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

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2008

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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 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), 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) / Uπ (Cr) 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) / Uπ (Cr). The image of Φ is also characterized in the paper. © 2007 Elsevier B.V. All rights reserved.

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