Global variants of four-dimensional gauge theories are specified by their spectrum of genuine Wilson-’t Hooft line operators. The choice of global variant has significant consequences when spacetime is taken to be $\textit{ℝ}$ × $\textit{S}$ . We focus on $$ \mathcal{N} $$ = 1 theories, which are closely connected to twisted elliptic Calogero-Moser systems. We establish, on general grounds, how this gauge-theoretic topological data manifests itself on the integrable system side by introducing a notion of global variants for complex many-body integrable systems associated with Lie algebras. Focusing on $$ \mathcal{N} $$ = 1 theories of type $\textit{A}$ and $\textit{B}$ , we elucidate the implications for the structure of gapped vacua, the emergent (generalized) symmetries realized in each vacuum, and the action of spontaneously broken modular invariance.

Argentina

Spain

Jeremías Damia

Belgium

Riccardo Argurio

France

Antoine Bourget

Germany

Valdo Tatitscheff

Romain Vandepopeliere

Global variants of $$ \mathcal{N} $$ = 1

<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>

Quentin Lamouret

Austria

Sinan Moura Soysüren

Marcus Sperling

Classification of minimal Abelian Coulomb branches

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.

United Kingdom

Zhenghao Zhong

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

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

Amihay Hanany

Rudolph Kalveks

China

A tale of N cones

We study the 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 RG-flows triggered by Fayet-Iliopoulos deformations in unitary quiver theories. These deformations can be implemented by a new quiver algorithm which contains at its heart a problem at the intersection of linear algebra and graph theory. When interpreted as magnetic quivers for SQFTs in various dimensions, our results provide a systematic way to explore RG-flows triggered by mass deformations and generalizations thereof. This is illustrated by case studies of SQCD theories and low rank 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs. A delightful by-product of our work is the discovery of an interesting new 3d mirror pair.

Italy

Simone Giacomelli

FI-flows of 3d N=4 theories

We study the full moduli space of vacua of 6d worldvolume SCFTs on M5 branes probing an A-type singularity, focusing on the geometric incarnation of the discrete gauging mechanism which acts as a discrete quotient on the Higgs branch fibered over the tensor branch. We combine insights from brane constructions and magnetic quiver techniques, in which discrete gauging is implemented through the concept of decoration introduced in [1]. We discover and characterize new transverse slices between phases of 6d SCFTs, identifying some of them with a family of isolated symplectic singularities recently discovered in [2], and conjecturing the existence of two new isolated symplectic singularities.

Fibrations and Hasse diagrams for 6d SCFTs

Hasse diagrams (or phase diagrams) for moduli spaces of supersymmetric field theories have been intensively studied in recent years, and many tools to compute them have been developed. The moduli space of instantons, despite being well studied, has proven difficult to deal with. In this note we explore the Hasse diagram of this moduli space from several perspectives — using the partial Higgs mechanism, using brane systems and using quiver subtraction — having to refine previously developed techniques. In particular we introduce the new concept of decorated quiver, which allows to deal with a large class of unitary quivers, including those with adjoint matter.

The Hasse diagram of the moduli space of instantons

We solve a long standing problem on the computation of the Higgs branch   <math> <mi>H</mi> </math>  $$ \mathcal{H} $$  of linear quivers with 8 supercharges and with both unitary and special unitary gauge nodes. The solution uses the concept of magnetic quivers, where components of   <math> <mi>H</mi> </math>  $$ \mathcal{H} $$  are described as 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Coulomb branches. When the starting quiver is good, there is a single component in   <math> <mi>H</mi> </math>  $$ \mathcal{H} $$  and the magnetic quiver is a 3d mirror. The magnetic quivers are obtained from studying the brane web for an auxiliary 5d theory (with only special unitary gauge groups), constrained by a new notion called brane locking, where some branes are required to move together. We view this as a computational tool rather than an operation in 5d. A detailed algorithm is provided.

Higgs branches of U/SU quivers via brane locking

We propose magnetic quivers for partial implosion spaces. Such partial implosions involve a choice of parabolic subgroup, with the Borel subgroup corresponding to the standard implosion. In the subregular case we test the conjecture by verifying that reduction by the Levi group gives the appropriate nilpotent orbit closure. In the case of a parabolic corresponding to a hook diagram we are also able to carry out this verification provided we work at nonzero Fayet-Iliopoulos parameters.