Simple Harish-Chandra modules over the superconformal current algebra

Abstract

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra $\hat{\mathfrak{g}}$, which is the semi-direct sum of the $N=1$ superconformal algebra with the affine Lie superalgebra $\dot{\mathfrak{g}}\otimes \mathcal{A}\oplus \mathbb{C}{C}_1$, where $\dot{\mathfrak{g}}$ 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.