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

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 tt∗tt∗-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(2r,R)/O(2p,2q) 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 GL(r,C)/U(p,q) of GL(2r,R)/O(2p,2q). This defines a map ΦΦ from harmonic bundles over M to pluriharmonic maps from M   to GL(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 NN into GL(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.