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

The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rankone theories with E-n flavour symmetry. We study their U-plane, which is the one-dimensional Coulomb branch of the theory on R-4 x S-1. 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 I9-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.

Automated summary

The study examines the U-plane and Seiberg-Witten (SW) curve of the simple non-trivial 5d superconformal field theories with E-n flavour symmetry. It classifies all possible Coulomb branch configurations for the E-n theories and their 4d descendants, and explores the global form of the flavour symmetry group. Special points in parameter space are investigated, such as those where flavour symmetry enhances and where specific theories appear.