Holography, Fractals and the Weyl Anomaly

TL;DR

This paper explores the connection between holographic renormalization, fractal geometry, and quantum field theory anomalies, proposing a novel relation between Weyl anomalies and fractal dimensions, with implications for entropy and c-theorems.

Contribution

It introduces a new correspondence linking Weyl anomalies to fractal dimensions and suggests an equivalence between entropy divergences and anomalies in quantum field theory.

Findings

Weyl anomaly relates to fractal dimension of path integral measure

Logarithmic UV divergence of Shannon entropy matches Weyl anomaly

Euclidean path integral measure's information dimension satisfies a c-theorem

Abstract

We study the large source asymptotics of the generating functional in quantum field theory using the holographic renormalization group, and draw comparisons with the asymptotics of the Hopf characteristic function in fractal geometry. Based on the asymptotic behavior, we find a correspondence relating the Weyl anomaly and the fractal dimension of the Euclidean path integral measure. We are led to propose an equivalence between the logarithmic ultraviolet divergence of the Shannon entropy of this measure and the integrated Weyl anomaly, reminiscent of a known relation between logarithmic divergences of entanglement entropy and a central charge. It follows that the information dimension associated with the Euclidean path integral measure satisfies a c-theorem.