The U-plane of rank-one 4d N=2 KK theories

2021/07/31 by Cyril Closset, Horia Magureanu
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Coulomb #Fibration #Global symmetry #Mathematical physics #Mathematics #Moduli space #Particle physics theoretical and experimental studies #Partition function (quantum field theory) #Physics #Pure mathematics #Quantum Chromodynamics and Particle Interactions #Quantum mechanics #Spontaneous symmetry breaking #Symmetry breaking #hep-th

paper · pdf · doi:10.21468/scipostphys.12.2.065

published as SciPost Phys. 12, 065 (2022) · 137 pages plus appendix, many figures. v2: added references, corrected typos and imprecisions; v3: corrected typos and references. SciPost version

arxiv created 2022/01/05 · openalex publication_date 2022/02/17 · arxiv updated 2022/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 ℝ⁴ × S¹. 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.

Citations

• On the Seiberg-Witten Approach to electric-magnetic Duality
• Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
• BPS Structure of Argyres-Douglas Superconformal Theories
• Geometric constraints on the space of N=2 SCFTs I: physical constraints on relevant deformations
• D-Branes And Mirror Symmetry
• Fractional branes and boundary states in orbifold theories
• The Strong-Coupling Spectrum of the Seiberg-Witten Theory
• Geometric realizations of BPS states in N = 2 theories
• Brane tilings and reflexive polygons
• Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD
• Geometric singularities and enhanced gauge symmetries
• Refined stable pair invariants for E-, M- and [p, q]-strings
• Non-simply-connected gauge groups and rational points on elliptic curves
• Twisted baryon number in N = 2 supersymmetric QCD
• Modular solutions to equations of generalized Halphen type
• Phases of N = 2 theories in two dimensions
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Generalized Global Symmetries
• N=2 Quantum Field Theories and Their BPS Quivers
• Topological quantum field theory
• Instanton counting on blowup. II. K-theoretic partition function
• 5d SCFTs from Decoupling and Gluing
• Attractor invariants, brane tilings and crystals
• Peacock patterns and new integer invariants in topological string theory
• Monstrous Moonshine