Thermodynamic Bethe Ansatz and Threefold Triangulations

TL;DR

This paper links the Thermodynamic Bethe Ansatz Y-system equations in 2D integrable quantum field theories to 3D manifold triangulations, providing new insights into their periodicity and solutions.

Contribution

It introduces a novel geometric interpretation of Y-systems via 3D triangulations, offering explicit solutions and explaining periodicity in ADE-related theories.

Findings

Explicit solutions for Y-systems in terms of a single function

Connection between Y-system periodicity and 3D manifold volume invariance

Dilogarithm equations relate to manifold triangulation deformations

Abstract

In the Thermodynamic Bethe Ansatz approach to 2D integrable, ADE-related quantum field theories one derives a set of algebraic functional equations (a Y-system) which play a prominent role. This set of equations is mapped into the problem of finding finite triangulations of certain 3D manifolds. This mapping allows us to find a general explanation of the periodicity of the Y-system. For the $A_N$ related theories and more generally for the various restrictions of the fractionally-supersymmetric sine-Gordon models, we find an explicit, surprisingly simple solution of such functional equations in terms of a single unknown function of the rapidity. The recently-found dilogarithm functional equations associated to the Y-system simply express the invariance of the volume of a manifold for deformations of its triangulations.