A Lie-theoretic description of the solution space of the tt*-Toda equations

TL;DR

This paper provides a Lie-theoretic framework to describe the solution space of the tt*-Toda equations, linking monodromy data to convex polytopes within Weyl alcoves, thus offering a new geometric perspective.

Contribution

It introduces a Lie-theoretic approach to characterize the solution space of tt*-Toda equations using monodromy data and Weyl alcoves, connecting differential equations with algebraic group theory.

Findings

Monodromy data parametrizes solutions via convex subsets of Weyl alcoves.

Lie-theoretic description simplifies understanding of the solution space.

Connection between Stokes data and conjugacy classes elucidates the structure of solutions.

Abstract

We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt$^*$-Toda equations) which were introduced by Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions can be parametrized by monodromy data of a certain flat $SL_{n+1}\mathbb{R}$-connection. Using Boalch's Lie-theoretic description of Stokes data, and Steinberg's description of regular conjugacy classes of a linear algebraic group, we express this monodromy data as a convex subset of a Weyl alcove of $SU_{n+1}$.