The super W_{1+\infty} algebra with integral central charge

TL;DR

This paper studies the super W_{1+ abla} algebra derived from the super circle's differential operators, revealing its structure at integral central charges, its minimal generating sets, and its relation to W-algebras and commutant vertex algebras.

Contribution

It identifies the minimal strong generating sets for super W_{1+ abla} algebras at integer central charges and connects them to W-algebras associated with gl(n|n).

Findings

V_n( ext{hat{SD}}) has a minimal strong generating set of 4n fields.

V_n( ext{hat{SD}}) is isomorphic to a W-algebra from a purely odd root system.

V_n( ext{hat{SD}}) can be realized as a limit of commutant vertex algebras.

Abstract

The Lie superalgebra SD of regular differential operators on the super circle has a universal central extension \hat{SD}. For each c\in C, the vacuum module M_c(\hat{SD}) of central charge c admits a vertex superalgebra structure, and M_c(\hat{SD}) \cong M_{-c}(\hat{SD}). The irreducible quotient V_c(\hat{SD}) of the vacuum module is known as the super W_{1+\infty} algebra. We show that for each integer n>0, V_n(\hat{SD}) has a minimal strong generating set consisting of 4n fields, and we identify it with a W-algebra associated to the purely odd simple root system of gl(n|n). Finally, we realize V_n(\hat{SD}) as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.