Kapranov's $L_\infty$ structures, Fedosov's star products, and one-loop exact BV quantizations on K\"ahler manifolds

TL;DR

This paper explores the relationship between Fedosov's star products, Kapranov's $L_$-algebra structures, and one-loop exact BV quantizations on K"ahler manifolds, providing new insights into their interconnected geometric and quantum properties.

Contribution

It constructs Fedosov's star products as quantizations of Kapranov's $L_$-algebra on K"ahler manifolds and demonstrates their associated BV quantizations are one-loop exact, simplifying the algebraic index formula.

Findings

Fedosov's star products quantize Kapranov's $L_$-algebra structures.

BV quantizations are one-loop exact, with higher-loop Feynman weights vanishing.

Derived a cochain level formula for the algebraic index in de Rham cohomology.

Abstract

We study quantization schemes on a K\"ahler manifold and relate several interesting structures. We first construct Fedosov's star products on a K\"ahler 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.