Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds

Abstract

In these lectures, we investigate generalizations of the ordinary Dirac operator to manifolds with additional structure. In particular, if the manifold comes equipped with a distribution and an associated Clifford algebra action on a bundle over the manifold, one may define a transversal Dirac operator associated to this structure. We investigate the geometric and analytic properties of these operators, and we apply the analysis to the settings of Riemannian foliations and of manifolds endowed with Lie group actions. Among other results, we show that although a bundle-like metric on the manifold is needed to define the basic Dirac operator on a Riemannian foliation, its spectrum depends only on the Riemannian foliation structure. Using these ideas, we produce a type of basic cohomology that satisfies Poincaré duality on transversally oriented Riemannian foliations. Also, we show that there is an Atiyah-Singer type theorem for the equivariant index of operators that are transversally elliptic with respect to a compact Lie group action. This formula relies heavily on the stratification of the manifold with group action and contains eta invariants and curvature forms. These notes contain exercises at the end of each subsection and are meant to be accessible to graduate students.2000 Mathematics Subject Classification. 53C12; 53C21; 58J50; 58J60. Key words and phrases. Riemannian foliation, Dirac operator, transverse geometry, index.for all sections s ∈ Γ (E) and vector fields V, W ∈ Γ (T M). We also require that Clifford multiplication of vectors is skew-adjoint with respect to the L 2 inner product, meaning thatIt can be shown that the expression for D above is independent of the choice of orthonormal frame of T M. In the case where E has the minimum possible rank k = 2 ⌊n/2⌋ , we call E a complex spinor bundle and D a spin c Dirac operator. If such a bundle exists over a smooth manifold M, we say that M is spin c . There is a mild topological obstruction to the existence of such a structure; the third integral Stiefel-Whitney class of T M must vanish.Often the bundle E comes equipped with a grading E = E + ⊕E − such that D maps Γ (E + ) to Γ (E − ) and vice-versa. In these cases, we often restrict our attention to D :Examples of ordinary Dirac operators are as follows:• The de Rham operator is defined to be d + δ : Ω even (M) → Ω odd (M) from even forms to odd forms. In this case, the Clifford multiplication is given bywhere v ∈ T x M and i (v) denotes interior product, and ∇ is the ordinary Levi-Civita connection extended to forms. • If M is even-dimensional, the signature operator is defined to befrom self-dual to anti-self-dual forms. This grading is defined as follows. Let * denote the Hodge star operator on forms, defined as the unique endomorphism of the bundle of forms such that * : Ω r (M) → Ω n−r (M) and α ∧ * β = (α, β) dvol, α, β ∈ Ω r (M) .Then observe that the operator ⋆ = i r(r−1)+ n 2 * : Ω r (M) → Ω n−r (M) satisfies ⋆ 2 = 1. Then it can be shown that d + δ anticommutes with ⋆ and thus maps the +1 ...