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The talk is concerned with the global asymptotic analysis of the tt* - Toda equation, 2(w i ) t ¯t = −e 2(w i+1 −w i ) + e 2(w i −w i+1 ) , where, for all i, w i = w i+n+1 (periodicity),w i = w i ( | t | ) (radial condi- tion), and w i + w −i−1 = 0 (“anti-symmetry”). The problem has been intensively studied since the early 90s work of Cecotti and Vafa. In these work a prominent role of the tt*- equations in the classification of supersymmetric field theories had been revealed and a series of important conjectures about their solutions has been formulated. Assuming n = 3 (the first case beyond the known Painlev´e III situation), we study the question using a combination of methods from p.d.e., isomonodromic deformations (Riemann-Hilbert method), and loop groups (Iwasawa factorization). 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 in 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 and prove some of Cecotti-Vafa conjectures. In the talk, the above mentioned results will be presented in detail with an outline of the key steps in their derivation. Also, the relation to 1998 work of Tracy and Widom on the tt*-Toda equation will be discussed. This is a joint work with Martin Guest and Chang-Shou Lin.

Part of Craig Tracy Workshop