SFB 647 I: Special Geometries and Fermionic Field Equations (Teilprojekt A 2)

Non-integrable special geometries were introduced in Riemannian geometry for dimensions $n \\le 8$ in the Seventies by A. Gray, who subsequently studied them with his collaborators. In the early Eighties, they turned out to play a crucial role in the investigation of eigenvalue estimates for the Dirac operator on a Riemannian manifold. More recently, non-integrable geometries became interesting in the context of string theory. Integrable geometries (Calabi-Yau manifolds, Joyce manifolds etc.) are exact solutions to Strominger s equations (1986) with vanishing B-field. Recently, solutions with non vanishing B became of interest, which may be obtained from non-integrable geometries. The goal of one topic of the project is to investigate this approach, with particular emphasis on its aspects linked to differential geometry, spectral theory of Dirac operators, and representation theory (holonomy concept, canonical connections with torsion and their Dirac operators). The second topic of the projects deals with solutions of the Einstein-Dirac equation, solutions of certain overdetermined fermionic field equations (twistor type equations), and the spectral theory of Dirac-type operators on special Riemannian manifolds. In case of an affine variety and an almost transitive action of an algebraic group, the Langlands-Robinson theory allows to study the semigroup of invariant operators and their spectr5al behaviour on the underlying Banach space representation. The goal is to derive character formulas of Lefschetz type in the described situation.