<p>We present an algorithm to extract the Coulomb branch Hasse diagram of orthosymplectic three-dimensional (3D) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> quiver gauge theories. The algorithm systematically predicts all descendant theories arising from Coulomb branch Higgsing, thereby detailing the stratification of the symplectic singularity defined by the initial Coulomb branch. Leveraging the Lie algebra isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="fraktur">su</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≅</mo><mi mathvariant="fraktur">so</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow></math>, we validate our algorithm via the 3D mirror of the 4D theories of class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">S</mi></math> of such type. This comparison involves moduli spaces that admit both orthosymplectic and unitary quiver realizations, the latter being well understood via standard techniques such as decay and fission. Higgsing on the Coulomb branch of the 3D mirror or the magnetic quiver translates to Higgs branch renormalization group flows of the corresponding higher-dimensional superconformal field theories. Thus, we benchmark our method via Higgsing 6D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> D-type orbi-instanton theories, predicting novel Higgsing patterns involving products of interacting fixed points, and class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">S</mi></math> theories of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="fraktur">so</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></math>, demonstrating Higgsing to products of theories of types specified by Levi subalgebras of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="fraktur">so</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></math>.</p>

Germany

Craig Lawrie

Lorenzo Mansi

Austria

Marcus Sperling

United Kingdom

Zhenghao Zhong

Pathway to decay and fission of orthosymplectic quiver theories

<p>Obtaining the classification of three-dimensional (3D) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> quivers whose Coulomb branches have an isolated singularity is an essential step in understanding moduli spaces of vacua of supersymmetric field theories with eight supercharges in any dimension. In this work, we derive a full classification for such Abelian quivers with arbitrary charges and identify all possible Coulomb branch geometries as quotients of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="double-struck">H</mi><mi>n</mi></msup></math> by U(1) or a finite cyclic group. We give two proofs, one which uses the “decay and fission” algorithm and another one relying only on explicit computations involving 3D mirror symmetry. In the process, we put forward a method for computing the 3D mirror of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>r</mi></msup></math> gauge theory, which is sensitive to discrete gauge factors in the mirror theory. This constitutes a confirmation for the decay and fission algorithm.</p>

France

Antoine Bourget

Quentin Lamouret

Sinan Moura Soysüren

Classification of minimal Abelian Coulomb branches

This paper classifies all Higgs branch Higgsing patterns for simply-laced unitary quiver gauge theories with eight supercharges (including multiple loops) and introduces a Higgs branch subtraction algorithm. All possible minimal transitions are given, identifying differences between slices that emerge on the Higgs and Coulomb branches. In particular, the algorithm is sensitive to global information including monodromies and Namikawa-Weyl groups. Guided by symplectic duality, the algorithm further determines the global symmetry on the Coulomb branch, and verifies the exclusion of <italic>C</italic> type or <math altimg='si1.svg'><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math> global symmetry for (simply-laced) unitary quiver gauge theories. The Higgs branches of some unitary quivers are verified to give slices in the nilpotent cones of exceptional simple Lie algebras.

Sam Bennett

Amihay Hanany

Guhesh Kumaran

Chunhao Li

Deshuo Liu

Quiver subtraction on the Higgs branch

An intriguing class of 6D supersymmetric theories are known as little strings theories, which exhibit a rich network of T-dualities. A robust feature of these theories are their Higgs branches. Focusing on the little string theories that are realized on a single curve of zero self-intersection, we utilize brane systems to derive the magnetic quivers. Using a variety of techniques (including branching rules, brane dynamics, F-theory geometry, quiver subtraction, and the decay and fission algorithm), we detail the Higgs branch Hasse diagram and determine the transverse slices for every elementary Higgs branch RG-flow. Building on these insights, we pursue two directions: first, we infer potential T-dual models by linking changes in 2-group structure constants along Higgs branch RG-flows to the RG-flow transition types encoded in the Hasse diagram. Second, we conjecture an algorithm for the non-Abelian flavor symmetry of the compactified little string theory by inspecting the magnetic quivers of all T-dual frames.

Unraveling <math><mi>T</mi></math> duality: Magnetic quivers in rank-zero little string theories

Magnetic quivers have been an instrumental technique for advancing our understanding of Higgs branches of supersymmetric theories with eight supercharges. In this work, we present the “decay and fission” algorithm for unitary magnetic quivers. It enables the derivation of the complete phase (Hasse) diagram and is characterized by the following key attributes: First and foremost, the algorithm is inherently simple, just relying on convex linear algebra. Second, any magnetic quiver can only undergo decay or fission processes; these reflect the possible Higgs branch RG flows (Higgsings), and the quivers thereby generated are the magnetic quivers of the new RG fixed points. Third, the geometry of the decay or fission transition (i.e., the transverse slice) is simply read off. As a consequence, the algorithm does not rely on a complete list of minimal transitions, but rather outputs the transverse slice geometry automatically. As a proof of concept, its efficacy is showcased across various scenarios, encompassing superconformal field theories from dimensions 3 to 6, instanton moduli spaces, and little string theories.

Higgs branch RG flows via decay and fission

In exploring supersymmetric theories with eight supercharges, the Higgs branches present an intriguing window into strong coupling dynamics. Magnetic quivers serve as crucial tools for understanding these branches. Here, we introduce the decay and fission algorithm for unitary magnetic quivers. It efficiently derives complete phase diagrams (Hasse diagrams) through convex linear algebra. It allows magnetic quivers to undergo decay or fission, reflecting Higgs branch RG flows in the theory. Importantly, the algorithm generates magnetic quivers for the RG fixed points and simplifies the understanding of transverse slice geometry with no need for a list of minimal transitions. In contrast, the algorithm hints to the existence of a new minimal transition, whose geometry and physics need to be explored.

Decay and Fission of Magnetic Quivers

This paper introduces two operations in quiver gauge theories. The first operation, collapse, takes a quiver with a permutation symmetry S  n  and gives a quiver with adjoint loops. The corresponding 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Coulomb branches are related by an orbifold of S  n . The second operation, multi-lacing, takes a quiver with n nodes connected by edges of multiplicity k and replaces them by n nodes of multiplicity qk. The corresponding Coulomb branch moduli spaces are related by an orbifold of type   <math> <msubsup> <mi>ℤ</mi> <mi>q</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </math>  $$ {\mathbb{Z}}_q^{n-1} $$ . Collapse generalises known cases that appeared in the literature [1–3]. These two operations can be combined to generate new relations between moduli spaces that are constructed using the magnetic construction.

Actions on the quiver: discrete quotients on the Coulomb branch

We study particular families of bad 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 quiver gauge theories, whose Higgs branches consist of many cones. We show the role of a novel brane configuration in realizing the Higgs moduli for each distinct cone. Through brane constructions, magnetic quivers, Hasse diagrams, and Hilbert series computations we study the intricate structure of the classical Higgs branches. These Higgs branches are both non-normal (since they consist of multiple cones) and non-reduced (due to the presence of nilpotent operators in the chiral ring). Applying the principle of inversion to the classical Higgs branch Hasse diagrams, we conjecture the quantum Coulomb branch Hasse diagrams. These Coulomb branches have several most singular loci, corresponding to the several cones in the Higgs branch. We propose the Hasse diagrams of the full quantum moduli spaces of our theories. The quivers we study can be taken to be 5d effective gauge theories living on brane webs. Their infinite coupling theories have Higgs branches which also consist of multiple cones. Some of these cones have decorated magnetic quivers, whose 3d Coulomb branches remain elusive.

Julius Grimminger

Rudolph Kalveks

China

A tale of N cones

The Higgs branches of the world-volume theories for multiple M5 branes on an A  k  or D  k -type ALE space are known to host a variety of fascinating properties, such as the small E 8 instanton transition or the discrete gauging phenomena. This setup can be further enriched by the inclusion of boundary conditions, which take the form of SU(k) or SO(2k) partitions, respectively. Unlike the A-type case, D-type boundary conditions are eventually accompanied by negative brane numbers in the Type IIA brane realisation. While this may seem discouraging at first, we demonstrate that these setups are well-suited to analyse the Higgs branches via magnetic quivers. Along the way, we encounter multiple models with previously neglected Higgs branches that exhibit exciting physics and novel geometric realisations. Nilpotent orbits, Słodowy slices, and symmetric products.

Magnetic quivers and negatively charged branes

We investigate orthosymplectic quivers that take the shape of D-type and B-type Dynkin diagrams. The D-type orthosymplectic quivers explored here contain a balanced “fork”, i.e. a balanced subquiver with a D-type bifurcation, whereas the B-type orthosymplectic quivers are obtained by folding the D-type quivers. The Coulomb branches of these quivers are products of two moduli spaces. In the second part, the relevant orthosymplectic quivers are shown to emerge as magnetic quivers for brane configurations involving ON0 planes. Notably, the appearance of ON0 plane clarifies the product nature of the theories in question. The derivation leads to the analysis of magnetic quivers from branes systems with intersecting Op, O(p + 2), and ON0 planes.