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}=2𝒩=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)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)SU(2) and SO(3)_±SO(3)± 4d \mathcal{N}=2𝒩=2 gauge theories. We propose that the 11-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}=2𝒩=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 00-form and/or 11-form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on \mathbb{R}^4× T^2ℝ4×T2.