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@article{49243,
author = {Engliš, Miroslav and Zhang, Genkai},
article_location = {Berlin (Germany)},
article_number = {February},
doi = {http://dx.doi.org/10.25537/dm.2020v25.189-217},
keywords = {Bergman space; bundle of Bergman spaces; Fock space; Fock bundle; Siegel domain; Chern connection and curvature; Toeplitz operator},
language = {eng},
issn = {1431-0643},
journal = {Documenta Mathematica},
title = {Connection and curvature on bundles on Bergman and Hardy spaces},
url = {https://www.elibm.org/article/10012026},
volume = {25},
year = {2020}
}
TY - JOUR
ID - 49243
AU - Engliš, Miroslav - Zhang, Genkai
PY - 2020
TI - Connection and curvature on bundles on Bergman and Hardy spaces
JF - Documenta Mathematica
VL - 25
IS - February
SP - 189-217
EP - 189-217
PB - Deutsche Mathematiker-Vereinigung e.V.
SN - 14310643
KW - Bergman space
KW - bundle of Bergman spaces
KW - Fock space
KW - Fock bundle
KW - Siegel domain
KW - Chern connection and curvature
KW - Toeplitz operator
UR - https://www.elibm.org/article/10012026
N2 - We consider a complex domain D x V in the space C-m x 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 del(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 [del((k))(Z), 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 [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.
ER -
ENGLIŠ, Miroslav a Genkai ZHANG. Connection and curvature on bundles on Bergman and Hardy spaces. \textit{Documenta Mathematica}. Berlin (Germany): Deutsche Mathematiker-Vereinigung e.V., 2020, roč.~25, February, s.~189-217. ISSN~1431-0643. doi:10.25537/dm.2020v25.189-217.
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