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Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

Abstract. We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL n R/SO n ) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solutions associated to nonlinear sigma models (quantum cohomology) or Landau-Ginzburg models (unfoldings of singularities), as conjectured by Cecotti and Vafa. In particular we establish the existence of a new family of pure and polarized TERP structures in the sense of [16], or noncommutative variations of Hodge structures in the sense of [19].

In [10] (part I) 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.

This paper, the third in a series, completes our description of all (radial) solutions on C * of the tt*-Toda equations 2(w i ) tt = −e 2(wi+1−wi) +e 2(wi−wi−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 nonresonant case.This allows us to give a complete picture of the monodromy data, holomorphic data, and asymptotic data of the global solutions.