Abstract
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.
Highlights
The moduli spaces of instantons remain the subject of much research and new constructions continue to be presented in the literature
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 have shown how the matching of extended Dynkin diagram U(N ) symmetries to those of T (SU(N )), leads to simple Highest Weight Generating (HWG) for reduced single instanton moduli space (RSIMS) in terms of modified Hall-Littlewood (mHL) polynomials
Summary
The moduli spaces of instantons remain the subject of much research and new constructions continue to be presented in the literature. A full way of describing the field content within a Hilbert series is to calculate the irreducible representations (“irreps”) within which the chiral operators transform under the global symmetry. Such a series can be presented in terms of the highest weight Dynkin labels of the irreps of the symmetry group(s) that occur within the theory, along with their multiplicities. This is a new technique [2], which permits the rich and systematic description of a wide range of moduli spaces. The algorithm used in the general monopole construction works, with U(N ) rather than U(1) symmetry
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