On the Stokes matrices of the $t{t}^{\ast }$-Toda equation

Abstract

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

18 pages; to appear in the Tokyo Journal of Mathematics. Comments & constructive criticism welcome!