Duality and Galois correspondence for vertex superalgebras

Abstract

Let V be a simple vertex superalgebra of countable dimension, G a finite automorphism group of V and T a positive integer. Let $S$ be a finite G-stable set of inequivalent irreducible $(V,T)$-modules. Then there is a finite dimensional semisimple associative algebra $A_{\alpha }(G,S)$ for a suitable 2-cocycle α naturally determined by the G-action on $S$ such that the actions of $A_{\alpha }(G,S)$ and $V^G$ on the direct sum of $(V,T)$-modules in $S$ form a Schur-Weyl type duality. As applications, we establish the following results: (1) Every irreducible g-twisted V-module is a completely reducible $V^G$-module for arbitrary $g\in G$ and irreducible $V^G$-modules appearing in different G-orbits are inequivalent; (2) The quantum Galois correspondence theorem in the context of vertex superalgebras is proved.