Trace of 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.

To appear in Communications in Math. Physics, 36 pages, tikz diagrams