Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional $$ \mathcal{N}=4 $$ theories

Abstract

Abstract The Coulomb branches of certain 3-dimensional $$ \mathcal{N}=4 $$ N = 4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional Lie algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler 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.