# Weyl Connections and their Role in Holography

## TL;DR

This paper extends the holographic dictionary by incorporating Weyl connections at the boundary, providing a clearer geometric interpretation of the Weyl anomaly and enriching the boundary theory with a Weyl current.

## Contribution

It introduces a modified Fefferman-Graham formalism that includes Weyl connections, preserving Weyl invariance and enhancing the holographic correspondence.

## Key findings

- The boundary geometry includes a Weyl connection instead of a Levi-Civita connection.
- The Weyl anomaly has a cohomological geometric interpretation.
- The boundary theory features a Weyl current that interacts with the stress tensor.

## Abstract

It is a well-known property of holographic theories that diffeomorphism invariance in the bulk space-time implies Weyl invariance of the dual holographic field theory in the sense that the field theory couples to a conformal class of background metrics. The usual Fefferman-Graham formalism, which provides us with a holographic dictionary between the two theories, breaks explicitly this symmetry by choosing a specific boundary metric and a corresponding specific metric ansatz in the bulk. In this paper, we show that a simple extension of the Fefferman-Graham formalism allows us to sidestep this explicit breaking; one finds that the geometry of the boundary includes an induced metric and an induced connection on the tangent bundle of the boundary that is a Weyl connection (rather than the more familiar Levi-Civita connection uniquely determined by the induced metric). Properly invoking this boundary geometry has far-reaching consequences: the holographic dictionary extends and naturally encodes Weyl-covariant geometrical data, and, most importantly, the Weyl anomaly gains a clearer geometrical interpretation, cohomologically relating two Weyl-transformed volumes. The boundary theory is enhanced due to the presence of the Weyl current, which participates with the stress tensor in the boundary Ward identity.