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

Abstract

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].

Boundary remnant of Yangian symmetry and the structure of rational reflection matrices

“…(Recall [1] that the admissible E locally parametrize G/H, although choosing k L = k R = 1 makes the specific choice E = diag(I M , −I N −M ).) They project onto (1, ) and ( , 1) respectively, so that the vector particle reflection matrix is as given in the previous section.…”

“…The vector multiplet bulk S-matrix has a simple pole at θ = 2iπ c A whose residue is the projector onto the second-rank antisymmetric representation of SU(N). This is interpreted according to the bootstrap principle as another particle multiplet, for which the reflection matrix is 1) or, diagrammatically,…”

“…(The reader is referred to [1] for further details.) Expressing the above diagram in this notation, we obtain…”

“…It was noted in [1] that the vector particle boundary scattering matrix K = θ 0 corresponding to the formation of a boundary bound state, transforming in ( , 1).…”

Introduction to Yangian Symmetry in Integrable Field Theory

“…the WZNW model on the half-line) [114]. Actually there is more freedom in such models, because the currents there have holomorphic components onlythe relation is described in [97].…”