# Self-dual and logarithmic representations of the twisted Heisenberg--Virasoro algebra at level zero

## TL;DR

This paper advances the understanding of the twisted Heisenberg-Virasoro algebra at level zero by providing explicit formulas for singular vectors, constructing self-dual and logarithmic modules, and exploring their extensions and relations.

## Contribution

It introduces new free field realizations of self-dual modules and constructs a broad class of logarithmic modules with potential links to projective covers.

## Key findings

- Explicit formulas for singular vectors in Verma modules
- Construction of self-dual modules via free field realization
- Existence of non-split self-extensions of irreducible modules

## 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.