Topological-antitopological fusion equations, pluriharmonic maps and special Kähler manifolds
Résumé
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 ${\rm GL}(r)/{\rm O}(p,q)$ 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 ${\rm Sp}(\bR^{2n})/{\rm U}(k,l) \subset {\rm SL}(2n)/{\rm SO}(p,q)\subset {\rm GL}(2n)/{\rm O}(p,q)$, where n = k +l.
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