Twisted regular representations for 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 σ 1 , σ 2 be commuting finite-order automorphisms of V and let σ = (σ 1 σ 2) − 1. Among the main results, for any σ -twisted V -module W and any nonzero complex number z , we construct a weak σ 1 ⊗ σ 2 -twisted V ⊗ V -module D σ 1 , σ 2 (z) (W) inside W ⁎. Let W 1 , W 2 be σ 1 -twisted, σ 2 -twisted V -modules, respectively. We show that P (z) -intertwining maps from W 1 ⊗ W 2 to W ⁎ are the same as homomorphisms of weak σ 1 ⊗ σ 2 -twisted V ⊗ V -modules from W 1 ⊗ W 2 into D σ 1 , σ 2 (z) (W). We also show that a P (z) -intertwining map from W 1 ⊗ W 2 to W ⁎ is equivalent to an intertwining operator of type ( W ′ W 1 W 2 ) , 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 D τ , τ − 1 (− 1) (V) and generate a τ ⊗ τ − 1 -twisted V ⊗ V -submodule isomorphic to M ⊗ M ′. [ABSTRACT FROM AUTHOR]