  In this paper, we study a new kind of vertex operator algebra related to the
twisted Heisenberg-Virasoro algebra, which we call the twisted
Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we
present some results concerning the relationship between the restricted module
categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and
several different kinds of module categories of their corresponding vertex
algebras. We also study fully the structures of the twisted Heisenberg-Virasoro
vertex operator algebra, give a characterization of it as a tensor product of
two well-known vertex operator algebras, and solve the commutant problem.
