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 \textit{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.

32A36, 47B35, 32Q20, 32L05, 47B10

Bergman space, bundle of Bergman spaces, Fock space, Fock bundle, Siegel domain, Chern connection and curvature, Toeplitz operator

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Mathematics Institute, Silesian University in Opava, 74601 Opava, and Mathematics Institute, Czech Academy of Sciences, 11567 Prague 1, Czech Republic

Mathematical Sciences, Chalmers University of Technology, and Göteborg University, SE-412 96 Göteborg, Sweden