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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.04878 |
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| _version_ | 1866917946665730048 |
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| author | Sakai, Kazuhiro |
| author_facet | Sakai, Kazuhiro |
| contents | We study the E-string theory on $\mathbb{R}^4\times T^2$ with Wilson lines. We consider two examples where interesting automorphisms arise. In the first example, the spectrum is invariant under the $F_4$ Weyl group acting on the Wilson line parameters. We obtain the Seiberg-Witten curve expressed in terms of Weyl invariant $F_4$ Jacobi forms. We also clarify how it is related to the thermodynamic limit of the Nekrasov-type formula. In the second example, the spectrum is invariant under the $D_4$ triality combined with modular transformations, the automorphism originally found in the 4d $\mathcal{N}=2$ supersymmetric $\mathrm{SU}(2)$ gauge theory with four massive flavors. We introduce the notion of triality invariant Jacobi forms and present the Seiberg-Witten curve in terms of them. We show that this Seiberg-Witten curve reduces precisely to that of the 4d theory with four flavors in the limit of $T^2$ shrinking to zero size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04878 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | E-strings, $F_4$, and $D_4$ triality Sakai, Kazuhiro High Energy Physics - Theory We study the E-string theory on $\mathbb{R}^4\times T^2$ with Wilson lines. We consider two examples where interesting automorphisms arise. In the first example, the spectrum is invariant under the $F_4$ Weyl group acting on the Wilson line parameters. We obtain the Seiberg-Witten curve expressed in terms of Weyl invariant $F_4$ Jacobi forms. We also clarify how it is related to the thermodynamic limit of the Nekrasov-type formula. In the second example, the spectrum is invariant under the $D_4$ triality combined with modular transformations, the automorphism originally found in the 4d $\mathcal{N}=2$ supersymmetric $\mathrm{SU}(2)$ gauge theory with four massive flavors. We introduce the notion of triality invariant Jacobi forms and present the Seiberg-Witten curve in terms of them. We show that this Seiberg-Witten curve reduces precisely to that of the 4d theory with four flavors in the limit of $T^2$ shrinking to zero size. |
| title | E-strings, $F_4$, and $D_4$ triality |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2304.04878 |