Abstract
The study of Coulomb branches of 3-dimensional mathcal{N}=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and 6-dimensional gauge theories with 8 supercharges. Recently, a conjecture connected different phases of 6-dimensional Higgs branches via gauging of a discrete global Snsymmetry. On the corresponding 3-dimensional Coulomb branch, this amounts to a geometric Sn-quotient. In this note, we prove the conjecture on Coulomb branches with unitary nodes and, moreover, extend it to Coulomb branches with other classical groups. The results promote discrete Sn-quotients to a versatile tool in the study of Coulomb branches.
Highlights
By associated 3-dimensional N = 4 theories, whose Coulomb branches M3Cd agree with the 6-dimensional Higgs branches M6Hd as algebraic varieties
These symmetry properties have been conjectured in two complementary studies: firstly, by computing the quantum corrections to the Coulomb branch metric in [21] and, secondly, by computing the Coulomb branch Hilbert series in [1]
In this note we have shown that discrete Sn-quotients on Coulomb branches of quivers with various bouquets are entirely local operations
Summary
One can generalise (1.6) by considering an arbitrary quiver coupled to one (or many) bouquet(s) of USp(2) ∼= SU(2) gauge nodes and provide a proof at the level of the monopole formula. A2 + a21 where the two contributions are treated as follows: Comparing this to the Hilbert series of USp(4) with a Λ2[1, 0] hypermultiplet and background charges, one has the conformal dimension (2.31) and the dressing factors [1]. A2 + a21 with ak := f (tk) , where the two contributions are treated as follows: Comparing this to the monopole formula of SO(5) with one Sym2[1, 0] hypermultiplet and background charges, the conformal dimension follows from (2.55) and the dressing factors read (2.60).
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