Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
Daniel L. Jafferis, Igor R. Klebanov, Silviu S. Pufu, Benjamin R. Safdi
TL;DR
The paper develops localization-based matrix models for 3D ${ m N}=2$ theories on $S^3$ and uses ${F}$-maximization to fix R-charges, revealing $F o N^{3/2}$ scaling in many models and linking ${F}$ to the gravity volumes of Sasaki–Einstein manifolds. It demonstrates a deep connection between field theory extremization and toric geometry via ${Z}$-minimization, with $F$ and ${Z}$ related across trial R-charges. The results provide strong tests of ${ m AdS}_4/{ m CFT}_3$ dualities, show universality in RG flows, and identify an $F$-theorem candidate in three dimensions, alongside an $N^{5/3}$ scaling regime when Chern–Simons levels do not sum to zero. Collectively, the work unifies localization, large-$N$ matrix models, toric geometry, and holography to enumerate and validate the landscape of ${ m N}=2$ M2-brane theories and their gravity duals.
Abstract
For 3-dimensional field theories with {\cal N}=2 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number of such large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the superpotential. In all our {\cal N}=2 superconformal examples, the local maximization of F yields answers that scale as N^{3/2} and agree with the dual M-theory backgrounds AdS_4 x Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the "F-theorem" that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N^{5/3} at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.