The Rational Higher Structure of M‐theory

Abstract

We review how core structures of string/M‐theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion‐subgroups in brane charges and focuses on tangent spaces of super space‐time. Already at this level, super homotopy theory discovers all super p‐brane species, their intersection laws, their M/IIA‐, T‐ and S‐duality relations, their black brane avatars at ADE‐singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D‐branes it recovers twisted K‐theory, rationally, but for the M‐branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M‐theory in super homotopy theory.

Higher Structures in M‐Theory

“…The category of real differential graded commutative superalgebras carries a projective model category structure with path space objects given by tensor product of algebras with , see [] and [, Theorems 1 & 6.5]. As pointed out in references [], this provides the homotopy theory of what in the supergravity literature are called FDA s or CIS s: local models for higher dimensional supergravity on superspace, see also [].…”

Parametrised Homotopy Theory and Gauge Enhancement

“…According to this picture, under double dimensional reduction M‐theory readily produces the fundamental string F1, the NS5‐brane and the D2 and D4‐branes of Type IIA string theory (whereas the D0‐brane is also obtained as the Chern class which classifies the 11‐dimensional background as an S 1 ‐fibration over the 10‐dimensional background). We emphasise that this schematic is a heuristic only; although there are many hints for this double dimensional reduction procedure in the literature, there is no derivation beyond the rational approximation …”

Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5

“…2]). We had initiated in [FSS13b] a program of attacking this problem, based on universal constructions in super-homotopy theory (see [FSS19a] for review), and used this to find firstpriniciples derivations of various aspects expected of M-theory (see [FSS16b][FSS19b] [FSS19c]). In this paper we look to carry this further and consider the following three major sub-problems: 1.…”

“…• This process culminates [FSS19a,p. 12] in the D = 11, N = 1 (hence N = 32) super-Minkowski spacetime, carrying the super M2-brane cocycle µ M2 and, on the corresponding higher extension, the super M5-brane cocycle [FSS13b] which is the curvature of the M5 Wess-Zumino (WZ) term [BLNPST97,(8)] [FSS15] (see [FSS19a] for review):…”

“…6.6) how ΩS 2 HW S 1 HW refines the naive circle action by taking the super-cocycle for the little-string in 6d into account. This is a 6d analog to capturing the form fields in 11d M-theory via the Cohomotopical 4-sphere coefficient [Sat13][FSS15], leading to a description of type IIA in ten dimensions using a refined variant of the loop space of S 4 , namely the cyclic loop space [FSS16a][FSS16b] (see [FSS19a] for overview).…”