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 $W^{*}$. 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 $M\otimes M'$.