ℂP^N sigma models via the SU(2) coherent states approach

2017/01/24 by Grundland, A. M., Strasburger, A., Dziewa-Dawidczyk, D.
#53B50 #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 81T45 #Secondary 53A07

paper · doi:10.48550/arxiv.1701.06994

Abstract

In this paper we present results obtained from the unification of SU(2) coherent states with ℂP^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 ℂP^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 ℂP^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.

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