Hyperbolic Superspaces and Super-Riemann Surfaces

TL;DR

This paper extends hyperbolic geometry concepts to supergeometry, relating super-Green functions on supermanifolds to supergeodesics in hyperbolic superspaces, thus broadening the mathematical framework of super-Riemann surfaces.

Contribution

It generalizes results from hyperbolic geometry to supergeometric settings, connecting super-Green functions with supergeodesics in hyperbolic superspaces.

Findings

Reexpression of super-Green functions using supergeodesics

Extension of hyperbolic geometry results to supergeometry

New insights into super-Riemann surfaces and superspaces

Abstract

In this paper, we will generalize some results in Manin's paper "Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry" to the supergeometric setting. More precisely, viewing $\mathbb{C}^{1|1}$ as the boundary of the hyperbolic superspace $\mathcal{H}^{3|2}$, we reexpress the super-Green functions on the supersphere $\hat{\mathbb{C}}^{1|1}$ and the supertorus $T^{1|1}$ by some data derived from the supergeodesics in $\mathcal{H}^{3|2}$.