# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1702.08108