Nappi–Witten vertex operator algebra via inverse quantum Hamiltonian reduction

The study of representation theory of the Nappi–Witten vertex operator algebra (VOA) was initiated in [Y. Bao, C. Jiang and Y. Pei, Representations of affine Nappi–Witten algebras, J. Algebra 342 (2011) 111–133; A. Babichenko, K. Kawasetsu, D. Ridout and W. Stewart, Representations of the Nappi–Witten vertex operator algebra, Lett. Math. Phys. 111 (2021) 131]. In this paper we use the technique of inverse quantum Hamiltonian reduction to investigate the representation theory of the Nappi–Witten VOA [Formula: see text]. We first prove that the quantum Hamiltonian reduction of [Formula: see text] is the Heisenberg–Virasoro VOA [Formula: see text] of level zero investigated in [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Canad. Math. Bull. 46(4) (2003) 529–537]. We invert the quantum Hamiltonian reduction in this case and prove that [Formula: see text] is realized as a vertex subalgebra of [Formula: see text], where [Formula: see text] is a certain lattice-like vertex algebra. Using such an approach we shall realize all relaxed highest weight modules which were classified in [A. Babichenko, K. Kawasetsu, D. Ridout and W. Stewart, Representations of the Nappi–Witten vertex operator algebra, Lett. Math. Phys. 111 (2021) 131]. We show that every relaxed highest weight module, whose top components is neither highest nor lowest weight [Formula: see text]-module, has the form [Formula: see text], where [Formula: see text] is an irreducible, highest weight [Formula: see text]-module and [Formula: see text] is an irreducible weight [Formula: see text]-module. Using the fusion rules for [Formula: see text]-modules from [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342] and the previously developed methods of constructing logarithmic modules from [D. Adamović and A. Milas, Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math. 15 (2009) 535–561] we are able to construct a family of logarithmic [Formula: see text]-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight [Formula: see text]-modules, so we expect that our logarithmic modules are also projective in a certain category of weight [Formula: see text]-modules.