Title: Self-dual and logarithmic representations of the twisted Heisenberg--Virasoro algebra at level zero,
Abstract: This paper is a continuation of arXiv:1405.1707. We present certain new
applications and generalizations of the free field realization of the twisted
Heisenberg-Virasoro algebra ${\mathcal H}$ at level zero.
We find explicit formulas for singular vectors in certain Verma modules. A
free field realization of self-dual modules for ${\mathcal H}$ is presented by
combining a bosonic construction of Whittaker modules from arXiv:1409.5354 with
a construction of logarithmic modules for vertex algebras. As an application,
we prove that there exists a non-split self-extension of irreducible self-dual
module which is a logarithmic module of rank two.
We construct a large family of logarithmic modules containing different types
of highest weight modules as subquotients. We believe that these logarithmic
modules are related with projective covers of irreducible modules in a suitable
category of ${\mathcal H}$-modules.