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

Understand

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.