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 V¹( h₄). We first prove that the quantum hamiltonian reduction of V¹( h₄) is the Heisenberg-Virasoro VOA LHVir of level zero investigated in arXiv:math/0201314 and arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and prove that V¹( h₄) is realized as a vertex subalgebra of LHVir ⊗ Π, where Π 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 h₄-module, has the form M₁ ⊗ Π₁ (λ) where M₁ is an irreducible, highest weight LHVir-module and Π₁ (λ) is an irreducible weight Π-module. Using the fusion rules for LHVir-modules and the previously developed methods of constructing logarithmic modules we are able to construct a family of logarithmic V¹( h₄)-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight Lₖ(sl(2))-modules, so we expect that our logarithmic modules are also projective in a certain category of weight V¹( h₄)-modules.