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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2308.10225 |
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| _version_ | 1866916263117193216 |
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| author | Closset, Cyril Magureanu, Horia |
| author_facet | Closset, Cyril Magureanu, Horia |
| contents | 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 $\mathbb{R}^4\times T^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10225 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories Closset, Cyril Magureanu, Horia High Energy Physics - Theory 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 $\mathbb{R}^4\times T^2$. |
| title | Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2308.10225 |