Integrable Extension of Nonlinear Sigma Model

We propose an integrable extension of nonlinear sigma model on the target space of Hermitian symmetric space (HSS). Starting from a discussion of soliton solutions of O(3) model and an integrally extended version of it, we construct general theory defined on arbitrary HSS by using the coadjoint orbit method. It is based on the exploitation of a covariantized canonical structure on HSS. This term results in an additional first-order derivative term in the equation of motion, which accommodates the zero curvature representation. Infinite conservation laws of nonlocal charges in this model are derived. PACS codes: 11.10.Lm, 11.10.Kk 1 E-mail: [email protected] The study of integrability of the nonlinear sigma model (NLSM) on the target space of homogeneous space G/H was initiated in the work of [1] where it was discovered that O(N) NLSM admits a zero curvature representation, and infinite family of local and nonlocal conserved charges [2] was derived along with one parameter family of Bäcklund transformation. This result was immediately generalized to the target space of CP (N) [3] and to the principal chiral SU(N) model where nontrivial soliton solutions were obtained by applying the inverse scattering technique [4]. Later, the complete integrability on the target space of Riemannian symmetric spaces was established through the works of [5], and various aspects of integrability were developed and applied to the survey of two dimensional quantum field theory [6]. In this Letter, we search for a possible integrable extension of the NLSM on the homogeneous space G/H. Our main result is that the NLSM allows an integrable extension when the target space is given by the Hermitian symmetric space (HSS) which is a symmetric space equipped with a complex structure acting on the coset [7]. In achieving this, a covariantized canonical structure term on HSS is added to the original action which results in a first-order total derivative in the equation of motion, and this is found to be completely integrable. In the process, we use the coadjoint orbit method [8] as an essential tool. We start from a discussion of an integrable extension of the simple O(3) model in this method and its soliton solution to give some motivation. Let us consider the action of O(3) NLSM: S = 1 2 ∫ dx∂μ ~ Q · ∂ μ ~ Q, ~ Q · ~ Q = 1. (1) We consider solution of the following equation; (with prime = ∂ ∂x , dot = ∂ ∂t ) ~ Q× ~ Q − ~ Q× ~̈ Q = 0. (2) 2 Note that cross product with ~ Q reproduces the formal equation of motion of (1), 2~ Q − ( ~ Q · 2~ Q) ~ Q = 0. We use the standard parameterization for S given by ~ Q = (sin θ cosφ, sin θ sin φ, cos θ). Substitution into (2) leads to 2 cos θθφ + sin θφ − 2 cos θθ̇φ̇− sin θφ̈ = 0, (3) θ − sin θ cos θφ ′2 − θ̈ + sin θ cos θφ̇ = 0. (4) Let us try solutions of the form θ ≡ θ(x− vt), φ ≡ φ̂(x− vt) + Ωt. We obtain 0 = 2(1− v) cos θθξφ̂ξ + (1− v ) sin θφ̂ξξ + 2Ωv cos θθξ, (5) 0 = (1− v)θξξ − (1− v ) sin θ cos θφ̂ξ − 2vΩ sin θ cos θφ̂ξ + Ω 2 sin θ cos θ, (6) with ξ = x− vt. Let χ = (1− v) sin θφ̂ξ (|v| < 1). Then, from (5) χξ = −θξ(2vΩ cos θ + χ cot θ), (7) which upon integration yields χ = vΩ(cos 2θ + c0) 2 sin θ . (8) From (6), we obtain θξ 2 + χ (1− v2)2 − Ω cos 2θ 2(1− v2) = E. (9) We choose c0 = −1, which simplifies χ to χ = −vΩ sin θ. Let us consider the effective potential θξ 2 + Veff(θ) = E given by Veff(θ) = − Ω cos θ (1− v2)2 + Ω(1 + v) 2(1− v2) . (10) For a soliton solution, we choose E = Ω 2(1+v2) 2(1−v2) , which yields θξ 2 = Ω (1− v2)2 cos θ. (11)