Simple smooth modules over the superconformal current algebra

Abstract

In this paper, we classify simple smooth modules over the superconformal current algebra $\mathfrak{g}$. More precisely, we first classify simple smooth modules over the Heisenberg-Clifford algebra, and then prove that any simple smooth $\mathfrak{g}$-module is a tensor product of such modules for the super Virasoro algebra and the Heisenberg-Clifford algebra, or an induced module from a simple module over some finite-dimensional solvable Lie superalgebras. As a byproduct, we provide characterizations for both simple highest weight $\mathfrak{g}$-modules and simple Whittaker $\mathfrak{g}$-modules. Additionally, we present several examples of simple smooth $\mathfrak{g}$-modules that are not tensor product of modules over the super Virasoro algebra and the Heisenberg-Clifford algebra.

Latex, 30pages, comments are welcome!