Connection and Curvature on Bundles of Bergman and Hardy Spaces
Miroslav EnglisMathematics Institute, Silesian University in Opava, 74601 Opava, and Mathematics Institute, Czech Academy of Sciences, 11567 Prague 1, Czech Republic
Genkai ZhangMathematical Sciences, Chalmers University of Technology, and Göteborg University, SE-412 96 Göteborg, Sweden
We consider a complex domain in the space and a family of weighted Bergman spaces on defined by a weight for a pluri-subharmonic function with a quantization parameter . The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain . We consider the natural covariant differentiation 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 for large and for the induced connection on Toeplitz operators . In the special case when the domain is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for 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 replaced by a general strictly pseudoconvex domain fibered over a domain . The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.
Cite this article
Miroslav Englis, Genkai Zhang, Connection and Curvature on Bundles of Bergman and Hardy Spaces. Doc. Math. 25 (2020), pp. 189–217DOI 10.4171/DM/744