Gelfand-Fuchs cohomology for affine superspaces A^(n,1)

TL;DR

This paper computes the Gelfand-Fuchs cohomology for the Lie superalgebra of vector fields on affine superspaces of the form A^{m,1}, extending previous results to this specific case.

Contribution

It provides the first explicit cohomology computation for vector fields on affine superspaces A^{m,1}, filling a gap in the existing literature.

Findings

Computed Gelfand-Fuchs cohomology for A^{m,1}

Extended known results from classical and special super-space cases

Provides new insights into the structure of vector field cohomology on superspaces

Abstract

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces $\mathbb{A}^{m,n}$ with trivial coefficients. Previously known results covered the classical case $\mathbb{A}^m = \mathbb{A}^{m,0}$ as well as the case $\mathbb{A}^{m,n}$ with $m \leq n$. In this paper we perform the computation for super-spaces $\mathbb{A}^{m,1}$.