The F-Theorem and F-Maximization

TL;DR

The work surveys the role of the three-sphere free energy $F$ ($F\equiv -\log|Z_{S^3}|$) in three-dimensional QFTs, articulating the $F$-theorem and the $F$-maximization principle for ${\cal N}=2$ SCFTs. It compiles exact and approximate computations of $F$ across free theories, perturbative regimes, holographic duals, and supersymmetric localization, illustrating monotonicity and the determination of the exact superconformal R-symmetry in concrete models such as the ${\cal N}=2$ Ising/Wess-Zumino systems, SQED, and ABJM-type theories. The chapter connects results from localization with gravity and bootstrap data, providing nonperturbative insights into IR dynamics and dualities. Open questions include a purely field-theoretic derivation of the $F$-theorem and potential generalizations to other dimensions or monotonic interpolating functions.

Abstract

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_UV in the ultraviolet to a conformal field theory CFT_IR, 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.