Algebraic operad of SUSY vertex algebra

TL;DR

This paper introduces algebraic operads that encode the structures of SUSY vertex algebras with N_W=N and N_K=N supersymmetry, extending the algebraic framework of chiral operads to supersymmetric contexts.

Contribution

It defines new algebraic operads for SUSY vertex algebras and studies their cohomology, providing a SUSY analogue of existing chiral operads.

Findings

Defined operads $\\mathcal{P}^{\text{ch}N_W=N}$ and $\mathcal{P}^{\text{ch}N_K=N}$ for SUSY vertex algebras.

Studied the cohomology theory associated with these operads.

Extended the algebraic translation of chiral operads to supersymmetric cases.

Abstract

We introduce algebraic operads $\mathcal{P}^{\text{ch}N_W=N}$ and $\mathcal{P}^{\text{ch}N_K=N}$ encoding the structures of $N_W=N$ and $N_K=N$ SUSY vertex algebras, and study the corresponding cohomology theory. Our operad is a SUSY analogue of the operad $P^{\text{ch}}$ introduced by Bakalov, De Sole, Heluani and Kac (2019) as a purely algebraic translation of the chiral operad of Beilinson and Drinfeld.