The Higgs mechanism — Hasse diagrams for symplectic singularities

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.

Magnetic quivers, Higgs branches, and 6d $$ \mathcal{N} $$ = (1, 0) theories — orthogonal and symplectic gauge groups

“…The leaves themselves admit a partial order via inclusion: L κ < L λ if and only if L κ ⊂ L λ . As argued in [26], the partial order of leaves is in one-to-one correspondence with the partial order among the set of subgroups {H κ }, such that the G gauge theory can be Higgsed to the corresponding H κ gauge theory, satisfying (3.2). In other words, L κ < L λ if and only if H κ > H λ , i.e.…”

“…Besides providing the Higgs branch description, one can moreover attempt to analyse the Higgs branch geometries understood as symplectic singularities [25]. As put forward in [26], the singularity structure can be encoded in a Hasse diagram. In many cases, the Hasse diagram can be derived either from the brane configuration using Kraft-Procesi transitions [39,40] or from the magnetic quiver description via quiver subtraction [23,41].…”

“…For instance, Coulomb branches have already been employed successfully to describe Higgs branches of Argyres-Douglas theories [18], as well as infinite coupling limits of 5-dimensional theories [19][20][21] and of 6-dimensional gauge theories [17,[22][23][24]. In fact, by interpreting these moduli spaces as symplectic singularities [25], the magnetic quiver techniques allowed to derive the Hasse diagrams for the various Higgs branches [26].…”

General prescription for globalU(1)’s in 6D SCFTs

“…For our present purposes of viewing all 6D SCFTs as fission and fusion products this additional class of flows will not play a role. See also references [45][46][47] for additional discussion of Higgs branch deformations of theories with eight real supercharges.…”

“…The tensor branch descriptions of 6D SCFTs corresponding to Hom(Γ, E 8 ) have been studied in [5,39,43,45,63,64], while the tensor branch descriptions of 6D SCFTs corresponding to Hom(su(2), g) have been studied in [37,39,43,46,111]. From these descriptions, we can use the match between 6D SCFTs and homomorphisms as a sort of "Rosetta Stone": U (1) symmetries appearing in the commutant H of a given homomorphism translate into U (1) symmetries of the SCFT quiver, which allows us to verify the field theory rules for U (1) symmetries in an SCFT quiver.…”