Geometric Holography, the Renormalization Group and the c-Theorem

TL;DR

Geometric Holography, the Renormalization Group and the c-Theorem addresses how a conformally invariant holographic projection of a diffeomorphism-invariant bulk theory can be realized through a geometric framework. The authors propose a holographic c-function defined by area elements along null geodesics and prove a c-theorem via Raychaudhuri's equation in an Einstein bulk, linking RG flow to bulk geometry. They explore both Lorentzian holography with a bulk ambient space and Penrose's conformal infinity, and extend to non-local observables such as Wilson loops, discussing conditions for well-posed holographic maps. The work identifies key boundary conditions (vanishing Bach/Cotton tensors) for unique holographic reconstructions and outlines future directions, including extensions to massive states and AdS with different signatures.

Abstract

In this paper the whole geometrical set-up giving a conformally invariant holographic projection of a diffeomorphism invariant bulk theory is clarified. By studying the renormalization group flow along null geodesic congruences a holographic version of Zamolodchikov's c-theorem is proven.

Figures (2)

• Figure 1: Geometrical representation of the space G corresponding to $M = S^1$, as the cone $\mathcal{C}$.
• Figure 2: $S_M$ in the simple case $M\equiv S^1$.