N=1 Neveu-Schwarz vertex operator superalgebras over Grassmann algebras and with odd formal variables

TL;DR

This paper develops the theory of N=1 Neveu-Schwarz vertex operator superalgebras over Grassmann algebras with odd formal variables, establishing their properties and categorical equivalences.

Contribution

It introduces and compares superalgebras with and without odd formal variables, proving key properties and their equivalence to the Jacobi identity.

Findings

Categories of such superalgebras are isomorphic

Weak supercommutativity and associativity are established

Supercommutativity and associativity are shown to be equivalent to the Jacobi identity

Abstract

The notions of N=1 Neveu-Schwarz vertex operator superalgebra over a Grassmann algebra and with odd formal variables and of N=1 Neveu-Schwarz vertex operator superalgebra over a Grassmann algebra and without odd formal variables are introduced, and we show that the respective categories of such objects are isomorphic. The weak supercommutativity and weak associativity properties for an N=1 Neveu-Schwarz vertex operator superalgebra with odd formal variables are established, and we show that in the presence of the other axioms, weak supercommutativity and weak associativity are equivalent to the Jacobi identity. In addition, we prove the supercommutativity and associativity properties for an N=1 Neveu-Schwarz vertex operator superalgebra with odd formal variables and show that in the presence of the other axioms, supercommutativity and associativity are equivalent to the Jacobi identity.