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

Paul Johnson, Fei Qi

TL;DR

This work generalizes cohomology theory from vertex algebras to grading-restricted vertex superalgebras by developing a Harrison-like cochain framework for supercommutative algebras and extending it to the vertex setting with $ar{W}$-valued rational functions. It defines a robust cochain complex incorporating the $ ext{D}$-derivative property, $ ext{d}$-conjugation, a composable condition, and a shuffle constraint to yield well-defined coboundaries. The main results identify $H^1$ with derivations and $H^2$ with square-zero extensions and first-order deformations, establishing a concrete deformation theory for vertex superalgebras and laying the foundation for future studies, including strongly generated W-algebras in nonsemisimple contexts.

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.

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

TL;DR

This work generalizes cohomology theory from vertex algebras to grading-restricted vertex superalgebras by developing a Harrison-like cochain framework for supercommutative algebras and extending it to the vertex setting with -valued rational functions. It defines a robust cochain complex incorporating the -derivative property, -conjugation, a composable condition, and a shuffle constraint to yield well-defined coboundaries. The main results identify with derivations and with square-zero extensions and first-order deformations, establishing a concrete deformation theory for vertex superalgebras and laying the foundation for future studies, including strongly generated W-algebras in nonsemisimple contexts.

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.
Paper Structure (17 sections, 11 theorems, 141 equations)

This paper contains 17 sections, 11 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Cohomology theory for supercommutative algebras
  3. Superalgebra and supermodules
  4. Shuffles and the super shuffle sum
  5. The super Harrison cochain
  6. Cohomology theory for vertex superalgebras
  7. Vertex superalgebras and their modules
  8. $\overline{W}$-valued rational functions
  9. Cochain construction I: $\mathcal{D}$-derivative property and $\mathbf{d}$-conjugation property
  10. Cochain construction II: composable condition
  11. Cochain construction III: shuffle condition
  12. Coface maps
  13. Coboundary map
  14. First and second cohomologies and first order deformations
  15. First cohomology and derivations
  16. ...and 2 more sections

Key Result

Theorem 2.6

For all $n\in\mathbb{N}$, $\partial^2 \widehat{C}^n=0$ (or simply $\partial^2=0$). Therefore, the collection $(\widehat{C}^n,\partial)_{n\in \mathbb{N}}$ forms a cochain complex, called the Hochschild cochain complex.

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 48 more