Topological-antitopological fusion equations, pluriharmonic maps and special Kähler manifolds

We introduce the notion of a tt*-bundle. It provides a simple definition, purely in terms of real differential geometry, for the geometric structures which are solutions of a general version of the equations of topological-antitopological fusion considered by Cecotti-Vafa, Dubrovin and Hertling. Then we give a simple characterization of the tangent bundles of special complex and special Kähler manifolds as particular types of tt*-bundles. We illustrate the relation between metric tt*-bundles of rank r and pluriharmonic maps into the pseudo-Riemannian symmetric space GL(r)/O(p, q) in the case of a special Kähler manifold of signature (p, q) = (2k, 2l). It is shown that the pluriharmonic map coincides with the dual Gauß map, which is a holomorphic map into the pseudo-Hermitian symmetric space Sp(R2n)/U(k, l) ⊂ SL(2n)/SO(p, q) ⊂ GL(2n)/O(p, q), where n = k + l. This work was supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche Forschungsgemeinschaft. Research of L. S. was supported by a joint grant of the ‘Deutscher Akademischer Austauschdienst’ and the CROUS Nancy-Metz.