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-Fré’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) mathcal{N} = (1, 0) 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 Hořava-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.
Highlights
By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within superhomotopy theory, we present an elegant joint solution to all three problems
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 superembedding formalism, that a Perry-Schwarz-type Lagrangian for single N = (1, 0) 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
Summary
For ease of reference and in order to introduce notation needed in later sections, we review here the bosonic part of the Perry-Schwarz-Lagrangian from [90], re-cast in coordinateindependent Cartan calculus and generalized to possibly non-trivial circle fibrations. Given an S1-bundle structure (2.4) on the worldvolume Σ6 and any choice of Ehresmann connection (2.7), the self-duality condition (2.2) is equivalently expressed in terms of the compactified fields (2.13) as:. If the worldvolume Σ6 is equipped with an S1-principal bundle structure (2.4) which is flat (2.10), the self-duality condition H = ∗H from (2.2), in its equivalent incarnation on compactified fields (2.20) implies the following differential equation: θ5 ∧ d H − Lv5 Bhor = 0. Due to proposition 2.8, one may regard the general bosonic Perry-Schwarz Lagrangian (definition 2.5) for the self-dual field on Σ6 compactified on S1 as that of (abelian) 5d Yang-Mills theory with a tower of KK-modes included, which is a perspective on the M5-brane theory later advanced in [35, 76] (see [77, 3.4.3])). We find this arise from the super-exceptional embedding construction below in theorem 7.3, see remark 7.5
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