Twisted regular representations for vertex operator algebras

Abstract

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ1,σ2 be commuting finite-order automorphisms of V and let σ=(σ1σ2)−1. Among the main results, for any σ-twisted V-module W and any nonzero complex number z, we construct a weak σ1⊗σ2-twisted V⊗V-module Dσ1,σ2(z)(W) inside W⁎. Let W1,W2 be σ1-twisted, σ2-twisted V-modules, respectively. We show that P(z)-intertwining maps from W1⊗W2 to W⁎ are the same as homomorphisms of weak σ1⊗σ2-twisted V⊗V-modules from W1⊗W2 into Dσ1,σ2(z)(W). We also show that a P(z)-intertwining map from W1⊗W2 to W⁎ is equivalent to an intertwining operator of type (W′W1W2), which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ-twisted V-module M with τ any finite-order automorphism of V, the coefficients of the q-graded trace function lie in Dτ,τ−1(−1)(V) and generate a τ⊗τ−1-twisted V⊗V-submodule isomorphic to M⊗M′.