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

Lars Schaefer (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 Rheinische Friedrich-Wilhelms-Universität Bonn

In this work we generalize the notion of a harmonic bundle of Simpson to the case of indefinite metrics. We show, that harmonic bundles are solutions of 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 ${\rm GL}(2r,\bR)/{\rm O}(2p,2q)$ 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 ${\rm GL}(r,\bC)/{\rm U}(p,q)$ of ${\rm GL}(2r,\bR)/{\rm O}(2p,2q).$ This defines a map $\Phi$ from harmonic bundles over M to pluriharmonic maps from M to ${\rm GL}(r,\bC)/{\rm U}(p,q)$. Its image is also characterized in the paper. This generalizes the correspondence of harmonic maps from a compact Kähler manifold N into $GL(r,\bC)/U(r)$ and harmonic bundles over N proven in Simpson's paper and explains the link between the pluriharmonic maps related to the two geometries.

Mots clés: Higgs bundles; harmonic bundles; tt*-geometry; pluriharmonic maps and pseudo-Riemannian symmetric spaces

Domaines: Géométrie différentielle [math.DG]

Lars Schaefer. Harmonic bundles, topological-antitopological fusion and the related pluriharmonic maps. Journal of Geometry and Physics, 2006, 56 (5), pp.830-842. ⟨hal-00143286⟩