SciPost Phys. 12, 065 (2022) ·
published 17 February 2022
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· pdf
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 $\mathbb{R}^4
\times S^1$. 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 $I_{9-n}$ 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.
