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.

We construct supersymmetric field theories on Riemannian three-manifolds M, focusing on N = 2 theories with a U (1) R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. The field theory on M possesses a single supercharge if and only if M admits an almost contact metric structure that satisfies a certain integrability condition. This may lead to global restrictions on M, even though we can always construct one supercharge on any given patch. We also analyze the conditions for the presence of additional supercharges. In particular, two supercharges of opposite R-charge exist on every Seifert manifold. We present general supersymmetric Lagrangians on M and discuss their flat-space limit, which can be analyzed using the Rcurrent supermultiplet. As an application, we show how the flat-space two-point function of the energy-momentum tensor in N = 2 superconformal theories can be calculated using localization on a squashed sphere.

We study Seiberg-like dualities in three dimensional N = 2 supersymmetric theories, emphasizing Chern-Simons terms for the global symmetry group, which affect contact terms in two-point functions of global currents and are essential to the duality map. We introduce new Seiberg-like dualities for Yang-Mills-Chern-Simons theories with unitary gauge groups with arbitrary numbers of matter fields in the fundamental and antifundamental representations. These dualities are derived from Aharony duality by real mass deformations. They allow to initiate the systematic study of Seiberg-like dualities in Chern-Simons quivers. We also comment on known Seiberg-like dualities for symplectic and orthogonal gauge groups and extend the latter to the Yang-Mills case. We check our proposals by showing that the localized partition functions on the squashed S 3 match between dual descriptions.

We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z M . Our primary focus is the dependence of Z M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N = 1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z M is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N = 2 theories with a U(1) R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that Z M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S 3 ×S 1 or S 2 ×S 1 , which are related to supersymmetric indices, and manifolds diffeomorphic to S 3 (squashed spheres). In examples where Z M has been calculated explicitly, our results explain many of its observed properties.

We study three-dimensional N = 2 supersymmetric gauge theories on Σ g × S 1 with a topological twist along Σ g , a genus-g Riemann surface. The twisted supersymmetric index at genus g and the correlation functions of half-BPS loop operators on S 1 can be computed exactly by supersymmetric localization. For g = 1, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simonsmatter theory, in terms of the associated Bethe equations for the theory on R 2 × S 1 . This also provides a powerful and simple tool to study 3d N = 2 Seiberg dualities. Finally, we study A-and B-twisted indices for N = 4 supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the S 2 × S 1 twisted indices and the Hilbert series of N = 4 moduli spaces.