A cohomology theory of supercommutative algebras and grading-restricted vertex superalgebras

TL;DR

This paper develops a cohomology theory for grading-restricted vertex superalgebras, extending previous theories and including foundational work on supercommutative associative algebras, to support future deformation studies.

Contribution

It introduces a new cohomology framework for vertex superalgebras and supercommutative associative algebras, generalizing existing theories and simplifying complex constructions.

Findings

Constructed cohomology theory for grading-restricted vertex superalgebras.

Included cohomology construction for supercommutative associative algebras.

Laid groundwork for deformation theory of vertex superalgebras.

Abstract

This paper constructs the cohomology theory for grading-restricted vertex superalgebras, generalizing Yi-Zhi Huang's cohomology theory of grading-restricted vertex algebras. To simplify the discussion, motivate the construction, and make it easier for the reader to understand the technical points, we also include the construction of the cohomology theory of supercommutative associative algebras, a generalization of the Harrison cohomology theory of a commutative algebra that has not been explicitly written down. The paper will serve as the foundation for many subsequent studies, especially, the deformation theory of vertex superalgebras.