The Fuzzy Supersphere

TL;DR

This paper constructs a fuzzy supersphere as a sequence of finite-dimensional noncommutative algebras, approximating the algebra of functions on a supersphere, and explores its geometric and cohomological properties.

Contribution

It introduces the fuzzy supersphere as a noncommutative algebraic approximation of the supersphere, including analogues of geometric maps and cohomology.

Findings

Fuzzy supersphere converges to the supersphere algebra in a suitable limit.

Super-deRham cohomology matches classical deRham cohomology at the fuzzy level.

Abstract

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative $Z_{2}$-graded algebras tending in a suitable limit to a dense subalgebra of the $Z_{2}$-graded algebra of ${\cal H}^{\infty}$-functions on the $(2| 2)$-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level".