Holomorphic Conformal Cartan Geometry: Conformal Gravity, Twistors, and Ambitwistors

Abstract

Four-dimensional conformal gravity, twistor theory, and ambitwistor theory are reviewed within the unified framework of holomorphic conformal Cartan geometry. Starting from the spinorial representation of the geometry, the normal $\mathfrak{sl}(4,\mathbb{C})$-valued Cartan connection, its curvature $2$-form, and the associated tractor bundles are systematically formulated. Evaluated on the real Lorentzian slice, the quadratic curvature action is shown to reproduce four-dimensional Weyl--Bach conformal gravity upon fixing a Weyl gauge. Local twistor and dual twistor bundles are constructed as associated spin tractor bundles, illustrating how parallel transport yields the classical incidence relations in flat space and chiral integrability constraints in curved backgrounds: anti-self-duality for $α$-surfaces and self-duality for $β$-surfaces. Furthermore, by pairing local twistors and dual twistors via an invariant canonical pairing, ambitwistor space is shown to parameterize complex null geodesics and symmetrically accommodate both left- and right-handed Weyl curvatures. This framework establishes a direct gauge-theoretic bridge linking conformal Cartan geometry to twistor and ambitwistor theories.