Abstract
We study Hasse diagrams of moduli spaces of 3d mathcal{N} = 4 quiver gauge theories. The goal of this work is twofold: 1) We introduce the notion of inverting a Hasse diagram and conjecture that the Coulomb branch and Higgs branch Hasse diagrams of certain theories are related through this operation. 2) We introduce a Hasse diagram to map out the entire moduli space of the theory, including the Coulomb, Higgs and mixed branches. For theories whose Higgs and Coulomb branch Hasse diagrams are related by inversion it is straight forward to generate the Hasse diagram of the entire moduli space. We apply inversion of the Higgs branch Hasse diagram in order to obtain the Coulomb branch Hasse diagram for bad theories and obtain results consistent with the literature. For theories whose Higgs and Coulomb branch Hasse diagrams are not related by inversion it is nevertheless possible to produce the Hasse diagram of the full moduli space using different methods. We give examples for Hasse diagrams of the entire moduli space of theories with enhanced Coulomb branches.
Highlights
The analysis of Coulomb and Higgs branches of 3d N = 4 quiver gauge theories has been of interest to both the physics community and the mathematics community since the advent of supersymmetry and lead to many interesting discoveries
The goal of this work is twofold: 1) We introduce the notion of inverting a Hasse diagram and conjecture that the Coulomb branch and Higgs branch Hasse diagrams of certain theories are related through this operation
Using the algorithm to obtain the Hasse diagram of the entire Moduli space, we see that for this entire oneparameter family, the picture of the Higgs branch, which classically intersects along a subvariety, becomes much more complicated in the quantum theory
Summary
There are Coulomb branches, which are symplectic singularities, for which no description as a hyper-Kahler quotient is known; for example the minimal nilpotent orbits of the E-type exceptional groups. HC and HH of the examples discussed are related by exchanging minimal nilpotent orbit closures with Kleinian singularities associated to the same group This is an example of inversion of a Hasse diagram, when there is only one transverse slice.
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