On a Grassmann odd analogue of Carrollian Manifolds
Andrew James Bruce
TL;DR
This work defines a Grassmann odd analogue of Carrollian manifolds by introducing super-null Riemannian manifolds: supermanifolds of dimension $n|1$ with an even degenerate metric whose kernel is generated by the odd supersymmetry generator $Q$ with $[Q,Q]=2P$. It develops the theory of degenerate metrics on supermanifolds, the reduced pseudo-Riemannian structure on $M_{red}$, and two notions of affine-connection compatibility (supersymmetry and metric compatibility); due to the non-integrable kernel, any such compatible connection must carry torsion. An Inönü–Wigner contraction of the ${ m R}^{4|4}$ supertranslation algebra provides a concrete example, with Shander coordinates simplifying local expressions for $Q$ and $P$. The super-null Lie algebra of infinitesimal automorphisms is shown to be finite-dimensional and closely tied to the Killing fields of the reduced metric, highlighting a sharp contrast with Carrollian geometries. The results lay groundwork for further mathematical and physical applications, including supersymmetric dynamics with evolving external parameters in a superspace setting.
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 Inönü--Wigner contraction of the supertranslation algebra on standard superspace $\mathbb{R}^{4|4}$.