Annales Henri Poincaré

, Volume 20, Issue 10, pp 3365–3387 | Cite as

\(\mathbb {C}P^{2S}\) Sigma Models Described Through Hypergeometric Orthogonal Polynomials

  • N. CrampeEmail author
  • A. M. Grundland


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.

Mathematics Subject Classification

81T45 53C43 35Q51 



This research was supported by the NSERC operating grant of one of the authors (A.M.G.). N.C. is indebted to the Centre de Recherches Mathématiques (CRM) for the opportunity to hold a CRM-Simons professorship.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  2. 2.Institut Denis PoissonUniversité de Tours - Université d’OrléansToursFrance
  3. 3.Université du QuébecTrois-RivièresCanada

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