Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II: Riemann–Hilbert Problem

Abstract

In Guest et al. (arXiv:1209.2045) (part I) we computed the Stokes data for the smooth solutions of the tt*-Toda equations whose existence we had previously established by p.d.e. methods. Here we formulate the existence problem as a Riemann–Hilbert problem, based on this Stokes data, and solve it under certain conditions (Theorem 5.4). In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data (Theorem 5.5). We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions (Theorem 4.1), clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, we resolve some conjectures of Cecotti and Vafa concerning the positivity of S + St (where S is the Stokes matrix) and the unimodularity of the eigenvalues of the monodromy matrix (Theorem 5.6). In particular, we show that “unitarity implies regularity” for the tt*-Toda equations.