Trace of the twisted Heisenberg Category

TL;DR

This paper establishes an isomorphism between the trace decategorification of the twisted Heisenberg category and a quotient of a subalgebra of $W_{1+ abla}$, extending previous work to a twisted setting.

Contribution

It introduces a twisted analogue of the relation between $W_{1+ abla}$ and the trace decategorification of the Heisenberg category.

Findings

Proves the isomorphism between the trace of the twisted Heisenberg category and a quotient of $W^-$.

Extends known relations to a twisted categorical setting.

Provides algebraic structure insights into the twisted Heisenberg category.

Abstract

We show that the trace decategorification, or zeroth Hochschild homology, of the twisted Heisenberg category defined by Cautis and Sussan is isomorphic to a quotient of $W^-$, a subalgebra of $W_{1+\infty}$ defined by Kac, Wang, and Yan. Our result is a twisted analogue of that by Cautis, Lauda, Licata, and Sussan relating $W_{1+\infty}$ and the trace decategorification of the Heisenberg category.