Simple Harish-Chandra modules over the superconformal current algebra
Y. He, D. Liu, Y. Wang
TL;DR
The paper addresses the problem of classifying simple Harish-Chandra modules over the $N=1$ superconformal current algebra $\widehat{\frak g}$. It employs the $\mathcal{A}$-cover method to reduce the problem to cuspidal modules and then constructs tensor modules $\Gamma(\lambda,V,b)$, along with parity twists, to capture all simples, showing they are either highest/lowest weight or quotients of these tensor modules. The main result provides a complete description of simple Harish-Chandra modules as either highest/lowest weight or $\Gamma(\lambda,V,b)$-type quotients (including $\Pi(\Gamma(\lambda,V,b))$), with an explicit parametrization by a finite-dimensional $\dot{\frak g}$-module $V$ and complex parameters $\lambda,b$. As an application, the framework yields a parallel classification for the $N=1$ Heisenberg-Virasoro algebra, including simple cuspidal modules $\Gamma(\lambda,b,d)$ and their simple quotients $\Gamma'(\lambda,b,d)$, thereby unifying the representation theory of this family of algebras and providing concrete, computable module realizations for physical contexts in superstring theory and related areas.
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.