Twisted regular representations of vertex operator algebras

TL;DR

This paper develops the theory of twisted regular representations for vertex operator algebras, constructing new modules and establishing correspondences with intertwining maps, advancing the understanding of twisted modules and their trace functions.

Contribution

It introduces twisted regular representations for VOAs, constructs associated modules, and relates intertwining maps to module homomorphisms, extending prior results to twisted settings.

Findings

Constructed weak -twisted modules inside dual modules.

Established equivalence between intertwining maps and module homomorphisms.

Showed trace function coefficients generate twisted submodules isomorphic to tensor products.

Abstract

This paper is to study what we call twisted regular representations for vertex operator algebras. Let $V$ be a vertex operator algebra, let $\sigma_1,\sigma_2$ be commuting finite-order automorphisms of $V$ and let $\sigma=(\sigma_1\sigma_2)^{-1}$. Among the main results, for any $\sigma$-twisted $V$-module $W$ and any nonzero complex number $z$, we construct a weak $\sigma_1\otimes \sigma_2$-twisted $V\otimes V$-module $\mathfrak{D}_{\sigma_1,\sigma_2}^{(z)}(W)$ inside $W^{*}$. Let $W_1,W_2$ be $\sigma_1$-twisted, $\sigma_2$-twisted $V$-modules, respectively. We show that $P(z)$-intertwining maps from $W_1\otimes W_2$ to $W^{*}$ are the same as homomorphisms of weak $\sigma_1\otimes \sigma_2$-twisted $V\otimes V$-modules from $W_1\otimes W_2$ into $\mathfrak{D}_{\sigma_1,\sigma_2}^{(z)}(W)$. We also show that a $P(z)$-intertwining map from $W_1\otimes W_2$ to $W^{*}$ is equivalent to an intertwining operator of type $\binom{W'}{W_1\; W_2}$, which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each $\tau$-twisted $V$-module $M$ with $\tau$ any finite-order automorphism of $V$, the coefficients of the $q$-graded trace function lie in $\mathfrak{D}_{\tau,\tau^{-1}}^{(-1)}(V)$, which generate a $\tau\otimes \tau^{-1}$-twisted $V\otimes V$-submodule isomorphic to $M\otimes M'$.