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F 4 Nilpotent Orbit Hasse Diagram. The left hand diagram is derived from Hilbert series and HWG inclusion relations. The right hand diagram is taken from the mathematical Literature [29, 30]. Yellow nodes indicate non-normal nilpotent orbits.
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A bstract
We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional gro...
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... is interesting to compare the inclusion relations obtained from this analysis of moduli spaces with the standard Hasse diagrams of nilpotent orbits in the mathematical Literature [10,29,30], which are based on earlier work in [31]. Figure 3 compares the Hasse diagram defined by the inclusion relations amongst the Hilbert series of nilpotent orbits g F 4 N O to the standard Hasse diagram. Unlike the case of Classical group nilpotent orbits, where there is an exact correspondence between the Hasse diagrams (omitted) based upon Hilbert series inclusion relations and the standard diagrams [10,11], there is a discrepancy involving the linking pattern between F 4 [2000] and F 4 [0010], where the restricted N ON method yields an inclusion relationship that is absent in the standard diagram. ...
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We study the moduli space of 3d $\mathcal{N}=4$ quiver gauge theories with unitary, orthogonal and symplectic gauge nodes, that fall into exceptional sequences. We find that both the Higgs and Coulomb branches of the moduli space factorise into decoupled sectors. Each decoupled sector is described by a single quiver gauge theory with only unitary g...
Citations
... Such theories are identified both by their Hasse diagrams and Hilbert series. Of course, Hasse diagrams are not unique to a moduli space, and thus explicit Hilbert series computations are required, alongside calculations involving the localisation formula [19,[57][58][59][60], to verify the identity of the moduli spaces presented in the following section. ...
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 C type or 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.
... • The unrefined Hilbert series presented in this work display palindromic numerators; for particularly cumbersome examples, ellipses · · · will be used after the last unique coefficient, with the understanding that for a numerator of degree 4n the coefficient of the term at t 2n+2k will be the same as that for t 2n−2k for 1 ≤ k ≤ n • Nilpotent orbits of exceptional algebras will be labelled by their Characteristic [70,71] and Bala-Carter label ...
... from which the Coulomb branch can be identified as C(Q 13b ) = Normalisation O (Bala-Carter labelà 2 ), where the normalisation of the nilpotent orbit closure is computed by the localisation formula [71]. The proposed intersection of the quivers Q 13a and Q 13b is Q 13c , with Coulomb branch Hilbert series ...
The technique of orthosymplectic quotient quiver subtraction is introduced. This involves subtraction of an orthosymplectic quotient quiver from a orthosymplectic quiver gauge theory which has the effect of gauging subgroups of the IR Coulomb branch global symmetry. Orthosymplectic quotient quivers for and are found and derived from Type IIA brane systems involving negatively charged branes for certain 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 and one of height three. A novel application is to find magnetic quivers and Type IIA brane systems for the worldvolume theory of two M5 branes probing Klein singularity and for conformal matter. These give a perturbative Lagrangian realisation to the dynamics of strongly interacting M5 branes. The magnetic quiver for conformal matter is star-shaped and can also be interpreted as a magnetic quiver for a class theory specified by algebra on a three-punctured sphere.
... One use of chain and cyclic polymerisation on magnetic quivers is to find new and nontrivial relationships between free field theories, nilpotent orbit closures [84,85], Slodowy slices [86] and Slodowy intersections [87]. ...
... Let A be the Cartan matrix of SU(k), then ∆(λ[n]) ≡ ∆[n] = −[n] · A −1 · 2. Alternatively, drawing on the Characteristics and weight maps ω of nilpotent orbits, ∆[n] can be found simply from the weight map of the regular orbit as, ∆[n] = −[n] · ω(reg.) [84,85]. ...
... (3 2 ,1) , which is a nilpotent orbit closure of height four. While unitary magnetic quivers for nilpotent orbits closures of height two or less are known for all semi-simple Lie algebras [85], very few magnetic quivers are given in the Literature for orbits of greater height. The Higgs branch of Q 44 is the Slodowy slice S C 3 N ,(2 3 ) , which is consistent with Lustig-Spaltenstein duality. ...
Two new diagrammatic techniques on quiver gauge theories, termed chain and cyclic quiver polymerisation are introduced. These gauge a diagonal 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 hyper-K\"ahler quotient. The polymerisation techniques build and generalise known composition methods from class . 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 instantons on to A-type singularities. Also a polymerisation construction of the magnetic quiver for the coming from two M5 branes probing an Klein singularity. We find a method of extending magnetic quivers for Class 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 . We explore the relationships between the Coulomb and Higgs branches of quivers under polymerisation.
... The closures of nilpotent orbits of semi-simple Lie algebras ("nilpotent orbits" or "orbits") have deep connections to the Coulomb branches of 3d N = 4 theories and many have constructions from magnetic quivers [14,15]. The theorem of Namikawa [16], applied to moduli spaces of 3d N = 4 theories, states that the chiral ring with generators at spin 1 of SU (2) R is a nilpotent orbit, which has been verified for the Coulomb branches of the corresponding magnetic quivers using the monopole formula. ...
... The Coulomb branch of a magnetic quiver is always a hyper-Kähler cone, so we can consider its hyper-Kähler quotient (HKQ) by some subgroup of its global symmetry. Physically, such an HKQ is a gauging [10] that incorporates the action of a moment map [15]. ...
... Nilpotent orbits have been studied extensively in the quiver gauge theory literature [14,15]. Notably, there exist unitary magnetic quivers for any orbit of A-type or any orbit of G that has a Characteristic height of 2. These are all balanced quivers of Dynkin type. ...
We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the SU(n) hyper-K\"ahler quotient (HKQ) of the Coulomb branch of a 3d 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 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.
... The reasoning can be also applied to other non-simply laced Coulomb branch quivers, even if there may not exist a known mirror. Such an example is the F 4 Coulomb branch quiver of [40]. Table 2 summarises the resulting theories after a suitable t n is gauged, following the prescriptions (2.58) and (2.56). ...
... The resulting Coulomb branch is the next-to-next-to minimal nilpotent Table 2: The F 4 Coulomb branch quiver and its q gaugings. The first row is the standard F 4 quiver proposed in [40]. Rows 2 -4 display different choices of gauging a t N of a U(N ) node in the Coulomb branch quiver for the minimal nilpotent orbit closure of F 4 . ...
The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d \mathcal{N}=4 𝒩 = 4 theories. In this paper, starting with known mirror pairs of 3d \mathcal{N}=4 𝒩 = 4 quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0-form, 1-form, and 2-group) and the mirror maps between them.
... Nilpotent orbits are of particular interest, both in mathematics and physics: see e.g. [98][99][100][101][102] for recent progress on complex nilpotent orbits. These orbits have been classified, and can be labeled by signed Young diagrams [103] (see also [104] for the classification of nilpotent orbits of the complex forms, and [95,Chap. ...
... See e.g. [98][99][100][101][102] for recent progress in the study of supersymmetric moduli spaces. Let us limit the scope of our discussion to the so C (n) case. ...
We derive manifestly covariant actions of spinning particles starting from coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show that the defining conditions of a classical Lie group can be treated as Hamiltonian constraints which generate the coadjoint orbits of another, dual, Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are (centrally-extended inhomogeneous) symplectic groups. This defines a symplectic dual pair correspondence between the coadjoint orbits of the isometry group and those of the dual Lie group, whose quantum version is the reductive dual pair correspondence \`a la Howe. We show explicitly how various particle species arise from the classification of coadjoint orbits of Poincar\'e and (A)dS symmetry. In the Poincar\'e case, we recover the data of the Wigner classification, which includes continuous spin particles, (spinning) tachyons and null particles with vanishing momenta, besides the usual massive and massless spinning particles. In (A)dS case, our classification results are not only consistent with the pattern of the corresponding unitary irreducible representations observed in the literature, but also contain novel information. In dS, we find the presence of partially massless spinning particles, but continuous spin particles, spinning tachyons and null particles are absent. The AdS case shows the largest diversity of particle species. It has all particles species of Poincar\'e symmetry except for the null particle, but allows in addition various exotic entities such as one parameter extension of continuous particles and conformal particles living on the boundary of AdS. Notably, we also find a large class of particles living in "bitemporal" AdS space, including ones where mass and spin play an interchanged role. We also discuss the relative inclusion structure of the corresponding orbits.
... There are fifteen nilpotent orbits of e 7 that are compatible with a Z 2 center flavor symmetry, as depicted in Table F.4. The sub-Hasse diagram formed by the subset of e 7 nilpotent orbits appears here almost matches the Hasse diagram for f 4 nilpotent orbits that appears in Hanany and Kalveks (2017). Similarly, the flavor symmetries surviving after the Stiefel-Whitney twist and those of the f 4 nilpotent orbits almost always match. ...
In this dissertation, we study the generalized symmetries in supergravities and superconformal field theories from the string theory perspective. Part one is devoted to the study of string universality in high spacetime dimensions. Answering this question requires us to combine the following two approaches. In the "top-down" approach, We focus on supergravity theories in 7, 8, and 9 dimensional spacetime with 16 supercharges. We emphasize two discrete aspects of these theories: generalized global symmetries and frozen singularities. We give an exhaustive classification of IIB supergravity theory in 8D, particularly emphasizing these two discrete aspects. In the "bottom-up" approach, we present a consistency condition of general 8D supergravity theories involving their higher-form symmetries use it to rule out many global structures of the gauge groups in 8D supergravity theories that do not admit string theory constructions. Part two studies the generalized global symmetries of geometrically-engineered quantum field theories via string theory. We examined branes wrapping on relative topological cycles that give heavy defects that are charged under generalized global symmetries, which can then be used to construct new lower-dimensional theories. By investigating the string theory origin of the topological operators, we provide a general construction of these topological operators in the context of geometric engineering as branes wrapped on the homological cycles in the asymptotic boundary of the internal geometry. We illustrate this proposal by determining non-invertible 2-form symmetries in 6D superconformal field theories. Furthermore, by wrapping type IIB 7-brane on the entire asymptotic boundary of the internal manifold, we explicitly give a unified string-theoretic construction of two different types of field-theoretic non-invertible duality defects.
... We consider again the two instanton case and turn on FI deformations which correspond to a mass term for the SU(2) global symmetry. Also in this case the deformation is implemented via quiver subtraction and we end up with the magnetic quivers associated with the next-to-minimal nilpotent orbits of E 6,7,8 [70]. ...
A bstract
We study the 3d 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 N = 2 SCFTs. A delightful by-product of our work is the discovery of an interesting new 3d mirror pair.
... We consider again the two instanton case and turn on FI deformations which correspond to a mass term for the SU(2) global symmetry. Also in this case the deformation is implemented via quiver subtraction and we end up with the magnetic quivers associated with the next-to-minimal nilpotent orbits of E 6,7,8 [70]. ...
We study the 3d 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 SCFTs. A delightful by-product of our work is the discovery of an interesting new 3d mirror pair.
... The reasoning can be also applied to other non-simply laced Coulomb branch quivers, even if there may not exist a known mirror. Such an example is the F 4 Coulomb branch quiver of [40]. Table 2 summarises the resulting theories after a suitable Z t n is gauged, following the prescriptions (2.58) and (2.56). ...
... The F 4 Coulomb branch quiver and its Z q gaugings. The first row is the standard F 4 quiver proposed in [40]. Rows 2 -4 display different choices of gauging a Z t N of a U(N ) node in the Coulomb branch quiver for the minimal nilpotent orbit closure of F 4 . ...
The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d theories. In this paper, starting with known mirror pairs of 3d quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0-form, 1-form, and 2-group) and the mirror maps between them.
