Simple Harish-Chandra modules over the superconformal current algebra

TL;DR

This paper classifies simple Harish-Chandra modules over a complex superconformal current algebra, extending understanding of representations in superconformal and affine Lie superalgebra contexts.

Contribution

It provides a new classification of simple Harish-Chandra modules over a specific superconformal current algebra, including applications to the N=1 Heisenberg-Virasoro algebra.

Findings

Complete classification of simple Harish-Chandra modules over the superconformal current algebra.

Direct derivation of classification results for the N=1 Heisenberg-Virasoro algebra.

Enhanced understanding of module structures in superconformal algebra representations.

Abstract

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra $\widehat{\frak g}$, which is the semi-direct sum of the $N=1$ superconformal algebra with the affine Lie superalgebra $\dot{\frak g} \otimes \mathcal{A}\oplus \mathbb CC_1$, where $\dot{\frak g}$ is a finite-dimensional simple Lie algebra, and $\mathcal{A}$ is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the $N=1$ Heisenberg-Virasoro algebra.