On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

Abstract

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 ...

On the Geometry of Almost -Manifolds

“…In [14] it is proved that a CR-integrable almost S-manifold admits a canonical connection analogous to the Tanaka-Webster connection of a strongly pseudoconvex CR manifold. For the relationship between this connection and the ∂ b operator of the corresponding tangential Cauchy-Riemann complex, as well as an application of this relationship to defining an analogue of geometric quantization for almost S-manifolds, see [7].…”

“…We can fix the coefficients of ξ by requiring that ξ be the Hamiltonian vector field associated to the constant function 1, as is standard for Jacobi structures (see [15]). It is easy to see that (7) then immediately forces us to take ξ = α j ξ j ; that is, the coefficients b j are equal the constants α j . Thus, ξ is essentially determined by the almost S structure, although η is constrained only by the condition η(ξ) = 1, so the Jacobi structure we define below cannot be considered entirely canonical (as one might expect).…”