Boundary S-matrix Research Articles

Wrapping corrections for non-diagonal boundaries in AdS/CFT

We consider an open string stretched between a Y = 0 brane and a Y θ = 0 brane. The latter brane is rotated with respect to the former by an angle θ, and is described by a non-diagonal boundary S-matrix. This system interpolates smoothly between the Y −Y (θ = 0) and the $$ Y-\overline{Y}\left(\theta =\pi /2\right) $$ systems, which are described by diagonal boundary S-matrices. We use integrability to compute the energies of one-particle states at weak coupling up to leading wrapping order (4, 6 loops) as a function of the angle. The results for the diagonal cases exactly match with those obtained previously.

Fusion for AdS/CFT boundary S-matrices

We propose a fusion formula for AdS/CFT worldsheet boundary S-matrices. We show that, starting from the fundamental Y=0 boundary S-matrix, this formula correctly reproduces the two-particle bound-state boundary S-matrices.

Yangian symmetry and bound states in AdS/CFT boundary scattering

We consider the problem of boundary scattering for Y = 0 maximal giant graviton branes. We show that the boundary S-matrix for the fundamental excitations has a Yangian symmetry. We then exploit this symmetry to determine the boundary Smatrix for two-particle bound states. We verify that this boundary S-matrix satisfies the boundary Yang-Baxter equations.

A note on open-chain transfer matrices from q-deformed su(2|2)S-matrices

In this note, we perform Sklyanin's construction of commuting open-chain/boundary transfer matrices to the q-deformed SU(2|2) bulk S-matrix of Beisert and Koroteev and a corresponding boundary S-matrix. This also includes a corresponding commuting transfer matrix using the graded version of the q-deformed bulk S-matrix. Utilizing the crossing property for the bulk S-matrix, we argue that the transfer matrix for both graded and non-graded versions contains a crucial factor which is essential for commutativity.

Boundary S-matrix in a (2, 0) theory ofAdS3supergravity

We will discuss two inequivalent generalizations of the standard (2, 0) supergravity action [1] to include gravitational Chern-Simons term. One is in the first order formalism where we treat ωMab as independent and the other is in the second order formalism where ωMab is determined in terms of other fields via a standard constraint equation. The two theories have different equations of motion and the solutions to the equations of motion of the first order theory spans only a subset of those of the second order theory. We will be interested in computing the boundary S-matrix, describing correlation functions in a dual conformal field theory, in this sector and hence we use the equations of motion coming out of the first order theory. We restrict ourselves to the gauge+fermionic sector of the theory to compute the boundary S-matrix. We will also look at the effect of higher derivative terms on the boundary S-matrix obtained using the standard supergravity action.

Open-chain transfer matrices for AdS/CFT

We extend Sklyanin's construction of commuting open-chain transfer matrices to the SU(2|2) bulk and boundary S-matrices of AdS/CFT. Using the graded version of the S-matrices leads to a transfer matrix of particularly simple form. We also find an SU(1|1) boundary S-matrix which has one free boundary parameter.

Non-diagonal reflection for the non-critical XXZ model

The most general physical boundary S-matrix for the open XXZ spin chain in the non-critical regime (cosh(η) > 1) is derived starting from the bare Bethe ansatz equations. The boundary S-matrix as expected is expressed in terms of Γq-functions. In the isotropic limit the corresponding results for the open XXX chain are also reproduced.

Generic boundary scattering in the open XXZ chain

The open critical XXZ spin chain with a general right boundary and a trivial diagonal left boundary is considered. Within this framework we propose a simple computation of the exact generic boundary S-matrix (with diagonal and non-diagonal entries), starting from the ‘bare’ Bethe ansatz equations. Our results as anticipated coincide with the ones obtained by Ghoshal and Zamolodchikov, after assuming suitable identifications of the bulk and boundary parameters.

Izergin–Korepin model with a boundary

The Izergin-Korepin model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the twisted quantum affine algebra U-q[((2))(2)]. We give the bosonization of the vacuum state with zero particle content. Excitation states are given by the action of the vertex operators on the vacuum state. We derive the boundary S-matrix. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy. (C) 2001 Elsevier Science B.V. All rights reserved.

Boundary S-matrix for the Gross–Neveu model

We study the scattering theory for the Gross–Neveu model on the half-line. We find the reflection matrices for the elementary fermions, and by fusion we compute the ones for the two-particle bound-states, showing that they satisfy non-trivial bootstrap consistency conditions. We also compute more general reflection matrices for the Gross–Neveu model and the nonlinear sigma model, and argue that they correspond to the integrable boundary conditions we identified in our previous paper [5].

Perturbation Method for Boundary S-Matrix in 2D Quantum Field Theory

We develop a perturbation theory for evaluating the boundary S-matrix in 2D quantum field theory. We apply this approach to calculate the one-loop boundary S-matrix for the elementary particle of the sine–Gordon model with a boundary interaction. Our perturbative result agrees with the exact expression of the S-matrix conjectured by Goshal; it also allows us to derive the perturbative relation between the parameter ϑ in the S-matrix and the free parameter M in the boundary action, in the particular case in which its other free parameter φ0 is zero.

Boundary ABF models

We diagonalize the transfer matrix of boundary ABF models using bosonized vertex operators. We compute the boundary S-matrix and check the scaling limit against known results for perturbed boundary conformal field theories.

The supersymmetrict-Jmodel with a boundary

An open supersymmetric t-J chain with boundary fields is studied by means of the Bethe Ansatz. Ground state properties for the case of an almost half-filled band and a bulk magnetic field are determined. Boundary susceptibilities are calculated as functions of the boundary fields. The effects of the boundary on excitations are investigated by constructing the exact boundary S-matrix. From the analytic structure of the boundary S-matrices one deduces that holons can form boundary bound states for sufficiently strong boundary fields.

Boundary S-matrix for the integrable q-Potts model

The 2D off-critical q-state Potts model with boundaries was studied as a factorizable relativistic scattering theory. The scattering S-matrices for particles reflecting off the boundaries were obtained for the cases of ``fixed'' and ``free'' boundary conditions. In the Ising limit, the computed results agreed with recent work[5].

Boundary energy and boundary states in integrable quantum field theories

We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of O( n) and minimal models.

Difference equations in spin chains with a boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the eight-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary S-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal-Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.

Boundary sine-Gordon interactions at the free fermion point

We study bosonization of the sine-Gordon theory in the presence of boundary interactions at the free fermion point. In this way we obtain the boundary S-matrix as a function of physical parameters in the boundary sine-Gordon Lagrangian. The boundary S-matrix can be matched onto the solution of Ghoshal and Zamolodchikov, thereby relating the formal parameters in the latter solution to the physical parameters in the lagrangian.

XXZ chain with a boundary

The XXZ spin chain with a boundary magnetic field h is considered, using the vertex operator approach to diagonalize the hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when h ⩾ 0, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary S-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the XXX model.

Direct calculation of the boundary S-matrix for the open Heisenberg chain

We calculate the boundary S-matrix for the open antiferromagnetic spin- 1/2 isotropic Heisenberg chain with boundary magnetic fields. Our approach, which starts from the model's Bethe-ansatz solution, is an extension of the Korepin-Andrei-Destri method. Our result agrees with the boundary S-matrix for the boundary sine-Gordon model with beta 2 to 8 pi and with 'fixed' boundary conditions.

Boundary S-matrix of the O( N)-symmetric non-linear sigma model

We conjecture that the O( N)-symmetric non-linear sigma model (with N⩾3) in the semi-infinite (1+1)-dimensional space is “integrable” with respect to the “free” and the “fixed” boundary conditions. We then derive, for both cases, the boundary S-matrix for the reflection of massive particles of this model off the boundary at x = 0.