Harmonic bundles, topological-antitopological fusion and the related pluriharmonic maps

Abstract

In this work we generalize the notion of a harmonic bundle of Simpson [C.T. Simpson, Higgs-bundles and local systems, Institut des hautes Etudes Scientifiques, Publication Mathematiques, N 75 (1992) 5–95] to the case of indefinite metrics. We show, that harmonic bundles are solutions of t t ∗ -geometry. Further we analyze the relation between metric tt*-bundles of rank r over a complex manifold M and pluriharmonic maps from M into the pseudo-Riemannian symmetric space GL ( 2 r , R ) / O ( 2 p , 2 q ) in the case of a harmonic bundle. It is shown, that in this case the associated pluriharmonic maps take values in the totally geodesic subspace G L ( r , C ) / U ( p , q ) of G L ( 2 r , R ) / O ( 2 p , 2 q ) . This defines a map Φ from harmonic bundles over M to pluriharmonic maps from M to G L ( r , C ) / U ( p , q ) . Its image is also characterized in the paper. This generalizes the correspondence of harmonic maps from a compact Kähler manifold N into G L ( r , C ) / U ( r ) and harmonic bundles over N proven in Simpson’s paper [C.T. Simpson, Higgs-bundles and local systems, Institut des hautes Etudes Scientifiques, Publication Mathematiques, N 75 (1992) 5–95] and explains the link between the pluriharmonic maps related to the two geometries.