Simple smooth modules over the Ramond algebra and applications to vertex operator superalgebras

TL;DR

The paper resolves the classification of simple smooth modules over the Ramond algebra $\mathcal{R}$ by proving every such module is either a simple highest weight module or an induced module from a simple $\mathfrak{b}^{(t)}$-module. The authors construct a broad family of simple induced $\mathcal{R}$-modules from simple $\mathfrak{b}$-modules and establish a precise equivalence between smoothness, local finiteness of $L_t$, and the two structural forms. They also classify simple $\mathfrak{b}^{(t)}$-modules for small $t$ and connect these Ramond-module classifications to weak $\psi$-twisted modules of certain vertex operator superalgebras. The results integrate with the theory of Whittaker and highest weight representations, and furnish explicit examples including high-order Whittaker modules, deepening the link between Ramond representation theory and VOAs.

Abstract

Simple smooth modules over the Virasoro algebra and one of the super-Virasoro algebras, named the Neveu-Schwarz algebra, have been classified. This problem remained unsolved for the other super-Virasoro algebra called the Ramond algebra.In this paper, all simple smooth modules over the Ramond algebra are classified. More precisely, we show that a simple smooth module over the Ramond algebra is either a simple highest weight module or isomorphic to an induced module from a simple module over a finite dimensional solvable Lie superalgebra.As an application we obtain all simple weak $ψ$-twisted modules over some vertex operator superalgebras.

Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof