Higher level twisted Zhu algebras

Abstract

The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by \documentclass[12pt]{minimal}\begin{document}$\Gamma /\mathbb {Z}$\end{document}Γ/Z for some subgroup Γ of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}$\end{document}R containing \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V), and we prove the following theorems: For each p, there is a bijection between the irreducible \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V)-modules and the irreducible Γ-twisted positive energy V-modules, and V is (Γ, H)-rational if and only if all its Zhu algebras \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \documentclass[12pt]{minimal}\begin{document}$\operatorname{Vir}^c$\end{document}Virc and the universal affine Kac-Moody vertex algebra \documentclass[12pt]{minimal}\begin{document}$V^k(\mathfrak {g})$\end{document}Vk(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.