Abstract
The objective of this paper is to establish a new relationship between the Veronese sequence of analytic solutions of the Euclidean 
sigma model in two dimensions and the orthogonal Krawtchouk polynomials. We show that such solutions of the 
model, defined on the Riemann sphere and having a finite action, can be explicitly parametrized in terms of these polynomials. We apply the obtained results to the analysis of surfaces associated with 
sigma models, defined using the generalized Weierstrass formula for immersion. We show that these surfaces are spheres immersed in the 𝔰𝔲(2s + 1) Lie algebra, and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a new connection between the 𝔰𝔲(2) spin-s representation and the 
model is explored in detail. It is shown that for any given holomorphic vector function in 
written as a Veronese sequence, it is possible to derive a sequence of analytic solutions of the 
model through algebraic recurrence relations which turn out to be simpler than the analytic relations known in the literature.
In honour of Decio Levi (University of Roma Tre)
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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), Université de Montréal for the opportunity to hold a CRM-Simons professorship.
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Crampé, N., Grundland, A.M. (2021). The Veronese Sequence of Analytic Solutions of the 
Sigma Model Equations Described via Krawtchouk Polynomials.
In: Paranjape, M.B., MacKenzie, R., Thomova, Z., Winternitz, P., Witczak-Krempa, W. (eds) Quantum Theory and Symmetries. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-55777-5_5
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