New boundary monodromy matrices for classical sigma models

Abstract

The 2d principal models without boundaries have G×G symmetry. The already known integrable boundaries have either H ×H Hereinafter, we look for consistent solutions for the equation (5). The most obvious ansatz for the reflection matrix is κ(λ) = U where U ∈ G is a constant matrix. Using this ansatz, the equation (5) is equivalent to the following two equations:Clearly, J 0 and J 1 are elements of the eigenspaces of the linear transformation Ad U : g → g with +1 and −1 eigenvalues. These are equivalent to dim(g) boundary conditions if and only if U 2 is proportional to 1. Thus, there is a Z 2 graded decomposition g = h ⊕ f where h and f are the +1 and −1 eigenspaces of the Ad U automorphism of g. Therefore the boundary conditions imply J 0 ∈ h and J 1 ∈ f. These are well known integrable boundary conditions [6]. In the

D-branes in λ-deformations

“…This section closely follows [22,23] for obtaining open string boundary conditions that preserve integrability based on a method first introduced by Cherdnik and Sklyanin [20,21] in the context of two-dimensional integrable systems. We add here a slightly more general procedure applicable to integrable sigma models which has not been clearly spelled out yet in the literature (however, see [28] for a recent usage hereof in the case of the PCM) but which can lead to distinct integrable D-brane configurations. Consider first a general (1 + 1)-dimensional field theory in a spacetime (or worldsheet) Σ parametrised by (τ, σ) on a periodic or infinite line with a global symmetry group G. The model is said to be classically integrable when its equations of motion can be recast in a zero curvature condition of a g C -valued Lax connection one-form L(z) depending on a spectral parameter z ∈ C [63],…”

“…We shall see that this quite naturally generalises the gluing conditions used in the case of the WZW model. This approach has its origins in [20,21] and has been used in a variety of contexts including the identification of integrable boundary conditions for strings in bosonic sigma models [22], in Green-Schwarz sigma models [23], 2 for the O(N ) sigma model [25,26], the principal chiral model [27,28], open spin chains (e.g. [29,30] although the literature is vast) and affine Toda field theories [31].…”

“…However the spectral parameter independence e.g. might be relaxed as in[28] where the objective is to map the boundary conditions to known boundary scattering matrices of the quantum theory containing a free parameter.…”

String integrability of defect CFT and dynamical reflection matrices

“…In principle, the reflection matrix can depend on the spectral parameter, or the dynamical variables, or both. The dynamical reflection matrices arise, for example, in O(N ) models with Robin boundary conditions [30,31]. Robin (i.e.…”

On the classification of rational K-matrices

“…which is the classical boundary Yang-Baxter equation for spectral parameter dependent κmatrix. In [Gom18a], there was derived some solutions of this equation in the defining representations of the matrix Lie-algebras. These solutions can be matched to the parameter dependent solutions of the bYBE.…”