Quotient Quiver Subtraction

TL;DR

This paper introduces a diagrammatic quiver subtraction method to compute hyper-K"ahler quotients of Coulomb branches in 3d $ abla$=4 theories, expanding the toolkit for analyzing moduli spaces in gauge theory.

Contribution

It develops a new quiver subtraction technique using 'bad' quotient quivers, enabling the calculation of hyper-K"ahler quotients for a broad class of theories, including those with complex moduli spaces.

Findings

Successfully computes Hilbert Series and HWGs for low-rank examples.

Conjectures HWGs for arbitrary rank in certain quiver families.

Demonstrates compatibility of subtraction with other diagrammatic techniques.

Abstract

We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the $\mathrm{SU}(n)$ hyper-K\"ahler quotient (HKQ) of the Coulomb branch of a $3d$ $\mathcal{N}=4$ unitary quiver theory. The target quivers are drawn from a wide range of theories, typically classified as ''good'' or ''ugly'', which satisfy identified selection criteria. Our subtraction procedure uses quotient quivers that are ''bad'', differing thereby from quiver subtractions based on Kraft-Procesi transitions. The procedure identifies one or more resultant quivers, the union of whose Coulomb branches corresponds to the desired HKQ. Examples include quivers whose Coulomb branches are moduli spaces of free fields, closures of nilpotent orbits of classical and exceptional type, and slices in the affine Grassmanian. We calculate the Hilbert Series and Highest Weight Generating functions for HKQ examples of low rank. For certain families of quivers, we are able to conjecture HWGs for arbitrary rank. We examine the commutation relations between quotient quiver subtraction and other diagrammatic techniques, such as Kraft-Procesi transitions, quiver folding, and discrete quotients.