Bimodules over twisted Zhu algebras and twisted fusion rules theorem for vertex operator algebras

TL;DR

The paper develops a comprehensive bimodule framework for twisted Zhu algebras associated with a strongly rational VOA $V$ and three commuting finite-order automorphisms $g_1,g_2,g_3$ with $g_3=g_1g_2$. It constructs a family of $A_{g_3,n}(V)$–$A_{g_2,m}(V)$-bimodules $\mathcal{A}_{g_3,g_2,n,m}(M^1)$ from a $g_1$-twisted module $M^1$ and uses these to build admissible $g_3$-twisted modules via tensor-type constructions; special cases recover the known bimodules $A_{g,n,m}(V)$. The authors then define tensor products of twisted modules through these bimodules and establish a twisted Frenkel–Zhu–Li fusion rules theorem, expressing the space of twisted intertwiners in terms of Hom spaces over twisted Zhu algebras. This work provides explicit associative-algebraic tools to study twisted representation theory and fusion in VOAs, enabling computation of twisted fusion rules and advancing the understanding of twisted module categories for strongly rational VOAs.

Abstract

Let $V$ be a strongly rational vertex operator algebra, and let $g_1, g_2, g_3$ be three commuting finitely ordered automorphisms of $V$ such that $g_1g_2=g_3$ and $g_i^T=1$ for $i=1, 2, 3$ and $T\in \N$. Suppose $M^1$ is a $g_1$-twisted module. For any $n, m\in \frac{1}{T}\N$, we construct an $A_{g_3, n}(V)$-$A_{g_2, m}(V)$-bimodule $\mathcal{A}_{g_3, g_2, n, m}(M^1)$ associated to the quadruple $(M^1, g_1, g_2, g_3)$. Given an $A_{g_2, m}(V)$-module $U$, an admissible $g_3$-twisted module $\mathcal{M}(M^1, U)$ is constructed. For the quadruple $(V, 1, g, g)$ with some finitely ordered $g\in \text{Aut}(V)$, $\mathcal{A}_{g, g, n, m}(V)$ coincides with the $A_{g, n}(V)$-$A_{g, m}(V)$-bimodules $A_{g, n, m}(V)$ constructed by Dong-Jiang, and $\mathcal{M}(V, U)$ is the generalized Verma type admissible $g$-twisted module generated by $U$. When $U=M^2(m)$ is the $m$-th component of a $g_2$-twisted module $M^2$ for some $m\in\frac{1}{T}\N$, we show that the submodule of $\M(M^1, M^2(m))$ generated by the $m$-th component satisfies the universal property of the tensor product of $M^1$ and $M^2$. Using this result, we obtain a twisted version of Frenkel-Zhu-Li's fusion rules theorem.

Theorems & Definitions (84)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Theorem 2.5
• Proposition 2.6
• Theorem 2.7
• Definition 2.8
• Definition 2.9
• Proposition 2.10