Classical O(N) nonlinear sigma model on the half line: a study on consistent Hamiltonian description

Abstract

The problem of consistent Hamiltonian structure for O(N) nonlinear sigma model in the presence of five different types of boundary conditions is considered in detail. For the case of Neumann, Dirichlet and the mixture of these two types of boundaries, the consistent Poisson brackets are constructed explicitly, which may be used, e.g., for the construction of current algebras in the presence of boundary. While for the mixed boundary conditions and the mixture of mixed and Dirichlet boundary conditions, we prove that there is no consistent Poisson brackets, showing that the mixed boundary conditions are incompatible with all nontrivial subgroups of O(N).

Classically integrable boundary conditions for symmetric-space sigma models

“…We have found that amongst the circles only the great circles give boundary conditions compatible with the Poisson bracket structure inherited from the model on the whole line, and then, from the existence of local conserved charges in involution, that these BCs are integrable. This accords with the results of Zhao and He [9], who find that there is no consistent set of Poisson brackets for the model with a general circle as the Dirichlet submanifold (their 'MD' condition). The case G = SO(N) was also studied in [2,3,9].…”

“…This accords with the results of Zhao and He [9], who find that there is no consistent set of Poisson brackets for the model with a general circle as the Dirichlet submanifold (their 'MD' condition). The case G = SO(N) was also studied in [2,3,9].…”

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.…”

“…The point is that with σ being the transposition operator in G × G, the only admissible nontrivial τ are (α, α) and σ(α, α) (with α being a freely chosen involution), and they respectively yield our two types of boundary condition: (2), (6) and (3), (7).…”

“…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].…”