Holographic RG flows on curved manifolds and the F-theorem

A bstract 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 Δ > 3/2 these F -functions correspond to an appropriately renormalized free energy. If instead the perturbing operator has dimension Δ < 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 (Δ = 2) and the free boson on S 3 (Δ = 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.