The Star Product on the Fuzzy Supersphere

TL;DR

This paper constructs a star-product on the fuzzy supersphere using coherent states, explores its classical limit, and extends it to bundle sections, providing insights into the geometric structure of supersymmetric noncommutative spaces.

Contribution

It introduces a new star-product for the fuzzy supersphere that preserves supersymmetry and extends to bundle sections, enriching the mathematical framework of fuzzy supersymmetric geometries.

Findings

Star-product constructed via OSp(2,1) coherent states

Classical limit matches the supersphere's commutative structure

Extended to sections of bundles on the fuzzy supersphere

Abstract

The fuzzy supersphere $S_F^{(2,2)}$ is a finite-dimensional matrix approximation to the supersphere $S^{(2,2)}$ incorporating supersymmetry exactly. Here the star-product of functions on $S_F^{(2,2)}$ is obtained by utilizing the OSp(2,1) coherent states. We check its graded commutative limit to $S^{(2,2)}$ and extend it to fuzzy versions of sections of bundles using the methods of [1]. A brief discussion of the geometric structure of our star-product completes our work.