Linearized Off-shell 4+7 Supergeometry of 11D Supergravity

TL;DR

This paper develops a linearized supergeometry framework for 11D supergravity with off-shell supersymmetries, using a $G_2$-structure and superfield prepotentials to describe geometric fluctuations.

Contribution

It introduces a novel linearized supergeometry of 11D supergravity incorporating a $G_2$-structure and provides both bottom-up and top-down formulations using superfield prepotentials.

Findings

Constructed supervielbein, spin connection, and super 3-form in terms of 4D, N=1 prepotentials.

Derived torsion and Bianchi identities systematically.

Identified torsion constraints leading to gauge field expressions.

Abstract

We describe the linearized supergeometry of eleven dimensional supergravity with four off-shell local supersymmetries. We start with a background Minkowski 11D, N=1 superspace, and an additional ingredient of a global, constant, $G_2$-structure which facilitates the definition of a $4|4+7$ background superspace. A bottom-up construction of linear fluctuations of the geometric constituents (such as supervielbein, spin connection, and the super 3-form of 11D supergravity) is given in terms of 4D, N=1 prepotential superfields. This is complemented by a top-down description of the linearized supergeometry of the $4|4+7$ superspace dealing directly with torsion, curvature, and Bianchi identities. Torsion constraints that (combined with the Bianchi identities) lead to the preceding prepotential expressions of the gauge fields are identified. All irreducible consequences of the torsion and 4-form Bianchi identities are systematically derived except for dimension 2 Bianchi identities of the 4-form, and dimension $\tfrac 52$ Bianchi identities of torsion, which set bosonic curls of components of one lower dimension to zero.