The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rank-one theories with E_nEn flavour symmetry. We study their UU-plane, which is the one-dimensional Coulomb branch of the theory on \mathbb{R}^4 \times S^1ℝ4×S1. The total space of the Seiberg-Witten (SW) geometry – the E_nEn SW curve fibered over the UU-plane – is described as a rational elliptic surface with a singular fiber of type I_{9-n}I9−n at infinity. A classification of all possible Coulomb branch configurations, for the E_nEn 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 UU-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 UU-plane, matching the infrared expectation for the couplings A(U)A(U) and B(U)B(U) to the UV computation using the Nekrasov partition function.

We study the Coulomb branches of four-dimensional supersymmetric quantum field theories with $$ \mathcal{N} $$ N = 2 supersymmetry, including the KK theories obtained from the circle compactification of the 5d $$ \mathcal{N} $$ N = 1 En Seiberg theories, with particular focus on the relation between their Seiberg-Witten geometries and rational elliptic surfaces. More attention is given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group PSL(2, ℤ), deriving closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups. Moreover, in such cases, we give a prescription for determining the BPS quivers from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This prescription can be also generalized to theories whose Coulomb branches contain ‘undeformable’ singularities, leading to known quivers of such theories.

We study 5d $$ \mathcal{N} $$ N = 1 supersymmetric field theories on closed five-manifolds $$ \mathcal{M} $$ M 5 which are principal circle bundles over simply-connected Kähler four-manifolds, $$ \mathcal{M} $$ M 4, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on $$ \mathcal{M} $$ M 5 through the insertion of a fibering operator, which introduces a non-trivial fibration over $$ \mathcal{M} $$ M 4, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such $$ \mathcal{M} $$ M 5, which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When $$ \mathcal{M} $$ M 4 is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.