Simple restricted modules over the $N=1$ Ramond algebra as weak modules for vertex operator superalgebras

TL;DR

This paper constructs new simple restricted modules over the $N=1$ Ramond algebra, classifies certain weak modules, and provides examples related to Whittaker modules, advancing the understanding of module structures in vertex operator superalgebras.

Contribution

It introduces a new class of simple modules induced from finite-dimensional solvable Lie superalgebras and classifies simple weak $ar W(0,c)$-modules under specific conditions.

Findings

Construction of new simple restricted modules over the $N=1$ Ramond algebra.

Classification of simple weak $ar W(0,c)$-modules in certain cases.

Examples of simple restricted $N=1$ Ramond modules as variants of Whittaker modules.

Abstract

In the present paper, a class of new simple modules over the $N=1$ Ramond algebra are constructed, which are induced from simple modules over some finite dimensional solvable Lie superalgebras. These new modules are simple restricted modules over the $N=1$ Ramond algebra. Combined with the result in \cite{L1}, a classification of simple weak $\psi$-twisted $\bar W(0,c)$-modules under certain conditions is also given. At last, some examples of simple restricted $N=1$ Ramond modules as various versions of Whittaker modules are presented (classical Whittaker modules were studied in \cite{LPX}).