Nappi-Witten vertex operator algebra via inverse Quantum Hamiltonian Reduction

TL;DR

The work identifies the minimal affine W-algebra for the Nappi–Witten VOA $V^1(rak h_4)$ as the Heisenberg–Virasoro algebra $L^{HVir}$ at level zero and constructs an inverse quantum Hamiltonian reduction embedding $V^1(rak h_4)\hookrightarrow L^{HVir}\otimes\Pi$, realized via a screening operator. This framework enables a complete realization of all irreducible relaxed highest-weight $V^1(rak h_4)$-modules as $L^{HVir}[x,y]\otimes\Pi_1(\lambda)$ and yields a wide class of logarithmic $V^1(rak h_4)$-modules by deforming modules along screenings, with explicit non-split extensions and Loewy diagrams echoing the projective modules of weight $L_k(\frak{sl}_2)$. The work also analyzes finite- and infinite-length logarithmic modules, establishes constructions $\mathcal P_r[x,y]$ and $\mathcal P_1(\lambda)$, and discusses potential projectivity within suitable subcategories, laying groundwork for further fusion and tensor-category study in non-reductive VOA settings.

Abstract

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 $L^{HVir} \otimes Π$, 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 $\mathfrak h_4$-module, has the form $M_1 \otimes Π_{1} (λ)$ where $M_1$ is an irreducible, highest weight $L^{HVir}$-module and $Π_{1} (λ)$ is an irreducible weight $Π$-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.

Theorems & Definitions (31)

• Definition 2.1
• Lemma 2.2
• Proposition 2.3
• Remark 3.1
• Theorem 4.1
• Proposition 5.1
• proof
• Theorem 6.1
• Theorem 6.2
• proof