Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits

We approach the topic of Classical group nilpotent orbits from the perspective of their moduli spaces, 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 HyperKahler 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.Comment: 67 pages, 11 figure

Highest Weight Generating Functions for Hilbert Series

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.Comment: 76 page

Construction and Deconstruction of Single Instanton Hilbert Series

Many methods exist for the construction of the Hilbert series describing the moduli spaces of instantons. We explore some of the underlying group theoretic relationships between these various constructions, including those based on the Coulomb branches and Higgs branches of SUSY quiver gauge theories, as well as those based on generating functions derivable from the Weyl Character Formula. We show how the character description of the reduced single instanton moduli space of any Classical or Exceptional group can be deconstructed faithfully in terms of characters or modified Hall-Littlewood polynomials of its regular semi-simple subgroups. We derive and utilise Highest Weight Generating functions, both for the characters of Classical or Exceptional groups and for the Hall-Littlewood polynomials of unitary groups. We illustrate how the root space data encoded in extended Dynkin diagrams corresponds to relationships between the Coulomb branches of quiver gauge theories for instanton moduli spaces and those for T(SU(N)) moduli spaces.Comment: 97 pages, 12 figure

Quiver Theories and Formulae for Nilpotent Orbits of Exceptional Algebras

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.Comment: 87 pages, 4 figure

SU(n)SU(n) hyper-K\"ahler quotients of 3d N=43d\,\mathcal N=4 Coulomb branches and quiver subtraction

We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the $SU(n)$ hyper-K\"ahler quotient (HKQ) of the Coulomb branch of a $3d$ $\mathcal{N}=4$ unitary quiver theory. The target quivers are drawn from a wide range of theories, typically classified as ''good'' or ''ugly'', which satisfy identified selection criteria. Our subtraction procedure uses quotient quivers that are ''bad'', differing thereby from quiver subtractions based on Kraft-Procesi transitions. The procedure identifies one or more resultant quivers, the union of whose Coulomb branches corresponds to the desired HKQ. Examples include quivers whose Coulomb branches are moduli spaces of free fields, closures of nilpotent orbits of classical and exceptional type, and slices in the affine Grassmanian. We calculate the Hilbert Series and Highest Weight Generating functions for HKQ examples of low rank. For certain families of quivers, we are able to conjecture HWGs for arbitrary rank. We examine the commutation relations between quotient quiver subtraction and other diagrammatic techniques, such as Kraft-Procesi transitions, quiver folding, and discrete quotients