On a Grassmann odd analogue of Carrollian Manifolds

TL;DR

This paper introduces a novel supermanifold structure called a Grassmann odd analogue of Carrollian manifolds, exploring its geometric properties and physical relevance through algebraic contractions.

Contribution

It defines the Grassmann odd Carrollian manifold, proves the existence of compatible affine connections with torsion, and relates the structure to superspace algebra contractions.

Findings

Reduced manifold is pseudo-Riemannian

Existence of affine connections with torsion

Application to supertranslation algebra contraction

Abstract

We define a Grassmann odd analogue of a Carrollian manifold as a supermanifold of dimension $n|1$ with an even degenerate metric such that the kernel is generated by a non-singular odd vector field that is a supersymmetry generator. Alongside other results, we establish that the reduced manifold is a pseudo-Riemannian manifold, and show that compatible affine connections always exist, albeit they must carry torsion. As a physically relevant example, we examine an In\"on\"u--Wigner contraction of the supertranslation algebra on standard superspace $\mathbb{R}^{4|4}$.