Veronese subsequent analytic solutions of the $\mathbb{C}P^{2s}$ sigma model equations described via Krawtchouk polynomials

TL;DR

This paper links Veronese solutions of the $ ext{CP}^{2s}$ sigma model to Krawtchouk polynomials, providing explicit parametrizations, geometric interpretations, and new algebraic recurrence relations for solutions.

Contribution

It introduces a novel relationship between $ ext{CP}^{2s}$ solutions and Krawtchouk polynomials, enabling explicit parametrizations and simpler recurrence relations for solutions.

Findings

Solutions parametrized by Krawtchouk polynomials

Associated surfaces are spheres in $ ext{su}(2s+1)$ Lie algebra

Recurrence relations simplify solution generation

Abstract

The objective of this paper is to establish a new relationship between the Veronese subsequent analytic solutions of the Euclidean $\mathbb{C}P^{2s}$ sigma model in two dimensions and the orthogonal Krawtchouk polynomials. We show that such solutions of the $\mathbb{C}P^{2s}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrised in terms of these polynomials. We apply the obtained results to the analysis of surfaces associated with $\mathbb{C}P^{2s}$ sigma models, defined using the generalized Weierstrass formula for immersion. We show that these surfaces are spheres immersed in the $\mathfrak{su}(2s+1)$ Lie algebra, and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a new connection between the $\mathfrak{su}(2)$ spin-s representation and the $\mathbb{C}P^{2s}$ model is explored in detail. It is shown that for any given holomorphic vector function in $\mathbb{C}^{2s+1}$ written as a Veronese sequence, it is possible to derive subsequent solutions of the $\mathbb{C}P^{2s}$ model through algebraic recurrence relations which turn out to be simpler than the analytic relations known in the literature.