Cuspidal modules over Superconformal algebras of rank \geq 1

TL;DR

This paper classifies all cuspidal modules over superconformal algebras of rank at least one, revealing new phenomena such as trivial central charge in most cases and detailed implications for related algebras.

Contribution

It provides an explicit classification of cuspidal modules over known superconformal algebras of rank ≥ 1, including their central extensions and new insights into their representation theory.

Findings

Central charge of cuspidal modules is trivial in most cases.

Unique non-trivial central extension for the contact algebra (4).

Impacts the understanding of (3), (6), and (2) algebras.

Abstract

According to V. Kac and J. van de Leur, the superconformal algebras are the simple $\Z$-graded Lie superalgebras of growth one which contains the Witt algebra. We describe an explicit classification of all cuspidal modules over the known supercuspidal algebras of rank $\geq 1$, and their central extensions. Our approach reveals some unnoticed phenomena. Indeed the central charge of cuspidal modules is trivial, except for one specific central extension of the contact algebra $\K(4)$. As shown in the paper, this fact also impacts the representation theory of $\K(3)$, $\CK(6)$ and $\K^{(2)}(4)$. Besides these four cases, the classification relies on general methods based on highest weight theory.