Geometric quantization, complex structures and the coherent state transform

Abstract

It is shown that the heat operator in the Hall coherent state transform for a compact Lie group $K$ is related with a Hermitian connection associated to a natural one-parameter family of complex structures on $T^{\ast }K$. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of $T^{\ast }K$ for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin and Axelrod, Della Pietra and Witten.

to appear in Journal of Functional Analysis

References in corpus (3)

• Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type
• On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group
• Coherent State Transforms and Vector Bundles on Elliptic Curves

Cited by in corpus (17)

• On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group
• A new look at adapted complex structures
• Hitchin's Connection in Half-Form Quantization
• Isometry theorem for the Segal-Bargmann transform on noncompact symmetric spaces of the complex type
• Complex structures adapted to magnetic flows
• Berezin-Toeplitz quantization on Lie groups
• Quantization of compact Riemannian symmetric spaces
• Adapted complex structures and the geodesic flow
• Degeneration of Kaehler structures and half-form quantization of toric varieties
• On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics
• Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds
• Coherent states for compact Lie groups and their large-N limits
• Generalised complex geometry in thermodynamical fluctuation theory
• Coherent state transforms and the Mackey-Stone-Von Neumann theorem
• Partial coherent state transforms, $G\times T$-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups
• Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics
• Fibering polarizations and Mabuchi rays on symmetric spaces of compact type