Root Systems and Boundary Bootstrap

Abstract

The principle of boundary bootstrap plays a significant role in the algebraic study of the purely elastic boundary reflection matrix $K_a(θ)$ for integrable quantum field theory defined on a space-time with a boundary. However, general structure of that principle in the form as was originally introduced by Fring and Köberle has remained unclear. In terms of a new matrix $J_a(θ)=\sqrt{K_a(θ)/K_{\bar{a}}(iπ+θ)}$, the boundary bootstrap takes a simple form. Incidentally, a hypothesised expression of the boundary reflection matrix for simply-laced $ADE$ affine Toda field theory defined on a half line with the Neumann boundary condition is obtained in terms of geometrical quantities of root systems à la Dorey.