The Higgs Mechanism -- Hasse Diagrams for Symplectic Singularities

TL;DR

This paper develops a geometric framework using Hasse diagrams to analyze the structure of Higgs branches in supersymmetric quantum field theories, extending understanding beyond classical cases.

Contribution

It introduces a method to construct Hasse diagrams for Higgs branches, applicable to both Lagrangian and non-Lagrangian theories, using brane systems and quiver techniques.

Findings

Hasse diagrams encode the symplectic leaf structure of Higgs branches.

The approach extends to non-Lagrangian theories and beyond nilpotent orbit closures.

Application to various theories reveals new geometric insights.

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.