Hyperk\"ahler Quotients, Mirror Symmetry, and F-theory

TL;DR

This paper explores the geometric and physical properties of hyperk"ahler quotients related to extended E_7 Dynkin diagrams, revealing connections to mirror symmetry and F-theory compactifications, and proposing a generalized Coulomb branch framework.

Contribution

It extends the graphical method for hyperk"ahler quotients to the E_7 case and suggests a generalized Coulomb branch concept for mirror symmetry in four dimensions.

Findings

Derived the full curve for the E_7 hyperk"ahler quotient.

Found the curve matches previous results by Minahan and Nemeschansky.

Proposed a generalized Coulomb branch framework for mirror symmetry.

Abstract

Using the graphical method developed in hep-th/9908082, we obtain the full curve corresponding to the hyperk\"ahler quotient from the extended E_7 Dynkin diagram. As in the E_6 case discussed in the same paper above, the resulting curve is the same as the one obtained by Minahan and Nemeschansky. Our results seem to indicate that it is possible to define a generalized Coulomb branch such that four dimensional mirror symmetry would act by interchanging the generalized Coulomb branch with the Higgs branch of the dual theory. To understand these phenomena, we discuss mirror symmetry and F-theory compactifications probed by D3 branes.