Magnetic Quivers, Higgs Branches, and 6d N=(1,0) Theories
Santiago Cabrera, Amihay Hanany, Marcus Sperling
TL;DR
This work introduces magnetic quivers as a robust bridge between 3d ${\mathcal N}=4$ Coulomb branches and 6d ${\mathcal N}=(1,0)$ Higgs branches in theories arising from multiple M5 branes near an M9 plane on ${\mathbb C^2/\mathbb Z_k}$. By mapping brane configurations to magnetic-quiver data, the authors systematically realize Higgs branches across finite and infinite gauge coupling, incorporating discrete gauging and small ${E_8}$ instanton transitions, and revealing exceptional symmetry ${E_8}$ directly from brane setups. The methodology yields explicit magnetic quivers for a wide array of embeddings ${\mathbb Z}_k\hookrightarrow E_8$ (including $k=1,2,3,4$) and general cases, and explains phase structure via dressed monopole operators. This framework unifies weakly coupled 6d quiver descriptions with strongly coupled Higgs branches, offering a practical tool for analyzing non-Lagrangian 6d theories through 3d Coulomb-branch methods and brane engineering. The results have potential implications for higher-dimensional theories and their 4d/5d reductions, as well as for understanding exceptional symmetries in string-brane constructions.
Abstract
The physics of M5 branes placed near an M9 plane on an A-type ALE singularity exhibits a variety of phenomena that introduce additional massless degrees of freedom. There are tensionless strings whenever two M5 branes coincide or whenever an M5 brane approaches the M9 plane. These systems do not admit a low-energy Lagrangian description so new techniques are desirable to shed light on the physics of these phenomena. The 6-dimensional N=(1,0) world-volume theory on the M5 branes is composed of massless vector, tensor, and hyper multiplets, and has two branches of the vacuum moduli space where either the scalar fields in the tensor or hyper multiplets receive vacuum expectation values. Focusing on the Higgs branch of the low-energy theory, previous works suggest the conjecture that a new Higgs branch arises whenever a BPS-string becomes tensionless. Consequently, a single theory admits a multitude of Higgs branches depending on the types of tensionless strings in the spectrum. The two main phenomena discrete gauging and small E8 instanton transition can be treated in a concise and effective manner by means of Coulomb branches of 3-dimensional N=4 gauge theories. In this paper, a formalism is introduced that allows to derive a novel object from a brane configuration, called the magnetic quiver. The main features are as follows: (i) the 3d Coulomb branch of the magnetic quiver yields the Higgs branch of the 6d system, (ii) all discrete gauging and E8 instanton transitions have an explicit brane realisation, and (iii) exceptional symmetries arise directly from brane configurations. The formalism facilitates the description of Higgs branches at finite and infinite gauge coupling as spaces of dressed monopole operators.