Two-dimensional generalized Toda lattice

Abstract

The zero curvature representation is obtained for the twodimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.

Quantum conserved currents in affine Toda theories

“…simply-laced theory (the results are easily extended to the simply-laced a (1) n case for any n). In general, two-dimensional power counting arguments and Lorentz invariance suggest that any violation at the quantum level of the classical conservation laws for a spin s current can occur at most at l = s − 1 loops, and from a restricted class of diagrams; for bosonic theories they are diagrams obtained by Wick-contracting the currents with just one factor of the interaction lagrangian.…”

“…β is the coupling constant, and µ sets the mass scale; we choose µ = 1. At the classical level these theories possess conserved currents of spins s equal to the exponents of the algebra modulo the Coxeter number [1].…”

“…3 and c (1) 2 theories, we prove conservation to all-loop order, thus establishing the existence of factorized S-matrices. For these theories, as well as the simply-laced a (…”

“…Because explicit formulas are not known for currents of high spin and also to avoid algebraic complexity, we have restricted ourselves to the lowest nontrivial spin s ≥ 2 (spin s = 1 corresponds to the stress tensor), for nonsimply-laced Toda theories with only two bosonic fields, namely the spin-3 current for the a (2) 3 and c (1) 2 theories. However, to set the notation and for comparison, we also consider the spin-2 current for the a (…”

“…Affine Toda field theories describe a class of massive systems which are classically integrable; from the identification of the field equations as compatibility conditions for a Lax pair follows the existence of an infinite number of classically conserved higher-spin currents J (s) + , J (s) − [1] ∂ − J (s) + + ∂ + J (s) − = 0 (1.1)…”