The Veronese Sequence of Analytic
Solutions of the CP 2s Sigma Model
Equations Described via Krawtchouk
Polynomials
Nicolas Crampé and Alfred Michel Grundland
In honour of Decio Levi (University of Roma Tre)
Abstract The objective of this paper is to establish a new relationship between
the Veronese sequence of analytic solutions of the Euclidean CP 2s sigma model in
two dimensions and the orthogonal Krawtchouk polynomials. We show that such
solutions of the CP 2s 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 CP 2s 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 CP 2s
model is explored in detail. It is shown that for any given holomorphic vector
function in C 2s+1 written as a Veronese sequence, it is possible to derive a sequence
of analytic solutions of the CP 2s model through algebraic recurrence relations
which turn out to be simpler than the analytic relations known in the literature.
Keywords Sigma models · Projector formalism · Integrable systems · Soliton
surfaces · Weierstrass formula for immersion · Spin matrices
Mathematical Subject Classification 81T45, 53C43, 35Q51
N. Crampé
Institut Denis Poisson, Université de Tours – Université d’Orléans, Orléans, France
A. M. Grundland ()
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC H3C 3J7, Canada
Université du Québec à Trois-Rivières, Trois-Rivières, QC G9A 5H7, Canada
e-mail: grundlan@crm.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_5
57
Solutions of the CP 2s Sigma Model
Equations Described via Krawtchouk
Polynomials
Nicolas Crampé and Alfred Michel Grundland
In honour of Decio Levi (University of Roma Tre)
Abstract The objective of this paper is to establish a new relationship between
the Veronese sequence of analytic solutions of the Euclidean CP 2s sigma model in
two dimensions and the orthogonal Krawtchouk polynomials. We show that such
solutions of the CP 2s 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 CP 2s 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 CP 2s
model is explored in detail. It is shown that for any given holomorphic vector
function in C 2s+1 written as a Veronese sequence, it is possible to derive a sequence
of analytic solutions of the CP 2s model through algebraic recurrence relations
which turn out to be simpler than the analytic relations known in the literature.
Keywords Sigma models · Projector formalism · Integrable systems · Soliton
surfaces · Weierstrass formula for immersion · Spin matrices
Mathematical Subject Classification 81T45, 53C43, 35Q51
N. Crampé
Institut Denis Poisson, Université de Tours – Université d’Orléans, Orléans, France
A. M. Grundland ()
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC H3C 3J7, Canada
Université du Québec à Trois-Rivières, Trois-Rivières, QC G9A 5H7, Canada
e-mail: grundlan@crm.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_5
57