Li filtrations of SUSY vertex algebras

TL;DR

This paper introduces a supersymmetric analogue of Li filtration for SUSY vertex algebras, demonstrating that its associated graded space forms a SUSY vertex Poisson algebra, and explores related algebraic structures.

Contribution

It extends the Li filtration concept to SUSY vertex algebras and establishes their associated graded spaces as SUSY vertex Poisson algebras, also discussing related algebraic notions.

Findings

Established SUSY Li filtration and its properties.

Proved the associated graded space forms a SUSY vertex Poisson algebra.

Discussed related structures like Zhu's $C_2$-Poisson superalgebras.

Abstract

Any vertex algebra has a canonical decreasing filtration, called Li filtration, whose associated graded space has a natural structure of a vertex Poisson algebra. In this note, we introduce an analogous filtration for any SUSY vertex algebra, which was introduced by Heluani and Kac as a superfield formalism of a supersymmetric vertex algebra. We prove that the associated graded superspace of our filtration has a structure of SUSY vertex Poisson algebras. We also introduce and discuss related notions, such as Zhu's $C_2$-Poisson superalgebras, associated superschemes and singular supports, for SUSY vertex algebras.