Non-Linear Integral Equations for complex Affine Toda associated to simply laced Lie algebras

TL;DR

This paper derives coupled non-linear integral equations for affine Toda models linked to simply laced Lie algebras, enabling analysis of excited states and extraction of conformal data, generalizing known equations for sine-Gordon models.

Contribution

It introduces a new set of integral equations for affine Toda models with quantum group symmetry, extending the Destri-De Vega equation to a broader class of theories.

Findings

Equations describe excited states of affine Toda models.

Central charge and conformal weights are obtained directly.

Quantum group truncation relates to RCFTs with extended symmetry.

Abstract

A set of coupled non-linear integral equations is derived for a class of models connected with the quantum group $U_q(\hat g)$ ($g$ simply laced Lie algebra), which are solvable using the Bethe Ansatz; these equations describe arbitrary excited states of a system with finite spatial length $L$. They generalize the Destri-De Vega equation for the Sine-Gordon/massive Thirring model to affine Toda field theory with imaginary coupling constant. As an application, the central charge and all the conformal weights of the UV conformal field theory are extracted in a straightforward manner. The quantum group truncation for $q$ at a root of unity is discussed in detail; in the UV limit we recover through this procedure the RCFTs with extended $W(g)$ conformal symmetry.