Hyperkahler quotients and algebraic curves

TL;DR

This paper introduces a graphical method to analyze polynomial invariants of gauge groups, enabling explicit derivation of algebraic curves for hyperkahler quotients, with applications to ALE spaces and various Lie algebra cases.

Contribution

It presents a novel graphical approach to determine algebraic curves associated with hyperkahler quotients, extending to complex cases like E_7 and E_8.

Findings

Derived explicit algebraic relations for A_k, D_k, and E_6 ALE spaces.

Connected deformations of curves to Fayet-Iliopoulos parameters.

Explored orbifold limits for E_7, E_8, and higher-dimensional examples.

Abstract

We develop a graphical representation of polynomial invariants of unitary gauge groups, and use it to find the algebraic curve corresponding to a hyperkahler quotient of a linear space. We apply this method to four dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the explicit relation between the deformations of the curves away from the orbifold limit and the Fayet-Iliopoulos parameters in the corresponding quotient construction. We work out the orbifold limit of E_7, E_8, and some higher dimensional examples.