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

## TL;DR

This paper introduces super-exceptional geometry, a novel framework combining homotopy theory and supergeometry, to address key open problems in M-theory, including M5-brane dynamics, gauge fields in heterotic M-theory, and supersymmetric M-geometry enhancement.

## Contribution

It provides a unified geometric approach to derive M5-brane Lagrangians, explain the origin of gauge fields, and enhance supersymmetry in M-theory using super-exceptional geometry.

## Key findings

- Derived the M5-brane Lagrangian as a super-exceptional trivialization.
- Connected charge quantization to the super-exceptional target space.
- Identified the extra fermion as the M-theoretic gaugino.

## Abstract

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals D'Auria-Fre's "hidden supergroup" of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the "half" M5-brane locus, super-exceptionally compactified on the Horava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field.