Michel Grundland, UQTR et CRM

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

Abstract: The objective of this talk is to establish a new relationship between the Veronese sequence of 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 non-intersecting 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 a sequence of analytic 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. Joint work with Nicolas Crampé.