Orbifold theory for vertex algebras and Galois correspondence

Abstract

Let $V$ be a simple vertex algebra of countable dimension, $G$ be a finite automorphism group of $V$ and $\sigma$ be a central element of $G$. Assume that $\mathcal{S}$ is a finite set of inequivalent irreducible $\sigma$-twisted $V$-modules such that $\mathcal{S}$ is invariant under the action of $G$. Then there is a finite dimensional semisimple associative algebra ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$ for a suitable $2$-cocycle $\alpha$ naturally determined by the $G$-action on $\mathcal{S}$ such that $({\mathcal{A}}_{\alpha }(G,\mathcal{S}),V^G)$ form a dual pair on the sum $\mathcal{M}$ of $\sigma$-twisted $V$-modules in $\mathcal{S}$ in the sense that (1) the actions of ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$ and $V^G$ on $\mathcal{M}$ commute, (2) each irreducible ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$-module appears in $\mathcal{M},$ (3) the multiplicity space of each irreducible ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$-module is an irreducible $V^G$-module, (4) the multiplicitiy spaces of different irreducible ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$-modules are inequivalent $V^G$-modules. As applications, every irreducible $\sigma$-twisted $V$-module is a direct sum of finitely many irreducible $V^G$-modules and irreducible $V^G$-modules appearing in different $G$-orbits are inequivalent. This result generalizes many previous ones. We also establish a bijection between subgroups of $G$ and subalgebras of $V$ containing $V^G.$

24 pages