Twisted representations of vertex operator algebras

TL;DR

This work develops a robust framework for twisted representations of vertex operator algebras by introducing the associative algebra $A_g(V)$ and two key functors, $L$ and $\Omega$, between $A_g(V)$-modules and admissible $g$-twisted $V$-modules. The central result is a bijection between simple objects, with $\Omega\circ L$ acting as the identity and $L$ providing a right inverse to $\Omega$, and, under complete reducibility, $L$ and $\Omega$ become inverse equivalences of categories. The construction leverages a Lie algebra $V[g]$ and generalized Verma modules to realize admissible $g$-twisted modules from $A_g(V)$-modules, yielding finite-dimensionality and semisimplicity results for $g$-rational VOAs and explicit correspondences of simple modules. The framework also yields existence results for twisted modules when $A_g(V)$ is finite-dimensional and has broad applications to orbifold theory, rationality questions, and Moonshine phenomena.

Abstract

Let $V$ be a vertex operator algebra and $g$ an automorphism of finite order. We construct an associative algebra $A_g(V)$ and a pair of functors between the category of $A_g(V)$-modules and a certain category of admissible $g$-twisted $V$-modules. In particular, these functors exhibit a bijection between the simple modules in each category. We give various applications, including the fact that the complete reducibility of admissible $g$-twisted modules implies both the finite-dimensionality of homogeneous spaces and the finiteness of the number of simple $g$-twisted modules.

Theorems & Definitions (43)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Theorem 2.4
• Remark 2.5
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.4
• Remark 3.5