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

Lars Schaefer (1, 2)

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

In this work we introduce the notion of a para-harmonic bundle, i.e. the generalization of a harmonic bundle to para-complex differential geometry. We show that para-harmonic bundles are solutions of the para-complex version of metric tt*-bundles introduced in former work. Further we analyze the correspondence between metric para-tt*-bundles of rank 2r over a para-complex manifold M and para-pluriharmonic maps from M into the pseudo-Riemannian symmetric space ${\rm GL}(r,\bR)/{\rm O}(p,q),$ which was shown in former work, 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 ${\rm GL}(r,C)/{\rm U}^{\pi}(C^r)$ of ${\rm GL}(2r,\bR)/{\rm O}(r,r).$ This defines a map $\Phi$ from para-harmonic bundles over M to ara-pluriharmonic maps from M to ${\rm GL}(r,C)/{\rm U}^{\pi}(C^r)$. The image of $\Phi$ is also characterized in the paper.

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

Domaines: Géométrie différentielle [math.DG]

Lars Schaefer. Harmonic bundle solutions of Topological-antitopological fusion in para-complex geometry. 2007. ⟨hal-00143290⟩