3d Coulomb branch and 5d Higgs branch at infinite coupling

TL;DR

The work resolves the long standing problem of describing the Higgs branch of 5d ${\cal N}=1$ SQCD theories at infinite coupling by proposing that its structure is captured by the Coulomb branches of 3d ${\cal N}=4$ quiver gauge theories, termed Exceptional Sequences. By leveraging global symmetry enhancements at the UV fixed point, the authors construct explicit 3d quivers whose Coulomb branches reproduce the Higgs branches ${\cal H}_\infty$ across a broad family of theories with varying Chern-Simons levels and flavors, including $E_8$, $E_7$, $E_6$, and $E_5$ sequences. They provide detailed 3d quiver data and highest weight generating functions that encode the representations of the enhanced flavor symmetries, and validate these structures through 5d instanton analyses and dual gauge theory descriptions. The results demonstrate the power of 3d Coulomb branch techniques to illuminate higher dimensional moduli spaces, offer concrete computational tools such as Hilbert series and HWGs, and establish a bridge between brane constructions, class S reductions, and hyperKähler geometry in the study of 5d SCFTs. The approach yields a coherent, testable framework for extracting the full operator content of ${\cal H}_\infty$ and underscores the deep interconnectedness of 3d and 5d supersymmetric dynamics.

Abstract

The Higgs branch of minimally supersymmetric five dimensional SQCD theories increases in a significant way at the UV fixed point when the inverse gauge coupling is tuned to zero. It has been a long standing problem to figure out how, and to find an exact description of this Higgs branch. This paper solves this problem in an elegant way by proposing that the Coulomb branches of three dimensional ${\cal N}=4$ supersymmetric quiver gauge theories, named "Exceptional Sequences", provide the solution to the problem. Thus, once again, 3d ${\cal N}=4$ Coulomb branches prove to be useful tools in solving problems in higher dimensions. Gauge invariant operators on the 5d side consist of classical objects such as mesons, baryons and gaugino bilinears, and non perturbative objects such as instanton operators with or without baryon number. On the 3d side we have classical objects such as Casimir invariants and non perturbative objects such as monopole operators, bare or dressed. The duality map works in a very interesting way.

Figures (3)

• Figure 1: Deriving the 3d quiver using the brane web. At the top left is the 5d brane web describing an $SU(n)_{\frac{1}{2}}+(2n+1)F$ gauge theory. Moving the top 7-brane to the bottom leads to the web on the top right. This one is in the form of Benini:2009aa and so reduces to the class S theory shown on the bottom right when compactified on a circle to 4d. Further compactification on a circle to 3d leads to the 3d quiver.
• Figure 2: Starting with the 5d SCFT described by the brane web in the top left, and reducing to 4d, it is conjectured by Ohmori:2015pia that we get the 4d theory on the top right, where here we take $n>k$. The small circle denotes an $SU(k)$ group and the arrows indicates it is gauging an $SU(k)$ global symmetry in both class S theories. Reducing further to 3d leads to the mirror quiver on the bottom.
• Figure 3: We conjecture that reducing the 5d SCFT described by the brane system on the left leads to a 3d theory with a mirror dual given by the quiver on the right.