Holographic c-theorems and higher derivative gravity

TL;DR

The paper extends holographic c-theorems to higher-curvature gravity by constructing a monotone $a(r)$ for Lovelock theories, proving $a'(r) \propto -(T^t_t-T^r_r)/(A')^d$ under the NEC, and thereby generalizing the boundary $a$-anomaly monotonicity to arbitrary Lovelock orders with second-order bulk EOM. It also analyzes $f(R)$ gravity as a toy model with higher derivatives, showing that monotonicity requires the additional condition $F''\ge0$ and can fail when higher-derivative effects dominate, hinting at potential ghost-related issues. The discussion connects the $a$-function to entanglement entropy and Wald entropy, and argues that a fully general holographic $c$-theorem in higher-derivative gravity likely cannot be proven without restricting to theories whose linearized equations of motion remain second order or without imposing extra unitarity constraints. Overall, the work clarifies when holographic monotonicity can be expected and illuminates the role of higher-derivative dynamics in AdS/CFT flows.

Abstract

In AdS/CFT, the holographic Weyl anomaly computation relates the a-anomaly coefficient to the properties of the bulk action at the UV fixed point. This universal behavior suggests the possibility of a holographic c-theorem for the a-anomaly under flows to the IR. We prove such a c-theorem for higher curvature Lovelock gravity, where the bulk equations of motion remain second order. We also explore f(R) gravity as a toy model where higher derivatives cannot be avoided. In this case, monoticity of the flow requires an additional condition related to the higher derivative nature of the theory. This is in contrast to the case of f(R) black hole entropy, where the second law follows from application of the full Einstein equations and the null energy condition.