Weighted Bergman kernels and $\star$-products
Andreas Sykora
TL;DR
This work develops a formal deformation-quantization framework for weighted Bergman spaces on domains in $\mathbb{C}^N$ with weights of the form $\rho = e^{-\alpha \phi}\mu g$, where $\phi$ is a Kähler potential and $g$ is the metric determinant. It introduces covariant Toeplitz operators with analytic symbols, a covariant triple symbol $S_\alpha$ for triple products, and a formal Berezin-Töplitz $\star$-product whose unit is the Bergman kernel symbol $k$, analyzing their associativity and asymptotics as $\hbar = 1/\alpha$. The paper derives explicit first-order expressions for the Berezin-Töplitz $\star$-product, its contravariant and covariant variants, and the Berezin transform, showing that the quantum corrections depend on the geometric data $(\Delta \mu, R, g^{i\overline{j}})$. It provides a coherent formalism to compute higher-order terms via recursion with the differential operators $R_n$, enabling systematic semiclassical analysis of quantized Kähler manifolds with general weights. This advances the understanding of symbol calculus and star-products in geometric quantization with general weight factors.
Abstract
We calculate the weighted Bergman kernel on a complex domain with a weight of the form $ρ=e^{-αφ}μg$, where $α$ is a positive real number, $φ$ is a Kähler potential, g is the determinant of the corresponding Kähler metric and $μ$ is a real-valued positive function. Several $\star$-products related to the Bergman kernel are determined. Explicit formulas are provided up to first order.