$N_K=1$ SUSY structure of chiral de Rham complex from the factorization structure

TL;DR

This paper explains how the factorization structure of the formal superloop space of a smooth algebraic variety induces an $N_K=1$ SUSY vertex algebra structure on the chiral de Rham complex, clarifying a remark from prior work.

Contribution

It provides a detailed explanation of the connection between superloop space factorization and SUSY vertex algebra structures in the chiral de Rham complex.

Findings

Establishes the link between superloop space factorization and SUSY vertex algebra.

Clarifies the remark in Kapranov-Vasserot (2011).

Shows the induced $N_K=1$ SUSY structure on the chiral de Rham complex.

Abstract

We elucidate the comment in (Kapranov-Vasserot, Adv.\ Math., 2011, Remark 5.3.4) that the $1|1$-dimensional factorization structure of the formal superloop space of a smooth algebraic variety $X$ induces the $N_K=1$ SUSY vertex algebra structure of the chiral de Rham complex of $X$.