Adapted complex structures and geometric quantization

Almost Complex Structures and Geometric Quantization

We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spinc quantization. We prove the analog of Kodaira vanishing for the Spinc Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semiclassically. We also introduce a new quantization scheme, based on a rescaled Lap...

Geometric Quantization, Complex Structures and the Coherent State Transform

It is shown that the heat operator in the Hall coherent state transform for a compact Lie group K [Ha1] is related with a Hermitian connection associated to a natural one-parameter family of complex structures on T ∗K. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of T ∗K for different choices of complex structures within the given fa...

Adapted complex structures and the geodesic flow

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real analytic Riemannian manifold. Motivated by the “complexifier” approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the “imaginary...

Geometric Quantization and Equivariant Cohomology Geometric Quantization and Equivariant Cohomology

Weyl Quantization from Geometric Quantization

In [23] a nice looking formula is conjectured for a deformed product of functions on a symplectic manifold in case it concerns a hermitian symmetric space of non-compact type. We derive such a formula for simply connected symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2-dimensional...