The Higgs Mechanism -- Hasse Diagrams for Symplectic Singularities

TL;DR

This work links the Higgs mechanism to the geometry of Higgs branches by treating them as symplectic singularities foliated into leaves, with closures organized by a Hasse diagram. It develops a unified framework using Kraft–Procesi transitions, magnetic quivers, and quiver subtraction to compute these diagrams for both Lagrangian and non-Lagrangian theories across 3–6 dimensions, including infinite-coupling regimes and Argyres–Douglas theories. The results reproduce known nilpotent-orbit structures, refine them in general symplectic settings, and offer concrete brane- and quiver-based tools to analyze complex moduli spaces. The approach provides new insights into the interplay between partial Higgsing and singularity structure, with broad applicability to 5d/6d SCFTs and non-Lagrangian theories, and establishes a foundation for extending these methods to more intricate gauge groups and representations.

Abstract

We explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are symplectic singularities, and as such admit a decomposition (or foliation) into so-called symplectic leaves, which are related to each other by transverse slices. We identify this foliation with the pattern of partial Higgs mechanism of the theory and, using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagram which encodes the structure of the foliation. While the unbroken gauge symmetry and the number of flat directions are obtainable by classical field theory analysis for Lagrangian theories, our approach allows us to characterise the geometry of the Higgs branch by a Hasse diagram with symplectic leaves and transverse slices, thus refining the analysis and extending it to non-Lagrangian theories. Most of the Hasse diagrams we obtain extend beyond the cases of nilpotent orbit closures known in the mathematics literature. The geometric analysis developed in this paper is applied to Higgs branches of several Lagrangian gauge theories, Argyres-Douglas theories, five dimensional SQCD theories at the conformal fixed point, and six dimensional SCFTs.

Figures (21)

• Figure 1: Hasse diagram encoding the poset structure of inclusions of the subsets of $\{x,y\}$.
• Figure 2: A schematic representation of the Higgs branch of $\mathrm{SU}(3)$ with $6$ fundamental hypermultiplets. At the $0$-dimensional origin the theory is unbroken. On the $5$-dimensional symplectic leaf, represented by the black line without the origin, the theory is broken to $\mathrm{SU}(2)$ with $4$ fundamental hypermultiplets and 5 neutral hypermultiplets. On the 10-dimensional symplectic leaf, represented by the plane without the black line and the origin, the gauge group is completely broken and the effective theory contains only 10 neutral hypermultiplets. The blue lines represent the transverse space of a point on the 5-dimensional symplectic leaf inside the 10-dimensional symplectic leaf. Locally, each individual transverse space looks like the Higgs branch of $\mathrm{SU}(2)$ with $4$ fundamental hypermultiplets.
• Figure 3: Hasse diagram obtained from partial Higgsing of $\mathrm{SU}(3)$ with 6 fundamentals. The effective gauge theory on each symplectic leaf is given by the quiver in the corresponding bracket and an extra number $N$ of neutral hypermultiplets, where $N$ is the number labelling the node. Note that $N$ is also the quaternionic dimension of the leaf.
• Figure 4: Hasse diagram with magnetic quivers representing the closures of symplectic leaves for the classical Higgs branch of $\mathrm{SU}(3)$ with $6$ fundamentals. The transverse slices between neighbouring symplectic leaves have been added in the edges of the Hasse diagram. This is a generalisation of the results by kraft1980minimalKraft1982 to a set of spaces different from nilpotent orbits.
• Figure 5: Depiction of the different 5-brane webs in the gauge enhancements up to $\mathrm{SU}(3)$ with $6$ fundamentals. The methods developed in Cabrera:2018jxt allow us to read magnetic quivers for the closure of all symplectic leaves in the Higgs branch as well as the transverse slices. This process can be translated into an operation between the magnetic quivers, called quiver subtraction. Coloured branes are assumed to be on different positions along the 7-branes.