On the Stokes matrices of the $tt^*$-Toda equation

TL;DR

This paper derives a formula for the signature of the symmetrized Stokes matrix in the $tt^*$-Toda equation, verifying a conjecture about its positive definiteness and characterizing the parameter space as an open convex polytope.

Contribution

It provides a new formula for the signature of the symmetrized Stokes matrix and characterizes the parameter space where it is positive definite.

Findings

Formula for the signature of $ ext{S}+ ext{S}^T$ derived.

Verification of Cecotti and Vafa's conjecture on positive definiteness.

Parameter space for Stokes matrices is an open convex polytope.

Abstract

We derive a formula for the signature of the symmetrized Stokes matrix $\cal{S}+\cal{S}^\mathrm{T}$ for the $tt^*$-Toda equation. As a corollary, we verify a conjecture of Cecotti and Vafa regarding when $\cal{S}+\cal{S}^\mathrm{T}$ is positive definite, reminiscent of a formula of Beukers and Heckmann for the generalized hypergeometric equation. The condition that $\cal{S}+\cal{S}^\mathrm{T}$ is positive definite is prominent in the work of Cecotti and Vafa on the $tt^*$ equation; we show that the Stokes matrices $\cal{S}$ satisfying this condition are parameterized by the points of an open convex polytope.