Quiver Polymerisation

Two new diagrammatic techniques on 3d\; N=4 quiver gauge theories, termed chain and cyclic quiver polymerisation are introduced. These gauge a diagonal SU/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 SU/U(k) hyper-K\"ahler quotient. The polymerisation techniques build and generalise known composition methods from class 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\;SU(N) instantons on C² to A-type singularities. Also a polymerisation construction of the magnetic quiver for the 6d\; N=(1,0) coming from two 1/2 M5 branes probing an E₆ Klein singularity. We find a method of extending magnetic quivers for Class 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 SO(7). We explore the relationships between the Coulomb and Higgs branches of quivers under polymerisation.