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Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories
by Cyril Closset, Horia Magureanu
Submission summary
| Authors (as registered SciPost users): | Cyril Closset |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2308.10225v3 (pdf) |
| Date submitted: | 2024-01-30 08:44 |
| Submitted by: | Closset, Cyril |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
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 $\mathbb{R}^4\times T^2$.
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Report
This is a sound, original, and well-written paper on a subject of interest to the quantum field theory community, especially those interested in supersymmetric field theories. It addresses, through detailed calculation in a few examples, the general problem of how and to what extent various symmetry properties of supersymmetric QFTs are reflected in their supersymmetry-protected low energy observables --- in this case the geometry of the moduli space of vacua of the theory. This gives evidence for an an interesting and compelling hypothesis relating the 1-form symmetry of the theory to the Mordell-Weil group of the Seiberg-Witten curve describing the Coulomb branch moduli geometry.