In this paper, we prove Kirchberg-type inequalities for any Kähler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kählerian twistor operators.

It is shown that the spectrum of a one-dimensional Dirac operator with a potential q tending to infinity at infinity, and such that the positive variation of 1}q is bounded, covers the whole real line and is purely absolutely continuous. An example is given to show that in general, pure absolute continuity is lost if the condition on the positive variation is dropped. The appendix contains a di...

Let (M, g, σ) be a compact Riemannian surface equipped with a spin structure σ. For any metric ḡ onM , we denote by ΣḡM the spinor bundle associated to ḡ. We let ∆ḡ be the Laplace-Beltrami operator acting on smooth functions of M and Dḡ be the Dirac operator acting on smooth spinor fields with respect to the metric ḡ. We also denote by μ1(ḡ) (resp. λ1(ḡ)) the smallest positive eigenvalue of ∆ḡ ...

Various approaches by the author and collaborators to define gravitational fluctuations associated with a noncommutative space are reviewed. Geometry of a noncommutative space is defined by the data (A, H,D) where A is a noncommutative involutive algebra, H is a separable Hilbert space and D a self-adjoint operator on H referred to as Dirac operator [1]. Geometry on Riemannian manifolds could b...

We elaborate an explicit example of dimensional reduction of the free massless Dirac operator with SU(3)-symmetry, defined on 12-dimensional manifold, which is the total space of principle SU(3)-bundle over 4-dimensional manifold. It turns out that after the dimensional reduction we obtain the “usual” Dirac operator, defined on 4-dimensional pseudo-Riemannian (non-flat) manifold but with mass t...

Let M be a smooth closed spin manifold. The higher index theorem computes the pairing between the group cohomology of π1(M) and the Chern character of the “higher” index of a Dirac-type operator on M. Using superconnections, we give a heat equation proof of this theorem on the level of differential forms on a noncommutative base space. As a consequence, we obtain a new proof of the Novikov conj...

In this paper, we prove Kirchberg-type inequalities for any Kähler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kählerian twistor operators.

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a projection under any residual trace depends only on the principal part of the proj...

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of Rn by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.

When chiral symmetry is spontaneously broken, the low-energy part of the Dirac operator spectrum can be computed analytically in the chiral limit. The tool is effective field theory or, equivalently in this case, Random Matrix Theory.