Universal relation between $C_{T}$ and the CFT Weyl anomaly

TL;DR

This paper proves a universal relation between the energy-momentum tensor two-point function coefficient and the Weyl anomaly coefficient in even-dimensional conformal field theories, using holographic and CFT methods.

Contribution

It establishes a new universal relation linking $C_T$ and the Weyl anomaly coefficient $c$ in generic even-dimensional CFTs, combining holographic and CFT derivations.

Findings

Validated the relation with known examples.

Connected holographic results with CFT renormalization group analysis.

Isolated the quadratic Weyl term unambiguously.

Abstract

We establish a universal relation between the coefficient $C_T$ of the energy momentum tensor two point function and the coefficient $c$ multiplying the term quadratic in the Weyl tensor in the Weyl anomaly of a generic even dimensional conformal field theory. Our first derivation combines long known holographic results for $C_T$ and for the Weyl anomaly in Einstein bulk gravity with a recently obtained Chern Gauss Bonnet formula for compact Einstein manifolds. This theorem isolates the Weyl squared contribution in the relation between the Euler density and the $Q$ curvature, allowing us to identify the relevant quadratic term unambiguously. We then provide a genuine CFT derivation based on the renormalization group running of the TT correlator with respect to the arbitrary but necessary mass scale $\mu$. Several known examples are revisited to illustrate and validate the general result.