I will explore aspects of the Coulomb-branch physics of five-dimensional superconformal field theories (SCFT). More precisely, I will consider the 5d SCFT on a circle, and describe the general structure of the Coulomb-branch BPS states as encoded in a "5d BPS quiver," which can be computed from standard string-theory geometric-engineering techniques. The interplay between 4d and 5d BPS quivers will play a central role in our story.

I will revisit the well-known construction of 5d SCFTs from M-theory on a CY3 singularity. Upon massive deformation, such 5d SCFTs are often expected to have 5d N=1 supersymmetric gauge theory descriptions at low energy. I will present a new way to study these 5d ``gauge theory phases'' systematically using type-IIA string theory, and I will comment on the phenomenon of "UV duality." Along the way, I will discuss some slightly subtle properties of the 5d N=1 Coulomb branch prepotential.

In the first part of my talk, I will present a general classification of Riemannian three-manifolds on which one can put 3d N=2 supersymmetric field theories while preserving some amount of supersymmetry. This formalism clarifies the relationship between the extra couplings necessary to preserve supersymmetry in curved space, on the one hand, and various operators of the flat space theory, on the other hand. In the second part of the talk I will present some simple applications of this formalism. In particular I will present exact results for various two-point functions of N=2 SCFTs which were hitherto out of reach.

I will explain how to put three-dimensional supersymmetric theories on curved three-manifolds preserving some supersymmetry. I will discuss some physical implications of this method, including an exact formula for the two-point function of the energy-momentum tensor in 3d N=2 superconformal theories.