Soliton surfaces associated with sigma models: differential and algebraic aspects

Goldstein, P. P.; Grundland, A. M.; Post, Sarah

doi:10.1088/1751-8113/45/39/395208

Cited by 8 publications

(14 citation statements)

References 35 publications

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“…Namely α k +ᾱ k = tr(∂P k∂ P k ) is the Lagrangian density in the action functional (1). Moreover, we have shown in [10] that tr(∂X k∂ X k ) = −tr(∂P k∂ P k ), which makes this quantity also the Lagrangian density for the surface immersion functions. It is also the non-diagonal element g 12 = g 21 of the metric tensor on the surface X k , while the diagonal elements of the metric tensor are zero [7].…”

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

confidence: 97%

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

confidence: 97%

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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“…The coefficients in (22) are the coordinates of the spin vector in the basis (17). For such matrices S z we have Proposition 1.…”

Section: Properties Of the Spin Matricesmentioning

confidence: 96%

“…According to Property 4, the spin matrices S z satisfy the E-L equations (5). Substituting (22) into those equations, we obtain the coordinates of (23) in the basis (17).…”

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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“…A direct connection was established between the CP 2s model and the spin-s su(2) representation [6,7]. The spin matrix S z is defined as a linear combination of the (2s+1) rank-1 Hermitian projectors P k , i.e.…”

Section: Projective Formalismmentioning

confidence: 99%

Veronese subsequent analytic solutions of the $\mathbb{C}P^{2s}$ sigma model equations described via Krawtchouk polynomials

Crampé¹,

Grundland²

2019

Preprint

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The objective of this paper is to establish a new relationship between the Veronese subsequent analytic solutions of the Euclidean CP 2s sigma model in two dimensions and the orthogonal Krawtchouk polynomials. We show that such solutions of the CP 2s model, defined on the Riemann sphere and having a finite action, can be explicitly parametrised in terms of these polynomials. We apply the obtained results to the analysis of surfaces associated with CP 2s sigma models, defined using the generalized Weierstrass formula for immersion.We show that these surfaces are spheres immersed in the su(2s + 1) Lie algebra, and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a new connection between the su(2) spin-s representation and the CP 2s model is explored in detail. It is shown that for any given holomorphic vector function in C 2s+1 written as a Veronese sequence, it is possible to derive subsequent solutions of the CP 2s model through algebraic recurrence relations which turn out to be simpler than the analytic relations known in the literature.

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Soliton surfaces associated with sigma models: differential and algebraic aspects

Cited by 8 publications

References 35 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

Veronese subsequent analytic solutions of the $\mathbb{C}P^{2s}$ sigma model equations described via Krawtchouk polynomials

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