How to superize Liouville equation

TL;DR

This paper introduces a new superization of the Liouville equation using Ramond superalgebra, providing explicit solutions and discussing open problems, expanding the mathematical framework of supersymmetric integrable systems.

Contribution

It presents a novel superLiouville equation based on Ramond superalgebra and offers explicit solutions, extending previous superizations that used Neveu--Schwarz superalgebra.

Findings

Explicit solutions for the superLiouville equation using Ramond superalgebra

Comparison with existing superizations for scalar and vector fields

Discussion of open problems in the superLiouville framework

Abstract

So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for $N\leq 4$) and for a vector-valued field (analogs of the Leznov--Saveliev equations) for N=1. Both superizations are performed with the help of Neveu--Schwarz superalgebra. We consider another version of these superLiouville equations based on the Ramond superalgebra, their explicit solutions are given by Ivanov--Krivonos' scheme. Open problems are offered.