Harmonic bundle solutions of topological–antitopological fusion in para-complex geometry

Abstract

In this work we introduce the notion of a para-harmonic bundle, i.e. the generalization of a harmonic bundle [C.T. Simpson, Higgs-bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992) 5–95] to para-complex differential geometry. We show that para-harmonic bundles are solutions of the para-complex version of metric t t ∗ -bundles introduced in [L. Schäfer, t t ∗ -bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60–89]. Further we analyze the correspondence between metric para- t t ∗ -bundles of rank 2 r over a para-complex manifold M and para-pluriharmonic maps from M into the pseudo-Riemannian symmetric space GL ( r , R ) / O ( p , q ) , which was shown in [L. Schäfer, t t ∗ -bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60–89], in the case of a para-harmonic bundle. It is proven, that for para-harmonic bundles the associated para-pluriharmonic maps take values in the totally geodesic subspace GL ( r , C ) / U π ( C r ) of GL ( 2 r , R ) / O ( r , r ) . This defines a map Φ from para-harmonic bundles over M to para-pluriharmonic maps from M to GL ( r , C ) / U π ( C r ) . The image of Φ is also characterized in the paper.