E-strings, $F_4$, and $D_4$ triality

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.