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 CFT U V in the ultraviolet to a conformal field theory CFT I R , the F -coefficient decreases: F U V > F I R . 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.