Quantum Trilogy: Discrete Toda, Y-System and Chaos

TL;DR

This paper introduces a discretized quantum Toda field theory linked to Lie algebras, revealing symmetries, connections to quantum chaos, and relations to quantum cluster algebras and higher Teichmuller theory.

Contribution

It generalizes discrete quantum Toda models to various Lie algebras, explores their symmetries, and connects them to quantum chaos and cluster algebra frameworks.

Findings

Discretization of quantum Toda theory for various Lie algebras.

Identification of a symmetry exchanging lattice size and algebra rank.

Relation to quantum chaos and quantum higher Teichmuller theory.

Abstract

We discuss a discretization of the quantum Toda field theory associated with a semisimple finite-dimensional Lie algebra or a tamely-laced infinite-dimensional Kac-Moody algebra $G$, generalizing the previous construction of discrete quantum Liouville theory for the case $G=A_1$. The model is defined on a discrete two-dimensional lattice, whose spatial direction is of length $L$. In addition we also find a "discretized extra dimension" whose width is given by the rank $r$ of $G$, which decompactifies in the large $r$ limit. For the case of $G=A_N$ or $A_{N-1}^{(1)}$, we find a symmetry exchanging $L$ and $N$ under appropriate spatial boundary conditions. The dynamical time evolution rule of the model is a quantizations of the so-called Y-system, and the theory can be well-described by the quantum cluster algebra. We discuss possible implications for recent discussions of quantum chaos, and comment on the relation with the quantum higher Teichmuller theory of type $A_N$.