Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

Lars Schaefer (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 Rheinische Friedrich-Wilhelms-Universität Bonn

In this paper we study para-tt*-bundles (TM,D,S) on the tangent bundle of an almost para-complex manifold $(M,\tau).$ We characterise those para-tt*-bundles with $\nabla=D+S$ induced by the one-parameter family of connections given by $\nabla^{\theta}=\exp(\theta \tau) \circ \nabla \circ\exp(-\theta \tau)$ and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-Kähler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds $(M,\tau)$ into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt*-bundles generalised para-pluriharmonic maps into $\mathrm{Sp}(\bR^{2n})/U^{\pi}(C^n),$ respectively into $ SO_0(n,n)/U^{\pi}(C^n),$ where $U^{\pi}(C^n)$ is the para-complex analogue of the unitary group.

Mots clés: Para-tt*-geometry and para-tt*-bundles; special para-complex and special para-Kähler manifolds; Nearly para-Kähler manifolds; para-pluriharmonic maps; pseudo-Riemannian symmetric spaces

Domaines: Géométrie différentielle [math.DG]

Lars Schaefer. Para-tt*-bundles on the tangent bundle of an almost para-complex manifold. 2007. ⟨hal-00143297⟩