Aspects of affine Toda field theory

Abstract

This paper describes affine Toda field theory which is a theory of r scalar fields in two-dimensional Minkowski space-time, where r is the rank of a compact semi-simple Lie algebra g. The classical field theory is determined by the lagrangian density L = 1/2 {partial derivative}{sub {rho}}{phi}{sup a}{partial derivative}{sup {mu}}{phi}{sup a} {minus} V({phi}) where V({phi}) = m{sup 2}/{beta}{sup 2} {Sigma}{sub 0}{sup r}n{sub i}e{sup {beta}{alpha}{sub i} {center dot} {phi}}. m and {beta} are real, classically unimportant constants, {alpha}{sub i} i = 1, . . . ,r are the simple roots of the Lie algebra g, and {alpha}{sub 0} = {Sigma}{sub 1}{sup 4} n{alpha}{sub i} is a linear combination of the simple roots; it corresponds to the extra spot on an extended Dynkin diagram for g. A reasonable question to ask is whether the classical integrability survives into the quantum field theory and, if so, what is the spectrum and to what extent is it possible to calculate explicitly quantities of interest such as S-matrices and form factors. The recent discoveries leave no doubt that these relatively simple models have much structure and their study (even in the {beta}{sup 2} {gt} 0 regime) will be informative. In this short review, the ADEmore » series of Lie algebras will be singled out for special attention.« less