Hasse diagrams for 3d $$ \mathcal{N} $$ = 4 quiver gauge theories — Inversion and the full moduli space

Abstract

We study Hasse diagrams of moduli spaces of 3d $$ \mathcal{N} $$ N = 4 quiver gauge theories. The goal of this work is twofold: 1) We introduce the notion of inverting a Hasse diagram and conjecture that the Coulomb branch and Higgs branch Hasse diagrams of certain theories are related through this operation. 2) We introduce a Hasse diagram to map out the entire moduli space of the theory, including the Coulomb, Higgs and mixed branches. For theories whose Higgs and Coulomb branch Hasse diagrams are related by inversion it is straight forward to generate the Hasse diagram of the entire moduli space. We apply inversion of the Higgs branch Hasse diagram in order to obtain the Coulomb branch Hasse diagram for bad theories and obtain results consistent with the literature. For theories whose Higgs and Coulomb branch Hasse diagrams are not related by inversion it is nevertheless possible to produce the Hasse diagram of the full moduli space using different methods. We give examples for Hasse diagrams of the entire moduli space of theories with enhanced Coulomb branches.

Towards a classification of rank r $$ \mathcal{N} $$ = 2 SCFTs. Part II. Special Kahler stratification of the Coulomb branch

“…We also discuss preliminary self-consistency conditions on the way the Coulomb, mixed, and Higgs branch structures come together. This idea has inspired a forthcoming work which revisits the rank 1 classification from the chiral algebra side [33], and was recently developed in the context of N = 4 three dimensional theories [30]. A tool that we will utilize to present these results, very much along the lines of [30], is the Hasse diagram, which is an efficient way to pictorially present a partially ordered set, in this case the poset provided by the stratification.…”

“…This idea has inspired a forthcoming work which revisits the rank 1 classification from the chiral algebra side [33], and was recently developed in the context of N = 4 three dimensional theories [30]. A tool that we will utilize to present these results, very much along the lines of [30], is the Hasse diagram, which is an efficient way to pictorially present a partially ordered set, in this case the poset provided by the stratification. We will introduce Hasse diagrams for Coulomb branches of four dimensional N = 2 SCFTs below and extend them, in a few specific examples, to the full moduli space Hasse diagrams.…”

“…The notion of elementary slice easily generalizes to the more general transverse slice to any stratum S These transverse slices are transverse in the sense of [39]. In the terminology of [29,30], two strata labelled by and k in the stratification poset are called neighboring strata if k covers , and the transverse slice between neighboring strata is an elementary slice. Recall that neighboring strata are ones which are connected by an edge of the Hasse diagram of the stratification.…”

“…The Hasse diagram for the Higgs branch of the (A 1 , D 6 ) AD theory. Following the convention of [30] we blue for Hasse diagrams of Higgs branches.…”

“…This theory is conjectured to be the second non-trivial entry of an infinite series of four dimensional N = 2 SCFTs with flavor symmetry sp(n + 3) (for n = 1 the theory is simply a bunch of free hypers) [68]. The Higgs branches of this infinite series are worked out in details using the techniques of magnetic quivers in [46] and in the context of 3d N = 4 in [30] where the corresponding theory is named X 6 . Our analysis perfectly matches with their predictions suggesting that this 4d theory flows to the X 6 upon circle compactification.…”