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For 3-dimensional field theories with {\cal N}=2 supersymmetry the Euclideanpath integrals on the three-sphere can be calculated using the method oflocalization; they reduce to certain matrix integrals that depend on theR-charges of the matter fields. We solve a number of such large N matrix modelsand calculate the free energy F as a function of the trial R-charges consistentwith the marginality of the superpotential. In all our {\cal N}=2superconformal examples, the local maximization of F yields answers that scaleas N^{3/2} and agree with the dual M-theory backgrounds AdS_4 x Y, where Y are7-dimensional Sasaki-Einstein spaces. We also find in toric examples that localF-maximization is equivalent to the minimization of the volume of Y over thespace of Sasakian metrics, a procedure also referred to as Z-minimization.Moreover, we find that the functions F and Z are related for any trialR-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 fieldtheories: the finite part of the free energy on the three-sphere decreasesalong RG trajectories and is stationary at RG fixed points. We also show thatin an infinite class of Chern-Simons-matter gauge theories where theChern-Simons levels do not sum to zero, the free energy grows as N^{5/3} atlarge N. This non-trivial scaling matches that of the free energy of thegravity duals in type IIA string theory with Romans mass.

Towards the F-Theorem: N=2 Field Theories on the Three-Sphere