Quiver Polymerisation

Abstract

Two new diagrammatic techniques on $3d\mathcal{N}=4$ quiver gauge theories, termed chain and cyclic quiver polymerisation are introduced. These gauge a diagonal $\mathrm{SU}/\mathrm{U}(k)$ subgroup of the Coulomb branch global symmetry of a quiver (or pair of quivers) with multiple legs. The action on the Coulomb branch is that of a $\mathrm{SU}/\mathrm{U}(k)$ hyper-Kähler quotient. The polymerisation techniques build and generalise known composition methods from class $\mathcal{S}$. Polymerisation is used to generate a wide range of magnetic quivers from various physical contexts. These include polymerisation constructions for Kronheimer-Nakajima quivers, which generalise the ADHM construction for the moduli space of $k\mathrm{SU}(N)$ instantons on ${\mathbb{C}}^2$ to A-type singularities. Also a polymerisation construction of the magnetic quiver for the $6d\mathcal{N}=(1,0)$ coming from two $\frac{1}{2}$ M5 branes probing an $E_6$ Klein singularity. We find a method of extending magnetic quivers for Class $\mathcal{S}$ theories to cure the incomplete Higgsing that arises when gluing punctures into the loops associated with higher genus theories. Other novel constructions include a unitary magnetic quiver for the closure of a height four nilpotent orbit of $\mathrm{SO}(7)$. We explore the relationships between the Coulomb and Higgs branches of quivers under polymerisation.

Typos corrected