Boundary reflection matrices for nonsimply laced affine Toda field theories

Abstract

The boundary reflection matrices for nonsimply laced affine Toda field theories defined on a half line with the Neumann boundary condition are investigated. The boundary reflection matrices for some pairs of the models are evaluated up to one loop order by perturbation theory. Then the exact boundary reflection matrices which are consistent with the one loop result are found under the assumption of ''duality'' and tested against algebraic consistency such as the boundary bootstrap equation and boundary crossing-unitarity relation.

Universal boundary reflection amplitudes

“…We obtain The solutions (4.36), (4.37) correspond precisely to those found by J.D. Kim in [56] after re-defining the effective coupling as B → B/2 and shifting θ by iπ. These solutions are especially trustworthy as they have also been double checked against perturbation theory.…”

“…For the simply laced case, it will turn out that our solutions coincide with the ones found by Kim [55] upon the use of the ambiguity (2.8) 2 . For the non-simply laced cases only two specific examples have been treated in [56]. On further solutions related to non-simply laced algebras we shall comment below.…”

Boundary reflection matrix foradeaffine Toda field theory

“…Parts of them have been already presented in Refs. [12,16] and are presented in the appendix of this paper.…”

“…In Ref. [16], we evaluated one loop boundary reflection matrix for d (1) 4 affine Toda field theory and showed a remarkable cancellation of non-meromorphic terms among themselves. This result also enabled us to determine the exact boundary reflection matrix uniquely under the assumption of the strong-weak coupling 'duality'.…”

Square root singularity in boundary reflection matrix

“…Classical boundary reflection matrices corresponding to the various choices of the integrable boundary condition ‡ have been constructed by linearising the equation of motion around a background solution in [14,15], where some conjectures on the corresponding exact boundary reflection matrices have been also made. A study on the boundary reflection matrix in quantum field theory has been initiated in the framework of the Feynman's perturbation theory in [22] and single particle reflection amplitudes for ATFT with the Neumann boundary condition were constructed in [23,24]. Quite recently, a geometric expression of the boundary reflection matrices in terms of root systems for simply-laced ATFT was otained in [25].…”