E-strings, F 4, and D 4 triality

Abstract We study the E-string theory on R4 × 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 N = 2 supersymmetric 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.