Restricted modules for gap-p Virasoro algebra and twisted modules for certain vertex algebras

Abstract

This paper studies restricted modules of gap-p Virasoro algebra gp and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted gp-modules of level ℓ_ and the category of twisted modules of vertex algebra VNp(ℓ_,0), where Np is a new Lie algebra, ℓ_:=(ℓ0,0,⋯,0)∈C[p2]+1, ℓ0∈C is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted gp-modules of level ℓ_. More explicitly, we give a uniform construction of simple restricted gp-modules as induced modules. We present several equivalent characterizations of simple restricted gp-modules, as locally nilpotent (equivalently, locally finite) modules with respect to certain positive part of gp. Moreover, simple restricted gp-modules of level ℓ_ are classified. They are either highest weight modules or simple induced modules. At the end, we exhibit several concrete examples of simple restricted gp-modules of level ℓ_ (including Whittaker modules).