Abstract

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} = 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.

Highlights

  • Supersymmetric gauge theories with 8 supercharges are central objects in string theory and quantum field theory

  • Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics

  • Due to the high amount of supersymmetry (SUSY), there are some families in which the field theory ingredients such as the gauge groups, matter content, and the superpotential interactions can be neatly encoded in so-called quiver diagrams [1, 2]

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Summary

Introduction

Supersymmetric gauge theories with 8 supercharges are central objects in string theory and quantum field theory. When studying the moduli space of unitary quivers derived from brane configurations [38], there always exists a trivially acting diagonal U(1)diag which may or may not be a subgroup of the gauge group. This U(1) describes the center of mass. That acts trivially on the matter content, the choice of GH or G affects the magnetic lattice and the moduli space This is supported by explicit Hilbert series computations. We present Coulomb branch Hilbert series of various families of unitary-orthosymplectic quiver theories where H equals the kernel of.

Coulomb branch and magnetic lattice
Main idea
Monopole formula
Generalities on lattices
Examples
Sums over magnetic sublattices
D4 example
E6 example Consider the unitary quiver whose Coulomb branch is Oem6in:
Orthosymplectic quivers for minimal En orbits
A tale of two Coulomb branches
E7 quiver
E5 quiver
E4 quiver
3.10 Higgs branch
3.11 Inequivalent embeddings
Rank 0 limit
Global symmetry
Conclusions and outlook

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