# $\mathbb{C}P^{2S}$ sigma models described through hypergeometric 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 Veronese subsequent analytical solutions of the $\mathbb {C}P^{2S}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrized in terms of these polynomials. We apply these results to the analysis of surfaces associated with $\mathbb {C}P^{2S}$ models defined using the generalized 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.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-02148242
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Submitted on : Wednesday, June 5, 2019 - 12:27:42 PM
Last modification on : Wednesday, October 9, 2019 - 11:39:41 PM

### Citation

N. Crampe, A.M. Grundland. $\mathbb{C}P^{2S}$ sigma models described through hypergeometric orthogonal polynomials. Annales Henri Poincare, 2019, 20 (10), pp.3365-3387. ⟨10.1007/s00023-019-00830-2⟩. ⟨hal-02148242⟩

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