Twisted regular representations of vertex operator algebras

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 $W^{\ast }$. 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 $W^{\ast }$ 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 $W^{\ast }$ is equivalent to an intertwining operator of type $\left(\genfrac{}{}{0px}{}{W^{\mathrm{\prime }}}{W_1{W}_2}\right)$, 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$V\otimes V$-submoduleisomorphicto$M\otimes M’$.