Analysis of \(\mathbb {C}P^{N-1}\) Sigma Models via Soliton Surfaces

  • P. P. Goldstein
  • A. M. GrundlandEmail author
Part of the CRM Series in Mathematical Physics book series (CRM)


In this paper we present results obtained from the study of an invariant formulation of completely integrable \(\mathbb {C}P^{N-1}\) Euclidean sigma models in two dimensions defined on the Riemann sphere, having finite actions. Surfaces connected with the \(\mathbb {C}P^{N-1}\) models, invariant recurrence relations linking the successive projection operators, and immersion functions of the surfaces are discussed in detail. We show that immersion functions of 2D-surfaces associated with the \(\mathbb {C}P^{N-1}\) model are contained in 2D-spheres in the \(\mathfrak {su}(N)\) algebra. Making use of the fact that the immersion functions of the surfaces satisfy the same Euler–Lagrange equations as the original projector variables, we derive surfaces induced by surfaces and prove that the stacked surfaces coincide with each other, which demonstrates the idempotency of the recurrent procedure. We also demonstrate that the \(\mathbb {C}P^{N-1}\) model equations admit larger classes of solutions than the ones corresponding to rank-1 Hermitian projectors. This fact allows us to generalize the Weierstrass formula for the immersion of 2D-surfaces in the \(\mathfrak {su}(N)\) algebra and to show that, in general, these surfaces cannot be conformally parametrized. Finally, we consider the connection between the structure of the projective formalism and the possibility of spin representations of the \(\mathfrak {su}(2)\) algebra in quantum mechanics.


Spin matrices Sigma models Soliton surfaces Integrable systems Weierstrass formula for immersion 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Theoretical Physics DivisionNational Centre for Nuclear ResearchWarsawPoland
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada

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