E-strings, F4, and D4 triality

Abstract

Abstract We study the E-string theory on ℝ4 × T2 with Wilson lines. We consider two examples where interesting automorphisms arise. In the first example, the spectrum is invariant under the F4 Weyl group acting on the Wilson line parameters. We obtain the Seiberg-Witten curve expressed in terms of Weyl invariant F4 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 D4 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 T2 shrinking to zero size.