Symplectic aspects of the tt*-Toda equations

Abstract

We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt*-Toda equations. This constant problem for the sinh–Gordon equation, which is the case n = 1 of the tt*-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297–311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.