Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bundle $E\rightarrow M$, whose equivariant index is well-defined as a generalized function on $G$, and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the $\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$ can be interpreted as the "quantization" of the contact manifold $(M,E)$; the character of the corresponding virtual $G$-representation is then given by the equivariant index of $\dirac$. By defining contact analogues of the algebra of observables, pre-quantum line bundle and polarization, we further extend the analogy by giving a contact version of the Kostant-Souriau approach to quantization, and discussing the extent to which this approach is reproduced by the index-theoretic method.Comment: 25 pages, references added, several corrections and clarifications and some reorganization of conten

Abstract. Given an elliptic action of a compact Lie group G on a co-oriented contact manifold (M, E) one obtains two naturally associated objects: A G-transversally elliptic operator D b / , and an equivariant differential form with generalized coefficients J (E, X) defined in terms of a choice of contact form on M .We explain how the form J (E, X) is natural with respect to the contact structure, and give a formula for the equivariant index of D b / involving J (E, X). A key tool is the Chern character with compact support developed by Paradan-Vergne [11,12].

We consider a consider the case of a compact manifold M , together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J (E, X) whose properties we describe.When E is equipped with a complex structure, we define a class of symbol mappings σ in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J (E, X) and standard equivariant characteristic classes. This formula generalizes the formula given in [9] for the case of a contact manifold.

An f -structure on a manifold M is an endomorphism field ϕ ∈ Γ(M, End(T M )) such that ϕ 3 + ϕ = 0. Any f -structure ϕ determines an almost CR structure E 1,0 ⊂ T C M given by the +i-eigenbundle of ϕ. Using a compatible metric g and connection ∇ on M , we construct an odd first-order differential operator D, acting on sections of S = ΛE * 0,1 , whose principal symbol is of the type considered in [Fit09b]. In the special case of a CR-integrable almost S-structure, we show that when ∇ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator D is given byis the tangential Cauchy-Riemann operator. We then describe two "quantizations" of manifolds with f -structure that reduce to familiar methods in symplectic geometry in the case that ϕ is a compatible almost complex structure, and to the contact quantization defined in [Fit10] when ϕ comes from a contact metric structure. The first is an index-theoretic approach involving the operator D; for certain group actions D will be transversally elliptic, and using the results in [Fit09b], we can give a Riemann-Roch type formula for its index. The second approach uses an analogue of the polarized sections of a prequantum line bundle, with a CR structure playing the role of a complex polarization. * Research supported by an NSERC postdoctoral fellowship f -structure is equivalent to an almost CR structure together with a choice of complement to the Levi distribution.In [Fit09b], we used almost CR structures to construct new examples of transversally elliptic symbols (in the sense of Atiyah [Ati74]), and gave a formula for their (cohomological) equivariant index. In this paper we will give a construction of a first-order differential operator whose principal symbol is of the type considered in [Fit09b]. Such an operator was introduced in the contact setting in [Fit09c], and the general approach first appeared in the author's thesis [Fit09a]. In [Fit09b] we required the existence of a subbundle E ⊂ T M of constant rank, and a group action on M such that the orbits of G are transverse to the subbundle E, in a sense we will make precise. While this construction does not produce the most general transversally elliptic operators, it does include many of the bestknown examples of transversally elliptic operators (or symbols) encountered, for example, in [Ati74, BV96b, Ver96].Given a manifold M with f -structure ϕ, it is always possible to find a compatible metric g and connection ∇ [Soa97] satisfying g(ϕX, Y ) + g(X, ϕY ) = 0 and ∇ϕ = ∇g = 0.The eigenvalues of ϕ (acting on T C M := T M ⊗ C) are 0 and ±i; we let E = im ϕ, and let E 1,0 ⊂ T C M denote the +i-eigenbundle of ϕ which, as noted above, defines an almost CR structure on M. We use the data (ϕ, g, ∇) to construct an odd first-order differential operator D acting on sections of S = ΛE * 0,1 , where E 0,1 = E 1,0 . The construction is based on the usual construction of a Dirac operator on an almost Hermitian manifold (see [BGV91,Nic05]): the metric g allows us to construct the bundle of Clifford algebras ...

An f -structure on a manifold M is an endomorphism field ϕ satisfying ϕ 3 + ϕ = 0.We call an f -structure regular if the distribution T = ker ϕ is involutive and regular, in the sense of Palais. We show that when a regular f -structure on a compact manifold M is an almost S-structure, it determines a torus fibration of M over a symplectic manifold. When rank T = 1, this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with S-structure or K-structure, we do not assume that the f -structure is normal. We also show that given an almost S-structure, we obtain an associated Jacobi structure, as well as a notion of symplectization. * Research supported by an NSERC postdoctoral fellowship