Discrete Gauging in Six Dimensions

TL;DR

The paper proposes that when $n$ M5-branes coincide on an A-type singularity $\mathbb{C}^2/\mathbb{Z}_k$, the discrete symmetry $S_n$ is gauged, so the six-dimensional Higgs branch at infinite coupling is the $S_n$-orbifold of the finite-coupling Higgs branch: ${\cal H}_\infty({\sf Q}_{n,k}) = {\cal H}_f({\sf Q}_{n,k}) / S_n$. The authors implement this idea via a network of 3d ${\cal N}=4$ Coulomb-branch quivers and their mirrors, showing that the various Higgs branches across the phases (as many as partitions of $n$) correspond to orbifolds ${\cal H}_{\{n_i\}}({\sf Q}_{n,k}) = {\cal H}_{\{1^n\}}({\sf Q}_{n,k}) / \prod_i S_{n_i}$ described by associated quivers ${\sf F}_{\{n_i\},k}$. They provide explicit tests in many cases (notably $n=2$, $k=2$; general $k$; and links to $SO(8)$, $G_2$, and $SU(3)$ moduli spaces) and connect the 6d physics to 3d mirrors, KP transitions, and discrete gauging. The work further extends to systems with an M9-plane, where small-instanton transitions (KP) interact with discrete gauging, yielding a rich web of Higgs-branch geometries and phase structures, all consistent with a unified discrete gauging picture. This framework offers a coherent explanation for the observed global and discrete symmetries and provides concrete predictions for moduli spaces and dualities across multiple brane configurations. The results have potential implications for understanding the non-Lagrangian 6d theories and their dimensional reductions, guiding future checks beyond the Higgs-branch sector.

Abstract

When $n$ M5 branes coincide on an A type singularity, $\mathbb{C}^2/\mathbb{Z}_k$, there is a multitude of tensionless strings which arise in the spectrum. The low energy theory when all M5 branes are separated at the singularity is given by a linear quiver with parameters $n$ and $k$. The theory has a multitude of phases, as many as partitions of $n$, each characterized by a different Higgs branch. Each such Higgs branch can be described by a Coulomb branch of a 3d $\mathcal{N}=4$ quiver. For example, at finite coupling, when all branes are separated, the quiver has a bouquet of $n$ $U(1)$ nodes connected to a single node. There is a natural discrete non Abelian $S_n$ global symmetry which acts in the theory by permuting $n$ identical objects. It acts in particular on the Higgs branch at the above finite coupling phase. It is conjectured that at the coincident point this discrete $S_n$ flavor symmetry is gauged, and at partial coincidence the corresponding subgroup of $S_n$ is gauged. This elegant and simple effect solves several problems which are raised recently on the physics of multiple M5 branes on an A type singularity. Similar results on multitude of phases are concluded for a system of $n$ M5 branes on an A type singularity next to an M9 plane.

Figures (3)

• Figure 1: The 6d quivers of the low-energy theory on the tensor branch, associated with the family of 6d SCFTs studied in this subsection. The form of the quiver differs depending on the value of $k$ mod $3$. All groups are of type $SU$ and in the second quiver the line from $SU(6)$ to itself stands for a half-hyper in the three index antisymmetric representation. Also the leftmost circle in the bottom quiver stands for the rank $1$ E-string theory, which is connected to the neighboring $SU(9)$ gauge group via gauging of $SU(9) \subset E_8$.
• Figure 2: The 4d theories conjectured by Ohmori:2015pia to result from the torus compactification of the 6d SCFTs living on $n$ M5-branes on a $\mathbb{C}^2/\mathbb{Z}_k$ singularity. The form of the theories differ depending on the relative size of $n$ and $k$. Here the pieces on the left and right correspond to two class S theories associated with a three punctured sphere and a one punctured torus respectfully, where beneath each class S theory, the type of $(2,0)$ theory of the class S theory is written. The two theories are connected through the middle gauge group, which gauges the punctures symmetries at the tip of the arrows coming out of it.
• Figure 3: An example of the 3d reduction when $n=4, k=3$. First, we reduce the 4d theory to 3d. As the $SU(3)$ gauging is IR free, we can just reduce the two class S theories, which have mirror Lagrangian duals, and connect them via the same gauging. The resulting theory is shown in the bottom right, where on the left is the star shaped mirror duals of $T_3$, and in the middle is the dual of the torus class S theory, where the circular line on the last node stands for an adjoint hyper. The two theories are connected by gauging the Coulomb branch $SU(3)$ symmetry associated with the quiver tail singled by the parenthesis. This gauging leads to the identification of the nodes connected to the quiver tails leading to the 3d quiver in the bottom right.

Theorems & Definitions (5)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Conjecture 4
• Conjecture 5