On the Integrability of Supersymmetric Versions of the Structural Equations for Conformally Parametrized Surfaces

TL;DR

This paper extends classical surface equations into supersymmetric forms, analyzes their symmetries, and explores conditions for integrability, including examples like the supersymmetric sine-Gordon equation.

Contribution

It introduces supersymmetric extensions of structural equations for surfaces and formulates a supersymmetric integrability conjecture with illustrative examples.

Findings

Supersymmetric versions of Gauss-Weingarten equations analyzed.

Symmetry properties of classical and supersymmetric equations compared.

Supersymmetric integrability conditions proposed and exemplified.

Abstract

The paper presents the bosonic and fermionic supersymmetric extensions of the structural equations describing conformally parametrized surfaces immersed in a Grasmann superspace, based on the authors' earlier results. A detailed analysis of the symmetry properties of both the classical and supersymmetric versions of the Gauss-Weingarten equations is performed. A supersymmetric generalization of the conjecture establishing the necessary conditions for a system to be integrable in the sense of soliton theory is formulated and illustrated by the examples of supersymmetric versions of the sine-Gordon equation and the Gauss-Codazzi equations.