On the surfaces associated with ℂPN−1models

Under the assumption that the CP N −1 sigma model is defined on the Riemann sphere and its action functional is finite, we derive surfaces induced by surfaces and we demonstrate that the stacked surfaces coincide with each other, which means idempotency of the recurrent procedure. Along the path to the solutions of the CP N −1 model equations, we demonstrate that the Euler-Lagrange equations for the projectors admit larger classes of solutions than the ones corresponding to rank-1 projectors.PACS numbers: primary 02.40Hw, 02.20Sv, secondary 02.30Ik AMS classification scheme numbers: 81T45, 53C43, 35Q51

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“…since the projectors P i are mutually orthogonal (9). The idempotency condition for the projector P requires that…”

Section: Higher-rank Projectors As Solutions Of the Euler-lagrange Eqmentioning

confidence: 99%

“…The main advantages of this approach are that this formulation preserves the conformal and scaling invariance of these quantities. It allows us to construct a regular algorithm for finding certain classes of solutions having a finite action functional [9,12]. A broad review of recent developments in this subject can be found in, e.g.…”

Section: Introductionmentioning

confidence: 99%

On a stack of surfaces obtained from the $\boldsymbol {\mathbb{C}P^{N-1}}$ sigma models

Goldstein

Grundland

2018

J. Phys. A: Math. Theor.

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“…These surfaces have no common points, except for CP 1 , where the only two surfaces X 0 and X 1 coincide [16].…”

Section: Basics Of the Cp 2s Sigma Modelsmentioning

confidence: 99%

“…Property 5 . It follows from Property 4 that the spin matrices S z are conditional stationary points of the same action integral (3) as the projectors, but the condition is (16) (rather than P 2 = P ).…”

Section: Properties Of the Spin Matricesmentioning

confidence: 99%

The $\mathfrak{su}(2)$ spin $s$ representations via $\mathbb{C}P^{2s}$ sigma models

Goldstein¹,

Grundland²,

Ruiz³

2020

Preprint

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We establish and analyze a new relationship between the matrices describing an arbitrary component of a spin s, where 2s ∈ Z + , and the matrices of CP 2s two-dimensional Euclidean sigma models. The spin matrices are constructed from the rank-1 Hermitian projectors of the sigma models or from the antihermitian immersion functions of their soliton surfaces in the su(2s + 1) algebra. For the spin matrices which can be represented as a linear combination of the generalized Pauli matrices, we find the dynamics equation satisfied by its coefficients. The equation proves to be identical to the stationary equation of a two-dimensional Heisenberg model. We show that the same holds for the matrices congruent to the generalized Pauli ones by any coordinate-independent unitary linear transformation. These properties open the possibility for new interpretations of the spins and also for application of the methods known from the theory of sigma models to the situations described by the Heisenberg model, from statistical mechanics to quantum computing.

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“…The su(N + 1)-valued matrix functions X k (ξ, ξ) constitute the generalized Weierstrass formula for the immersion of 2D surfaces into R N (N +2) , isomorphic to the Lie algebra su(N + 1), [19]. The matrix-valued functions X k satisfy the following cubic matrix equations (the minimal polynomial identity) [20],…”

Section: Solutions Of Cp N Models Expressed In Terms Of Projectorsmentioning

confidence: 99%

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

Grundland

Strasburger

Dziewa-Dawidczyk

2017

Banach Center Publ.

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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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On the surfaces associated with ℂP^N−1models

Cited by 11 publications

References 15 publications

On a stack of surfaces obtained from the $\boldsymbol {\mathbb{C}P^{N-1}}$ sigma models

On a stack of surfaces obtained from the $\boldsymbol {\mathbb{C}P^{N-1}}$ sigma models

The $\mathfrak{su}(2)$ spin $s$ representations via $\mathbb{C}P^{2s}$ sigma models

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

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