Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional N=4 theories

TL;DR

This paper demonstrates how Coulomb branch constructions of 3D N=4 theories can explicitly resolve nilpotent orbit closures, revealing their symplectic resolutions and Mukai flops through the monopole formula.

Contribution

It shows that monopole formulas with background charges accurately describe resolutions of height two nilpotent orbit closures, including symplectic resolutions and Mukai flops.

Findings

Monopole formula reproduces existence of symplectic resolutions.

Resolutions are given by cotangent bundles of Hermitian symmetric spaces.

All resolutions are realized via unitary Coulomb branch quivers.

Abstract

The Coulomb branches of certain 3-dimensional N=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-K\"ahler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic resolution, (ii) the form of the symplectic resolution, and (iii) the Mukai flops in the case of multiple resolutions. Moreover, the (characteristic) height two nilpotent orbit closures are resolved by cotangent bundles of Hermitian symmetric spaces and the unitary Coulomb branch quiver realisations exhaust all the possibilities.