Symplectic aspects of the tt*-Toda equations

TL;DR

This paper explicitly evaluates asymptotic constants for solutions of the tt*-Toda equations, introduces symplectic structures on data spaces, and demonstrates their preservation under the Riemann-Hilbert correspondence.

Contribution

It provides explicit asymptotic constant evaluations and introduces symplectic structures that are preserved under the Riemann-Hilbert correspondence for a broad class of solutions.

Findings

Explicit asymptotic constant ratios for tt*-Toda solutions.

Introduction of symplectic structures on asymptotic and monodromy data.

Symplectic structures are preserved by the Riemann-Hilbert correspondence.

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 C. A. Tracy. 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.