Twisted regular representations and bimodules in vertex operator algebra theory
Yiyi Zhu
TL;DR
This work extends Li's twisted regular representation theory to the setting of mutually commuting finite-order automorphisms by developing and analyzing twisted bimodule structures over twisted Zhu algebras. It introduces and studies two families of twisted bimodules, $A^{\dag}_{g_2^{-1}, g_1, n, m}(W)$ and $A^{\diamond}_{g_2^{-1}, g_1, n, m}(W)$, establishing their $A_{g_2^{-1}, n}(V)$-$A_{g_1, m}(V)$-bimodule structures and identifying their duals with twisted regular-regime modules $\Omega_{m,n}(\mathfrak{D}_{g_1, g_2}^{(-1)}(W))$ and $\Omega^{\diamond}_{m, n}(\mathfrak{D}_{g_1, g_2}^{(-1)}(W))$. The approach generalises Li's untwisted results to the twisted setting and provides a concrete mechanism to recover and extend Dong–Jiang’s bimodule theory. A central achievement is confirming Dong and Jiang's conjecture in the twisted case, showing that $O_{g_1, n, m}(V)=O'_{g_1, n, m}(V)$, which clarifies the structure of twisted Zhu algebras and their bimodules. The results enhance computability and conceptual understanding of twisted representations and their algebraic underpinnings.
Abstract
In this paper, we use the twisted regular representation theory of vertex operator algebras to construct bimodules over twisted Zhu algebras, extending Haisheng Li's work in untwisted scenarios. Moreover, a conjecture of Dong and Jiang on bimodule theory is confirmed.