# On geometric aspects of the SUSY Fokas-Gel'fand immersion formula

## TL;DR

This paper introduces a new geometric framework for supersymmetric soliton supermanifolds, utilizing linear spectral problems to analyze their immersion into Lie superalgebras and compute curvatures, with applications to the supersymmetric sine-Gordon equation.

## Contribution

It develops a novel geometric characterization for supersymmetric Fokas-Gel'fand immersions, including techniques to derive multiple spectral problems and analyze supermanifold geometry.

## Key findings

- Derived new linear spectral problems for supersymmetric integrable systems.
- Computed mean and Gaussian curvatures of supermanifolds.
- Applied the framework to the supersymmetric sine-Gordon equation.

## Abstract

In this paper, we develop a new geometric characterization for the supersymmetric versions of the Fokas--Gel'fand formula for the immersion of soliton supermanifolds with two bosonic and two fermionic independent variables into Lie superalgebras. In order to do so, from a linear spectral problem of a supersymmetric integrable system using the covariant fermionic derivative, we provide a technique to obtain two additional linear spectral problems for that integrable system, one using the bosonic variable derivatives and the other using the fermionic variable derivatives. This allows us to investigate, through the first and second fundamental forms, the geometry of the ($1+1\vert2$)-supermanifolds immersed in Lie superalgebras. Whenever possible, the mean and Gaussian curvatures of the supermanifolds are calculated. These theoretical considerations are applied to the supersymmetric sine-Gordon equation.