Classical Integrability of the O(N) Nonlinear Sigma Model on a Half-Line

Abstract

The classical integrability the O(N) nonlinear sigma model on a half-line is examined, and the existence of an infinity of conserved charges in involution is established for the free boundary condition. For the case N = 3 other possible boundary conditions are considered briefly.

Quantum integrability of certain boundary conditions

“…In that paper Ghoshal solved the boundary Yang-Baxter equation consistent with this choice (and this is the main argument for its integrability!). The classical integrability of the Neumann condition for the O(N) nlσ model was established by means of a generalization of the Lax pair to the half-line by Corrigan and Sheng in [7].…”

“…In this section we prove the integrability of Neumann (∂ 1 n| x=0 = 0), Dirichlet ( n| x=0 = n 0 a constant, or equivalently ∂ 0 n| x=0 = 0), and a mixed boundary condition (where some components of n satisfy Neumann and the others Dirichlet). We also analyze the boundary condition proposed by Corrigan and Sheng in [7] for the O(3) nlσ model. For the GN model we show that the spin 4 charge discussed in the previous section, with the boundary condition ψ a + | x=0 = ǫ a ψ a − | x=0 , ǫ a = ±1, provides a conserved charge in the boundary case.…”

Boundary integrability of nonlinear sigma models

“…The next example (also in [9]) is SO(3)/SO (2), for which some detailed results exist [3], [6]. Allowing τ to be an arbitrary involution and thus allowing H τ to be an arbitrary SO(2) would give a D-submanifold {σ(g 0 l)(g 0 l) −1 | l ∈ SO(2) } ⊂ S 2 , which is an arbitrary circle.…”

“…But relatively little attention has been given to the nonlinear sigma model (i.e., harmonic maps from the half-plane), mostly limited to the O(N ) model (the model on SO(N )/SO(N − 1)) [3]- [6]. Here, we report work that began with the principal chiral field (on a compact Lie group G) [7], [8] and was then generalized to compact symmetric spaces G/H (with the principal chiral field realized as the symmetric space G × G/G) [9].…”

Supersymmetric WZW Σ Model on Infinite and Half-Plane

“…The classical integrability of the O(N ) nonlinear σ model on a half-line was examined by Corrigan and Sheng [10], where they established the existence of an infinity of conserved charges in involution for the free boundary condition. Moriconi and De Martino [11] have studied the quantum integrability of the O(N ) nonlinear σ model and the O(N ) Gross-Neveu model on the half-line and found that the first model is integrable with Neumann, Dirichlet and a mixed boundary conditions and that the latter model is integrable under a certain condition.…”