Beta super-functions on super-Grassmannians

TL;DR

This paper introduces a super-analogue of the Euler beta function by incorporating odd variables, extending Gelfand's geometric interpretation of hypergeometric functions on super-Grassmannians.

Contribution

It constructs a novel beta super-function with added odd variables and relates it to gamma and hypergeometric functions, broadening the geometric framework.

Findings

Defined a super-analogue of the beta function

Connected the super-beta function to gamma and hypergeometric functions

Extended Gelfand's geometric interpretation to super-Grassmannians

Abstract

Israel M. Gelfand gave a geometric interpretation for general hypergeometric functions as sections of the tautological bundle over a complex Grassmannian $G_{k,n}$. In particular, the beta function can be understood in terms of $G_{2,3}$. In this manuscript, we construct one of the simplest generalizations of the Euler beta function by adding arbitrary-many odd variables to the classical setting. We also relate the beta super-function to the gamma and the hypergeometric function.