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

This paper studies restricted modules of gap-$p$ Virasoro algebra $\L$ and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted $\L$-modules of level $\el$ and the category of twisted modules of vertex algebra $V_{\mathcal{N}_{p}}(\el,0)$, where $\mathcal{N}_{p}$ is a new Lie algebra, $\el:=(\ell_{0},0,\cdots,0)\in\C^{\halfp+1}$, $\ell_{0}\in\C$ is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted $\L$-modules of level $\el$. More explicitly, we give a uniform construction of simple restricted $\L$-modules as induced modules. We present several equivalent characterizations of simple restricted $\L$-modules, as locally nilpotent (equivalently, locally finite) modules with respect to certain positive part of $\L$. Moreover, simple restricted $\L$-modules of level $\el$ are classified. They are either highest weight modules or simple induced modules. At the end, we exhibit several concrete examples of simple restricted $\L$-modules of level $\el$ (including Whittaker modules).