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

Abstract

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces ${\mathbb{A}}^{m,n}$ with trivial coefficients. In this paper we extend the methodology developed in the previous paper (arXiv:2210.16585) to perform the calculation for arbitrary dimensions $(m,n)$ with $m\ge n$. This combined with previously known results for other dimensions $(m,n)$ fully settles the local problem of Gelfand-Fuchs cohomology with trivial coefficients for super-manifolds. As part of the proof we also compute the cohomology of Lie superalgebras $\mathfrak{gl}(m,n)$ with coefficients given by Schur functors of the standard representation.

17 pages