  The quantum cohomology of CP^1 provides a distinguished solution of the third
Painleve equation. S. Cecotti and C. Vafa discovered this from a physical
viewpoint. We derive it from a differential geometric viewpoint, using the
theory of harmonic maps and in particular the generalized Weierstrass
representation (DPW representation) for spacelike surfaces of constant mean
curvature in Minkowski space. The nontrivial aspects are the characterization
of the solution, and its global behaviour. From our point of view, the latter
property says that the extended harmonic map remains entirely within a single
Iwasawa orbit of the appropriate loop group.
