Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories

Abstract

We study how the global structure of rank-one 4d $\mathcal{N}=2$ supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the $\mathfrak{su}(2)$ gauge theory, we distinguish between relative and absolute Seiberg-Witten curves. For instance, we discuss in detail the three distinct absolute curves for the $SU(2)$ and $SO(3{)}_{\pm }$ 4d $\mathcal{N}=2$ gauge theories. We propose that the $1$-form symmetry of an absolute theory is isomorphic to a torsion subgroup of the Mordell-Weil group of sections of the absolute curve, while the full defect group of the theory is encoded in the torsion sections of a so-called relative curve. We explicitly show that the relative and absolute curves are related by isogenies (that is, homomorphisms of elliptic curves) generated by torsion sections -- hence, gauging a one-form symmetry corresponds to composing isogenies between Seiberg-Witten curves. We apply this approach to Kaluza-Klein (KK) 4d $\mathcal{N}=2$ theories that arise from toroidal compactifications of 5d and 6d SCFTs to four dimensions, uncovering an intricate pattern of 4d global structures obtained by gauging discrete $0$-form and/or $1$-form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on ${\mathbb{R}}^4\times T^2$.

arXiv comment62 pages plus appendix; v2: added section 2.3 on non-invertible symmetry in the SO(3) theory, fixed typos and references; v3: minor rewriting of appendix A.5

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• Candidate Phases for SU(2) Adjoint QCD${}_4$ with Two Flavors from $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
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• Disconnected gauge groups in the infrared
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings