The Normal Conformal Cartan Connection and the Bach Tensor

TL;DR

This paper links the Bach tensor in four-dimensional conformal geometry to the Yang-Mills current of the Cartan normal conformal connection, providing new insights into its structure and solutions, especially for Fefferman metrics.

Contribution

It expresses the Bach tensor via the CNC connection's Yang-Mills current and analyzes its properties for special conformal geometries like Fefferman metrics.

Findings

Bach tensor identified with the Yang-Mills current of the CNC connection.

Degenerate Bach tensor occurs when the CNC connection is reducible.

The unique homogeneous solution to the Yang-Mills equations is the Nurowski-Plebanski metric.

Abstract

The goal of this paper is to express the Bach tensor of a four dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang-Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach tensor. As an example we study the case of a CNC connection which admits a twisting covariantly constant twistor field. This class of conformal geometries of this property is known as given by the Fefferman metric tensors. We use our result to calculate the Bach tensor of an arbitrary Fefferman metric and show it is proportional to the tensorial square of the four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang-Mills equations imposed on the CNC connection for all the homogeneous Fefferman metrics. The only solution is the Nurowski-Plebanski metric.