The Higgs mechanism \u2014 Hasse diagrams for symplectic singularities

Antoine Bourget,Santiago Cabrera,Julius F Grimminger,Marcus Sperling,Amihay Hanany,Zhenghao Zhong,Anton Zajac

doi:10.1007/jhep01(2020)157

Antoine Bourget, Santiago Cabrera + Show 5 more

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https://doi.org/10.1007/jhep01(2020)157

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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.

Highlights

The Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism for the Abelian case [1,2,3] and for the non-Abelian case [4], called Higgs mechanism for short, by which a gauge group is broken to a subgroup by the vacuum expectation value (VEV) of scalar fields, has played a central role in theoretical physics in the last few decades
We can compute Hasse diagrams, we compare them against the prediction from the Higgs mechanism, and we find agreement
The remainder of this paper is organised as follows: in section 2 we demonstrate how the predictions from classical Higgs mechanism match with the Hasse diagram obtained by assuming the KP transitions

Summary

Introduction

The Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism for the Abelian case [1,2,3] and for the non-Abelian case [4], called Higgs mechanism for short, by which a gauge group is broken to a subgroup by the vacuum expectation value (VEV) of scalar fields, has played a central role in theoretical physics in the last few decades. In the context of supersymmetric gauge theories with 8 supercharges, the part of the moduli space called the Higgs branch refers to this old idea since it is parametrised by the VEVs of the scalar components which trigger this mechanism. On specific loci of positive codimension in the Higgs branch, the VEVs can leave a certain subgroup unbroken. Iterating this process, we see an interesting pattern of partial Higgsing, which can be characterised by the various subspaces of the Higgs branch. We see an interesting pattern of partial Higgsing, which can be characterised by the various subspaces of the Higgs branch These subspaces are naturally partially ordered by inclusion of their closures, and as such can be arranged into a Hasse diagram.

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