Surfaces obtained from CP^(N-1) sigma models

TL;DR

This paper employs the Weierstrass technique to construct surfaces immersed in Euclidean spaces from CP^(N-1) sigma models, providing explicit examples and analyzing their geometric properties based on solutions of the models.

Contribution

It introduces a method to derive conformally parametrized surfaces from CP^(N-1) models using the Weierstrass formula, with explicit constructions for su(3) and specific solutions.

Findings

Surfaces associated with meron solutions are semi-infinite cylinders.

Holomorphic and mixed Veronese solutions produce surfaces in R^8 and R^3.

The method applies to finite-action solutions on S^2, linking sigma models to surface geometry.

Abstract

In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2 and the associated action functional of this model is finite, then the generalized Weierstrass formula for immersion describes conformally parametrized surfaces in the su(N) algebra. In particular, for any holomorphic or antiholomorphic solution of this model the associated surface can be expressed in terms of an orthogonal projector of rank (N-1). The implementation of this method is presented for two-dimensional conformally parametrized surfaces immersed in the su(3) algebra. The usefulness of the proposed approach is illustrated with examples, including the dilation-invariant meron-type solutions and the Veronese solutions for the CP^2 model. Depending on the location of the critical points (zeros and poles) of the first fundamental form associated with the meron solution, it is shown that the associated surfaces are semi-infinite cylinders. It is also demonstrated that surfaces related to holomorphic and mixed Veronese solutions are immersed in R^8 and R^3, respectively.