Abstract
The Coulomb branches of certain 3-dimensional mathcal{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.
Highlights
The U(1) flavour symmetry naively suggests that no symplectic resolution exists, it is known that the symplectic resolution is of the form T ∗ (G2/U(2)) → O{2,0}, see [35]
In this paper we have examined to what extent the prescription of the monopole formula with background charges is suitable to study the resolutions of certain Coulomb branches, which are nilpotent orbit closures of height two
For the examples considered with T ∗(G/P ) → O such that G/P ∼= Oμss, the Highest Weight Generating functions (HWG) takes a remarkably simple form
Summary
As we are concerned with quiver gauge theories whose Coulomb branches are closures of nilpotent orbits, we review the necessary ingredients. Nilpotent orbits for the classical Lie algebras sl(n), sp(n), and so(n), can be labelled by partitions ρ = (Bn) If g = so(2n+1), there exists a bijection between N (g) and the set of partitions ρ of 2n + 1 such that even parts have even multiplicity. We restrict to nilpotent orbits of ht(O) = 2 for the following two reasons: (i) the closure of a nilpotent orbit of height ht(O) ≤ 2 always admits a unitary Coulomb branch quiver realisation and (ii) the observation from [3, 4] that height two orbit closures have a simple Hilbert series or Highest Weight Generating function. The height of exceptional nilpotent orbits can be calculated following [26, section 2]; the definition agrees for classical algebras with the partition data given above
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