A covariant Poisson deformation quantization with separation of variables up to the third order

TL;DR

This paper derives a covariant deformation quantization formula up to third order on Kähler and complex manifolds, extending previous methods by expressing the third-order operator in terms of the Poisson tensor.

Contribution

It introduces a modified third-order operator for deformation quantization that is covariant and expressed via the Poisson tensor, applicable to arbitrary complex manifolds.

Findings

Derived a simple formula for the third-order operator C_3

Modified C_3 to be covariant and expressed in terms of the Poisson tensor

Extended deformation quantization to arbitrary complex manifolds with Poisson bivectors

Abstract

We give a simple formula for the operator C_3 of the standard deformation quantization with separation of variables on a K\"ahler manifold M. Unlike C_1 and C_2, this operator can not be expressed in terms of the K\"ahler-Poisson tensor on M. We modify C_3 to obtain a covariant deformation quantization with separation of variables up to the third order which is expressed in terms of the Poisson tensor on M and thus can be defined on an arbitrary complex manifold endowed with a Poisson bivector field of type (1,1).