Weyl invariant polynomial and deformation quantization on Kahler manifolds

TL;DR

This paper characterizes Weyl invariant polynomials on Kähler manifolds, providing explicit formulas for covariant differential operators and star products, advancing the understanding of deformation quantization in complex geometry.

Contribution

It introduces a simple criterion for invariance of polynomial differential operators and derives explicit formulas for star products on Kähler manifolds.

Findings

Criterion for invariance of polynomial differential operators

Explicit composition formula for invariant differential operators

General explicit formula for star products on Kähler manifolds

Abstract

Given a polynomial P of partial derivatives of the Kahler metric, expressed as a linear combination of directed multigraphs, we prove a simple criterion in terms of the coefficients for $P$ to be an invariant polynomial, i.e. invariant under the transformation of coordinates. As applications, we prove an explicit composition formula for covariant differential operators under a canonical basis, also known as invariant differential operators in the case of bounded symmetric domains. We also prove a general explicit formula of star products on Kahler manifolds.