Connection and Curvature on Bundles of Bergman and Hardy Spaces

  • Miroslav Englis

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

    Mathematical Sciences, Chalmers University of Technology, and Göteborg University, SE-412 96 Göteborg, Sweden
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We consider a complex domain D×VD\times V in the space Cm×Cn\mathbb{C}^m\times \mathbb{C}^n and a family of weighted Bergman spaces on VV defined by a weight ekϕ(z,w)e^{-k\phi(z, w)} for a pluri-subharmonic function ϕ(z,w)\phi(z, w) with a quantization parameter kk. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain DD. We consider the natural covariant differentiation Z\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)R^{(k)}(Z, Z) for large kk and for the induced connection [Z(k),Tf(k)][\nabla_Z^{(k)}, T_f^{(k)}] on Toeplitz operators TfT_f. In the special case when the domain DD is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [Z(k),Tf(k)][\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×VD\times V replaced by a general strictly pseudoconvex domain VCm×Cn\mathcal{V}\subset\mathbb{C}^m\times\mathbb{C}^n fibered over a domain DCmD\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.

Cite this article

Miroslav Englis, Genkai Zhang, Connection and Curvature on Bundles of Bergman and Hardy Spaces. Doc. Math. 25 (2020), pp. 189–217

DOI 10.4171/DM/744