Holographic phase space: $c$-functions and black holes as renormalization group flows

TL;DR

<3-5 sentence high-level summary> The paper introduces the N-function for Lovelock gravity as a unifying holographic c-function that tracks RG flows in domain-wall spacetimes and encodes horizon physics for black holes. A flow equation ties the radial variation of N to a gravitational potential, yielding monotonic behavior that matches the Euler anomaly in AdS and relates to black hole entropy at the horizon; planar horizons exhibit a divergent N signaling a large N^2 growth of effective degrees of freedom. It recasts N as S/(4πΩ_eff), where S is Wald entropy and Ω_eff is an effective phase-space volume, linking holographic information content to a Verlinde-type entropic gravity picture and yielding a dual field-theory momentum cutoff Λ_eff. The work also develops a second, area-based phase-space proposal and a thermodynamic interpretation using the proper radial distance β, suggesting a partition-function-like structure for AdS geometries and outlining implications for entanglement entropy and non-vacuum RG flows in holography.

Abstract

We construct a $\mathcal N$-function for Lovelock theories of gravity, which yields a holographic $c$-function in domain-wall backgrounds, and seemingly generalizes the concept for black hole geometries. A flow equation equates the monotonicity properties of $\mathcal N$ with the gravitational field, which has opposite signs in the domain-wall and black hole backgrounds, due to the presence of negative/positive energy in the former/latter, and accordingly $\mathcal N$ monotonically decreases/increases from the UV to the IR. On $AdS$ spaces the $\mathcal N$-function is related to the Euler anomaly, and at a black hole horizon it is generically proportional to the entropy. For planar black holes, $\mathcal N$ diverges at the horizon, which we interpret as an order $N^2$ increase in the number of effective degrees of freedom. We show how $\mathcal N$ can be written as the ratio of the Wald entropy to an effective phase space volume, and using the flow equation relate this to Verlinde's notion of gravity as an entropic force. From the effective phase space we can obtain an expression for the dual field theory momentum cut-off, matching a previous proposal in the literature by Polchinski and Heemskerk. Finally, we propose that the area in Planck units counts states, not degrees of freedom, and identify it also as a phase space volume. Written in terms of the proper radial distance $β$, it takes the suggestive form of a canonical partition function at inverse temperature $β$, leading to a "mean energy" which is simply the extrinsic curvature of the surface. Using this we relate this definition of holographic phase space with the effective phase space appearing in the $\mathcal N$-function.