Conformally parametrized surfaces associated with CP^(N-1) sigma models

TL;DR

This paper explores the geometry of surfaces associated with CP^(N-1) sigma models, providing explicit formulas for their fundamental properties and illustrating the approach with examples in low-dimensional cases.

Contribution

It introduces a method to construct and analyze conformally parametrized surfaces linked to CP^(N-1) sigma models, including explicit formulas for geometric invariants.

Findings

Explicit formulas for fundamental forms, curvatures, and Willmore functional.

Computed Lie-point symmetries for arbitrary N.

Illustrated the approach with examples in low-dimensional su(N) algebras.

Abstract

Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in R^(N^2-1). The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the CP^2 model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras.