Microscopic justification of the magnetic-quiver/monopole-formula method for 5d Higgs branches

Establish a microscopic proof that the computation of the Hilbert series via the monopole formula for the Coulomb branch of the three-dimensional N=4 magnetic quiver associated to a five-dimensional braneweb correctly reproduces the Higgs-branch chiral ring of the original five-dimensional N=1 gauge theory, thereby justifying the step from classical 't Hooft–Polyakov monopoles to monopole operators in this context.

Background

The paper relies on the magnetic-quiver/monopole-formula framework to compute Hilbert series and extract generators and relations of the Higgs-branch chiral ring for 5d N=1 theories engineered by brane webs.

The authors acknowledge a logical gap between classical 't Hooft–Polyakov monopoles and the monopole-operator formalism used in the monopole formula and note that a microscopic proof of this identification is still lacking, even though the method matches alternative computations in many cases.

References

Strictly speaking, there is a logical leap from considering 't Hooft Polyakov monopoles, which are classical solutions, to magnetic monopole operators, which are local disoder operators in the path integral. A proof justifying microscopically this procedure is still lacking.

— Chiral ring along the RG flow in 5d $\mathcal{N}=1$  (2510.15635 - Hanany et al., 17 Oct 2025) in Section 5 (Case studies), footnote after the paragraph introducing the monopole-formula approach

The magnetic quiver extension for the NS5 branch (\ie \MQNS) is conjectured to be

— Generalised Symmetries, Anomalies, and Maximal Branches of 3d Chern-Simons Matter Theories  (2608.30349 - Marino et al., 31 Aug 2026) in Section "Magnetic quivers and 1-form symmetry", subsection "Magnetic quiver extension: non-simply laced edges", paragraph "$\MQNS$ extension" (Section \ref{subsec:MQ_1form})