Nappi-Witten vertex operator algebra via inverse Quantum Hamiltonian Reduction

The representation theory of the Nappi-Witten VOA was initiated in arXiv:1104.3921 and arXiv:2011.14453. In this paper we use the technique of inverse quantum hamiltonian reduction to investigate the representation theory of the Nappi-Witten VOA $ V^1(\mathfrak h_4)$. We first prove that the quantum hamiltonian reduction of $ V^1(\mathfrak h_4)$ is the Heisenberg-Virasoro VOA $L^{HVir}$ of level zero investigated in arXiv:math/0201314 and arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and prove that $V^1(\mathfrak h_4)$ is realized as a vertex subalgebra of $L^{HVir} \otimes \Pi$, where $\Pi$ is a certain lattice-like vertex algebra. Using such an approach we shall realize all relaxed highest weight modules which were classified in arXiv:2011.14453. We show that every relaxed highest weight module, whose top components is neither highest nor lowest weight $\mathfrak h_4$-module, has the form $M_1 \otimes \Pi_{1} (\lambda)$ where $M_1$ is an irreducible, highest weight $L^{HVir}$-module and $\Pi_{1} (\lambda)$ is an irreducible weight $\Pi$-module. Using the fusion rules for $L^{HVir}$-modules and the previously developed methods of constructing logarithmic modules we are able to construct a family of logarithmic $V^1(\mathfrak h_4)$-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight $L_k(\mathfrak{sl}(2))$-modules, so we expect that our logarithmic modules are also projective in a certain category of weight $ V^1(\mathfrak h_4)$-modules.