Bargmann-Fock sheaves on K\"ahler manifolds

TL;DR

This paper extends Fedosov's deformation quantization method to construct a sheaf of Bargmann-Fock modules on K"ahler manifolds, linking geometric quantization with sheaf-theoretic deformation techniques.

Contribution

It provides an explicit analytic construction of Bargmann-Fock sheaves over K"ahler manifolds with Fedosov connections, connecting deformation quantization and geometric quantization.

Findings

Constructed sheaf of Bargmann-Fock modules over K"ahler manifolds.

Showed sheaf of flat sections forms a module over deformation quantization algebra.

Linked sheaf expansion to powers of a prequantum line bundle.

Abstract

Fedosov used flat sections of the Weyl bundle on a symplectic manifold to construct a star product $\star$ which gives rise to a deformation quantization. By extending Fedosov's method, we give an explicit, analytic construction of a sheaf of Bargmann-Fock modules over the Weyl bundle of a K\"ahler manifold $X$ equipped with a compatible Fedosov abelian connection, and show that the sheaf of flat sections forms a module sheaf over the sheaf of deformation quantization algebras defined $(C^\infty_X[[\hbar]], \star)$. This sheaf can be viewed as the $\hbar$-expansion of $L^{\otimes k}$ as $k \to \infty$, where $L$ is a prequantum line bundle on $X$ and $\hbar = 1/k$.