On the Stokes Matrices of the $tt^*$-Toda Equation

Abstract

We derive a formula for the signature of the symmetrized Stokes matrix $\mathcal{S}+\mathcal{S}^\mathrm{T}$ for the $tt^*$-Toda equation, reminiscent of a formula of Beukers and Heckmann for the generalized hypergeometric equation. The condition $\mathcal{S}+\mathcal{S}^\mathrm{T} > 0$ is prominent in the work of Cecotti and Vafa on the $tt^*$ equation; using our formula, we show that the Stokes matrices $\mathcal{S}$ satisfying this condition are parameterized by the points of an open convex polytope.