Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories

We consider three-dimensional N=2 superconformal field theories on a three-sphere and analyze their free energy F as a function of background gauge and supergravity fields. A crucial role is played by certain local terms in these background fields, including several Chern-Simons terms. The presence of these terms clarifies a number of subtle properties of F. This understanding allows us to prove the F-maximization principle. It also explains why computing F via localization leads to a complex answer, even though we expect it to be real in unitary theories. We discuss several corollaries of our results and comment on the relation to the F-theorem.