tt∗tt∗-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps

We introduce the notion of a para-tt∗tt∗-bundle, the generalization of a tt∗tt∗-bundle (compare [V. Cortés, L. Schäfer, Topological–antitopological fusion equations, pluriharmonic maps and special Kähler manifolds, in: O. Kowalski, E. Musso, D. Perrone (Eds.), Proceedings of the Conference “Curvature in Geometry” organized in Lecce in honor of Lieven Vanhecke, Progress in Mathematics, vol. 234, Birkhäuser, 2005] and [L. Schäfer, tt∗tt∗-geometry and pluriharmonic maps, Ann. Global Anal. Geom., in press]) in para-complex geometry. The main result is the definition of a map Φ   from the space of metric para-tt∗tt∗-bundles of rank r over a para-complex manifold M to the space of para-pluriharmonic maps from M   to GL(r)/O(p,q)GL(r)/O(p,q) where (p,q)(p,q) is the signature of the metric and the description of the image of this map Φ  . Then we recall and prove some results known in special complex and special Kähler geometry in the setting of para-complex geometry, which we use in the sequel to give a simple characterization of the tangent bundle of a special para-complex and special para-Kähler manifold as a particular type of tt∗tt∗-bundles. For the case of a special para-Kähler manifold it is shown that the para-pluriharmonic map coincides with the dual Gauß map, which is a para-holomorphic map into the symmetric space Sp(R2n)/Uπ(Cn)⊂SL(2n)/SO(n,n)⊂GL(2n)/O(n,n)Sp(R2n)/Uπ(Cn)⊂SL(2n)/SO(n,n)⊂GL(2n)/O(n,n).