Abstract
The singularity structure of the Coulomb and Higgs branches of good 3d mathcal{N}=4 circular quiver gauge theories (CQGTs) with unitary gauge groups is studied. The central method employed is the Kraft-Procesi transition. CQGTs are described as a generalisation of a class of linear quivers. This class degenerates into the familiar class Tρσ(SU(N)) in the linear case, however the circular case does not have the degeneracy and so the class of CQGTs contains many more theories and much more structure. We describe a collection of good, unitary, CQGTs from which the entire class can be found using Kraft-Procesi transitions. The singularity structure of a general member of this collection is fully determined, encompassing the singularity structure of a generic CQGT. Higher-level Hasse diagrams are introduced in order to write the results compactly. In higher-level Hasse diagrams, single nodes represent lattices of nilpotent orbit Hasse diagrams and edges represent traversing structure between lattices. The results generalise the case of linear quiver moduli spaces which are known to be nilpotent varieties of mathfrak{s}{mathfrak{l}}_n .
Highlights
The sets of zero energy configurations, or moduli spaces of vacua, of supersymmetric quantum field theories possess rich algebro-geometric structure
We describe a collection of good, unitary, circular quiver gauge theories (CQGTs) from which the entire class can be found using KraftProcesi transitions
The effects of Kraft-Procesi transitions on the brane configurations whose low energy dynamics are described by the CQGTs are investigated
Summary
The sets of zero energy configurations, or moduli spaces of vacua, of supersymmetric quantum field theories possess rich algebro-geometric structure. We use Kraft-Procesi transitions on circular quiver gauge theories to uncover the singularity structure of their moduli space branches. The effects of Kraft-Procesi transitions on the brane configurations whose low energy dynamics are described by the CQGTs are investigated This allows the identification of a set of theories whose moduli space branches contain the branches of any CQGT as subvarieties. Procesi transitions remove transverse slices from the moduli space varieties, the singularity structure of every circular quiver in the class πρσ(M, N1, N2, L) can be found inside that of an appropriately formulated maximal theory through the application of Kraft-Procesi transitions. We present the general higher-level Hasse diagram for a generic member of the minimal set of maximal theories This diagram encompasses the singularity structure of any CQGT in the class πρσ(M, N1, N2, L). Kraft-Procesi transitions are powerful tools for performing a local analysis of the moduli spaces, being able to use their results to inform a global analysis would provide a new method for investigations into global moduli space structures
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