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

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.