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The quantum cohomology of CP^1 provides a distinguished solution of the thirdPainleve equation. S. Cecotti and C. Vafa discovered this from a physicalviewpoint. We derive it from a differential geometric viewpoint, using thetheory of harmonic maps and in particular the generalized Weierstrassrepresentation (DPW representation) for spacelike surfaces of constant meancurvature in Minkowski space. The nontrivial aspects are the characterizationof the solution, and its global behaviour. From our point of view, the latterproperty says that the extended harmonic map remains entirely within a singleIwasawa orbit of the appropriate loop group.

The tt* structure of the quantum cohomology of CP^1 from the viewpoint of differential geometry