Nonlinear Sigma Models on a Half Plane

Mourad, M. F.; Sasaki, Ryu

doi:10.1142/s0217751x96001504

Cited by 8 publications

(15 citation statements)

References 22 publications

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“…For integrability we require a great deal of R. As we have said, there are two infinite sets of charges. The Yangian charges appear no longer to be conserved on the half-line, even with pure Neumann BCs [19] (naively, at least, it seems that there are remnants only, as we shall see). However, we believe they are not essential for integrability, because precisely half of the local charges remain conserved, with either q s + q −s or q s − q −s surviving.…”

Section: Boundary Conditions For the Model On The Half-linementioning

confidence: 83%

Boundary Scattering, Symmetric Spaces and the Principal Chiral Model on the Half-Line

MacKay

Short

2003

Communications in Mathematical Physics

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We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric 1 summarize: a connection between boundary integrability and symmetric spaces emerges naturally in two very different ways: by seeking classically integrable boundary conditions, and by solving the BYBE.The plan of the paper is as follows. In section two, building naturally on the results of [3,4] for the bulk PCM, we discuss boundary conditions which lead naturally to conservation of local charges. As mentioned, there are two classes of BC, which we call 'chiral' and 'non-chiral'. In section three we find minimal boundary S-matrices, by making ansätze for the BYBE solutions and applying the conditions of crossing-unitarity, hermitian analyticity and R-matrix unitarity, and explain how these are related to symmetric spaces. This section is necessarily rather long and involved, and many of the details appear in appendices. From these, in section four, we construct boundary S-matrices for the PCM, and find that these correspond naturally to the chiral BCs. The key statements of our results for the boundary S-matrices can be found in section 3.4 (for the minimal case, without physical strip poles) and section 4.2 (for the full PCM S-matrices). This paper supersedes the preliminary work of [6].

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Section: Boundary Conditions For the Model On The Half-linementioning

confidence: 83%

Boundary Scattering, Symmetric Spaces and the Principal Chiral Model on the Half-Line

MacKay

Short

2003

Communications in Mathematical Physics

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“…Therefore, we conclude that there is no solution to the equations (27), (28) with W = M −1 . This implies that the AM boundaries are not compatible with the bulk O(N) symmetry, which has been used to obtain the Poisson brackets (5)-(7) upon which the equations (27), (28) are based. Therefore, we shall temporarily restrict ourselves to the cases (a) and (b).…”

Section: O(n) Symmetric Boundaries Ad and Anmentioning

confidence: 84%

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

Zhao

2003

Physics Letters B

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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).

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“…vanishes only on h, so the G-symmetry is broken to H. A similar calculation for Q (1) gives 6) which vanishes neither on h nor on m. At first it was thought that this meant that nonlocal charges were not essential for integrability [98], but it was later noticed that a modified set of nonlocal charges is conserved [99], as follows.…”

Section: Boundary Remnant Of Yangian Chargesmentioning

confidence: 99%