Canonical surfaces associated with projectors in grassmannian sigma models

TL;DR

This paper explores the construction of higher-dimensional surfaces from harmonic maps into Grassmannians, analyzing their properties and curvature, with specific focus on Veronese sequence mappings.

Contribution

It introduces two methods for constructing surfaces from projectors in Grassmannian sigma models and analyzes their geometric properties, including curvature behavior.

Findings

Gaussian curvature is generally non-constant for these surfaces.

Surfaces from Veronese sequence maps have constant curvature, varying with the specific map.

The study provides insights into the geometric structure of Grassmannian sigma model surfaces.

Abstract

We discuss the construction of higher-dimensional surfaces based on the harmonic maps of $S^2$ into $CP^{N-1}$ and other grassmannians. We show that there are two ways of implementing this procedure - both based on the use of the relevant projectors. We study various properties of such projectors and show that the Gaussian curvature of these surfaces, in general, is not constant. We look in detail at the surfaces corresponding to the Veronese sequence of such maps and show that for all of them this curvature is constant but its value depends on which mapping is used in the construction of the surface.