The F-theorem and F-maximization

This contribution contains a review of the role of the three-sphere free energy F in recent developments related to the F-theorem and F-maximization. The F-theorem states that for any Lorentz-invariant RG trajectory connecting a conformal field theory CFTUV in the ultraviolet to a conformal field theory CFTIR, the F-coefficient decreases: F-UV > F-IR. I provide many examples of CFTs where one can compute F, approximately or exactly, and discuss various checks of the F-theorem. F-maximization is the principle that in an N = 2 SCFT, viewed as the deep IR limit of an RG trajectory preserving N = 2 supersymmetry, the superconformal R-symmetry maximizes F within the set of all R-symmetries preserved by the RG trajectory. I review the derivation of this result and provide examples.