Communications in Number Theory and Physics

Volume 16 (2022)

Number 2

Kapranov’s $L_\infty$ structures, Fedosov’s star products, and one-loop exact BV quantizations on Kähler manifolds

Pages: 299 – 351

DOI: https://dx.doi.org/10.4310/CNTP.2022.v16.n2.a2

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

We study quantization schemes on a Kähler manifold and relate several interesting structures. We first construct Fedosov’s star products on a Kähler manifold $X$ as quantizations of Kapranov’s $L_\infty$-algebra structure. Then we investigate the Batalin–Vilkovisky (BV) quantizations associated to these star products. A remarkable feature is that they are all one-loop exact, meaning that the Feynman weights associated to graphs with two or more loops all vanish. This leads to a succinct cochain level formula in de Rham cohomology for the algebraic index.

Keywords

$L_\infty$ structure, deformation quantization, BV quantization, algebraic index theorem

2010 Mathematics Subject Classification

Primary 53D55, 58J20. Secondary 81Q30, 81T15.

Full Text (PDF format)

K. Chan was supported by a grant of the Hong Kong Research Grants Council (Project No. CUHK14303019) and direct grant (No. 4053395) from CUHK. N. C. Leung was supported by grants of the Hong Kong Research Grants Council (Project No. CUHK14301117 & CUHK14303518) and direct grant (No. 4053400) from CUHK. Q. Li was supported by a grant from National Natural Science Foundation of China (Project No. 12071204), and Guangdong Basic and Applied Basic Research Foundation (Project No. 2020A1515011220).

Received 16 July 2020

Accepted 17 January 2022

Published 27 April 2022