  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).

Representation Theory;Quantum Algebra