Real ADE-equivariant (co)homotopy and Super M-branes

TL;DR

This paper proposes real equivariant cohomotopy as a rational framework to identify hidden degrees of freedom at ADE-singularities in M-theory, linking super-cocycles, black branes, and equivariant homotopy theory.

Contribution

It introduces a novel approach using equivariant cohomotopy to classify M-brane super-cocycles and analyze ADE-singularities within a homotopical and super-geometric context.

Findings

Revealed ADE-singularities as data of equivariant 4-sphere super-cocycles.

Classified super-cocycles leading to an enhanced brane scan with intersecting black M-branes.

Connected brane instanton actions to homotopy data in the orbit category.

Abstract

A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show that real equivariant cohomotopy on superspaces is a consistent such choice, at least rationally. After explaining this new approach, we demonstrate how to use Elmendorf's theorem in equivariant homotopy theory to reveal ADE-singularities as part of the data of equivariant 4-sphere-valued super-cocycles on 11d super-spacetime. We classify these super-cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super-cocycles on strata of intersecting black M-brane species. We find that on each singular stratum the black brane's instanton contribution, namely its super Nambu-Goto/Green-Schwarz action, appears as the homotopy datum associated to the morphisms in the orbit category.