Quiver origami: discrete gauging and folding

Abstract

We study two types of discrete operations on Coulomb branches of 3d$$ \mathcal{N} $$ N = 4 quiver gauge theories using both abelianisation and the monopole formula. We generalise previous work on discrete quotients of Coulomb branches and introduce novel wreathed quiver theories. We further study quiver folding which produces Coulomb branches of non-simply laced quivers. Our methods explicitly describe Coulomb branches in terms of generators and relations including mass deformations.

(5d RG-flow) trees in the tropical rain forest

“…These quivers were studied in [78,[80][81][82][83] in which the computation of their Hilbert series and heighest weight generating functions (HWGs) are performed, thus identifying the corresponding nilpotent orbits and Hasse diagrams. The simplest occurrence of this phenomenon can be traced back to a the observation by Kostant and Brylinski [84] that a Z 2 quotient of the closure of the minimal nilpotent orbit of d n+1 yields the closure of the next-to-minimal nilpotent orbit of b n , combined with the fact that this quotient can be implemented on the quiver using the bouquet construction [85] or wreathing [83]. Note that magnetic quivers for the closures of the non-extremal leaves in the diagrams of table 7 are non-simply-laced, possibly with additional identifications (see section 5.2.4 in [83] for a detailed discussion of this subtle point).…”

“…The simplest occurrence of this phenomenon can be traced back to a the observation by Kostant and Brylinski [84] that a Z 2 quotient of the closure of the minimal nilpotent orbit of d n+1 yields the closure of the next-to-minimal nilpotent orbit of b n , combined with the fact that this quotient can be implemented on the quiver using the bouquet construction [85] or wreathing [83]. Note that magnetic quivers for the closures of the non-extremal leaves in the diagrams of table 7 are non-simply-laced, possibly with additional identifications (see section 5.2.4 in [83] for a detailed discussion of this subtle point). In the following, we do not represent these non-simply-laced quivers, and we apply the quiver subtraction algorithm using directly the non elementary slices of table 7.…”

S-fold magnetic quivers

“…A feature of all magnetic quivers coming from S-folds studied in the following is the appearance of non-simply laced edges, where the order of the non-simply laced edges corresponds to the order of the S-fold (the folding parameter). The 3d N = 4 Coulomb branches of non-simply laced quivers were studied in [44][45][46][47]. The non-simply laced quivers can be obtained from simply laced ones through an operation called folding [48][49][50].…”

Discrete gauging and Hasse diagrams

“…When this is the case, the quiver is called a magnetic quiver for that Higgs branch [46][47][48][49][50]. Our second main result is a magnetic quiver for the Higgs branch of SU(N ) I theories, in the form of a wreathed quiver, as introduced in [39]. As a check of our conjecture we compute the 3d N = 4 Coulomb branch Hilbert series of that quiver and find perfect agreement with the Higgs branch Hilbert series of the corresponding SU(N ) I theory that was computed in [1,28].…”

“…This partial Higgs mechanism is naturally described by a partial order diagram, called the Hasse diagram, where each node of the diagram is related to the subgroup of the initial gauge group that is left unbroken by the Higgs mechanism. The systematic study of the Higgs branch of theories with 8 supercharges using Hasse diagrams was initiated in [34] and further analysed in [35][36][37][38][39][40][41][42][43][44][45]. The Higgs branch Hasse diagram in turns reveals the geometric structure of the Higgs branch as a symplectic singularity, the nodes being in correspondence with symplectic leaves, and the links representing elementary transverse slices.…”