Tt*-Geometry and Pluriharmonic Maps

Abstract

In this paper we use the real differential geometric definition of a metric (a unimodular oriented metric) tt*-bundle of Cortes and the author (Topological-anti-topological fusion equations, pluriharmonic maps and special Kahler manifolds) to define a map Φ from the space of metric (unimodular oriented metric) tt*-bundles of rank r over a complex manifold M to the space of pluriharmonic maps from M to {GL}(r)/O(p,q) (respectively {SL}(r)/SO(p,q)), where (p,q) is the signature of the metric. In the sequel the image of the map Φ is characterized. It follows, that in signature (r,0) the image of Φ is the whole space of pluriharmonic maps. This generalizes a result of Dubrovin (Comm. Math. Phys. 152 (1992; S539–S564).