Integrable QFT(2) Encoded on Products of Dynkin Diagrams

TL;DR

This paper shows that the Thermodynamic Bethe Ansatz equations for certain integrable 2D quantum field theories can be represented using products of ADE Dynkin diagrams, revealing new dilogarithm sum rules and conformal dimension formulas.

Contribution

It establishes that the graphs encoding these equations are restricted to ADE types and introduces new dilogarithm sum rules and formulas for conformal dimensions based on ADE properties.

Findings

Graphs are restricted to ADE types for encoding TBA equations.

New dilogarithm sum rules related to ADE×ADE are proposed.

A simple formula for UV conformal dimensions in terms of ADE ranks and Coxeter numbers.

Abstract

A large class of Thermodynamic Bethe Ansatz equations governing the Renormalization Group evolution of the Casimir energy of the vacuum on the cylinder for an integrable two-dimensional field theory, can often be encoded on a tensor product of two graphs. We demonstrate here that in this case the two graphs can only be of $ADE$ type. We also give strong numerical evidence for a new large set of Dilogarithm sum Rules connected to $ADE\times ADE$ and a simple formula for the ultraviolet perturbing operator conformal dimensions only in terms of rank and Coxeter numbers of $ADE\times ADE$. We conclude with some remarks on the curious case $ADE\times D$. [Talk given by F.R. at the Cargese Workshop "New Developments in String Theory, Conformal Models and Topological Field Theory" (May 1993)]