ flux by twisted 3-cohomotopy, with a differential refinement to recover gauge potentials. It develops explicit constructions showing that null-concordances map to the usual gauge potentials $C_{3},C_{6},B_{2}$ and that concordances encode gauge transformations, first in tangentially twisted cohomotopy, then in twistorial cohomotopy, and finally in equivariant twistorial cohomotopy for orbifolds. Throughout, the modified Bianchi identities induced by gravity and orbifold twists (e.g., $\mathrm{d}(2\mathrm{CS}(\omega))=p_{1}(\omega)$ and $\mathrm{d}H_{3}=\tilde{G}_{4}-\frac{1}{2}p_{1}(\omega)$, with analogous twistorial/equivariant versions) are checked for consistency with the surjections. The results show that familiar local M5-brane formulas arise as global consequences of a cohomotopical flux quantization framework, enabling global completion of higher gauge data in twisted and equivariant settings with potential implications for heterotic M‑theory and orbifold compactifications.
Abstract
In this article, we work out some variations on the discussion of the C-field flux densities in the Sati-Schreiber program. We start by explaining the need for global completion of the field content: the fluxes, the gauge potentials and the gauge transformations on the worldvolume of a single M5 brane in eleven-dimensional supergravity, and how this is encoded by the choice of flux quantization law. Assuming Hypothesis H that the 4-flux in M-Theory is flux quantized in a non-abelian cohomology theory called 4-cohomotopy, and the three-flux on the M5 brane worldvolume in (twisted) 3-cohomotopy, we generalize some previous calculations known in the literature to include twisting by background gravity and placing M5 branes on orbifolds. We show that the null concordances of cohomotopically charged fluxes give rise to the traditional gauge potentials and the null concordances of concordances give rise to the corresponding gauge transformations via surjections, in the cases of tangentially twisted cohomotopy, twistorial cohomotopy and equivariant twistorial cohomotopy. We construct the surjections explicitly for these cases and check the consistency relations with the corresponding Bianchi identities. Thus, we show how the traditional formulas for the local gauge potentials on M5 brane worldvolume on curved spacetimes and orbifolds are indeed reproduced by the homotopy theory and, as such, become amenable to global completion in cohomotopical charge quantization.