Bargmann-Fock sheaves on K\"ahler manifolds

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 $\hbar = 1/k$.