The U-plane of rank-one 4d N=2 KK theories

The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rank-one theories with E_n flavour symmetry. We study their U -plane, which is the one-dimensional Coulomb branch of the theory on ℝ⁴ × S¹ . The total space of the Seiberg-Witten (SW) geometry -- the E_n SW curve fibered over the U -plane -- is described as a rational elliptic surface with a singular fiber of type I 9 − n at infinity. A classification of all possible Coulomb branch configurations, for the E_n theories and their 4d descendants, is given by Persson's classification of rational elliptic surfaces. We show that the global form of the flavour symmetry group is encoded in the Mordell-Weil group of the SW elliptic fibration. We study in detail many special points in parameters space, such as points where the flavour symmetry enhances, and/or where Argyres-Douglas and Minahan-Nemeschansky theories appear. In a number of important instances, including in the massless limit, the U -plane is a modular curve, and we use modularity to investigate aspects of the low-energy physics, such as the spectrum of light particles at strong coupling and the associated BPS quivers. We also study the gravitational couplings on the U -plane, matching the infrared expectation for the couplings A(U) and B(U) to the UV computation using the Nekrasov partition function.