Abstract
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.
Highlights
Exceptional group.1 Nilpotent orbits are increasingly being recognised as being relevant to many topics, ranging from supergravity (“SUGRA”) theories involving G/H coset spaces, whose field content can be characterised by nilpotent orbits of G [5], to counting massive vacua in N = 1 Super Yang-Mills (“SYM”) theory [6], where the number of vacua is derived from the structure of the nilpotent orbits of the gauge group
We choose to approach the topic of nilpotent orbits from the perspective of their moduli spaces and Hilbert series, which we analyse using the tools of the Plethystics Program [8, 9]
Each such Hilbert series counts holomorphic functions on the closure of a nilpotent orbit [10]. It appears that all the nilpotent orbits of any Classical group G correspond to the moduli spaces of particular SUSY quiver gauge theories that can be constructed on the root lattice of G and which are determined by the canonical parameters associated with the orbits
Summary
We will limit ourselves to a brief summary that is pertinent to the enumeration of nilpotent orbits for Classical groups; the reader is referred to [4] for a full exposition. We start from the Jacobson Morozov Theorem, which states that the nilpotent orbits of a group G are in one to one correspondence, up to conjugation, with the homomorphisms ρ from SU(2) to G
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