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

Abstract

In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). 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, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt ^*-bundles generalised para-pluriharmonic maps into ${{{\rm Sp}(\mathbb{R}^{2n})/U^{\pi}(C^n)}}$ , respectively, into SO ₀(n,n)/U ^π(C ⁿ), where U ^π(C ⁿ) is the para-complex analogue of the unitary group.