SciPost Phys. 12, 065 (2022) ·
published 17 February 2022

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The simplest nontrivial 5d superconformal field theories (SCFT) are the
famous rankone theories with $E_n$ flavour symmetry. We study their $U$plane,
which is the onedimensional Coulomb branch of the theory on $\mathbb{R}^4
\times S^1$. The total space of the SeibergWitten (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 $I_{9n}$ 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 MordellWeil 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 ArgyresDouglas and MinahanNemeschansky 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 lowenergy 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.
SciPost Phys. 6, 052 (2019) ·
published 1 May 2019

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We revisit the correspondence between CalabiYau (CY) threefold isolated
singularities $\mathbf{X}$ and fivedimensional superconformal field theories
(SCFTs), which arise at low energy in Mtheory on the spacetime transverse to
$\mathbf{X}$. Focussing on the case of toric CY singularities, we analyze the
"gaugetheory phases" of the SCFT by exploiting fiberwise Mtheory/type IIA
duality. In this setup, the lowenergy gauge group simply arises on stacks of
coincident D6branes wrapping 2cycles in some ALE space of type $A_{M1}$
fibered over a real line, and the map between the K\"ahler parameters of
$\mathbf{X}$ and the Coulomb branch parameters of the field theory (masses and
VEVs) can be read off systematically. Different type IIA "reductions" give rise
to different gauge theory phases, whose existence depends on the particular
(partial) resolutions of the isolated singularity $\mathbf{X}$. We also comment
on the case of nonisolated toric singularities. Incidentally, we propose a
slightly modified expression for the Coulombbranch prepotential of 5d
$\mathcal{N}=1$ gauge theories.