Supersymmetric version of the equations of conformally parametrized surfaces

TL;DR

This paper develops a supersymmetric extension of classical surface equations, deriving new symmetry structures and explicit solutions for supersymmetric conformally parametrized surfaces in Grassmann superspace.

Contribution

It introduces a supersymmetric formulation of Gauss-Weingarten and Gauss-Codazzi equations, expanding classical surface theory into superspace with novel symmetry analysis.

Findings

Derived six supersymmetric Gauss-Codazzi equations

Classified symmetry algebras and subalgebras for classical and supersymmetric cases

Obtained explicit solutions and geometric interpretations of supersymmetric surfaces

Abstract

In this paper, we formulate a supersymmetric extension of the Gauss-Weingarten and Gauss-Codazzi equations for conformally parametrized surfaces immersed in a Grassmann superspace. We perform this analysis using a superspace-superfield formalism together with a supersymmetric version of a moving frame on a surface. In constrast to the classical case, where we have three Gauss-Codazzi equations, we obtain six such equations in the supersymmetric case. We determine the Lie symmetry algebra of the classical Gauss-Codazzi equations to be infinite-dimensional and perform a subalgebra classification of the one-dimensional subalgebras of its largest finite-dimensional subalgebra. We then compute a superalgebra of Lie point symmetries of the supersymmetric Gauss-Codazzi equations and classify the one-dimensional subalgebras of this superalgebra into conjugacy classes. We then use the symmetry reduction method to find invariants, orbits and reduced systems for two one-dimensional subalgebras in the classical case and three one-dimensional subalgebras in the supersymmetric case. Through the solutions of these reduced systems, we obtain explicit solutions and surfaces of the classical and supersymmetric Gauss-Codazzi equations. We provide a geometrical interpretation of the results.