Hidden Relations of Central Charges and OPEs in Holographic CFT

TL;DR

This paper explores the relationships between central charges and OPE coefficients in holographic CFTs, revealing hidden relations and invariance properties through higher-order gravity corrections and differential operators.

Contribution

It introduces a new holographic derivation of central charges and OPE coefficients, uncovering hidden relations and invariance under metric field redefinitions in higher-derivative gravity theories.

Findings

Derived holographic central charges and OPE coefficients with higher-order corrections.

Discovered a hidden relation expressing two OPE coefficients in terms of the third.

Proved the validity of a key relation between central charges and effective AdS radius in general Riemann invariants.

Abstract

It is known that the $(a,c)$ central charges in four-dimensional CFTs are linear combinations of the three independent OPE coefficients of the stress-tensor three-point function. In this paper, we adopt the holographic approach using AdS gravity as an effect field theory and consider higher-order corrections up to and including the cubic Riemann tensor invariants. We derive the holographic central charges and OPE coefficients and show that they are invariant under the metric field redefinition. We further discover a hidden relation among the OPE coefficients that two of them can be expressed in terms of the third using differential operators, which are the unit radial vector and the Laplacian of a four-dimensional hyperbolic space whose radial variable is an appropriate length parameter that is invariant under the field redefinition. Furthermore, we prove that the consequential relation $c=1/3 \ell_{\rm eff}\partial a/\partial\ell_{\rm eff}$ and its higher-dimensional generalization are valid for massless AdS gravity constructed from the most general Riemann tensor invariants.