Schur-Weyl Duality for Heisenberg Cosets

TL;DR

This paper establishes a Schur-Weyl duality framework for Heisenberg cosets within vertex operator algebras, demonstrating how modules relate and extending understanding of their structure and rationality.

Contribution

It proves a Schur-Weyl duality for modules over Heisenberg cosets, introduces new vertex algebra extensions, and links rationality properties of the involved algebras.

Findings

Duality holds for simple and indecomposable modules

Every simple coset module appears in some V-module

Under certain conditions, the coset C is rational and C2-cofinite

Abstract

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of ${C}$ are found and every simple ${C}$-module is shown to be contained in at least one ${V}$-module. A corollary of this is that if ${V}$ is rational and $C_2$-cofinite and CFT-type, and $\text{Com}\left( {C}, {V}\right)$ is a rational lattice vertex operator algebra, then so is ${C}$. These results are illustrated with many examples and the $C_1$-cofiniteness of certain interesting classes of modules is established.