Integrable systems

In the context of Boundary Quantum Field Theory, spatial boundaries are unavoidable in the description of realistic, physical processes. Yet, even the study of a particularly simple class of two-dimensional Quantum Field Theories (QFTs) called integrable QFTs, when restricted to the half-line, is not fully developed. In collaboration with Ed Corrigan (York) ,  we have calculated non-perturbatively the reflection factors for one of the simplest integrable systems (the sinh-Gordon model, whose spectrum consists of a single scalar particle)  restricted to a half-line by the most general integrable boundary conditions, which depend on two arbitray parameters. The essential idea , already present in a previous work of Ed Corrigan (York) and Gustav Delius (York)   (hep-th/9909145)  was to obtain the energy spectrum of boundary breathers in two independent  ways. Firstly by using the boundary bootstrap, and secondly by quantising the classical solutions corresponding to boundary breathers. The elegant relationship uncovered between the boundary parameters appearing in the classical sinh-Gordon Lagrangian and the two parameters occurring in the quantum reflection factors is the result of a rather sophisticated calculation whose technicalities had proven a challenge for many years. An interesting outcome of this work is a generalisation of a weak-strong coupling duality from the bulk theory to the theory with integrable boundary conditions. Such dualities are common in integrable or solvable models, but to our knowledge, our result is the first of this kind involving boundary parameters  (hep-th/0008237) .

This work paves the way for the long-term goal of finding a complete classification and understanding of the pattern of integrable conditions discovered previously. In the meantime, our successful analysis is being tested on the next simplest integrable model with a single scalar field, based on the twisted affine a2(2) algebra, where the unique boundary parameter arises in two distinct ways.