Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 1

A geometric construction of representations of the Berezin–Toeplitz quantization

Pages: 1 – 36

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n1.a1

Authors

Kwokwai Chan (Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Naichung Conan Leung (Institute of Mathematical Sciences and Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Qin Li (Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, China)

Abstract

For a Kähler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin–Toeplitz deformation quantization algebra $(C^\infty (X) [[\hbar]], \star_{BT})$ parametrized by points $z_0 \in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^0 (X, L^{\otimes m})$ around $z_0$ in the large volume limit.

Published 21 October 2022