Explicit Entropic Proofs of Irreversibility Theorems for Holographic RG Flows

TL;DR

This paper provides a new entropic proof of irreversibility theorems for holographic renormalization group flows, using only the Null Energy Condition and entanglement entropy, applicable across various dimensions.

Contribution

It introduces a holographic $c$-function based on minimal surface distances, proving monotonicity for flows within and across dimensions using entropic methods.

Findings

Reproves $c$-, $F$-, and $a$-theorems holographically in 2, 3, and 4 dimensions.

Establishes monotonicity of flows from $ ext{AdS}_{D+1}$ to $ ext{AdS}_3$.

Proves the novel case of flows from $ ext{AdS}_5$ to $ ext{AdS}_4$.

Abstract

We revisit the existence of monotonic quantities along renormalization group flows using only the Null Energy Condition and the Ryu-Takayanagi formula for the entanglement entropy of field theories with anti-de Sitter gravity duals. In particular, we consider flows within the same dimension and holographically reprove the $c$-, $F$-, and $a$-theorems in dimensions two, three, and four. We focus on the family of maximally spherical entangling surfaces, define a quasi-constant of motion corresponding to the breaking of conformal invariance, and use a properly defined distance between minimal surfaces to construct a holographic $c$-function that is monotonic along the flow. We then apply our method to the case of flows across dimensions: There, we reprove the monotonicity of flows from $\mathrm{AdS}_{D+1}$ to $\mathrm{AdS}_3$ and prove the novel case of flows from $\mathrm{AdS}_5$ to $\mathrm{AdS}_4$.