Connection and Curvature on Bundles of Bergman and Hardy Spaces

Abstract

We consider a complex domain $D\times V$ in the space $\mathbb{C}^m\times \mathbb{C}^n$ and a family of weighted Bergman spaces on $V$ defined by a weight $e^{-k\phi(z, w)}$ for a pluri-subharmonic function $\phi(z, w)$ with a quantization parameter $k$. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain $D$. We consider the natural covariant differentiation $\nabla_Z$ on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures $R^{(k)}(Z, Z)$ for large $k$ and for the induced connection $[\nabla_Z^{(k)}, T_f^{(k)}]$ on Toeplitz operators $T_f$. In the special case when the domain $D$ is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for $[\nabla_Z^{(k)}, T_f^{(k)}]$ as Toeplitz operators. This generalizes earlier work of J. E. Andersen in [Commun. Math. Phys. 255, No. 3, 727--745 (2005; Zbl 1079.53136)]. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of $D\times V$ replaced by a general strictly pseudoconvex domain $\mathcal{V}\subset\mathbb{C}^m\times\mathbb{C}^n$ fibered over a domain $D\subset\mathbb{C}^m$. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.