Quiver theories and Hilbert series of classical Slodowy intersections

Abstract

We build on previous studies of the Higgs and Coulomb branches of SUSY quiver theories having 8 supercharges, including 3d N = 4, and Classical gauge groups. The vacuum moduli spaces of many such theories can be parameterised by pairs of nilpotent orbits of Classical Lie algebras; they are transverse to one orbit and intersect the closure of the second. We refer to these transverse spaces as "Slodowy intersections". They embrace reduced single instanton moduli spaces, nilpotent orbits, Kraft-Procesi transitions and Slodowy slices, as well as other types. We show how quiver subtractions, between multi-flavoured unitary or ortho-symplectic quivers, can be used to find a complete set of Higgs branch constructions for the Slodowy intersections of any Classical group. We discuss the relationships between the Higgs and Coulomb branches of these quivers and T ρ σ theories in the context of 3d mirror symmetry, including problematic aspects of Coulomb branch constructions from ortho-symplectic quivers. We review Coulomb and Higgs branch constructions for a subset of Slodowy intersections from multi-flavoured Dynkin diagram quivers. We tabulate Hilbert series and Highest Weight Generating functions for Slodowy intersections of Classical algebras up to rank 4. The results are confirmed by direct calculation of Hilbert series from a localisation formula for normal Slodowy intersections that is related to the Hall Littlewood polynomials.

Magnetic quivers from brane webs with O5 planes

“…While the Hasse diagram derivation for the unitary quivers is straightforward from the results of [70,74], the algorithm for orthosymplectic quivers is more subtle. Quiver subtraction with (framed) orthosymplectic quivers have been used in [65,81] and the first Hasse diagrams derived in this class have been presented in [61], in the context of 6d theories.…”

“…Following the spirit of [70] one can derive the Higgs branch Hasse diagrams of the theories considered by using the brane web construction of section 3. In principle, one could also use the algorithm of quiver subtraction, as introduced in [73,74] and further developed for unitary quivers in [70] and for framed (flavoured) orthosymplectic quivers in [81]. However, the orthosymplectic quivers appearing in table 1 are not framed, and may involve unitary gauge groups, and as a consequence the currently known algorithm needs to be extended.…”

Hasse diagrams for 3d $$ \mathcal{N} $$ = 4 quiver gauge theories — Inversion and the full moduli space

“…An analysis of effective theories was performed in the realm of nilpotent orbits and Slodowy slices in [40,44], their 'descendants' or 'Slodowy intersection' figures contain the information of the effective massless interacting theory on each leaf in the full moduli space, or in other words the theories associated to all transverse spaces inside the full moduli space. It should be noted, that our analysis is not restricted to nilpotent orbits, see for example (3.7)-(3.9).…”

“…In this appendix we review the operation called quiver subtraction developed in [17,25] for unitary quivers and give some additional insight for the non-unitary case, which was discussed in the realm of nilpotent orbits in [44]. We focus only on quiver subtraction of an elementary slice, when the slice is a Kleinian singularity or the closure of a minimal nilpotent orbit.…”

“…gauge nodes one has to subtract a U(1) piece with the same number of flavours and for the SU (2) and Sp(n) case one has to subtract SU(2)=Sp(1) pieces with the same number of flavours. Rules for quiver subtraction of framed orthosymplectic quivers are given for 'Slodowy intersections' in [44]. Unframed orthosymplectic quivers are studied in [45,62].…”