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

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 tt∗tt∗-bundles introduced in [L. Schäfer, tt∗tt∗-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-tt∗tt∗-bundles of rank 2r over a para-complex manifold M and para-pluriharmonic maps from M   into the pseudo-Riemannian symmetric space GL(r,R)/O(p,q)GL(r,R)/O(p,q), which was shown in [L. Schäfer, tt∗tt∗-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)/Uπ(Cr)GL(r,C)/Uπ(Cr) of GL(2r,R)/O(r,r)GL(2r,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)/Uπ(Cr)GL(r,C)/Uπ(Cr). The image of Φ is also characterized in the paper.