Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations

TL;DR

This paper introduces a Lie-theoretic framework for the tt*-Toda equations, analyzing their Stokes data through Kostant and Steinberg theories, and computes canonical Stokes data for specific solutions.

Contribution

It provides a novel Lie-theoretic approach to the tt*-Toda equations and describes the structure of their Stokes data using advanced algebraic theories.

Findings

Stokes data has a remarkable structure related to Lie algebra theory.

The framework allows visualization of root orbit structures under Coxeter actions.

Canonical Stokes data can be explicitly computed from solution asymptotics.

Abstract

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra $\mathfrak{g}$, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. Exploiting a framework introduced by Boalch, we show that this data has a remarkable structure, which can be described using Kostant's theory of Cartan subalgebras in apposition and Steinberg's theory of conjugacy classes of regular elements. A by-product of this is a convenient visualization of the orbit structure of the roots under the action of a Coxeter element. As an application, we compute canonical Stokes data of certain solutions of the tt*-Toda equations in terms of their asymptotics.