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

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 results concern the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. First, by exploiting a framework introduced by Boalch, we show that this data has a remarkable structure. It can be described using Kostant’s theory of Cartan subalgebras in apposition and Steinberg’s theory of conjugacy classes of regular elements, and it can be visualized on the Coxeter Plane. Second, we compute canonical Stokes data for a certain family of solutions of the tt*-Toda equations in terms of their asymptotics. To do this, we compute the Stokes data of an auxiliary meromorphic connection, related to the original meromorphic connection by a loop group Iwasawa factorization.