Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa III: Iwasawa Factorization and Asymptotics

Abstract

This paper, the third in a series, completes our description of all (radial) solutions on $${\mathbb {C}}^*$$ of the tt*-Toda equations $$2(w_i)_{ {t{\bar{t}}} }=-e^{2(w_{i+1}-w_{i})} + e^{2(w_{i}-w_{i-1})}$$, using a combination of methods from p.d.e., isomonodromic deformations (Riemann–Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data, holomorphic data, and asymptotic data of the global solutions.