# $\mathbb{C}P^{2S}$ sigma models described through hypergeometric orthogonal polynomials

## TL;DR

This paper establishes a novel link between $ ext{CP}^{2S}$ sigma model solutions and Krawtchouk orthogonal polynomials, enabling explicit parametrization and geometric analysis of associated surfaces.

## Contribution

It introduces a new parametrization of $ ext{CP}^{2S}$ sigma model solutions using hypergeometric orthogonal polynomials, specifically Krawtchouk polynomials, and explores their geometric and algebraic properties.

## Key findings

- Solutions parametrized by Krawtchouk polynomials.
- Surfaces are homeomorphic to spheres in $ ext{su}(2s+1)$.
- Geometrical characteristics expressed via orthogonal polynomials.

## Abstract

The main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb{C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any such projector 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 these results to the analysis of surfaces associated with $\mathbb{C}P^{2S}$ models defined using the generalised Weierstrass formula for immersion. We show that these surfaces are homeomorphic to spheres in the $\mathfrak{su}(2s+1)$ algebra, and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a connection between the $\mathfrak{su}(2)$ spin-s representation and the $\mathbb{C}P^{2S}$ model is explored in detail.