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

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.

42 pages; v2, published version

References in corpus (1)

• A geometric construction of representations of the Berezin-Toeplitz quantization

Cited by in corpus (4)

• Quantization of Kähler manifolds
• Bargmann-Fock sheaves on Kähler manifolds
• Quantizable functions on Kähler manifolds and non-formal quantization
• Geometric quantization and quantum moment maps on coadjoint orbits and Kähler-Einstein manifolds