The Veronese Sequence of Analytic Solutions of the CP 2s Sigma Model. . .
65
The Kähler angles are given by
tan
1
2
θ k (m)
=
df k (m)(∂/∂ξ − )
df k (m)(∂/∂ξ + )
,
m∈ S
2
and have constant positive values
cos θ k =
s − k
2sk + s − k 2 .
The Euler–Poincaré characters of the 2D-surfaces X k are the integer Δ k = 2 for all
k such that 0 ≤ k ≤ 2s. This means that all 2D-surfaces associated with the CP 2s
model are non-intersecting spheres with radius R k given by (32).
The technique for obtaining surfaces via projective structures and their links with
orthogonal polynomials, elaborated from the CP 2s models, can be extended to
different types of Grassmannian manifolds. An analysis of these manifolds can
provide us with much more diverse types of surfaces.
Acknowledgments 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.
References
1. W.J. Zakrzewski, Low Dimensional Sigma Models (Hilger, Bristol, 1989)
2. P.P. Goldstein, A.M. Grundland, Invariant recurrence relations for CP N −1 models. J. Phys. A
Math. Theor. 43(265206), 1–18 (2010)
3. P.P. Goldstein, A.M. Grundland, S. Post, Soliton surfaces associated with sigma models:
differential and algebraic aspects. J. Phys. A Math. Theor. 45(395208), 1–19 (2012)
4. P.P. Goldstein, A.M. Grundland, On a stack of surfaces obtained from the CP N −1 sigma model.
J. Phys. A Math. Theor. 51(095201), 1–13 (2018)
5. J. Bolton, G.R. Jensen, M. Rigoli, L.M. Woodward, On conformal minimal immersion of S 2
into CP N . Math. Ann. 279, 599–620 (1988)
6. N. Crampé, A.M. Grundland, CP 2S sigma models described through hypergeometric orthogonal polynomials. Ann. Henri Poincaré 20(10), 1–23 (2019)
7. P.P. Goldstein, A.M. Grundland, Analysis of CP N −1 sigma models via soliton surfaces, in A
Volume in Honour of V. Hussin, ed. by S. Kuru, et al. CRM series in Math. Phys. (Springer,
New York, 2019), pp. 341–358
8. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1998)
9. T. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic
interpretation. SIAM J. Math. Anal. 13, 10111023 (1982)
10. Y.I. Granovski, A. Zhedanov, Orthogonal polynomials in the Lie algebras. Sov. Phys. J. 29,
387393 (1986)
11. A.M. Grundland, I. Yurdusen, On analytic descriptions of two-dimensional surfaces associated
with the CP N −1 model. J. Phys. A Math. Theor. 42(172001), 1–5 (2009)
65
The Kähler angles are given by
tan
1
2
θ k (m)
=
df k (m)(∂/∂ξ − )
df k (m)(∂/∂ξ + )
,
m∈ S
2
and have constant positive values
cos θ k =
s − k
2sk + s − k 2 .
The Euler–Poincaré characters of the 2D-surfaces X k are the integer Δ k = 2 for all
k such that 0 ≤ k ≤ 2s. This means that all 2D-surfaces associated with the CP 2s
model are non-intersecting spheres with radius R k given by (32).
The technique for obtaining surfaces via projective structures and their links with
orthogonal polynomials, elaborated from the CP 2s models, can be extended to
different types of Grassmannian manifolds. An analysis of these manifolds can
provide us with much more diverse types of surfaces.
Acknowledgments 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.
References
1. W.J. Zakrzewski, Low Dimensional Sigma Models (Hilger, Bristol, 1989)
2. P.P. Goldstein, A.M. Grundland, Invariant recurrence relations for CP N −1 models. J. Phys. A
Math. Theor. 43(265206), 1–18 (2010)
3. P.P. Goldstein, A.M. Grundland, S. Post, Soliton surfaces associated with sigma models:
differential and algebraic aspects. J. Phys. A Math. Theor. 45(395208), 1–19 (2012)
4. P.P. Goldstein, A.M. Grundland, On a stack of surfaces obtained from the CP N −1 sigma model.
J. Phys. A Math. Theor. 51(095201), 1–13 (2018)
5. J. Bolton, G.R. Jensen, M. Rigoli, L.M. Woodward, On conformal minimal immersion of S 2
into CP N . Math. Ann. 279, 599–620 (1988)
6. N. Crampé, A.M. Grundland, CP 2S sigma models described through hypergeometric orthogonal polynomials. Ann. Henri Poincaré 20(10), 1–23 (2019)
7. P.P. Goldstein, A.M. Grundland, Analysis of CP N −1 sigma models via soliton surfaces, in A
Volume in Honour of V. Hussin, ed. by S. Kuru, et al. CRM series in Math. Phys. (Springer,
New York, 2019), pp. 341–358
8. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1998)
9. T. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic
interpretation. SIAM J. Math. Anal. 13, 10111023 (1982)
10. Y.I. Granovski, A. Zhedanov, Orthogonal polynomials in the Lie algebras. Sov. Phys. J. 29,
387393 (1986)
11. A.M. Grundland, I. Yurdusen, On analytic descriptions of two-dimensional surfaces associated
with the CP N −1 model. J. Phys. A Math. Theor. 42(172001), 1–5 (2009)