Holographic RG flows on curved manifolds and the $F$-theorem

TL;DR

This paper develops a holographic framework to define and analyze $F$-functions on curved manifolds, demonstrating their monotonic behavior along RG flows and connecting them to free energy and entanglement entropy, with applications to free theories.

Contribution

It introduces a method to construct good $F$-functions from holographic on-shell actions and entanglement entropy, applicable to various operator dimensions and beyond holography.

Findings

$F$-functions decrease monotonically along RG flows.

Different $F$-functions are suitable depending on operator dimension.

Monotonicity holds for free fermion and scalar theories on $S^3$.

Abstract

We study $F$-functions in the context of field theories on $S^3$ using gauge-gravity duality, with the radius of $S^3$ playing the role of RG scale. We show that the on-shell action, evaluated over a set of holographic RG flow solutions, can be used to define good $F$-functions, which decrease monotonically along the RG flow from the UV to the IR for a wide range of examples. If the operator perturbing the UV CFT has dimension $\Delta > 3/2$ these $F$-functions correspond to an appropriately renormalized free energy. If instead the perturbing operator has dimension $\Delta < 3/2$ it is the quantum effective potential, i.e. the Legendre transform of the free energy, which gives rise to good $F$-functions. We check that these observations hold beyond holography for the case of a free fermion on $S^3$ ($\Delta=2$) and the free boson on $S^3$ ($\Delta=1$), resolving a long-standing problem regarding the non-monotonicity of the free energy for the free massive scalar. We also show that for a particular choice of entangling surface, we can define good $F$-functions from an entanglement entropy, which coincide with certain $F$-functions obtained from the on-shell action.