Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem

TL;DR

This paper solves a Riemann-Hilbert problem to compute the connection matrix for smooth solutions of tt*-Toda equations, completing monodromy data analysis and confirming conjectures on positivity and eigenvalue unimodularity.

Contribution

It provides an alternative proof of solution existence via Riemann-Hilbert methods and fully determines the monodromy data for tt*-Toda equations, including connection formulas.

Findings

Computed the connection matrix for all smooth solutions.

Established connection formulas relating asymptotics at zero and infinity.

Confirmed conjectures on positivity of S+S^t and eigenvalue unimodularity.

Abstract

In Part I (arXiv:1209.2045) we computed the Stokes data, though not the "connection matrix", for the smooth solutions of the tt*-Toda equations whose existence we established by p.d.e. methods. Here we give an alternative proof of the existence of some of these solutions by solving a Riemann-Hilbert problem. In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data. We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions, clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, for the tt*-Toda equations, we resolve some conjectures of Cecotti and Vafa concerning the positivity of S+S^t (where S is the Stokes matrix) and the unimodularity of the eigenvalues of the monodromy matrix.