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

DOI10.1142/S0219199718500086zbMATH Open1439.17026arXiv1703.00531MaRDI QIDQ4630365

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Publication date: 26 March 2019

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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 $mathcalH$ at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for $mathcalH$ 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 $mathcalH$-modules.

Full work available at URL: https://arxiv.org/abs/1703.00531

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