Quiver Theories and Hilbert Series of Classical Slodowy Intersections

TL;DR

This paper explores the geometric and algebraic structures of Slodowy intersections within classical Lie algebras using quiver theories, providing explicit constructions, relationships, and Hilbert series calculations for these moduli spaces.

Contribution

It introduces a comprehensive quiver subtraction method to construct all Higgs branches of Slodowy intersections for classical groups and analyzes their relation to Coulomb branches and mirror symmetry.

Findings

Complete set of Higgs branch constructions for Slodowy intersections

Tabulated Hilbert series and generating functions up to rank 4

Confirmed results with localization formulas and Hall Littlewood polynomials

Abstract

We build on previous studies of the Higgs and Coulomb branches of SUSY quiver theories having 8 supercharges, including $3d~{\cal N}=4$, 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 $T_{\sigma}^{\rho}$ 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.