Transversal Dirac families in Riemannian foliations

Abstract

We describe a family of differential operators parametrized by the transversal vector potentials of a Riemannian foliation relative to the Clifford algebra of the foliation. This family is non-elliptic but in certain ways behaves like a standard Dirac family in the absolute case as a result of its elliptic-like regularity properties. The analytic and topological indices of this family are defined as elements of K-theory in the parameter space. We indicate how the cohomology of the parameter space is described via suitable maps to Fredholm operators. We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument. A determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation. With respect to a generalized coupled Dirac-Yang-Mills system, we indicate how chiral anomalies are located relative to the foliation.

Natural equivariant transversally elliptic Dirac operators

“…The leaves of Riemannian foliations with bundle-like metrics are locally equidistant. There is a natural construction of transversal Dirac operators (see [8] , [11] , [15]) on Riemannian foliations, which is a special case of the construction in this section. Choose a frame field {e 1 , .…”

The index of a transverse Dirac-type operator: the case of abelian Molino sheaf

“…With this approach, the natural inner product on the holonomy-invariant sections involves the volume form of g M . In order to obtain a self-adjoint basic Dirac-type operator with this approach, one must assume that the mean curvature form κ of the foliated manifold (M, F ) is a basic one-form [21]. Note that the mean curvature form κ, which lives on M, is distinct from the mean curvature form τ in this paper, which lives on T .…”

“…Then there is a notion of a "basic" Dirac-type operator D, a first-order differential operator that acts on the holonomy-invariant sections of a normal Clifford module. It was shown by El Kacimi [18] and Glazebrook-Kamber [21] that D is Fredholm and hence has a well-defined index Index(D) ∈ Z. (In fact, this is true for any basic transversally elliptic operator [18].)…”

Lower bounds for the eigenvalues of the basic Dirac operator on a Riemannian foliation

“…Then the transverse spin structure is a principal Spin(q)-bundle P spin (F ) associated with it which is a fiberwise non-trivial double covering of P so (F ). Let S(F ) be the foliated spinor bundle [5,8,10] associated with P spin (F ). Then the transversal Dirac operator D tr is locally defined [2,5] by…”