On a stack of surfaces obtained from the $\mathbb{C}P^{N-1}$ sigma models

TL;DR

This paper investigates surfaces derived from the $ ext{CP}^{N-1}$ sigma model on the Riemann sphere, showing that stacked surfaces are identical and exploring broader classes of solutions to the model's equations.

Contribution

It demonstrates the idempotency of the surface stacking process and identifies larger solution classes for the model's Euler–Lagrange equations.

Findings

Stacked surfaces from the model are identical, confirming idempotency.

Broader classes of solutions to the Euler–Lagrange equations are found.

Surfaces are derived under the assumption of finite action on the Riemann sphere.

Abstract

Under the assumption that the $\mathbb{C}P^{N-1}$ sigma model is defined on the Riemann sphere and its action functional is finite, we derive surfaces induced by surfaces and we demonstrate that the stacked surfaces coincide with each other, which means idempotency of the recurrent procedure. Along the path to the solutions of the $\mathbb{C}P^{N-1}$ model equations, we demonstrate that the Euler--Lagrange equations for the projectors admit larger classes of solutions than the ones corresponding to rank-1 projectors.