Abstract:We approach the topic of Classical group nilpotent orbits from the perspective of the moduli spaces of quivers, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series nilpotent orbits. We present systematic constructions for BCD series nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKähler quotients, between quivers. We analyse all Classical group nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.

We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups. The method draws on plethystic functions and character generating functions along with Weyl integration. We give explicit examples showing how the use of such highest weight generating functions ("HWGs") permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and also of the moduli spaces of instantons. We identify how the HWGs of gauge invariant operators of a selection of classical and exceptional SQCD theories result from the interaction under symmetrisation between a product group and the invariant tensors of its gauge group. In order to calculate HWGs, we derive and tabulate character generating functions for low rank groups by a variety of methods, including a general character generating function, based on the Weyl Character Formula, for any classical or exceptional group.

For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the choice of the precise gauge group becomes crucial when the magnetic lattice of the theory is considered. This question is addressed in the context of Coulomb branches for 3d $$ \mathcal{N} $$ N = 4 quiver gauge theories, which are moduli spaces of dressed monopole operators. We compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers for different choices of gauge groups, including diagonal quotients of the product gauge group of individual factors, where the quotient is by a trivially acting subgroup. Choosing different such diagonal groups results in distinct Coulomb branches, related as orbifolds. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions 4, 5, and 6. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.

Abstract:We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.

We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra g. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone S XN of g. We calculate refined Hilbert series for Classical algebras up to rank 4 (and A 5 ), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections S X N ; such dual quiver theories exist for the Slodowy slices of A algebras, but are limited to a subset of the Slodowy slices of BCD algebras. The analysis opens new questions about the extent of 3d mirror symmetry within the class of SCFTs known as T ρ σ pGq theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the SU p2q embedding into G of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.