On the surfaces associated with $\mathbb{C}P^{N-1}$ models

TL;DR

This paper explores the geometric and algebraic properties of surfaces linked to $ ext{CP}^{N-1}$ models, revealing linear dependencies, polynomial relations, and invariant angles in their associated Lie algebra and spectral problem wave functions.

Contribution

It introduces new algebraic relations and geometric invariants for surfaces and wave functions in $ ext{CP}^{N-1}$ models, extending understanding of their structure and symmetries.

Findings

Immersion functions span an (N-1)-dimensional subspace of $su(N)$.

Minimal polynomials of immersion functions are cubic, quadratic for special solutions.

Angles between position vectors are independent of the independent variables.

Abstract

We study certain new properties of 2D surfaces associated with the $\mathbb{C}P^{N-1}$ models and the wave functions of the corresponding linear spectral problem. We show that $su(N)$-valued immersion functions expressed in terms of rank-1 orthogonal projectors are linearly dependent, but they span an $(N-1)$-dimensional subspace of the Lie algebra $su(N)$. Their minimal polynomials are cubic, except for the holomorphic and antiholomorphic solutions, for which they reduce to quadratic trinomials. We also derive the counterparts of these relations for the wave functions of the linear spectral problems. In particular, we provide a relation between the wave functions, which results from the partition of unity into the projectors. Finally, we show that the angle between any two position vectors of the immersion functions, corresponding to the same values of the independent variables, does not depend on those variables.