Lie algebraic structures in integrable models, affine Toda field theory

TL;DR

This paper explores the Lie algebraic structures underlying affine Toda field theories, developing new S-matrix formulas, analyzing high-energy behavior, and connecting to conformal field theories and coset models.

Contribution

It introduces universal Lie algebraic formulas for affine Toda S-matrices and extends methods to color valued S-matrices, providing new insights into their high-energy limits and connections to conformal models.

Findings

Derived universal Lie algebraic formulas for S-matrices.

Calculated central charges using thermodynamic Bethe ansatz.

Identified staircase pattern in scaling functions of Homogeneous Sine-Gordon models.

Abstract

The most prominent class of integrable quantum field theories in 1+1 dimensions is affine Toda theory. Distinguished by a rich underlying Lie algebraic structure these models have in recent years attracted much attention not only as test laboratories for non-perturbative methods in quantum field theory but also in the context of off-critical models. After a short introduction the mathematical preliminaries such as root systems, Coxeter geometry, dual algebras, q-deformed Coxeter elements and q-deformed Cartan matrices are introduced. Using this mathematical framework the bootstrap analysis of the affine Toda S-matrices with real coupling is performed and several universal Lie algebraic formulae proved. The Lie algebraic methods are then extended to define a new class of colour valued S-matrices and also here universal expressions are derived. The second part of the thesis presents a detailed analysis of the high-energy regime of the integrable models discussed in the first part. By means of the thermodynamic Bethe ansatz the central charges of the ultraviolet conformal field theories are calculated and in case of affine Toda theories also the first order term in the scaling function is analytically obtained. For the colour valued S-matrices the connection to WZNW coset models is discussed. A particular subclass of them, the so-called Homogeneous Sine-Gordon models, is investigated in some detail and it is found that the presence of unstable particles in these theories gives rise to a staircase pattern in the corresponding scaling function. This thesis summarizes the results of previously published articles listed in the introduction.