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

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.

137 pages plus appendix, many figures. v2: added references, corrected typos and imprecisions; v3: corrected typos and references. SciPost version

References in corpus (33)

• Generalized Global Symmetries
• The Refined Topological Vertex
• Webs of five-branes and N=2 superconformal field theories
• 5-dim Superconformal Index with Enhanced En Global Symmetry
• F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds
• Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory
• 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities
• Holomorphic Anomaly in Gauge Theories and Matrix Models
• Higher-form symmetries of 6d and 5d theories
• Brane Tilings and Reflexive Polygons
• Higher-Form Symmetries and Their Anomalies in M-/F-Theory Duality
• On Rationally Parametrized Modular Equations
• (5d RG-flow) Trees in the Tropical Rain Forest
• Higher Form Symmetries of Argyres-Douglas Theories
• Instantons, Symmetries and Anomalies in Five Dimensions
• (Symplectic) Leaves and (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest
• Peacock patterns and new integer invariants in topological string theory
• 1-Form Symmetry, Isolated N=2 SCFTs, and Calabi-Yau Threefolds
• Flavor Symmetry of 5d SCFTs, Part 2: Applications
• Bootstrapping BPS spectra of 5d/6d field theories
• Flavor Symmetry of 5d SCFTs, Part 1: General Setup
• S-folds, String Junctions, and 4D $\mathcal{N}=2$ SCFTs
• Instanton counting on blowup. II. $K$-theoretic partition function
• Topological Strings and (Almost) Modular Forms
• On the BPS spectrum of 5d SU(2) super-Yang-Mills
• 5D BPS Quivers and KK Towers
• On the compactification of 5d theories to 4d
• The Cat's Cradle: Deforming the higher rank $E_1$ and ${\tilde{E}}_1$ theories
• BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations
• Elliptic Loci of SU(3) Vacua
• Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD
• A new rank-2 Argyres-Douglas theory
• Fibre-base duality of 5d KK theories

Cited by in corpus (4)

• 2-Group Symmetries in Class S
• Connecting 5d Higgs Branches via Fayet-Iliopoulos Deformations
• The Characteristic Dimension of Four-dimensional $\mathcal{N}=2$ SCFTs
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories