The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories

TL;DR

The paper presents a comprehensive study of the one-dimensional Coulomb (U) plane for rank-one 4d N=2 KK theories obtained from circle compactifications of 5d E_n SCFTs. It encodes the full Seiberg–Witten geometry as a rational elliptic surface fibered over the U-plane, with the fiber at infinity constrained to an I_{9-n} type, and provides a Persson/Mirranda-based classification of all Coulomb-branch configurations. A central result is that the global flavor symmetry and one-form symmetry are read from the Mordell–Weil group of the SW elliptic fibration, with torsion encoding the non-abelian global form and free sections generating abelian factors, while the MW lattice orthogonal to the 7-brane root lattice ties to the physical charge lattice. The work also highlights modularity: for many massless configurations, the U-plane is a modular curve for a finite index subgroup of PSL(2,Z), enabling explicit identification of BPS spectra, quivers, and singularity structures; this modular viewpoint is employed to study strong-coupling dyons, Argyres–Douglas points, and the corresponding quivers. Additionally, gravitational couplings A(U) and B(U) on toric U-planes are matched to Nekrasov-function-based UV computations, extending SW-geometric insights to 5d/4d KK contexts. Overall, the paper provides a unifying geometric framework linking RES, MW structure, modularity, and BPS quivers to the IR physics of rank-one KK theories and their 4d descendants, while mapping rich interplays between geometry, string/F-theory intuition, and field theory data.

Abstract

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.

Figures (23)

• Figure 1: Paths ${\boldsymbol{\gamma}}_v$ generating the fundamental group of the $U$-plane. The path around infinity is equal to minus the sum of all the other paths, ${\boldsymbol{\gamma}}_\infty=-({\boldsymbol{\gamma}}_1+ \cdots+{\boldsymbol{\gamma}}_k)$.
• Figure 2: Examples of affine Dynkin diagrams corresponding to resolved Kodaira fibers. These are the ones that correspond to the semi-simple $E_n$ Lie algebras. The affine node $\Theta_{v, 0}$ is indicated in dark red, and the nodes with unit multiplicity ($\widehat{m}_{v, i}=1$) are all the nodes in (dark or light) red. The multiplicities $\widehat{m}_{v, i}>1$ are indicated next to the nodes.
• Figure 3: The $u$-plane of 4d $\mathcal{N}=2$$SU(2)$ with $N_f$ massless flavours.
• Figure 4: Fundamental domains for $\Gamma^0(4)$. Figure (a) shows a standard choice, with width one cusps at $\tau = 0$ and $2$, while in figure (b) the cusp at $\tau = \pm 2$ is split, with the branch cut of the periods indicated by the dashed line.
• Figure 5: Fundamental domains for 4d $SU(2)$ theories with $N_f = 1, 2, 3$ flavours.