The technique of $\textit{orthosymplectic quotient quiver subtraction}$ is introduced for framed orthosymplectic quivers. This involves subtracting an $\textit{orthosymplectic quotient quiver}$ from a framed orthosymplectic 3$\textit{d}$ $$ \mathcal{N} $$ = 4 quiver gauge theory which has the effect of gauging an SO($\textit{n}$) or Sp($\textit{n}$) subgroup of the IR Coulomb branch global symmetry with complete Higgsing. The orthosymplectic quotient quivers take the form of magnetic quivers for class $$ \mathcal{S} $$ theories on cylinders with (twisted) maximal punctures of simply laced algebras, similar to the case of unitary quotient quiver subtraction. This gives a set of quotient quivers for all classical groups. Notably, quotient quiver subtraction for framed and unframed orthosymplectic quivers are different procedures.

United Kingdom

Sam Bennett

Amihay Hanany

Guhesh Kumaran

Orthosymplectic quotient quiver subtraction. Part II. Framed quivers

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.

Chunhao Li

Deshuo Liu

Austria

Marcus Sperling

Quiver subtraction on the Higgs branch

We study Higgs branches of field theories with 8 supercharges in 5 and 6 dimensions, focusing on theories realised on 5-brane webs in Type IIB with an O7 plane, or a D6-D8-NS5 brane system in Type IIA in the presence of an O8 plane. We find magnetic quivers for the Higgs branches of these theories. The main consequence of the presence of the orientifold is that it renders the magnetic quiver to be non-simply-laced. We propose a contribution of the O7 to the usual stable intersection number of 5-branes from tropical geometry, and show that it is consistent with Fayet-Iliopoulos deformations of magnetic quivers which represent mass deformations of 5d SQFTs. From the magnetic quivers, we compute phase diagrams and highest weight generating functions for the Higgs branches, enabling us to identify the global form of the flavour symmetry for several families of 5d SQFTs; among them Bhardwaj’s rank-1 theory. For 6d theories realised on a 4 curve, we observe the appearance of an additional slice on top of the phase diagram as one goes to the tensionless limit.

Japan

Mohammad Akhond

Guillermo Arias-Tamargo

Federico Carta

Julius Grimminger

On brane systems with O

The technique of orthosymplectic quotient quiver subtraction is introduced. This involves subtraction of an orthosymplectic quotient quiver from a 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 orthosymplectic quiver gauge theory which has the effect of gauging subgroups of the IR Coulomb branch global symmetry. Orthosymplectic quotient quivers for SU(2), SU(3), G 2, and SO(7) are found and derived from Type IIA brane systems involving negatively charged branes for certain 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) gauge theories. Orthosymplectic quotient quiver subtraction is applied to magnetic quivers for nilpotent orbit closures providing new orthosymplectic counterparts to known unitary quivers. New Coulomb branch constructions are found such as for two height four nilpotent orbit closures of F 4 and one of height three. A novel application is to find magnetic quivers and Type IIA brane systems for the 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) worldvolume theory of two   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>M</mi> <mn>5</mn> </math>  $$ \frac{1}{2}\textrm{M}5 $$  branes on E 6 Klein singularity and for 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) (E 6, E 6) conformal matter. These give a perturbative Lagrangian realisation to the dynamics of strongly interacting M5 branes. The magnetic quiver for 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) (E 6, E 6) conformal matter is star-shaped and can also be interpreted as a magnetic quiver for a class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  theory specified by SO(26) algebra on a three-punctured sphere.

Orthosymplectic quotient quiver subtraction

Two new diagrammatic techniques on 3$\textit{d}$ $$ \mathcal{N} $$ = 4 quiver gauge theories, termed $\textit{chain}$ and $\textit{cyclic quiver polymerisation}$ are introduced. These gauge a diagonal SU/U($\textit{k}$) subgroup of the Coulomb branch global symmetry of a quiver (or pair of quivers) with multiple legs. The action on the Coulomb branch is that of a SU/U($\textit{k}$) hyper-Kähler quotient. The polymerisation techniques build and generalise known composition methods from class $$ \mathcal{S} $$ . Polymerisation is used to generate a wide range of magnetic quivers from various physical contexts. These include polymerisation constructions for Kronheimer-Nakajima quivers, which generalise the ADHM construction for the moduli space of $\textit{k}$ SU($\textit{N}$) instantons on $\textit{ℂ}$ to A-type singularities. Also a polymerisation construction of the magnetic quiver for the 6$\textit{d}$ $$ \mathcal{N} $$ = (1, 0) theory coming from two $$ \frac{1}{2} $$ M5 branes probing an $\textit{E}$ Klein singularity. We find a method of extending magnetic quivers for Class $$ \mathcal{S} $$ theories to cure the incomplete Higgsing that arises when gluing punctures into the loops associated with higher genus theories. Other novel constructions include a unitary magnetic quiver for the closure of a height four nilpotent orbit of SO(7). We explore the relationships between the Coulomb and Higgs branches of quivers under polymerisation.

Rudolph Kalveks

Quiver polymerisation

We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the <math><mrow><mi>SU</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math> hyper-Kähler quotient (HKQ) of the Coulomb branch of a 3d <math><mi>N</mi><mo>=</mo><mn>4</mn></math> unitary quiver theory. The target quivers are drawn from a wide range of theories, typically classified as “good” or “ugly”, which satisfy identified selection criteria. Our subtraction procedure uses quotient quivers that are “bad”, differing thereby from quiver subtractions based on Kraft-Procesi transitions. The simple diagrammatic procedure identifies one or more resultant quivers, the union of whose Coulomb branches corresponds to the desired HKQ. Examples include quivers whose Coulomb branches are moduli spaces of free fields, closures of nilpotent orbits of classical and exceptional type, and slices in the affine Grassmanian. We calculate the Hilbert Series and Highest Weight Generating functions for HKQ examples of low rank. For certain families of quivers, we are able to conjecture HWGs for arbitrary rank. We examine the commutation relations between quotient quiver subtraction and other diagrammatic techniques, such as Kraft-Procesi transitions, quiver folding, and discrete quotients.

Quotient quiver subtraction

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.

France

Antoine Bourget

China

Zhenghao Zhong

A tale of N cones

We construct Poisson bracket relations between the operators which generate the chiral ring of the Coulomb branch of certain 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 quiver gauge theories. In the case where the Coulomb branch is a free space, ADE Klein singularity, or the minimal A 2 nilpotent orbit, we explicitly compute the Poisson brackets between the generators using either inherited properties of the abstract Coulomb branch variety, or the expected charges of these operators under the global symmetry (known through use of the monopole formula). We also conjecture Poisson brackets for Higgs branches that originate from 6d theories with tensionless strings or 5d theories with massless instantons for which the HWG is known, based on representation theoretic and operator content constraints known from the study of their magnetic quiver.

Kirsty Gledhill

Poisson brackets for some Coulomb branches

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.