# A family of non-weight modules over the super-Virasoro algebras

## TL;DR

This paper constructs and classifies a family of non-weight modules over super-Virasoro algebras, providing conditions for simplicity and complete classifications for Ramond and Neveu-Schwarz types.

## Contribution

It introduces a new family of non-weight modules over super-Virasoro algebras, with a full classification and simplicity criteria.

## Key findings

- Modules are free of rank 1 over Ramond algebra's Cartan subalgebra.
- Modules are free of rank 2 over Neveu-Schwarz algebra's Cartan subalgebra.
- Complete classification of these modules over super-Virasoro algebras.

## Abstract

In this paper, we construct a family of non-weight modules over the super-Virasoro algebras. Those modules when regarded as modules of the Ramond algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{h}$ are free of rank $1$, while when regarded as modules of the Neveu-Schwarz algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{H}$ are free of rank $2$. We obtain a sufficient and necessary condition for such modules to be simple. Moreover, we determine the isomorphism classes of these modules. Finally, we show that these modules constitute a complete classification of free $U(\mathfrak{h})$-modules of rank $1$ over the super-Virasoro algebra of Ramond type, and also constitute a complete classification of free $U(\mathfrak{H})$-modules of rank $2$ over the super-Virasoro algebra of Neveu-Schwarz type.