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$\mathbb CP^{N}$ sigma models via the $\mathbf{SU}(2)$ coherent states approach

Volume 113 / 2017

A. M. Grundland, A. Strasburger, D. Dziewa-Dawidczyk Banach Center Publications 113 (2017), 169-191 MSC: 53C42, 53C43, 70S10, 81R30. DOI: 10.4064/bc113-0-10


In this paper we present results obtained from the unification of ${\bf SU}(2)$ coherent states with $\mathbb{C}P^{N}$ sigma models defined on the Riemann sphere having finite actions. The set of coherent states generated by a vector belonging to a carrier space of an irreducible representation of the group gives rise to a map from the sphere into the set of rank-1 Hermitian projectors in that space. The map can be identified with a particular solution of the $\mathbb{C}P^{N}$ sigma model, where $N+1$ is equal to the dimension of the representation space. In particular a choice of the generating vector as the highest weight vector of the representation gives rise to the map known as a Veronese immersion. Using a description of the matrix elements of these representations in terms of Jacobi polynomials, we obtain an explicit parametrization of the solutions of the $\mathbb{C}P^{N}$ models, which has not been previously found. We relate the analytical properties of the solutions, which are known to belong to separate classes—holomorphic, anti-holomorphic and various types of mixed ones—to the weight corresponding to the chosen weight vector. Some examples of the described constructions are elaborated in detail in this paper.


  • A. M. GrundlandCentre de Recherches Mathématiques
    Université de Montréal
    Montreal, C.P. 6128 (QC) H3C 3J7, Canada
    Department of Mathematics and Computer Sciences
    Université du Québec
    Trois-Rivières, CP 500 (QC) G9A 5H7, Canada
  • A. StrasburgerInstitute of Mathematics and Cryptology, Faculty of Cybernetics
    Military University of Technology — WAT
    ul. S. Kaliskiego 2
    00-908 Warsaw, Poland
  • D. Dziewa-DawidczykDepartment of Applied Mathematics
    Warsaw University of Life Sciences (SGGW)
    ul. Nowoursynowska 159
    02-787 Warszawa, Poland

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