SIGMA 22 (2026), 103, 36 pages      arXiv:2510.17293      https://doi.org/10.3842/SIGMA.2026.103
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

A Cohomology Theory of Supercommutative Algebras and Grading-Restricted Vertex Superalgebras

Paul Johnson ^a and Fei Qi ^b
a) Department of Mathematics, University of Denver, Denver, CO 80210, USA
^b) School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, P.R. China

Received October 25, 2025, in final form October 15, 2026; Published online October 29, 2026

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.

Key words: vertex operator superalgebras; cohomology theory; square-zero extensions; first-order deformations.

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