Z-extremization and F-theorem in Chern-Simons matter theories

The three dimensional exact R symmetry of N = 2 SCFTs extremizes the partition function localized on a three sphere. Here we verify this statement at weak coupling. We give a detailed analysis for two classes of models. The first one is an SU(N)(k) gauge theory at large k with both fundamental and adjoint matter fields, while the second is a flavored version of the ABJ theory, where the CS levels are large but they do not necessarily sum up to zero. We study in both cases superpotential deformations and compute the R charges at different fixed points. When these fixed points are connected by an RG flow we explicitly verify that the free energy, computed from the partition function localized on S-3, decreases at the endpoints of the flow between the fixed points, corroborating the conjecture of an F-theorem in three dimensions.