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

This paper studies restricted modules of gap-p Virasoro algebra g(p) and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted g(p)-modules of level (l) under bar and the category of twisted modules of vertex algebra V-Np((l) under bar, 0), where N-p is a new Lie algebra, (l) under bar :=(l(0), 0, center dot center dot center dot, 0) is an element of C[p/2]+1, l(0) is an element of C is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted g(p)-modules of level (l) under bar. More explicitly, we give a uniform construction of simple restricted g(p)-modules as induced modules. We present several equivalent characterizations of simple restricted g(p)-modules, as locally nilpotent (equivalently, locally finite) modules with respect to certain positive part of g(p). Moreover, simple restricted g(p)-modules of level (l) under bar are classified. They are either highest weight modules or simple induced modules. At the end, we exhibit several concrete examples of simple restricted g(p)-modules of level (l) under bar (including Whittaker modules). (c) 2023 Elsevier B.V. All rights reserved.