Normal Conformal Cartan Connection Research Articles

Symplectic charges in the Yang-Mills theory of the normal conformal Cartan connection: Applications to gravity

Symplectic charges in the Yang-Mills theory of the normal conformal Cartan connection: Applications to gravity

The Main Theorem for (Anti-)Self-Dual Conformal Torsion-Free Connection on a Four-Dimensional Manifold

In this paper we obtain results that occur on a four-manifold with conformai torsion-free connection for all possible signatures of angular metric. It is proved that three of the four terms of the formula for the decomposition of the basic tensor are equidual, one is anti-dual. Based on this result we find conditions for (anti-)self-duality of exterior 2-forms Φ i j , i, j = 1, 2, 3, 4, which are part of components of the conformal curvature matrix. With the help of the latter result, the main theorem is proved: a conformal torsion-free connection on a four-manifold with the signatures of the angular metric s = ±4; 0 is (anti-)self-dual if and only if the Weyl tensor of the angular metric and the exterior 2-form Φ 0 0 are (anti-)self-dual and Einstein’s and Maxwell’s equations are satisfied. In particular, the normal conformal Cartan connection is (anti-)self-dual if and only if such is also the Weyl tensor of the angular metric.

The ambient obstruction tensor and conformal holonomy

For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurowski and Bryant.

Weyl gravity and Cartan geometry

We point out that the Cartan geometry known as the second-order conformal structure provides a natural differential geometric framework underlying gauge theories of conformal gravity. We are concerned by two theories: the first one will be the associated Yang-Mills-like Lagrangian, while the second, inspired by~\cite{Wheeler2014}, will be a slightly more general one which will relax the conformal Cartan geometry. The corresponding gauge symmetry is treated within the BRST language. We show that the Weyl gauge potential is a spurious degree of freedom, analogous to a Stueckelberg field, that can be eliminated through the dressing field method. We derive sets of field equations for both the studied Lagrangians. For the second one, they constrain the gauge field to be the `normal conformal Cartan connection'. Finally, we provide in a Lagrangian framework a justification of the identification, in dimension $4$, of the Bach tensor with the Yang-Mills current of the normal conformal Cartan connection, as proved in \cite{Korz-Lewand-2003}.

Residual Weyl symmetry out of conformal geometry and its BRST structure

The conformal structure of second order in $m$-dimensions together with the so-called (normal) conformal Cartan connection, is considered as a framework for gauge theories. The dressing field scheme presented in a previous work amounts to a decoupling of both the inversion and the Lorentz symmetries such that the residual gauge symmetry is the Weyl symmetry. On the one hand, it provides straightforwardly the Riemannian parametrization of the normal conformal Cartan connection and its curvature. On the other hand, it also provides the finite transformation laws under the Weyl rescaling of the various geometric objects involved. Subsequently, the dressing field method is shown to fit the BRS differential algebra treatment of infinitesimal gauge symmetry. The dressed ghost field encoding the residual Weyl symmetry is presented. The related so-called algebraic connection supplies relevant combinations found in the literature in the algebraic study of the Weyl anomaly.

Conformal holonomy of bi-invariant metrics

We discuss in this paper the conformal geometry of bi-invariant metrics on compact semisimple Lie groups. For this purpose, we develop an invariant Cartan calculus. Our main goal is to derive an iterative formula for the holonomy algebra of the normal conformal Cartan connection of a bi-invariant metric. As an example, we demonstrate the application of our invariant calculus to the computation of the conformal holonomy of SO(4). Its conformal holonomy algebra is so(7).

The normal conformal Cartan connection and the Bach tensor

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 that 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–Plebański metric.

Essential conformal vector fields

A geometrical proof is provided of a necessary and sufficient condition for a conformal Killing vector field with fixed point on a pseudo-Riemannian manifold to be homothetic or Killing with respect to a locally defined metric conformally related to g. The result relies on the existence of normal metrics associated with the normal conformal Cartan connection.

Twistor connection and normal conformal Cartan connection

It is shown how the normal conformal Cartan connection, used by Schmidt [1] to define conformal infinity of space-time, is related to the connection on the vector bundle of local twistors.