BV construction of SUSY vertex algebras from SUSY factorization algebras

Abstract

We construct $N=1$ supersymmetric (SUSY) vertex algebras from supersymmetric enhancements of Costello--Gwilliam factorization algebras on super Riemann surfaces. Introducing SUSY factorization algebras defined on embedded SUSY disks together with natural symmetry conditions, we prove a SUSY analogue of the Costello--Gwilliam extraction theorem. As an application, we study the holomorphic sigma model in the BV formalism. For a linear target, we obtain the free $bc$-$\beta\gamma$ system and recover its structure as a SUSY vertex algebra. For general complex targets, we describe the descent of the theory under coordinate changes and identify the resulting SUSY vertex algebra with the chiral de Rham complex. We further show that Ricci-flat K\"ahler and hyperk\"ahler targets give rise to $N=2$ and $N=4$ supersymmetric enhancements introduced by Ben-Zvi--Heluani--Szczesny.