Quiver Theories and Formulae for Slodowy Slices of Classical Algebras

Abstract

We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra g. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone S∩N of g. We calculate refined Hilbert series for Classical algebras up to rank 4 (and A5), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections S∩N; such dual quiver theories exist for the Slodowy slices of A algebras, but are limited to a subset of the Slodowy slices of BCD algebras. The analysis opens new questions about the extent of 3d mirror symmetry within the class of SCFTs known as Tσρ(G) theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the SU(2) embedding into G of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.

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Nuclear Physics B 939 (2019) 308–357
www.elsevier.com/locate/nuclphysb
Quiver theories and formulae for Slodowy slices of
classical algebras
Santiago Cabrera, Amihay Hanany, Rudolph Kalveks ∗
Theoretical Physics Group, The Blackett Laboratory, Imperial College London, Prince Consort Road,
London SW7 2AZ, United Kingdom
Received 3 October 2018; received in revised form 6 December 2018; accepted 17 December 2018
Available online 21 December 2018
Editor: Clay Córdova
Abstract
We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces trans-
verse to the nilpotent orbits of a Lie algebra g. We analyse classes of quiver theories, with Classical gauge
and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the
nilpotent cone S∩Nof g. We calculate refined Hilbert series for Classical algebras up to rank 4(andA5),
and find descriptions of their representation matrix generators as algebraic varieties encoding the relations
of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections
S∩N; such dual quiver theories exist for the Slodowy slices of Aalgebras, but are limited to a subset of the
Slodowy slices of BCD algebras. The analysis opens new questions about the extent of 3dmirror symme-
try within the class of SCFTs known as Tρ
σ(G) theories. We also give simple group theoretic formulae for
the Hilbert series of Slodowy slices; these draw directly on the SU(2)embedding into Gof the associated
nilpotent orbit, and the Hilbert series of the nilpotent cone.
©2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.
*Corresponding author.
E-mail addresses: santiago.cabrera13@imperial.ac.uk (S. Cabrera), a.hanany@imperial.ac.uk (A. Hanany),
rudolph.kalveks09@imperial.ac.uk (R. Kalveks).
https://doi.org/10.1016/j.nuclphysb.2018.12.022
0550-3213/©2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 309
1. Introduction
The relationships between supersymmetric (“SUSY”) quiver gauge theories, the Hilbert series
(“HS”) of their Higgs and Coulomb branches, and the nilpotent orbits (“NO”) of simple Lie
algebras gwere analysed in two recent papers [1,2]. Closures of classical nilpotent orbits appear
as Higgs branches on N=2 quiver theories in 4d, and also as Coulomb branches on N=4
quiver theories in 2 +1 dimensions. Both these types of theory have 8 supercharges.
The aim herein is to examine systematically the relationships between these SUSY quiver
gauge theories and the spaces transverse to nilpotent orbits, known as Slodowy slices. The focus
herein is the Slodowy slices of the nilpotent orbits of Classical algebras, which are associated
with a rich array of 3dN=4 quiver theories and dualities. The relationships between SUSY
quiver gauge theories and the Slodowy slices of nilpotent orbits of Exceptional algebras remain
to be treated.
The mathematical study of Slodowy slices has its roots in [3], which built on earlier work
by Brieskorn [4], Grothendieck and Dynkin [5]. This showed that each nilpotent orbit Oρof a
Lie algebra gof a Classical group Ghas a transverse slice, or Slodowy slice Sρ, lying within
the algebra g.1There is a variety defined by the intersection between the Slodowy slice and the
nilpotent cone Nof the algebra: SN,ρ ≡N∩Sρ. In this paper, we deal almost entirely with
these intersections SN,ρ and refer to them loosely as Slodowy slices (except where the context
requires otherwise). Each such slice is a singularity that can be characterised by a sub-algebra f
of gthat commutes with (or stabilises) the su(2). In the case of the sub-regular nilpotent orbit
SN,ρ is a Kleinian singularity of type ADE.2
The connection between nilpotent orbits and their Slodowy slices, and instanton moduli
spaces, i.e. the solutions of self dual Yang–Mills equations, was made in [7]. The use of Dynkin
diagrams and quiver varieties to define instantons on ALE spaces was discussed in [8]. The rel-
evance of nilpotent orbits and Slodowy slices to 3dN=4 quiver theories was later explored in
detail in [9] and [10]. In this context, they appear as effective gauge theories describing the brane
dynamics of a system in Type IIB string theory. Brane systems of the type of [11]are relevant
for the Aseries and systems with three dimensional orientifold planes [12]f
or the BCD series.3
In the course of these latter papers, aclass of superconformal field theories (“SCFT”) was
proposed, with moduli spaces defined by the intersections between Slodowy slices and nilpotent
orbits. These are termed Tρ
σ(G) theories, where Gis a Lie group. Several types of Classical
quiver theories were identified, along with associated brane configurations, including theories
whose Higgs or Coulomb branches correspond to certain varieties SN,ρ, and a relationship
between S-duality and dualities arising from the 3dmirror symmetry [17]of Classical quiver
theories was conjectured.4
For example, in the case of an Aseries nilpotent orbit Oρ, where ρdescribes the embedding of
su(2)into su(n)that defines the nilpotent orbit, and ρ=(1N)corresponds to the trivial nilpotent
1ρidentifies the embedding of su(2)into gthat defines the nilpotent orbit.
2For general background on nilpotent orbits the reader is referred to [6].
3Note that these brane systems can explicitly realize the transverse slices developed by Brieskorn and Slodowy [3,4].
A systematic analysis of transverse slices was carried out by Kraft and Procesi [13]and the physics realization was
studied in [14,15]. The concept of transverse slices can be further extended as an operation of subtractions between two
quivers [16].
4In the case of nilpotent orbits of C and D type, the precise match between quivers and orbits was given in [18]. Subse-
quently, [19] described the relation for B type and unified all classical cases via the introduction of the Barbasch–Vogan
map [20].

310 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
orbit, these dualities entail that the Higgs branch of a linear quiver based on a partition ρT, yields
the closure of the nilpotent orbit ¯
Oρ, while the Coulomb branch of a linear quiver based on the
partition ρgives its Slodowy slice SN,ρ. The application of 3dmirror symmetry to this pair
of linear quivers yields a pair of “balanced” quivers, with the Coulomb branch of the former
yielding ¯
Oρand the Higgs branch of the latter yielding SN,ρ .5
More recently, in [19] and [21], nilpotent orbits and Slodowy slices have been used in the study
of 6dN=(2, 0)theories on Riemann surfaces. Relationships between diagram automorphisms
of quiver varieties and Slodowy slices are explored in [22]. In [23]the algebras of polynomial
functions on Slodowy slices were shown to be related to classical (finite and affine) W-algebras.
Each Slodowy slice of a sub-algebra fof ghas a ring of holomorphic functions transforming
in irreps of the sub-group Fof G. Our approach is to compute the Hilbert series of these rings.
Presented in refined form, such Hilbert series faithfully encode the class function content of
Slodowy slices, and can be subjected to further analysis using the tools of the Plethystics Program
[24–26].
Importantly, following a result in [3], the Hilbert series of Slodowy slices SN,ρ are always
complete intersections, i.e. quotients of geometric series. It was shown in [27]how the HS of the
Slodowy slices of Aseries and certain BCD series algebras can be calculated from the Coulomb
branches of linear quivers (or from the Higgs branches of their 3dmirror duals). [27]also iden-
tified a relationship between Slodowy slices and the (modified) Hall Littlewood polynomials of
g, under the mapping g →su(2)⊗f.
Methods of calculating Hilbert series for Tρ
σ(G) theories with multi-flavoured quivers of
unitary or alternating O/USp type were developed in [28], using both Coulomb branch and
Higgs branch methods. As elaborated in [29], the calculation of Coulomb branches of quivers of
O/USp type requires close attention to the distinction between SO and Ogroups.
This paper builds systematically on such methods to calculate the Hilbert series of Slodowy
slices of closures of nilpotent orbits of low rank Classical Lie algebras and to identify relevant
generalisations to arbitrary rank.
In Section 2we summarise some facts about nilpotent orbits and review the relationship be-
tween a Slodowy slice SN,ρ and the homomorphism ρdefining the embedding of su(2)into g
(and thus of the mapping of irreps of Ginto the irreps of SU(2)) associated with a nilpotent
orbit Oρ. We give some simple representation theoretic formulae for calculating the dimensions
and Hilbert series of a Slodowy slice, given a homomorphism ρ.
In Section 3we treat Aseries Slodowy slices, summarising the relevant Higgs branch and
Coulomb branch formulae, describing the quivers upon which they act, and tabulating the com-
mutant global symmetry group and the Hilbert series of Slodowy slices for all nilpotent orbits up
to rank 5. We also build upon the language of Tρ
σ(SU (N)) theories to summarise the known exact
Aseries dualities between quiver theories for Slodowy slices and nilpotent orbits. We find matrix
formulations for the generators of Aseries Slodowy slices and their relations, which explicate
the mechanism of symmetry breaking and the residual symmetries of the parent group.
In Section 4we extend this analysis to Slodowy slices of BCD series algebras up to rank 4.
We find a complete set of refined Hilbert series, by working with the Higgs branches of multi-
flavoured alternating O/USp quivers with appropriately balanced gauge nodes. As in the case of
BCD nilpotent orbits [1], calculation of these Higgs branches requires taking Z2averages over
5The notation in the Literature regarding partitions and their dual maps has changed a great deal; see [15, sec. 4] for a
summary of the different maps that are relevant to our study and an explicit review of the different conventions used in
mathematics and physics.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 311
the SO and O−forms of Ogroup characters. We also identify a limited set of Higgs branch
constructions based on Dseries Dynkin diagrams. We summarise the restricted set of Coulomb
branch monopole constructions that are available for SN,ρ, which are based on alternating
SO/USp linear quivers. We highlight apparent restrictions on 3dmirror symmetry between
Higgs and Coulomb branches of BCD quiver theories; these include the requirements that the
nilpotent orbit Oρshould be special, and that the O/USp quivers should not be “bad” [10] due to
containing monopole operators with zero conformal dimension. We find matrix formulations for
the Higgs branch generators of BCD series Slodowy slices, and their relations, which explicate
the mechanism of symmetry breaking and the residual symmetries of the parent group.
Take n together with other recent studies [1,29], this analysis of Hilbert series is relevant for the
understanding of Tρ
σ(G) theories of type BCD. It highlights the difference between orthogonal
O(n) and special orthogonal SO(n) nodes and the surrounding problems associated with 3d
mirror symmetry between orthosymplectic quivers.
The final Section summarises conclusions, discusses open questions and identifies areas for
further work. Some notational conventions are detailed in Appendix A.
2. Slodowy slices
2.1. Relationship to nilpotent orbits
Each nilpotent orbit Oρof a Lie algebra gis defined by the conjugacy class gXof nilpotent
elements X∈gunder the group action [6]. Each nilpotent element Xforms part of a standard
su(2)triple {X, Y, H}and, following the Jacobson Morozov theorem, the conjugacy classes are
in one to one correspondence with the equivalence classes of embeddings of su(2)into g, de-
scribed by some homomorphism ρ. The closure of each orbit ¯
Oρ, can, as discussed in [1,2], be
described as a moduli space, by a refined Hilbert series of representations of G, graded according
to the degree of symmetrisation of the underlying nilpotent element.
The closures ¯
Oρof the nilpotent orbits of gform a poset, ordered according to their inclusion
relations.6The closure of the maximal (also termed principal or regular) nilpotent orbit is called
the nilpotent cone N; it contains all the orbits Oρand has dimension |N|equal to that of the
rootspace of g. The poset of nilpotent orbits contains a number of canonical orbits. These include
the trivial nilpotent orbit (described by the Hilbert series 1 with dimension zero), a minimal
(lowest dimensioned non-trivial) nilpotent orbit, a sub-regular orbit of dimension |N|−2 and
the maximal nilpotent orbit:
{0}=Otrivial ⊂¯
Omini mal ...⊂¯
Osub−regular ⊂¯
Omaxi mal =N.(2.1)
All nilpotent orbits have an even (complex) dimension and are HyperKähler cones.
The closure of the minimal nilpotent orbit of gcorresponds to the reduced single G-instanton
moduli space [7,30]. As discussed in [1], the Hilbert series of the nilpotent cone has a simple
expression in terms of the symmetrisations of the adjoint representation of G, modulo Casimir
operators, or equivalently in terms of (modified) Hall Littlewood polynomials:
gN
HS =PEχG
adjointt2−
r

i=1
t2di,
gN
HS =mH LG
singlet t2,
(2.2)
6See for example the Hasse diagrams in [13].

312 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
where tis a counting fugacity, χG
adjoint is the character of the adjoint representation and
{d1, ..., dr}are the degrees of the symmetric Casimirs of G, which is of rank r.
Slodowy slices are defined as slices Sρ⊆gthat are transverse in the sense of [3]to the
orbit Oρ. The varieties SN,ρ that concern the present study are slices inside the nilpotent cone N.
They can be constructed as:
SN,ρ ≡Sρ∩N.(2.3)
Naturally, the slice SN,ρ transverse to the trivial nilpotent orbit is the entire nilpotent cone N
and the slice SN,ρ transverse to the maximal nilpotent orbit is trivial. In between these limiting
cases, however, the Slodowy slices do not match any nilpotent orbit. Consequently we have a
complementary poset of Slodowy slices:
N=Strivial >Smini mal ... >Ssub−regular >Smax imal ={0}.(2.4)
2.2. Dimensions and symmetry groups
The dimensions of a Slodowy slice SN,ρ plus those of the nilpotent orbit Oρcombine to the
dimensions of the nilpotent cone N:
SN,ρ +Oρ=|N|=|g|−rank[g].(2.5)
The elements of the Slodowy slice SN,ρ lie in a subalgebra f, which is the centraliser of the
nilpotent element X∈g, so that [X, f]=0, and fis often termed the commutant of su(2)in g.
The structure of fand the dimensions of SN,ρ and Oρcan be determined by analysing the
embedding of su(2)→g.
Following [5], ahomomorphism ρcan be described by a root space map from gto su(2),
and this is conveniently encoded in a Characteristic set of Dynkin labels.7The Characteristic
[q1...q
r]provides a map from the simple root fugacities {z1, ..., zr}of gto the simple root
fugacity {z}of su(2):
ρ[q1...q
r]:{z1,...,z
r}→z
q1
2,...,zqr
2,(2.6)
where the labels qi∈{0, 1, 2}. This induces corresponding weight space maps under which
each representation of Gof dimension Nbranches to representations [n]of SU(2)at
some multiplicity mn. This branching is conveniently described using partition notation,
(|[N−1]|mN−1,...,|[n]|mn,...,1m0), which lists the dimensions of the SU(2)irreps, using ex-
ponents to track multiplicities. These partitions are tabulated in [1]for the key irreps of Classical
groups up to rank 5, identifying each nilpotent orbit by its Characteristic.
For example, the homomorphism ρwith Characteristic [202], which generates the 10 dimen-
sional nilpotent orbit of A3, induces the following maps:
ρ[202]:{z1,z
2,z
3}→{z, 1,z
},
ρ[202]:[1,0,1]→[4]+3⊗[2]+[0]⇐⇒ χA3
adjoint →5,33,1,
ρ[202]:[1,0,0]→[2]+[0]⇐⇒ χA3
fundamental →(3,1).
(2.7)
7A Characteristic G[...]is distinct from highest weight Dynkin labels [..., ...]G.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 313
These SU(2)partitions are invariant under the symmetry group F⊆Gof the Slodowy slice and
hence the multiplicities encode representations of F.
Under the branching, the adjoint representation of Gdecomposes to representations of the
product group SU(2) ⊗Fwith branching coefficients anm:
χG
adjoint →
[n][m]
anm χSU(2)
[n]χF
[m].(2.8)
Other than for the trivial nilpotent orbit (in which the adjoint of Gbranches to itself times an
SU(2)singlet), the adjoint of SU(2)and the adjoint (if any) of Feach appear separately in the
decomposition, so that rank[G] rank[F] 0. Along with the requirement that any multiplic-
ities mnappearing in a partition must be dimensions of representations of F, this often makes it
possible to determine the Lie algebra fof the Slodowy slice directly from the partition data. In
Example 2.7 the presence of a single SU(2)singlet in the partition of the adjoint of A3entails
that the symmetry group of the Slodowy slice to the [202]orbit is simply U(1).
The adjoint partition data also permits direct calculation of the complex dimensions of a
Slodowy slice or nilpotent orbit, by summing multiplicities of SU(2)irreps or, equivalently,
dimensions of Firreps:
Sρ=
[n][m]
anm χF
[m],
Oρ=|G|−Sρ,
SN,ρ =Sρ−rank[G].
(2.9)
2.3. Hilbert series
The branching of the adjoint representation of Gdetermines the generators of the Slodowy
slice. If the decomposition (2.8)is known, the Hilbert series for the Slodowy slice can be derived
from the HS of the nilpotent cone by substitution under a particular choice of grading. Consider
the map ˜ρof the adjoint that is obtained from (2.8)b
y the replacement of SU(2)irreps by their
highest weight fugacities χSU(2)
[n]→tn, taking the particular choice of tfrom (2.2)as the counting
variable:
˜ρ:χG
adjoint →
[n][m]
anmχF
[m]tn.(2.10)
When the adjoint map (2.10)is applied to the generators of the nilpotent cone (2.2), the re-
placement of the SU(2)representations [n]by the counting fugacity tnentails that the resulting
Hilbert series only transforms in the symmetry group of the Slodowy slice. Thus, gSN,ρ
HS =gN|˜ρ
HS ,
or, written more explicitly:
gSN,ρ
HS (x, t ) =PEχG
adjoint˜ρt2−
r

i=1
t2di
=PE⎡
⎣
[n][m]
anmχF
[m]tn+2−
r

i=1
t2di⎤
⎦.
(2.11)

314 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 1
Sub-regular Slodowy slices of classical groups.
Group Singularity Dimension Hilbert series
Arˆ
Ar≡C2/Zr+12PE2tr+1+t2−t2r+2
Brˆ
A2r−1≡C2/Z2r2PE2t2r+t2−t4r
Cr>1ˆ
Dr+1≡C2/Dicr−12PEt2r−2+t2r+t4−t4r
Dr>2ˆ
Dr≡C2/Dicr−22PEt2r−4+t2r−2+t4−t4r−4
The dicyclic group of order 4kis denoted as Dick.
The expression (2.11)gives the refined Hilbert series of the Slodowy slice in terms of its genera-
tors, which are representations of the Slodowy slice symmetry group F, at some counting degree
in t, less its relations, which are set by the degrees of the Casimirs of G.8
Importantly, an unrefined Hilbert series, with representations of Freplaced by their dimen-
sions, mn=
m
anm|χF
[m]|, can be calculated directly from the adjoint partition under ρ, without
knowledge of the precise details of the embedding (2.8):
gSN,ρ
HS (1,t)=PE
n
mntn+2−
r

i=1
t2di.(2.12)
Finally, the freely generated Hilbert Series for the canonical Slodowy slices Sρare related to
those of their nilpotent intersections SN,ρ simply by the exclusion of the Casimir relations:
gSρ
HS(x, t ) ≡gSN,ρ
HS (x, t ) P E r

i=1
t2di=PE⎡
⎣
[n][m]
anmχF
[m]tn+2⎤
⎦.(2.13)
In Sections 3and 4we set out the quiver constructions that provide a comprehensive method
for identifying the decomposition (2.8) and for calculating the refined Hilbert series of the
Slodowy slices SN,ρ.
2.4. Sub-regular singularities
As shown in [3,4], the Slodowy slices of sub-regular orbits SN,subregular take the form of
ADE type singularities, C2/, where is a finite group of type ADE. Under the nilpotent orbit
grading by t2used herein, these take the forms in Table 1. The intersection SN,subr egul ar is
an example of a transverse slice between adjacent nilpotent orbits; all such transverse slices of
Classical algebras were classified by Kraft and Procesi in [13].
This known pattern of singularities amongst the Slodowy slices of subregular orbits, along
with the known forms of trivial and maximal Slodowy slices and dimensions, provide consistency
checks on the grading methods and constructions given herein.
8This construction for Slodowy slices is simpler, but equivalent to the Hall Littlewood method presented in [28].

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 315
Fig. 1. ASeries linear and balanced quiver types. In the canonical linear quiver, the unitary gauge nodes (blue/round)
are in descending order with decrements in non-increasing order. In the balanced quiver, the unitary gauge nodes are all
balanced by their attached gauge nodes (blue/round) and flavour nodes (red/square).
3. Aseries quiver constructions
3.1. Quiver types
The constructions for the Slodowy slices of Aseries nilpotent orbits draw upon the same
two quiver types as the constructions for the closures of the nilpotent orbits. These are shown in
Fig. 1:
1. Linear quivers based on partitions. These quivers LA(ρ) consist of a SU(N0)flavour node
connected to a linear chain of U(Ni)gauge nodes, where the decrements between nodes,
ρi=Ni−1−Ni, constitute an ordered partition of N0, ρ≡{ρ1, ..., ρk}, where ρiρi+1
and k
i=1ρi=N0.
2. Balanced quivers based on Dynkin diagrams. These quivers BA(Nf)consist of a linear chain
of U(Ni)gauge nodes (in the form of an Aseries Dynkin diagram), with each gauge node
connected to a flavour node of rank Nfi, where Nfi0. The ranks of the gauge nodes are
chosen such that each gauge node is balanced (as explained below), after taking account of
any attached flavour nodes.
On the Higgs branch, the flavour nodes of both types of quiver define an overall S(⊗
i
UNfi)global
symmetry, while on the Coulomb branch, the global symmetry group follows from the Dynkin
diagram formed by any balanced gauge nodes in the quiver.9
The balance of a unitary gauge node is defined as the sum of the ranks of its adjacent gauge
nodes, plus the number of attached flavours, less twice its rank:
Balance(i) ≡Nfi+
j=i±1
Nj−2Ni.(3.1)
For the Aseries balanced theories, the balance condition is B ≡{Balance(i)} =0, and (3.1) can
be simplified as:
Nf=A·N,(3.2)
where the flavour and gauge nodes have been written as vectors Nf≡(Nf1, ..., Nfk)and N ≡
(N1, ..., Nk), and Ais the Cartan matrix of Ak.
9The concept of balance was used in [9], in order to distinguish between (a) those Coulomb branch monopole operators
that are “good”, with unit conformal dimension and act as root space operators, (b) those that are “ugly” with half-integer
conformal dimension and act as weight space operators, and (c) those that are “bad” with zero or negative conformal
dimension, which lead to divergences.

316 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Aseries nilpotent orbits are in bijective correspondence with the partitions of N, and the
linear quivers provide a complete set of Higgs branch constructions. The balanced quivers also
provide a complete set of Coulomb branch constructions under the unitary monopole formula.
Both types of quiver are thus in bijective correspondence with Aseries orbits and can be related
by 3dmirror symmetry [17].
For Slodowy slices, the roles of these quiver types are reversed: the linear Aseries quivers
provide a complete set of Coulomb branch constructions, while the balanced Aseries quivers
provide a complete set of Higgs branch constructions.
When quivers of linear type are used to calculate Slodowy slices, via their Coulomb branches,
the lack of balance of such quivers generally breaks the symmetry of SU(N0)to a subgroup,
which becomes the isometry group of the Slodowy slice; this subgroup is in turn defined by the
Dynkin diagram of the subset of gauge nodes in the linear quiver that remain balanced.
The identification of quivers for Slodowy slices follows directly from the partition data dis-
cussed in section 2.2. For the Aseries, it is convenient to write the SU(2)partition of the
fundamental representation under ρas:
ρ[1,0,...
]A=NNfN,...,n
Nfn,...,1Nf1,(3.3)
so that the multiplicities of partition elements, which may be zero, are mapped to the flavour
vector Nf. The linear quiver LA(ρ) can be extracted simply by writing ρ[fund.]in long form.
The ranks Nof the gauge nodes of the balanced quiver BA(Nf(ρ)) can be found from Nfby
inverting (3.2). Alternatively, the balanced quivers BA(Nf(ρ)) can be obtained by applying 3d
mirror symmetry transformations to the linear quivers LA(ρ), and vice versa.
We can use the notation above to express the key relationships and dualities involving Aseries
quivers for the Slodowy slices of nilpotent orbits:
SN,ρ =HiggsBA(Nf(ρ))=Coulomb [LA(ρ)],
¯
Oρ=HiggsLA(ρ T)=Coulomb BA(Nf(ρT)),(3.4)
or, taking the transpose of ρ:
SN,ρT=HiggsBA(Nf(ρT))=CoulombLA(ρT),
¯
OρT=Higgs[LA(ρ )]=Coulomb BA(Nf(ρ)).
(3.5)
The quivers for Aseries slices are related to the quivers for the underlying orbits simply by
the transpose of the partition ρ, combined with exchange of Coulomb and Higgs branches. This
transposition of partitions, which is an order reversing involution on the poset of Aseries orbits,
is known as the Lusztig–Spaltenstein map [31].
These linear or balanced quiver types correspond to the limiting cases of Tρ
σ(SU (N)) theories
[10,28], where one of the partitions σor ρis taken as the trivial partition:
LA(ρ) ⇔T(1,1,...,1)
ρ(SU (N)) ,
BA(Nf(ρ)) ⇔Tρ
(1,1,...,1)(SU (N)) .(3.6)
Those quivers, whose Higgs or Coulomb branches yield Slodowy slices of Aseries groups up
to rank 5, are tabulated in Figs. 2and 3, labelled by the nilpotent orbit, giving the partition ρof

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 317
Fig. 2. Quivers for A1to A4Slodowy slices. The Higgs quivers are of type BA(Nf(ρ)) and the Coulomb quivers are of
type LA(ρ).
the fundamental, the dimensions of the Slodowy slice, and the residual symmetry group.10 The
balanced quivers used in the Higgs branch construction always have gauge nodes equal in number
to the rank of G =SU(N), while the linear quivers used in the Coulomb branch constructions
always have a number of flavours equal to the fundamental dimension of G =SU(N). The
quivers LA((1N)) and BA(Nf(1N)) for the Higgs and Coulomb branch constructions of the
Slodowy slice to the trivial nilpotent orbit are identical.
10 We describe a U(1)symmetry as D1if the characters qnof U(1)irreps always appear paired with their conjugates
in representations (qn+q−n).

318 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Fig. 3. Quivers for A5Slodowy slices. The Higgs quivers are of type BA(Nf(ρ)) and the Coulomb quivers are of type
LA(ρ).
3.2. Higgs branch constructions
The calculation of Higgs branch Hilbert series from the balanced quivers draws on similar
methods to those used in the calculation of the Higgs branches of the linear quivers for Aseries
nilpotent orbits, as elaborated in [1]. Pairs of bi-fundamental fields (and their complex conju-
gates) connect adjacent gauge nodes and, in addition, each non-trivial flavour node gives rise
to a pair of bi-fundamental fields connected to its respective gauge node. The characters of all
these fields are included in the PE symmetrisations. A HyperKähler quotient is taken once for
each gauge node, exactly as in the case of a linear quiver, and the Wey l integrations are then
carried out over the gauge groups. The order of Weyl integrations can be chosen to facilitate
computation.
The general Higgs branch formula for Aseries Slodowy slices is:
gHiggs[BA(Nf(ρ))]
HS
=
U(N1)⊗...U(Nk)
dμ
k

n=1
PE[fund.]U(Nn)⊗[anti.]U(Nfn)+[anti.]U(Nn)⊗[fund.]U(Nfn),t
PE[adjoint]U(Nn),t2
×
k−1

n=1
PE[fund.]U(Nn)⊗[anti.]U(Nn+1)+[anti.]U(Nn)⊗[fund.]U(Nn+1),t,
(3.7)

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 319
where dμ is the Haar measure for the U(N1)⊗...U(Nk)product group. Note that the bifun-
damental fields are symmetrised with the fugacity t, while the HyperKähler quotient (“HKQ”) is
symmetrised with t2.
The Higgs branch formula can be simplified, by drawing on the dimensions of the bi-
fundamentals and the gauge groups, to give a rule for the dimensions of an Aseries Slodowy
slice, and this can be simplified further by the balance condition (3.2):
gHiggs[BA(Nf(ρ))]
HS =2Nf(ρ)·N(ρ)−N(ρ)·A·N(ρ)
=Nf(ρ)·N(ρ).
(3.8)
For further details of the calculation methodology the reader is referred to the Plethystics Pro-
gram Literature. The same Hilbert series can in principle also be obtained algebraically by
working with matrix generators and relations, as in section 3.5.
3.3. Coulomb branch constructions
The monopole formula, which was introduced in [32], provides a systematic method for the
construction of the Coulomb branches of particular SUSY quiver theories, being N=4su-
perconformal gauge theories in 2 +1 dimensions. The Coulomb branches of these theories are
HyperKähler manifolds. The formula draws upon a lattice of monopole charges, defined by the
linked system of gauge and flavour nodes in a quiver diagram.
Each gauge node carries adjoint valued fields from the SUSY vector multiplet and the links
between nodes correspond to complex bi-fundamental scalars within SUSY hypermultiplets. The
monopole formula generates the Coulomb branch of the quiver by projecting charge configura-
tions from the monopole lattice into the root space lattice of G, according to the monopole flux
over each gauge node, under a grading determined by the conformal dimension of each overall
monopole flux q.
The conformal dimension (equivalent to R-charge or the highest weight of the SU(2)Rglobal
symmetry) of a monopole flux is given by applying the following general schema [10]to the
quiver diagram:
(q)=1
2
r

i=1
ρi∈Ri
|ρi(q)|
 
contributi on of N =4
hyper multiplets
−
α∈+
|α(q)|
 
contributi on of N =4
vector multiplets
.(3.9)
The positive R-charge contribution in the first term comes from the bi-fundamental matter fields
that link adjacent nodes in the quiver diagram. The second term captures a negative R-charge
contribution from the vector multiplets, which arises due to symmetry breaking, whenever the
monopole flux qover a gauge node contains a number of different charges.
The calculation of Hilbert series for Coulomb branches of Atype quivers draws on the unitary
monopole formula, which follows from specialising (3.9)to unitary gauge groups. Each U(Ni)
gauge node carries a monopole flux qi≡(qi,1, ..., qi,Ni)comprising one or more monopole
charges qi,m. The fluxes are assigned the collective coordinate q≡(q1, ..., qr). Each flavour
node carries Nfiflavours of zero charge.11
11 Flavour nodes may also carry non-zero charges, although these are not required by the Slodowy slice (or nilpotent
orbit) constructions.

320 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
With these specialisations, the conformal dimension (q) associated with a flux qyields the
formula:
(q)=1
2
r

j>i1
m,n qi,mAij −qj,nAji
 
gauge - gauge hypers
+1
2
i
m
Nfiqi,m
 
gauge - flavour hypers
−
r

i=1
m>n qi,m −qi,n
 
gauge vplets
,
(3.10)
where (i) the summations are taken over all the monopole charges within the flux qand (ii) the
linking pattern between nodes is defined by the Aij off-diagonal ArCartan matrix terms, which
are only non-zero for linked nodes.12
With conformal dimension defined as above, the unitary monopole formula for a Coulomb
branch HS is given by the schema [32]:
gCoulomb
HS z, t 2≡
q
PU(N)
qt2zqt2(q),(3.11)
where:
1. The limits of summation for the monopole charges are ∞ qi,1...q
i,m ...q
i,Ni−∞
for i=1, ...r.
2. The monopole flux over the gauge nodes is counted by the fugacity z≡(z1, ..., zr), where
the ziare fugacities for the simple roots of Ar.
3. The monomial zqcombines the monopole fluxes qiinto total charges for each ziand is
expanded as zq≡
r

i=1
z
Ni

m=1
qi,m
i.
4. The term PU(N)
qencodes the degrees di,j of the Casimirs of the residual U(N) symmetries
that remain at the gauge nodes under a monopole flux q:
PU(N)
q(t2)≡
i,j
1
1−t2di,j (q) =
r

i=1
Ni

j=1
λij (qi)

k=1
1
1−t2k.(3.12)
Recalling that a U(N) group has Casimirs of degrees 1 through N, the residual symme-
tries can be determined as in [32].13 Alternatively, the residual symmetries for a flux qican
be fixed from the sub-group of U(Ni)identified by the Dynkin diagram formed by those
12 For theories with simply laced quivers of ADE type, Aij =0or −1, for i= j.
13 We construct a partition of Nifor each node, which counts how many of the charges qi,m are equal, such that
λ(qi) =(λi,1, ..., λi,Ni), where Ni

m=1
λi,m =Niand λi,m λi,m+10. The non-zero terms λi,j in the partition give
the ranks of the residual U(N)symmetries associated with each node, so that it is a straightforward matter to compound
the terms in the degrees of Casimirs. For example, if qi,m =qi,n for all m, n, then {di,1, ...d
i,Ni} ={1, ..., Ni} and if
qi,m = qi,n for all m, n, then {di,1, ...d
i,Ni} ={1, ..., 1}.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 321
monopole charges that equal their successors {qi,m :qi,m =qi,m+1}, (or equivalently, corre-
spond to zero Dynkin labels).
The exact calculation of a Coulomb branch HS can be carried out by evaluating (3.11)as a geo-
metric series over each sub-lattice of monopole charges q, for which both conformal dimension
(q) and the symmetry factors PU(N)
qare linear (rather than piecewise or step) functions, and
then summing the many resulting polynomial quotients. These sub-lattices of monopole charges
form a hypersurface and care needs to be taken to avoid duplications at edges and intersections.
3.4. Hilbert series
The Hilbert series of the Slodowy slices of algebras A1to A4, calculated as above, are sum-
marised in Table 2, and those of A5are summarised in Table 3. Both the Higgs and Coulomb
branch calculations lead to identical refined Hilbert series, up to choice of CSA coordinates or
fugacities.
The Hilbert series are presented in terms of their generators, or PL[HS], using character
notation [n1, ..., nr]to label Arirreps. Symmetrisation of these generators using the PE recov-
ers the refined Hilbert series. The underlying adjoint maps (2.10) can readily be recovered from
the generators by inverting (2.11). The HS can be unrefined by replacing representations of the
global symmetry groups by their dimensions.
Several observations can be made about the Hilbert series.
1. As expected, (i) the Slodowy slice to the trivial nilpotent orbit SN,(1N)has the same Hilbert
series as the nilpotent cone, (ii) the slice to the sub-regular orbit SN,(N−1,1)has the Hilbert
series of a Kleinian singularity of type ˆ
AN−1, and (iii) the slice to the maximal nilpotent
orbit SN,(N) is trivial.
2. As expected, the Slodowy slices SN,ρ are all complete intersections.
3. The global symmetry groups of the Slodowy slice generators include mixed SU and unitary
groups, and descend in rank as the dimension of the Slodowy slice reduces. Sometimes
different Slodowy slices share the same symmetry group, with inequivalent embeddings of
Finto G.
4. Complex representations always appear combined with their conjugates to give real repre-
sentations.
5. The adjoint maps often contain singlet generators at even powers of tup to the (twice the)
degree of the highest Casimir of g; these generators may be cancelled by one or more Casimir
relations.
Many of these observations have counterparts amongst the Slodowy slices of BCD series, al-
though these also raise several new intricacies, as will be seen in section 4.
3.5. Matrix generators for unitary quivers
A Hilbert series over the class functions of a Classical group can be viewed in terms of matrix
generators (or operators), and this perspective makes it possible to identify the generators of a
Slodowy slice directly from the partition data or its Higgs branch quiver.

322 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 2
Hilbert series for Slodowy slices of A1, A2, A3and A4.
Nilpotent orbit Dimension |SN,ρ |Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[0]2A1[2]t2−t4(1−t4)
(1−t2)3
[2]0∅01
[00]6A2[1,1]t2−t4−t6(1−t4)(1−t6)
(1−t2)8
[11]2D1t2+(1)q1/q2t3−t6(1−t6)
(1−t2)(1−t3)2
[22]0∅01
[000]12 A3[1,0,1]t2−t4−t6−t8(1−t4)(1−t6)(1−t8)
(1−t2)15
[101]6A1⊗D1t2+[2]t2+[1](1)q1/q2t3−t6−t8(1−t6)(1−t8)
(1−t2)4(1−t3)4
[020]4A1[2]t2+[2]t4−t6−t8(1−t6)(1−t8)
(1−t2)3(1−t4)3
[202]2D1t2+(1)q1/q3t4−t8(1−t8)
(1−t2)(1−t4)2
[222]0∅01
[0000]20 A4[1,0,0,1]t2−t4−t6−t8−t10 (1−t4)(1−t6)(1−t8)(1−t10)
(1−t2)24
[1001]12 A2⊗U(1)t
2+[1,1]t2+[1,0]q1/q2t3+[0,1]q2/q1t3−t6−t8−t10 (1−t6)(1−t8)(1−t10)
(1−t2)9(1−t3)6
[0110]8A1⊗D1t2+[2]t2+[2]t4+[1](1)q1/q2t3−t6−t8−t10 (1−t6)(1−t8)(1−t10)
(1−t2)4(1−t3)4(1−t4)3
[2002]6A1⊗D1t2+[2]t2+[1](1)q1/q3t4−t8−t10 (1−t8)(1−t10 )
(1−t2)4(1−t4)4
[1111]4D1t2+(1)t3+t4+(1)q2/q3t5−t8−t10 (1−t8)(1−t10 )
(1−t2)(1−t3)2(1−t4)(1−t5)2
[2112]2D1t2+(1)q1/q4t5−t10 (1−t10)
(1−t2)(1−t5)2
[2222]0∅01
N.B. (n)qdenotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 323
Tab l e 3
Hilbert series for Slodowy slices of A5.
Nilpotent orbit Dimension |SN,ρ |Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[00000]30 A5[1,0,0,0,1]t2−t4−t6−t8−t10 −t12 (1−t4)(1−t6)(1−t8)(1−t10)(1−t12 )
(1−t2)35
[10001]20 A3⊗U(1)
t2+[1,0,1]t2+([1,0,0]q1/q2+[0,0,1]q2/q1)t 3
−t6−t8−t10 −t12
(1−t6)(1−t8)(1−t10)(1−t12 )
(1−t2)16(1−t3)8
[01010]14 A1⊗A1⊗D1
t2+[2][0]t2+[0][2]t2+[1][1](1)q1/q2t3
+[2][0]t4−t6−t8−t10 −t12
(1−t6)(1−t8)(1−t10)(1−t12 )
(1−t2)7(1−t3)8(1−t4)3
[00200]12 A2[1,1]t2+[1,1]t4−t6−t8−t10 −t12 (1−t6)(1−t8)(1−t10)(1−t12)
(1−t2)8(1−t4)8
[20002]12 A2⊗U(1)t2+[1,1]t2+[1,0]q1/q3t4+[0,1]q3/q1t4
−t8−t10 −t12
(1−t8)(1−t10)(1−t12 )
(1−t2)9(1−t4)6
[11011]8U(1)⊗U(1)2t2+((1)q1/q2+(1)q2/q3))t3+t4+(1)q1/q3t4
+(1)q2/q3t5−t8−t10 −t12
(1−t8)(1−t10)(1−t12 )
(1−t2)2(1−t3)4(1−t4)3(1−t5)2
[02020]6A1[2]t2+[2]t4+[2]t6−t8−t10 −t12 (1−t8)(1−t10 )(1−t12)
(1−t2)3(1−t4)3(1−t6)3
[21012]6A1⊗D1t2+[2]t2+[1](1)q1/q4t5−t10 −t12 (1−t10 )(1−t12)
(1−t2)4(1−t5)4
[20202]4D1t2+t4+(1)q2/q4t4+(1)q2/q4t6−t10 −t12 (1−t10)(1−t12 )
(1−t2)(1−t4)3(1−t6)2
[22022]2D1t2+(1)q1/q5t6−t12 1−t12
(1−t2)(1−t6)2
[22222]0∅01
N.B. (n)qdenotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).

324 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
3.5.1. Fundamental decomposition
From (3.3), it follows that the character of the fundamental representation of Gdecomposes
into fundamental representations of a unitary product group:
ρ:χG
fund. →
[n]
[n]ρχ
SUNfn+1
[fund.]qn+1,(3.13)
where [n]ρare irreps of the SU(2)associated with the nilpotent orbit embedding ρ, and the U(1)
charges qion the flavour nodes satisfy the overall gauge condition
k

i=1
iNfiqi=1.14 Once this
decomposition has been identified, the mapping of the adjoint of Ginto matrix generators fol-
lows, by taking the product of the fundamental and antifundamental characters, and eliminating
a singlet. This can be checked against the adjoint partition ρ:χG
adjoint.
3.5.2. Generators from quiver paths
Alternatively the operators can be read from a quiver of type BA(Nf(ρ)), following the pre-
scription:
1. Draw the chiral multiplets explicitly as arrows in the quiver:
N1
↑↓
Nf1

N2
↑↓
Nf2

N3
↑↓
Nf3
···
Nk
↑↓
Nfk
(3.14)
2. Every path in the quiver that starts and ends on a flavor node corresponds to an operator in
the chiral ring of the Higgs branch.
3. There is a one to one correspondence between paths that appear as generators in the PL[HS]
of the Higgs branch and the paths of the type Pij (a), defined as below.
4. The operator Pij (a) transforms under the fundamental representation of U(Nfi)and the
antifundamental representation of U(Nfj)and sits on an irrep of SU(2)Rwith spin s=A/2,
where Ais the number of arrows in the path that defines Pij (a). This means that it appears
in the plethystic logarithm of the refined Hilbert series as the character of the corresponding
representation multiplied by the fugacity tA.
5. Therefore, there is a one to one correspondence between operators Pij (a) and irreducible
representations in the decomposition of the adjoint representation of Akin (2.10).
Definition Pij(a). Let Pij (a) be an operator Pij (a) with i, j∈{1, 2, ..., k}and a∈{1, 2, ...,
min(i, j)}. Pij (1)is defined as the operator formed by products of operators represented by
arrows in the shortest possible path that starts at node Nfiand ends at node Nfj(note that i
and jcould be equal). Pij (2)represents a path that differs from Pij (1)only in that it has been
extended to incorporate the arrows between the gauge nodes Nmin(i,j) and Nmin(i,j )−1. Pij (3)
differs from Pij (2)in that it also includes arrows between the gauge nodes Nmin(i,j)−1and
Nmin(i,j )−2. In this way Pij (a) is defined recursively as an extension of Pij (a −1).
14 This corresponds to viewing the fields in a centre of mass frame.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 325
Tab l e 4
Generators for Slodowy slice to A[101].
Pij (a) Quiver path Generator
P1,1(1)2
↑↓
2
2
1
[2]t2
P2,2(1)2
2
↑↓
1
1
t2
P1,2(1)2
↑
2
→2
↓
1
1
[1]qt3
P2,1(1)2
↓
2
←2
↑
1
1
[1]q−1t3
P2,2(2)2

2
↑↓
1
1
t4
Example 1 Let us start with the balanced A3quiver based on the fundamental partition ρ=
(2, 12), whose Higgs branch is the Slodowy slice SN,(2,12)to the nilpotent orbit A[101]. The
quiver is:
BA(Nf(2,12)) =2
|
2
−2
|
1
−1
.(3.15)
From Table 2, the Hilbert series is:
gHiggs[BA(Nf(2,12))]
HS =PE[t2+[2]t2+[1](q +1/q)t 3−t6−t8].(3.16)
To obtain this using the prescription in section 3.5.1, we first identify the fugacity map for the
group decomposition using (3.13):
SU(4)→SU(2)ρ⊗SU(2)⊗U(1),
[1,0,0]→[1]ρq1/2+[1]q−1/2,
[0,0,1]→[1]ρq−1/2+[1]q1/2,
[1,0,1]→([2]+1)[0]ρ+[1](q +1/q )[1]ρ+[2]ρ.
(3.17)
Next the irreps [n]ρof SU(2)ρare mapped to the fugacity tn+2, giving the generators:
[1,0,1]→[2]t2+t2+[1](q +1/q)t3+t4.(3.18)
Subtracting the relations −t4−t6−t8, corresponding to Casimirs of A3, we obtain:
PL[gHiggs[BA(Nf(2,12))]
HS ]=[2]t2+t2+[1](q +1/q)t 3−t6−t8.(3.19)
The generators in (3.19) can be understood as operators from paths in the quiver (3.15) (Table 4).
The irrep of each generator corresponds with the flavor nodes where the path starts and ends.
The U(1)fugacity q≡q1/q2. The exponent of the fugacity tcorresponds to the length of the
path.

326 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 5
Generators for Slodowy slice to A[1111].
Pij (a) Quiver path Generator
P2,2(1)1
2
↑↓
1
2
1
t2
P3,3(1)1
2
2
↑↓
1
1
t2
P2,3(1)1
2
↑
1
→2
↓
1
1
q2/q3t3
P3,2(1)1
2
↓
1
←2
↑
1
1
q3/q2t3
P2,2(2)1

2
↑↓
1
2
1
t4
P3,3(2)1
2

2
↑↓
1
1
t4
P2,3(2)1

2
↑
1
→2
↓
1
1
q2/q3t5
P3,2(2)1

2
↓
1
←2
↑
1
1
q3/q2t5
P3,3(3)1

2

2
↑↓
1
1
t6
Example 2 Now consider the balanced quiver based on the A4partition (3, 2), whose Higgs
branch is the Slodowy slice SN,(3,2)to the nilpotent orbit A[1111]:
BA(Nf(3,2)) =1
−2
|
1
−2
|
1
−1
.(3.20)
The group decomposition is:
SU(5)→SU(2)ρ⊗S(U(1)⊗U(1)). (3.21)
The paths in the quiver can be used to predict the generators in Table 5. Subtracting relations
−
5
i=1t2i, corresponding to the special condition in (3.21), which eliminates one of the U(1)
symmetries, and the Casimirs of A4, and substituting qfor q2/q3gives the expected PL[HS]:
PL[gHiggs[BA(Nf(3,2))]
HS ]=t2+q+1
qt3+t4+q+1
qt5−t8−t10,(3.22)
in accordance with Table 2.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 327
3.5.3. Matrices and relations
In this section we offer a reinterpretation of the previous results for Slodowy slices SN,ρ as
sets of matrices that satisfy specific relations. The aim of this analysis is to build a bridge between
the algebraic definition of the nilpotent cone SN,(1N)=Nand that of the Kleinian singularity
SN,(N−1,1)=C2/ZN.
First, let us remember that the Kleinian singularity SN,(N−1,1)=C2/ZNcan be defined as
the set of points parametrized by three complex variables x, y, z∈C, subject to one relation:
xN=yz. (3.23)
Secondly, the nilpotent cone SN,(1N)=Ncan be defined as a set of complex variables ar-
ranged in a N×Nmatrix M∈CN×N, subject to the following relations:
tr(Mp)=0∀p=1,2,...,N. (3.24)
We want to show that a Slodowy slice SN,ρ can be viewed as an intermediate case between
these two descriptions. In order to do this we build examples of varieties described by sets of
complex matrices, choose relations among them and compute the (unrefined) Hilbert series of
their coordinate rings, utilizing the algebraic software Macaulay2 [33]. These Hilbert series can
be checked to be the same as those in Table 2.
The specific matrices that generate the coordinate rings are chosen according to the operators
Pij (a) found in the balanced quivers BA(Nf(ρ)). For example, let us study the balanced quiver
whose Higgs branch is the Slodowy slice SN,(2,13):
BA(Nf(2,13)) =3
|
3
−3
|
1
−2
−1
.(3.25)
One can assemble the generators Pij (a) into three different complex matrices M, Aand Bof
dimensions 3 ×3, 3 ×1 and 1 ×3 respectively. Let us show how this can be done explicitly.
There are five paths of the form Pij (a): P11(1), P22 (1), P22(2), P12 (1), P21(1). Out of these
five sets of operators P22(1)can be removed by the relation −t2that removes the centre of mass
and P22(2)by the first Casimir invariant relation −t4. This means that there is a remaining set of
generators transforming in the following irreps:
P11(1)→([1,1]+[0,0])t2,
P12(1)→([1,0]q)t3,
P21(1)→([0,1]1
q)t3.
(3.26)
One can now assemble these generators in three complex matrices that transform in the usual
way under the global symmetry U(3):
([1,1]+[0,0])t 2→M3×3,
([1,0]q)t3→A1×3,
([0,1]1
q)t3→B3×1.
(3.27)
The chiral ring is then parametrized by the set of all matrices {M, A, B}, subject to one relation
at order t6, another relation at order t8and a final relation at order t10. These relations are
invariant under the global U(3)symmetry. One can choose the following set of relations:

328 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 6
A1, A2and A3varieties, generated by complex matrices M, Aand Band their relations, with
Hilbert series calculated by Macaulay2 to match Slodowy slices SN,ρ . Note that SN,(12)has two
alternative descriptions, one as the nilpotent cone and one as the subregular Kleinian singularity.
Orbit Partition Dimension Generators; Degree Relations
[0] (12)2
M2×2;2tr(M) =0
tr(M2)=0
M1×1;2
A1×1;2
B1×1;2
tr(M2)=AB
[2] (2)0– –
[00] (13)6M3×3;2
tr(M) =0
tr(M2)=0
tr(M3)=0
[11] (2,1)2
M1×1;2
A1×1;3
B1×1;3
tr(M3)=AB
[22] (3)0– –
[000] (14)12 M4×4;2
tr(M) =0
tr(M2)=0
tr(M3)=0
tr(M4)=0
[101] (2,12)6
M2×2;2
A1×2;3
B2×1;3
tr(M3)=AB
tr(M4)=AMB
[020] (22)4M2×2;2
N2×2;4
tr(M) =0
tr(N) =0
tr(M3)=tr(MN)
tr(M4)=tr(N2)
[202] (3,1)2
M1×1;2
A1×1;4
B1×1;4
tr(M4)=AB
[222] (4)0– –
tr(M3)=AB, (3.28)
tr(M4)=AMB , (3.29)
tr(M5)=AM2B. (3.30)
Note that these look like corrections to the equations of the nilpotent cone (3.24). The Hilbert
series of the coordinate ring is then computed using Macaulay2 to be:
HS =(1−t6)(1−t8)(1−t10)
(1−t2)9(1−t3)6.(3.31)
This is the same Hilbert series as that of the variety SN,(2,13)computed in Table 2.
In Tables 6and 7we provide a set of algebraic varieties described by matrices such that their
HS have been computed to be identical to those of the corresponding Slodowy slices SN,ρ . Note
that we rewrite the Kleinian singularity in terms of 1 ×1 matrices, to clarify the connection with
the algebraic description of the other Slodowy slices.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 329
Tab l e 7
A4varieties, generated by complex matrices M, Aand Band their relations, with Hilbert series
calculated by Macaulay2 to match Slodowy slices SN,ρ .
Orbit Partition Dimension Generators; Degree Relations
[0000] (15)20 M5×5;2
tr(M) =0
tr(M2)=0
tr(M3)=0
tr(M4)=0
tr(M5)=0
[1001] (2,13)12
M3×3;2
A1×3;3
B3×1;3
tr(M3)=AB
tr(M4)=AMB
tr(M5)=AM2B
[0110] (22,1)8
M2×2;2
A1×2;3
B2×1;3
N2×2;4
tr(M3)=AB
tr(M4)+tr(N2)=AMB
tr(M5)=A(M2+N)B
tr(N) =0
[2002] (3,12)6
M2×2;2
A1×2;4
B2×1;4
tr(M4)=AB
tr(M5)=AMB
[1111] (3,2)4
M1×1;2
A1×1;3
B1×1;3
N1×1;4
C1×1;5
D1×1;5
tr(M4)+tr(N2)=AMB +AD
+BC
tr(M5)=CD
[2112] (4,1)2
M1×1;2
A1×1;5
B1×1;5
tr(M5)=AB
[2222] (5)0– –
4. BCD series quiver constructions
4.1. Quiver types
The constructions for the Slodowy slices of BCD algebras draw upon a different set of quiver
types to the A series.
1. Linear orthosymplectic quivers. These quivers LB/C/D(σ ) consist of a B, Cor Dse-
ries flavour node of vector irrep dimension N0connected to an alternating linear chain of
(S)O / U Sp(Ni)gauge nodes of non-increasing vector dimension. For a subset of these lin-
ear quivers, the decrements, σi=Ni−1−Ni, between nodes constitute an ordered partition
of N0, σ≡{σ1, ..., σk}, where σiσi+1and k
i=1σi=N0. More generally, however, the
σiform a sequence of non-negative integers, subject to k
i=1σi=N0, and to selection rules,
such that USp nodes of odd dimension do not arise.
2. Balanced orthosymplectic quivers. These quivers BB/C/D(Nf)consist of an alternating
linear chain of O/USp(Ni)nodes, with each gauge node connected to a flavour node
O/USp(Nfi), where Nfi0. The ranks of the gauge nodes are chosen such that, taking

330 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Fig. 4. BCD linear and balanced quiver types. In the linear quivers LBC, LCD and LDC , the ranks and fundamental
dimensions of the gauge nodes (blue/round) are in non-increasing order L to R and the quivers are in the form of
alternating B−Cor D−Cchains. In the balanced quivers, BB/C/D, the gauge nodes (blue/round) inherit their balance,
taking account of attached gauge and flavour nodes (red/square), from a quiver for the nilpotent cone. Nodes labelled Cr
represent the group USp(2r). Nodes labelled Brand Drrepresent SO/O(2r+1)and SO/O(2r) respectively. Nodes
labelled BC, BD or DC indicate a group of one of the two types, subject to the alternation rule and to balance.
account of any attached flavour nodes, each gauge node inherits its balance B(via (4.2))
from that of a canonical quiver (as defined below).
3. Dynkin diagram quivers. These quivers DG(Nf)consist of a chain of U(Ni)gauge nodes in
the form of a simply laced Dynkin diagram, with each gauge node connected to Nfiflavours,
where Nfi0. Nfmatches the Characteristic G[...]of a nilpotent orbit, and the ranks of
the gauge nodes are chosen such that each is balanced (similarly to the Aseries quivers in
section 3.1). These constructions are limited to certain Slodowy slices of ADE algebras, as
the Higgs branch construction is not available on non-simply laced Dynkin diagrams.
Recall, the nilpotent orbits of a BCDalgebra correspond to a subset of the partitions ρof N, once
these have been subjected to selection rules,15 and linear quivers LB/C/D(ρT)provide a complete
set of Higgs branch constructions. Also, balanced quivers BB/C/D(Nf)provide Coulomb branch
constructions, using the O/USp monopole formula, for the unrefined Hilbert series of certain
nilpotent orbits of orthogonal groups, as discussed in [29]. The linear and balanced quivers can
partially be related by 3dmirror symmetry, as discussed further in section 5. Many of these linear
quivers have “Higgs equivalent” quivers, LB/C/D(σ ), with a different choice of orthogonal gauge
node dimensions, but the same Higgs branches; these are generally described by sequences σ
rather than partitions ρT: a USp −O−USp subchain with the sub-partition (..., n, n, ...) has
the Higgs equivalent sequence (. . . , σi, σi+1, ...) =(. . . , n −1, n +1, ...), in which the vector
dimension of the central Onode is increased by 1 [1].
Returning to Slodowy slices, the roles of these quiver types are essentially reversed: balanced
quivers BB/C/D provide a complete set of Higgs branch refined Hilbert series constructions,
while linear quivers LB/C/D provide Coulomb branch constructions for the unrefined HS of
certain Slodowy slices. Within the general classes of linear and balanced quiver types, those that
are most relevant to the construction of Slodowy slices are shown in Fig. 4.
15 In a valid Bor Dpartition ρeach even integer appears at an even multiplicity; in a valid Cpartition each odd integer
appears at an even multiplicity [6].

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 331
Tab l e 8
Higgs branch quivers for nilpotent cones.
Group Canonical linear quiver for NGauge node balance
AU(N−1)
|
SU(N)
−U(N−2)
−...−U(2)
−U(1)
0forall
B
USp(2n)
|
SO(2n+1)
−O(2n−1)
−...−USp(2)
−O(1)
0forall
CO(2n)
|
USp(2n)
−USp(2n−2)
−...−USp(2)
−O(2)
USp :+2
O(even) :−2
D
USp(2n−2)
|
SO(2n)
−O(2n−2)
−...−USp(2)
−O(2)
USp :+2
O(even) :−2
We refer to the quivers of type LBC, LCD or LDC, which contain pure B−C, C−Dor
D−Cchains, as canonical linear quivers. On the Higgs branch, the flavour nodes (of either type
of quiver) identify the overall global symmetry, although it is necessary to distinguish within
the Band Dseries between Oand SO groups. However, it is not easy to identify the global
symmetry of the Coulomb branch of a O/USp quiver.
It is important to explain how the specific quivers used in the construction of the Hilbert
series for BCD Slodowy slices arise from the partition of the vector representation of Gunder
the homomorphism ρ.
The balanced quivers BB/C/D(Nf(ρ)) are found via a modification of the Aseries method
explained in section 3.1. Firstly, the SU(2)partition of a BCD series vector representation under
ρcan be used to define a vector Nf(ρ) of alternating O/USp flavour nodes, similarly to (3.3):
ρ[1,0,...
]B/C/D =NNfN,...,n
Nfn,...,1Nf1.(4.1)
Next, consider linear quivers, whose Higgs branches match the nilpotent cone N. In the case of
BCD groups, these quivers can be chosen, using Higgs equivalences, to be of canonical type.
The balances Bof their gauge nodes can be calculated by applying (3.1)to vectors Nfand N
defined from the vector/fundamental dimensions of the fields, as shown in Table 8.
These canonical quivers obey the generalisation of (3.2):
Nf=A·N+B,(4.2)
where B =(0, ..., 0)for the Aand Bseries canonical quivers, B =(−2, 2, ..., −2)for the C
series and B =(2, −2, ..., −2)for the Dseries canonical quivers.
By fixing B, the gauge node balance condition (4.2) can be extended from Nto general
Slodowy slices SN,ρ , permitting the calculation of each gauge node vector Nfrom its flavour
node vector Nf. In effect, the quivers BB/C/D(Nf(ρ)) descend from the canonical linear quivers
for N, through a series of transitions that leave the balance vector Binvariant. These canonically
balanced quivers provide Higgs branch constructions for BCD Slodowy slices. They are tabu-

332 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Fig. 5. Quivers for B1to B3Slodowy Slices. The Higgs quivers are of type BB/C/D Nf(ρ)and the Coulomb quivers
are of type LCD dBV (ρ)TCD . Gauge nodes of Bor Dtype are evaluated as Onodes on the Higgs branch and
SO nodes on the Coulomb branch.  =0 indicates a diagram for which the monopole formula contains zero conformal
dimension.
lated in Figs. 5–10, along with the partitions of the fundamental, the dimensions of the Slodowy
slices, and their residual symmetry groups.16
On the other hand, the identification of linear quivers LB/C/D(σ ) for Coulomb branch con-
structions of BCD series Slodowy slices poses a number of complications.
1. There is no bijection between partitions of Nand nilpotent orbits of O(N) or USp(N).
So the quiver LB/C/D(ρ) is valid only for partitions ρof special nilpotent orbits; in the
other cases LB/C/D(ρ) (unlike LB/C/D(ρT)) would contain USp(N)vectors of odd dimen-
sion N.
2. In the case of Coulomb branch constructions, GNO duality [34]i
s relevant. This indicates
that, since the non-simply laced Band Cgroups are GNO dual to each other, partitions of B
16 Note that other quivers whose Higgs branches match Ncould be taken to define B; each leads to a different family
of quivers, whose Higgs branches match the Slodowy slice Hilbert series. The canonical choice, however, best illustrates
the Higgs–Coulomb quiver dualities.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 333
Fig. 6. Quivers for B4Slodowy slices. The Higgs quivers are of type BB/C/D Nf(ρ)and the Coulomb quivers are of
type LCD dBV(ρ)TCD. Gauge nodes of Bor Dtype are evaluated as Onodes on the Higgs branch and SO nodes
on the Coulomb branch.  =0 indicates a diagram for which the monopole formula contains zero conformal dimension.
type will be necessary to produce quivers whose Coulomb branches generate Slodowy slices
of Calgebras, and vice versa.
3. A quiver LB/C/D(ρT)may have several Higgs equivalent quivers LB/C/D(σ ), in which σ
is a sequence of non-negative integers, rather than an ordered partition. Such quivers have
the same Higgs branch refined HS, but generally have different ranks of gauge groups, and
therefore different Coulomb branch dimensions.
4. Any candidate quiver for a Slodowy slice must have the correct Hilbert series dimension.
Since the Coulomb branch monopole construction leads to a HS with complex dimension
equal to twice the sum of the gauge group ranks in the quiver, this limits the candidates
amongst Higgs equivalent quivers.
5. The Coulomb branches of quivers with Ogauge groups differ from those with SO gauge
groups; a correct choice of orthogonal gauge groups needs to be made [29].
6. When the orthosymplectic Coulomb branch monopole formula is applied to a quiver, the
conformal dimension of all monopole operators must be positive for the Hilbert series to be
well formed.

334 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Fig. 7. Quivers for C1to C3Slodowy slices. The Higgs quivers are of type BB/C/D Nf(ρ)and the Coulomb quivers
are of type LBC dBV(ρ)TBC. Gauge nodes of Bor Dtype are evaluate d as Onodes on the Higgs branch and SO
nodes on the Coulomb branch.  =0 indicates a diagram for which the monopole formula contains zero conformal
dimension.
Leaving the discussion of conformal dimension to section 4.3, it is remarkable that a procedure
exists for a partial resolution of these complexities, and indeed forms the basis for Coulomb
branch constructions for the unrefined Hilbert series of nilpotent orbits of special orthogonal
groups in [29]. The method draws on the Barbasch–V
ogan map17 dBV (ρ) [20], which provides
a bijection between the partitions of real vector representations associated with Bseries special
nilpotent orbits and those of pseudo-real vector representations associated with Cseries special
nilpotent orbits. By making use of Higgs equivalences, to select canonical linear quivers of type
LBC, LCD or LDC , which can be done for all special nilpotent orbits, the dBV (ρ) map can
be extended to identify candidates for Coulomb branch constructions of Hilbert series of BCD
Slodowy slices, in each case starting from a homomorphism ρ.
17 A particularly clear description of this map is given in equation (5) of [35]. A summary of dual maps between
partitions and their appearance in the literature can be found in [15, sec. 4].

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 335
Fig. 8. Quivers for C4Slodowy slices. The Higgs quivers are of type BB/C/D Nf(ρ)and the Coulomb quivers are of
type LBC dBV(ρ)TBC . Gauge nodes of Bor Dtype are evaluated as Onodes on the Higgs branch and SO nodes
on the Coulomb branch.  =0 indicates a diagram for which the monopole formula contains zero conformal dimension.
The specific transformations from the partitions ρTto the sequences σare summarised in
Table 9. Within these; ρTindicates the transpose of a partition; ρN→N±1indicates increment-
ing (decrementing) the first (last) term of a partition by 1; ρB, ρC, or ρDindicates collapsing
a partition to a lower partition that is a valid B, C, or Dpartition [6]; |BC or |CD indicates
shifting Dor Bnodes in a linear quiver to a ‘Higgs equivalent’ quiver that consists purely
of B−Cor of C−Dpairs of nodes. The transformations can be written more concisely as
σ=dBV (ρ)Tcanoni cal . The resulting linear quivers, LCD (σ), LBC (σ)and LDC (σ), whose
Coulomb branches are candidates for Slodowy slices of BCD groups up to rank 4, are included
in Figs. 5through 10.
The quivers LDC((1N)) and BD(Nf(1N)) for the Higgs and Coulomb branch constructions
of the Slodowy slice to the trivial nilpotent orbit are the same. These tables also include iden-

336 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Fig. 9. Quivers for D2to D3Slodowy slices. The Higgs balanced quivers are of type BB/C/D Nf(ρ)and the Coulomb
quivers are of type LDC dBV (ρ)TDC . The Dynkin type quivers DD(Nf)are identified by Aseries isomorphisms.
Gauge nodes of Bor Dtype are evaluated as Onodes on the Higgs branch and SO nodes on the Coulomb branch.  =0
indicates a diagram for which the monopole formula contains zero conformal dimension.
tified quivers of type DG(Nf([dBV (ρ)])), whose Higgs branch Hilbert series match those of
BB/C/D(Nf(ρ)).
Type IIB string theory brane systems Note that all the resulting quivers, presented in Figs. 5
through 10 represent 3dN=4 gauge theories that admit an embedding in Type IIB string the-
ory. They correspond to the effective gauge theory living on the world-volume of D3-branes
suspended along one spatial direction between NS5-branes and D5-branes. This is achieved by
taking the construction of [11] and introducing O3-planes [12]. This type of system was further
explored in [10] where the Coulomb branches and Higgs branches were described in terms of
nilpotent orbits and Slodowy slices, and the label Tρ
σ(G) was introduced to denote the SCFT at
the superconformal fixed point. These brane systems and 3dquivers were also studied in [15],
finding the physical realization of transverse slices between closures of nilpotent orbits that are
adjacent in their corresponding Hasse diagrams. This phenomenon has been named the Kraft–
Procesi transition.
4.2. Higgs branch constructions
In the case of the balanced unitary quivers DD(Nf), based on Dseries Dynkin diagrams,
the calculation of Higgs branch Hilbert series proceeds similarly to the Aalgebras. This leads

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 337
Fig. 10. Quivers for D4Slodowy slices. The Higgs balanced quivers are of type BB/C/D Nf(ρ)and the Coulomb
quivers are of type LDC dBV (ρ)TDC . The Dynkin quivers of type DD(Nf([dBV (ρ)])), are those that have Higgs
branches matching the balanced quivers. Gauge nodes of Bor Dtype are evaluate d as Onodes on the Higgs branch and
SO nodes on the Coulomb branch.  =0 indicates a diagram for which the monopole formula contains zero conformal
dimension.
to a Higgs branch formula that is comparable to (3.7), modified to include the connection of
three pairs of bifundamental fields to the central node. The dimension formula (3.8) remains
unchanged.
In the case of orthosymplectic quivers of type BB/C/D(Nf), modifications to the Aseries
Higgs branch formula are required. The O/USp alternating chains are taken to comprise bifun-
damental (half) hypermultiplet fields that transform in vector representations [1, 0, ..., 0]B/D ⊗
[1, 0, ..., 0]C. Also, it is necessary to average the integrations over the disconnected SO and
O−components of the Ogauge groups; this requires precise choices both of the character for
the vector representation of O−and of the HKQ associated with the integration over O−, as
explained in [1].18
18 The effect of non-connected Ogroup components is also discussed in [36].

338 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 9
Coulomb branch quiver candidates for Slodowy slices.
Group ¯
OρTransformation SN,ρ
AHiggs
LA(ρT)ρ=ρTTCoulomb[LA(ρ)]
BHiggs
LB(ρT)σ≡ρTN→N−1CTCD
Coulomb LCD(σ )
CHiggs
LC(ρT)σ≡ρTN→N+1BTBC
Coulomb LBC(σ )
DHiggs
LD(ρT)σ=ρTDTDC
Coulomb LDC(σ )
In other respects, the calculation of the Higgs branch of a balanced BCD quiver follows a
similar Wey l integration to the Aseries. The general Higgs branch formula for BCD series
Slodowy slices is:
gHiggs[BB/C/D(Nf(ρ))]
HS
=1
2#O
O±
G1(N1)⊗...Gk(Nk)
dμ
k

n=1
PE[vect or]Gn(Nn)⊗[vect or]Gfn(Nfn),t
HKQ[Gn(Nn), t]
×
k−1

n=1
PE[vect or]Gn(Nn)⊗[vect or]Gn+1(Nn+1),t.
(4.3)
In (4.3), Gnalternates between O(N) and USp(N), dμ is the Haar measure for the G1(N1)⊗
...G
k(Nk)product group, HKQ
[Gn(Nn), t ]is the HyperKähler quotient for a gauge node, and
the summation indicates that the calculation is averaged over the non-connected SO and O−
components of Ogauge groups [1].
The character [vector ]O(2r)−=[vect or]O(2)−⊕[vector ]USp(2r−2), where [vector ]O(2)−
is (the trace of) a diagonal matrix with eigenvalues {1, −1}. The HKQ is given by
HKQ
[Gn(Nn), t ]=PE[adjoint]Gn,t2, where for the O(2r)−component of an O(2r)
group, [adjoint]O(2r)−≡2[vector]O(2r)−.19
The structure of the Higgs branch formula can be used to identify the dimensions of the
Hilbert series. In essence, each bifundamental field contributes HS generators according to its
dimensions (being the product of the dimensions of the Oand USp vectors), and each gauge
group offsets the generators by HS relations numbering twice the dimension of the gauge group
(once for the Wey l integration and once for the HKQ). This gives a rule for the dimensions of a
Slodowy slice calculated from a balanced BB/C/D(Nf(ρ)) quiver:
gHiggs[B(Nf(ρ))]
HS =Nf(ρ) ·N(ρ) −1
2N(ρ) ·A·N(ρ ) +N(ρ) ·K,(4.4)
where Kn= +1ifGn=B/D
−1ifGn=Cand (4.2)is used to calculate N(ρ).
19 For further detail, see [1].

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 339
Tab l e 10
Monopole and Dynkin label lattices.
Group Monopole lattice Basis transformations Dynkin labels [n1,...,n
r]
U(r) ∞>q
1...q
i...q
r>−∞ qi≡r
j=inj ∞>n
i<r 0
∞>n
r>−∞
Ar∞>q
1...q
i...q
r0qi≡r
j=inj∞>n
i0
Br∞>o
1...o
i...o
r0oi≡r−1
j=inj+nr/2 ∞>n
i0
nr=2k
Cr∞>s
1...s
i...s
r0si≡r
j=inj∞>n
i0
Dr∞>o
1...o
i...|or|0!oi<r ≡r−2
j=inj+nr−1+nr/2
or=−nr−1+nr/2 ∞>n
i0
nr+1+nr=2k
4.3. Coulomb branch constructions
While the O/USp version of the monopole formula (4.5) derives from (3.9)by following
similar general principles to the unitary monopole formula (3.11), there are several aspects and
subtleties that require discussion:
gCoulomb
HS (f,t2)≡
o,s
PG
o/s (t2)fo/s t2(o,s) .(4.5)
1. Monopole lattice. The lattice of monopole charges depends on the symmetry group. For
SU and USp groups, points in the monopole charge lattice correspond to sets of ordered
integers and are in bijective correspondence with highest weight Dynkin labels. However, the
monopole charge lattices of orthogonal groups only span the vector sub-lattices and exclude
weight space states whose spinor Dynkin labels sum to an odd number. Labelling monopole
charges as q ≡(q1, ..., qr)for unitary nodes, s ≡(s1, ..., sr)for symplectic nodes and o ≡
(o1, ..., or)for orthogonal nodes, the relationships between monopole and integer weight
space lattices can be summarised as in Table 10.
2. Characters. The definition of conformal dimension draws on the characters of the bifunda-
mental scalar fields in the hyper multiplets and of the adjoint scalars in the vector multiplets:
the weights of the fugacities in the characters become coefficients of the monopole charges
qin (q). These characters take a relatively simple form in the monopole lattice basis, com-
pared with the weight space integer (Dynkin label) basis, as shown in Tables 11 and 12.
CSA fugacities are taken as {x1, ..., xr}in the weight space integer (Dynkin label) basis, or
{y1, ..., yr}in the monopole basis.
3. Conformal dimension. The contributions to conformal dimension of the O/USp bifunda-
mental fields linking gauge or flavour nodes, and of the O/USp gauge nodes, follow from
(3.9)in a similar manner to the unitary Case (3.10), starting from the relevant characters: the
coefficients {0, ±1, ±2}of the monopole charges {o, s}in the conformal dimension formula
match the weights (exponents) of the yfugacities in the characters of the respective bifunda-
mental or adjoint representations in the monopole basis. Tables 13 and 14 show the resulting
contributions from the various types of gauge node and bifundamental field.
4. Symmetry factors. The residual symmetries for a flux (whether o, s, or q) over a gauge node
can be fixed from the sub-group of the O/USp/U gauge group identified by the Dynkin
diagram formed by those monopole charges that equal their successors (or, equivalently,

340 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 11
Vector/fundamental characters.
Group Monopole basis vector/Fundamental Weight space basis vector/Fundamental
U(r) r
i=1yix1+r
i=2xi/xi−1
Arr
i=1yi+r
i=11/yix1+r
i=2xi/xi−1+1/xr
Br1+r
i=1yi+r
i=11/yi
1+1/x1+x1
+r−1
i=2xi−1/xi+xi/xi−1+xr−1/x2
r+x2
r/xr−1
Crr
i=1yi+r
i=11/yi1/x1+x1+r
i=2xi−1/xi+xi/xi−1
Drr
i=1yi+r
i=11/yi1/x1+x1+r−2
i=2xi−1/xi+xi/xi−1
+xr−2/(xr−1xr)+xr−1/xr+xr/xr−1+(xr−1xr)/xr−2
Tab l e 12
Adjoint characters.
Group Monopole basis
U(r) r +i=jyi/yj
Arr+r
i=11/yir
j=11/yj+r
i=1yir
j=1yi+i=jyi/yj
Brr+r
i=1(yi+1/yi)+i<j yiyj+yi/yj+yj/yi+1/(yiyj)
Crr+r
i=1y2
i+1/y2
i+i<j yiyj+yi/yj+yj/yi+1/(yiyj)
Drr+i<j yiyj+yi/yj+yj/yi+1/(yiyj)
Tab l e 13
Gauge node conformal dimensions.
Gauge group (Node)
U(r) −1i<jr|qi−qj|
Br−r
i=1|oi|−1i<jr|oi±oj|
Cr−2r
i=1|si|−1i<jr|si±sj|
Dr−1i<jr|oi±oj|
Tab l e 14
Bifundamental conformal dimensions.
Gauge groups (Bifundamental)
U(r1)−U(r2)1
2
r1

i=1
r2

j=1
|q1,i −q2,j |
Br1−Cr2
1
2
r2

j=1
|sj|+ 1
2
r1

i=1
r2

j=1
|oi±sj|
Dr1−Cr2
1
2
r1

i=1
r2

j=1
|oi±sj|

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 341
Tab l e 15
Quiver chain unit conformal dimensions.
Gauge group chain r(1,0...0)
Cr1−Dr−Cr2r1+r2−2r+2
Cr1−Br−Cr2r1+r2−2r+1
Br1−Cr−Br2r1+r2−2r+1
Br1−Cr−Dr2r1+r2−2r+1/2
Dr1−Cr−Dr2r1+r2−2r
correspond to zero Dynkin labels). Note that the symmetry factors may belong to a sub-group
from a different series to the gauge node.
5. Ovs SO gauge nodes. Both the characters of vector irreps and symmetry factors depend
on whether a Dseries gauge node is taken as SO or as O. As noted in [28], the Casimirs
of an O(2n) symmetry group are the same as those of SO(2n +1), due to the absence of
a Pfaffian in O(2n) (since the determinant of representation matrices can be of either sign).
The Coulomb branch calculations for Slodowy slices herein are based entirely on SO gauge
nodes. This is a choice consistent with the results in [29]. When these results are translated
to the brane configurations, the action of the Lusztig’s Canonical Quotient ¯
A(O)related to
each quiver can be seen in terms of collapse transitions [15] performed in the branes. Each
time a collapse transition moves two half D5-branes away from each other all magnetic
lattices of the orthogonal gauge nodes in between are acted upon by a diagonal Z2action.
The brane configurations [10,12,15]for linear quivers LB/C/D(σ ) do not present this effect,
and therefore all gauge nodes are SO.
6. Fugacities. In the unitary monopole formula, zin (3.11)can be treated as a fugacity for the
simple roots of the group for which the quiver is a balanced Dynkin diagram. As discussed
in [27], such a treatment cannot be extended to the O/USp monopole formula due to the
non-unitary gauge groups involved. Thus, while it can be helpful to include fugacities f ≡
(f1,...,f
r)during the calculation of Coulomb branches, their interpretation is unclear. Such
issues do not affect the validity of the unrefined Hilbert series ultimately obtained by setting
∀fi:fi→1.
In order for a Coulomb branch Hilbert series not to lead to divergences when the fugacities
fare set to unity, it is necessary that no sub-lattice of the monopole lattice (other than the ori-
gin) should have a conformal dimension of zero (to ensure that the fugacities fonly appear as
generators when coupled with tk, where k>0). A necessary (albeit not always sufficient) condi-
tion on O/USp quivers can be formulated by examining unit shifts away from the origin of the
monopole lattice. This is similar to the “good or ugly, but not bad” balance condition on unitary
quivers [10].
In Table 15 we examine the unit conformal dimensions that result, based on Tables 13 and 14,
from setting a single monopole charge (o1, or s1) on a central gauge node in a chain of three
nodes to unity, depending on the ranks of the nodes involved. We can use this table to check that
no gauge node in a quiver is necessarily “bad”. For example, the central gauge node in the chain
D2−C1−D1is assigned a unit conformal dimension of 1 and is a “good” node. Quivers with
zero conformal dimension are identified as such in Figs. 5through 10. Their Hilbert series clearly
do not match those of the Higgs branch constructions for Slodowy slices, and are not tabulated
here.

342 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Providing (i) a nilpotent orbit Oρis special (so that the Barbasch–Vogan map can be uniquely
applied), and (ii) that the quiver LBC/CD/DC(σ (ρ )) does not suffer from zero conformal di-
mension, the O/USp monopole formula (4.5) can be used to calculate unrefined Hilbert series
for Slodowy slices; these match those calculated on the Higgs branch of BB/C/D(Nf(ρ)) using
(3.7).
4.4. Hilbert series
The Hilbert series of the Slodowy slices of algebras B1to B4, C1to C4and D2to D4, cal-
culated as above, are summarised in Tables 16, 17, 18, 19 and 20. The refined Hilbert series are
based on the Higgs branches of the balanced quivers BB/C/D(Nf(ρ)).
Whenever the flavour symmetry groups are from the Bor the Dseries, a choice has to be
made between the characters of SO(N) or O(N)−. In the tables, B/D flavour nodes have been
taken as SO type, with the exception of B0where the O(1)fugacity ki=±1 has been used (with
indices dropped where no ambiguity arises).20
The Hilbert series are presented in terms of their generators, or PL[HS], using character
notation [n1, ..., nr]Gto label irreps. Symmetrisation of these generators using the PE recovers
the refined Hilbert series. The underlying adjoint maps (2.10) can readily be recovered from
the generators by inverting (2.11). The HS can be unrefined by replacing irreps of the global
symmetry groups by their dimensions.
Many observations can be made about these Hilbert series.
1. As expected, (i) the Slodowy slice to the trivial nilpotent orbit SN,(1N)has the same Hilbert
series as the nilpotent cone, (ii) the slice to the sub-regular orbit has the Hilbert series of a
Kleinian singularity of type ˆ
A2r−1for the Bseries, ˆ
Dr+1for the Cseries, and ˆ
Drfor the D
series, and (iii) the slice to the maximal nilpotent orbit is trivial.
2. The Slodowy slices SN,ρ are all complete intersections, giving a good answer to the question
posed in [37].
3. The adjoint maps can contain singlet generators at even powers of tup to (twice) the degree
of the highest Casimir of G; these generators may be cancelled by one or more Casimir
relations.
4. The global symmetry groups of the Slodowy slice generators include mixed BCD Lie groups
(or Aseries isomorphisms), as well as finite groups of type B0, and descend in rank as
the dimension of the Slodowy slice reduces. Different Slodowy slices may share the same
symmetry group, while having inequivalent embeddings into G.
5. The sub-regular Slodowy slices of non-simply laced algebras match those of specific simply
laced algebras, in accordance with their Kleinian singularities, as listed in Table 1. In the
case of Slodowy slices of Cnnilpotent orbits with vector partitions of type (2n −k, k), it
was identified in [22] that these isomorphisms with Dn+1extend further down the Hasse
diagram: SN,C(2n−k,k) ≡SN,D(2n−k+1,k+1). This occurs due to matching chains of Kraft–
Procesi transitions [13] within such slices.
6. We have not attempted an exhaustive analysis of Z2factors associated with the choice of
SO vs Oflavour groups and the ensuing subtleties.
20 Note that if one wishes to read the generators of the chiral ring from the quiver as described in Section 4.5.2, then all
fugacities kineed to be set to 1.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 343
Tab l e 16
Hilbert series for Slodowy slices of B1, B2, and B3.
Nilpotent orbit Dimension |SN,ρ |Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[0]2B1[2]t2−t4(1−t4)
(1−t2)3
[2]0∅01
[00]8B2[0,2]t2−t4−t8(1−t4)(1−t8)
(1−t2)10
[01]4C1⊗B0[2]t2+[1]kt3−t8(1−t8)
(1−t2)3(1−t3)2
[20]2D1⊗B0t2+(1)kt 4−t8(1−t8)
(1−t2)(1−t4)2
[22]0∅01
[000]18 B3[0,1,0]t2−t4−t8−t12 (1−t4)(1−t8)(1−t12)
(1−t2)21
[010]10 B1⊗C1[2]Bt2+[2]Ct2+[2]B[1]Ct3−t8−t12 (1−t8)(1−t12 )
(1−t2)6(1−t3)6
[200]8D2⊗B0[2,0]t2+[0,2]t2+[1,1]kt4−t8−t12 (1−t8)(1−t12 )
(1−t2)6(1−t4)4
[101]6C1⊗B0[2]t2+[1]kt3+t4+[1]kt 5−t8−t12 (1−t8)(1−t12)
(1−t2)3(1−t3)2(1−t4)(1−t5)2
[020]4D1⊗B0t2+(1)kt 4+(2)t4+t6−t8−t12 (1−t8)(1−t12)
(1−t2)(1−t4)4(1−t6)
[220]2D1⊗B0t2+(1)kt 6−t12 (1−t12)
(1−t2)(1−t6)2
[222]0∅01
N.B. (n) denotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).
kdenotes the character ±1if B0→O(1)or 1if B0→SO(1).

344 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 17
Hilbert series for Slodowy slices of B4.
Nilpotent orbit Dimension |SN,ρ |Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[0000]32 B4[0,1,0,0]t2−t4−t8−t12 −t16 (1−t4)(1−t8)(1−t12)(1−t16 )
(1−t2)36
[0100]20 B2⊗C1[0,2]t2+[2]t2+[1,0][1]t3−t8−t12 −t16 (1−t8)(1−t12 )(1−t16)
(1−t2)13(1−t3)10
[2000]18 D3⊗B0[0,1,1]t2+[1,0,0]kt4−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)15(1−t4)6
[0001]16 C2⊗B0[2,0]t2+[1,0]kt3+[0,1]t4−t8−t12 −t16 (1−t8)(1−t12 )(1−t16 )
(1−t2)10(1−t3)4(1−t4)5
[1010]12 C2⊗D1⊗B0[2]t2+[1](1)t3+[1]kt3+(1)kt 4+t4+[1]kt5+t2−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)4(1−t3)6(1−t4)3(1−t5)2
[0200]10 B1⊗D1[2]t2+(2)t4+[2](1)t4+t2+t6−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)4(1−t4)8(1−t6)
[0020]8B1[2]t2+[4]t4+[2]t6−t8−t12 −t16 (1−t8)(1−t12 )(1−t16)
(1−t2)3(1−t4)5(1−t6)3
[2200]8D2⊗B0[2,0]t2+[0,2]t2+[1,1]kt6−t12 −t16 (1−t12 )(1−t16 )
(1−t2)6(1−t6)4
[0201]6C1⊗B0[2]t2+[1]kt5+[2]t6−t12 −t16 (1−t12)(1−t16)
(1−t2)3(1−t5)2(1−t6)3
[2101]6C1⊗B0[2]t2+[1]kt5+[1]kt7+t4−t12 −t16 (1−t12)(1−t16)
(1−t2)3(1−t4)(1−t5)2(1−t7)2
[2020]4B0⊗B0⊗B0(k1k3+k3k5)t4+(k1k5+k3k5)t 6+k3k5t8+t4−t12 −t16 (1−t12 )(1−t16)
(1−t4)3(1−t6)2(1−t8)
[2220]2D1⊗B0(1)kt 8+t2−t16 1−t16
(1−t2)(1−t8)2
[2222]0∅01
N.B. (n) denotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).
kdenotes the character ±1if B0→O(1)or 1if B0→SO(1).

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 345
Tab l e 18
Hilbert series for Slodowy slices of C1, C2, and C3.
Nilpotent orbit Dimension |SN,ρ |Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[0]2C1[2]t2−t4(1−t4)
(1−t2)3
[2]0∅01
[00]8C2[2,0]t2−t4−t8(1−t4)(1−t8)
(1−t2)10
[10]4C1⊗B0[2]t2+[1]kt3−t8(1−t8)
(1−t2)3(1−t3)2
[02]2D1t2+(1)t4−t8(1−t8)
(1−t2)(1−t4)2
[22]0∅01
[000]18 C3[2,0,0]t2−t4−t8−t12 (1−t4)(1−t8)(1−t12)
(1−t2)21
[100]12 C2⊗B0[2,0]t2+[1,0]kt3−t8−t12 (1−t8)(1−t12)
(1−t2)10(1−t3)4
[010]8C1⊗D1[2]t2+[1](1)t3+(2)t4+t2−t8−t12 (1−t8)(1−t12)
(1−t2)4(1−t3)4(1−t4)2
[002]6B1[2]t2+[4]t4−t8−t12 (1−t8)(1−t12)
(1−t2)3(1−t4)5
[020]4C1[2]t2+[2]t6−t8−t12 (1−t8)(1−t12)
(1−t2)3(1−t6)3
[210]4C1⊗B0[2]t2+[1]kt5−t12 1−t12
(1−t2)3(1−t5)2
[202]2B0⊗B0t4+k2k4t4+k2k4t6−t12 1−t12
(1−t4)2(1−t6)
[222]0∅01
N.B. (n) denotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).
kdenotes the character ±1if B0→O(1)or 1if B0→SO(1).

346 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 19
Hilbert series for Slodowy slices of C4.
Nilpotent orbit Dimension |SN,ρ|Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[0000]32 C4[2,0,0,0]t2−t4−t8−t12 −t16 (1−t4)(1−t8)(1−t12)(1−t16 )
(1−t2)36
[1000]24 C3⊗B0[2,0,0]t2+[1,0,0]kt3−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)21(1−t3)6
[0100]18 C2⊗D1[2,0]t2+[1,0](1)t3+(2)t4+t2−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)11(1−t3)8(1−t4)2
[0010]14 C1⊗B1[2]Bt2+[2]Ct2+[2]B[1]Ct3+[4]Bt4−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)6(1−t3)6(1−t4)5
[0002]12 D2[2,0]t2+[0,2]t2+[2,2]t4−t8−t12 −t16 (1−t8)(1−t12 )(1−t16)
(1−t2)6(1−t4)9
[2100]12 C2⊗B0[2,0]t2+[1,0]kt5−t12 −t16 (1−t12)(1−t16 )
(1−t2)10(1−t5)4
[0200]10 C1⊗C1[2]t2+[2]t2+[1][1]t4+[2]t6−t8−t12 −t16 (1−t8)(1−t12)(1−t16 )
(1−t2)6(1−t4)4(1−t6)3
[0110]8C1⊗B0[2]t2+[1]kt3+[1]kt5+[2]t6+t4−t8−t12 −t16 (1−t8)(1−t12 )(1−t16)
(1−t2)3(1−t3)2(1−t4)(1−t5)2(1−t6)3
[2010]8C1⊗B0⊗B0[2]t2+[1]k2t3+[1]k4t5+k2k4t4+k2k4t6+t4−t12 −t16 (1−t12 )(1−t16)
(1−t2)3(1−t3)2(1−t4)2(1−t5)2(1−t6)
[2002]6D1⊗B0(1)kt 4+(2)t4+(1)kt 6+t2+t4−t12 −t16 (1−t12)(1−t16 )
(1−t2)(1−t4)5(1−t6)2
[0202]4D1(2)t4+(2)t 8+t6−t12 −t16 (1−t12 )(1−t16)
(1−t2)(1−t4)2(1−t6)(1−t8)2
[2210]4C1⊗B0[2]t2+[1]kt7−t16 1−t16
(1−t2)3(1−t7)2
[2202]2B0⊗B0k2k6t6+k2k6t8+t4−t16 1−t16
(1−t4)(1−t6)(1−t8)
[2222]0∅01
N.B. (n) denotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).
kdenotes the character ±1if B0→O(1)or 1if B0→SO(1).

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 347
Tab l e 20
Hilbert series for Slodowy slices of D2, D3, and D4.
Nilpotent orbit Dimension |SN,ρ|Symmetry FGenerators of HS ≡PL[HS] Unrefined HS
[00]4D2[2,0]t2+[0,2]t2−2t4(1−t4)2
(1−t2)6
[20]2C1∼
=A1[2]t2−t41−t4
(1−t2)3
[02]2C1∼
=A1[2]t2−t41−t4
(1−t2)3
[22]0∅01
[000]12 D3[0,1,1]t2−t4−t6−t8(1−t4)(1−t6)(1−t8)
(1−t2)15
[011]6C1⊗D1[2]t2+[1](1)t3+t2−t6−t8(1−t6)(1−t8)
(1−t2)4(1−t3)4
[200]4B1⊗B0[2]t2+[2]kt4−kt6−t8(1−t6)(1−t8)
(1−t2)3(1−t4)3
[022]2D1(2)t4+t2−t81−t8
(1−t2)(1−t4)2
[222]0∅01
[0000]24 D4[0,1,0,0]t2−t4−2t8−t12 (1−t4)(1−t8)2(1−t12)
(1−t2)28
[0100]14 D2⊗C1[2,0]t2+[0,2]t2+[2]t2+[1,1][1]t3−2t8−t12 (1−t8)2(1−t12)
(1−t2)9(1−t3)8
[0002]12 C2[2,0]t2+[0,1]t4−2t8−t12 (1−t8)2(1−t12)
(1−t2)10(1−t4)5
[0020]12 C2[2,0]t2+[0,1]t4−2t8−t12 (1−t8)2(1−t12)
(1−t2)10(1−t4)5
[2000]12 B2⊗B0[0,2]t2+[1,0]kt4−kt8−t8−t12 (1−t8)2(1−t12)
(1−t2)10(1−t4)5
[1011]8C1⊗B0⊗B0[2]t2+[1](k1+k3)t3+[1]k1t5+t4+k1k3t4−k1k3t8−t8−t12 (1−t8)2(1−t12)
(1−t2)3(1−t3)4(1−t4)2(1−t5)2
[0200]6D1⊗D12t2+(1)(1)t4+(2)t 4+t6−2t8−t12 (1−t8)2(1−t12 )
(1−t2)2(1−t4)6(1−t6)
[0202]4C1[2]t2+[2]t6−t8−t12 (1−t8)(1−t12)
(1−t2)3(1−t6)3
[0220]4C1[2]t2+[2]t6−t8−t12 (1−t8)(1−t12)
(1−t2)3(1−t6)3
[2200]4B1⊗B0[2]t2+[2]kt6−kt8−kt12 (1−t8)(1−t12 )
(1−t2)3(1−t6)3
[2022]2B0⊗B0k3k5t4+t4+k3k5t6−t12 1−t12
(1−t4)2(1−t6)
[2222]2∅01
N.B. (n) denotes the character of the D1≡SO(2)reducible representation qn+q−nof U(1).
kdenotes the character ±1if B0→O(1)or 1if B0→SO(1).

348 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
For example, the slices SN,B[20]and SN,C[02]have the global symmetries D1⊗B0and D1,
respectively, with the Bseries Slodowy slice having an extra B0fugacity (k=±1), notwith-
standing the isomorphism between the Band CLie algebras.
Similarly, in the case of D4, the spinor pair slices, respectively SN,D[0020]/SN,D[0002]or
SN,D[0220]/SN,D[0202], only carry a C1or2 series symmetry, while the corresponding vector
slices of the same dimension, SN,D[2000]or SN,D[2200], carry a B0⊗B1or2 symmetry.
Whilst Higgs branch constructions based on the balanced quivers of type BB/C/D(Nf(ρ)) are
available for all Slodowy slices, Coulomb branch constructions based on LBC/CD/DC quivers or
Higgs branch constructions based on the quivers of type DG(Nf)are not generally available:
1. In the cases calculated, the slice to a sub-regular nilpotent orbit always has a Coulomb branch
construction.
2. Many BCDSlodowy slices do not have Coulomb branch constructions as LBC/CD/DC quiv-
ers, either because their underlying nilpotent orbits are not special, or due to zero conformal
dimension problems under the O/USp monopole formula. While the issue of zero confor-
mal dimension ( =0) is less prevalent for low dimension Slodowy slices, the problem is
inherent in maximal Br−Cr−Br−1sub-chains, and so affects many Cseries Slodowy
slices; certain other quivers are also problematic.
3. Other than Aseries isomorphisms, the quivers of type DG(Nf)only provide Higgs branch
constructions for Dseries Slodowy slices of low dimension. The nilpotent orbits underlying
these Slodowy slices are dual, under the Barbasch–Vogan map, to (minimal or near-to-
minimal) nilpotent orbits of Characteristic height 2, for which Coulomb branch constructions
using the unitary monopole formula are known [2], plus some others, such as SN,D[0200].
These Dynkin diagram quivers have S(U ⊗...U) flavour nodes and their refined Hilbert
series may not replicate all the possible combinations of orthogonal group characters.
These matters are discussed further in the concluding section.
4.5. Matrix generators for orthosymplectic quivers
In the case of BCD series, prescriptions are similarly available for obtaining the generators
of the chiral ring corresponding to a Slodowy slice directly from the partition data or from the
Higgs branch quiver.
4.5.1. Vector decomposition
From (4.1)and the alternating nature of the quiver, it follows that the character of the vec-
tor representation of Gdecomposes into vector representations of an O/USp product group,
tensored with the SU(2)embedding:
ρ:χO(N)
vector →⊕
[n]
bosonic
[n]ρχ
ONfn+1
vector ⊕
[n]
fermionic
[n]ρχ
USpNfn+1
vector ,
ρ:χUSp(N)
vector →⊕
[n]
bosonic
[n]ρχ
USpNfn+1
vector ⊕
[n]
fermionic
[n]ρχ
ONfn+1
vector ,
(4.6)
where [n]ρare bosonic (odd dimension) or fermionic (even dimension) irreps of the SU(2)asso-
ciated with the nilpotent orbit embedding ρ. The requirement that the partition ρobeys the BCD

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 349
selection rules ensures that the USp irreps are all of even dimension. Once this decomposition
has been identified, the mapping of the adjoint of Ginto matrix generators (2.8) follows, either
by symmetrising the USp vector, or by antisymmetrising the Ovector. This can be checked
against the adjoint partition ρ:χG
adjoint. Note that a choice can be made whether to use the SO
form of orthogonal group characters or the O−form.
4.5.2. Generators from quiver paths
For orthosymplectic quivers, the method in section 3.5.2 can be applied, with a few changes.
An operator Pij (a) formed from a path in the quiver is defined identically. However, for or-
thosymplectic quivers, Pij (a) =Pji(a)T, and a path yields only one generator when i= j.
Other differences follow from the irreducible representations of the operators Pij (a) and the
gauge group invariants. There are two cases:
1. i= j. The operator transforms in the defining representation of the initial flavour group and
the defining representation of the final flavour group. For example, if the flavour node at
position iis O(7)and the flavour node at position jis USp(4), Pij (a) transforms in the
irrep of dimension 7 ×4.
2. i=j. The operator has two indices that transform under the flavour group at position i.
They are symmetrized if the gauge node at the mid point of the path is of O-type, or anti-
symmetrized if the gauge node is of USp-type.
The set of operators Pij (a) gives us all the generators of the chiral ring. The relations are inher-
ited from those of the nilpotent cone N, and for SN,ρ are always the Casimir invariants of G.
Now, an O(Nfi)flavour node (of rank >0) always contributes (at least) a path Pii(1)of length
2 that starts at O(Nfi), goes to the gauge node USp(Ni)and comes back to O(Nfi). Since the
gauge node in the middle of the path is USp, the operator transforms in the second antisym-
metrization 2[fund.]O=[adjoint]O. Similarly, aUSp(Nfi)flavour node always contributes
(at least) a path Pii(1)of length 2 that starts at USp(Nfi), goes to the gauge node O(Ni)and
comes back to USp(Nfi). Since the gauge node in the middle of the path is O, the operator
transforms in the second symmetrization Sym2[fund.]USp =[adjoint]USp. Consequently, the
adjoint of every flavour group appears as a generator at path length 2.
Example Consider the balanced quiver based on the partition (22, 14), whose Higgs branch is
the Slodowy slice SN,(4,2)to the nilpotent orbit D[0100]:
BD(Nf(22,14)) =USp(4)
|
O(4)
−O(6)
|
USp(2)
−USp(4)
−O(4)
−USp(2)
−O(2)
.(4.7)
The decomposition of Gto SU(2) ⊗Fis:
SO(8)→SU(2)ρ⊗O(4)⊗USp(2). (4.8)
The Hilbert series of the chiral ring of operators in the Higgs branch has generators Pij (a) given
by the quiver paths in Table 21. For D4the Casimirs give relations, −t4−2t8−t12, therefore,
the PL[HS] read directly from the quiver is:
PL[gHiggs[BD(Nf(22,14))]
HS ]=[2,0]t2+[0,2]t2+[2]t2+[1,1][1]t3−2t8−t12 .(4.9)

350 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 21
Generators for Slodowy slice to D[0100].
Pij (a) Quiver path Generator
P11(1)
USp(4)
↑↓
O(4)
O(6)
USp(4)
O(4)
USp(2)
O(2)
2([1,1]t) =[2,0]t2+[0,2]t2
P2,2(1)
USp(4)
O(6)
↑↓
USp(2)
USp(4)
O(4)
USp(2)
O(2)
Sym2([1]t) =[2]t2
P2,2(2)
USp(4)

O(6)
↑↓
USp(2)
USp(4)
O(4)
USp(2)
O(2)
2([1]t2)=[0]t4
P1,2(1)
USp(4)
↑
O(4)
→
O(6)
↓
USp(2)
USp(4)
O(4)
USp(2)
O(2)
[1,1][1]t3
4.5.3. Matrices and relations
Finally, in Tables 22–24 we provide a set of algebraic varieties described by matrices such
that their HS have been computed to be identical to those of the corresponding Slodowy slices
SN,ρ of B1to B3nilpotent orbits. The analysis can in principle be continued to higher rank.
5. Discussion and conclusions
Higgs branch We have presented constructions for quivers whose Higgs branches yield Hilbert
series corresponding to the Slodowy slices of the nilpotent orbits of A1to A5plus BCD al-
gebras up to rank 4. There are essentially two families of quivers, the balanced unitary type
{BA=DA, DD}and the canonically balanced orthosymplectic type {BB/C/D}. The balanced
unitary quivers have gauge nodes in the pattern of the parent algebra Dynkin diagram and yield
constructions for Slodowy slices of simply laced algebras, including all Aseries slices and D
series slices of low dimension. The orthosymplectic quivers yield constructions of all BCD
Slodowy slices.
The global symmetry Fof a Slodowy slice descends from that of the parent group G(in the
case of the slice to the trivial nilpotent orbit), via subgroups of G, down to Z2symmetries (for the
slices of some near maximal nilpotent orbits). The grading of the Hilbert series is such that (i) the
sets of Slodowy slices and nilpotent orbits intersect at the nilpotent cone and at the origin and
(ii) the sub-regular slices match the known singularities [3,4,15]. In between, we have shown how
the Slodowy slice symmetry groups and mappings of Grepresentations to SU(2) ⊗Ffollow, via
the Higgs branch formula, from the SU(2)homomorphisms into Gof the associated nilpotent
orbits.
We anticipate that these results generalise to Classical groups of arbitrary rank.
Coulomb branch As is known, in the case of the Aseries, the existence of a bijection be-
tween partitions and their transposes (the Lusztig–Spaltenstein map) leads to a complete set of
Coulomb branch constructions for Slodowy slices; these yield the same set of Hilbert series as
the Higgs branch constructions. The Coulomb branch constructions are based on applying the
unitary monopole formula to linear quivers LA, which are not generally balanced.

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 351
Tab l e 22
B1, B2and B3varieties, generated by complex matrices M, N, O, Aand Band their relations, which have Hilbert series
matching Slodowy slices SN,ρ . The matrices M=−MTand O=−OTare antisymmetric, N=NTis symmetric and
represents a square matrix that is antisymmetric and invariant under the action of USp(2n).
Orbit Partition Dim. Generators; Degree Relations
B[0] (13)2M3×3;2tr(M2)=0
B[2] (3)0– –
B[00] (15)8M5×5;2tr(M2)=0
tr(M4)=0
B[01] (22,1)4N2×2;2
A2×1;3tr ((N )4)=ATN A
B[20] (3,12)2M2×2;2
A2×1;4tr(M4)=ATA
B[22] (5)0– –
B[000] (17)18 M7×7;2
tr(M2)=0
tr(M4)=0
tr(M6)=0
B[010] (22,13)10
M3×3;2
N2×2;2
A3×2;3
tr(M4)+tr((N )4)=tr(AATM)+tr(ATAN)
tr(M6)+tr((N )6)=t r ((AAT)2)
B[200] (3,14)8M4×4;2
A4×1;4
tr(M4)=ATA
tr(M6)=ATM2A
B[101] (3,22)6
N2×2;2
A2×1;3
M2×2;4
B2×1;5
tr ((N )4+(M )2)=BTA
tr ((N )6+(M )3)=BTMA
B[020] (32,1)4
M2×2;2
A2×1;4
N2×2;4
O2×2;6
tr(N) =0
tr(M4+N2)=ATA
tr(M6+N3+O2)=ATNA
B[220] (5,12)2M2×2;2
A2×1;6tr(M6)=ATA
B[222] (7)0– –
In the case of the BCD series, however, other than for accidental isomorphisms with the A
series, this study has clarified that (i) the existence of suitable linear orthosymplectic quivers
{LBC, LCD, LDC}is limited to the Slodowy slices of special nilpotent orbits, (ii) within these,
the applicability of the Coulomb branch orthosymplectic monopole formula is restricted to those
quivers that have positive conformal dimension, and (iii) the resulting Hilbert series are only
available in unrefined form.
Slodowy slice formula The refined Hilbert series of a Slodowy slice can also be obtained di-
rectly from the mapping of the adjoint representation of Ginto SU(2) ⊗F, using (2.11). This
mapping follows from the decomposition of the fundamental/vector of G →SU(2) ⊗Funder
(3.13)or (4.6).
Dualities and 3dmirror symmetry The Aseries findings verify the known 3dmirror symme-
try relations (3.4) and (3.5). Under these, linear or balanced quivers based on partitions ρcan
be used either for Higgs branch or Coulomb branch constructions; one combination yields a

352 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 23
C1, C2and C3varieties, generated by complex matrices M, N, O, P, Aand Band their relations, which have Hilbert
series matching Slodowy slices SN,ρ . The matrices M=−MTand O=−OTare antisymmetric, N=NTand P=
PTare symmetric and represents a square matrix that is antisymmetric and invariant under the action of USp(2n).
Orbit Partition Dim. Generators; Degree Relations
C[0] (12)2N2×2;2tr(N2)=0
C[2] (2)0– –
C[00] (14)8N4×4;2tr ((N )2)=0
tr ((N )4)=0
C[10] (2,12)4N2×2;2
A2×1;3tr ((N )4)=ATN A
C[02] (22)2M2×2;2
N2×2;4
tr(N) =0
tr(M4)=tr(N2)
C[22] (4)0– –
C[000] (16)18 N6×6;2
tr ((N )2)=0
tr ((N )4)=0
tr ((N )6)=0
C[100] (2,14)12 N4×4;2
A4×1;3
tr ((N )4)=ATN A
tr ((N )6)=AT(N )3A
C[010] (22,12)8
N2×2;2
M2×2;2
A2×2;3
P2×2;4
tr(P) =0
tr ((N )4+M4+P2)=tr(ATNA)
tr ((N )6+M6+P3)=tr(AT(N)2A)
C[002] (23)6M3×3;2
N3×3;4
tr(N) =0
tr(M4)=tr(N2)
tr(M6)=tr(N3)
C[020] (32)4
N2×2;2
M2×2;4
P2×2;6
tr(M) =0
tr(P) =0
tr ((N )4+(M )2)=0
tr ((N )6+(M )3+(P )2)=0
C[210] (4,12)4N2×2;2
A2×1;5tr ((N )6)=ATN A
C[202] (4,2)2
N1×1;4
A1×1;4
P1×1;6
tr(NA2)=tr(P2)
C[222] (6)0– –
Slodowy slice and the other combination yields a (generally different) dual nilpotent orbit under
the Lusztig–Spaltenstein map ρT, as illustrated in Fig. 11.
The analysis of BCDseries quivers shows, however, that such a picture of dualities [10] does
not extend to the BCD series, other than in a limited way, due to the various restrictions on
Coulomb branch constructions, discussed above. The refined (i.e. faithful) HS relationships for
nilpotent orbits of the BCD series can be summarised:
SN,ρ =HiggsBB/C/D(Nf(ρ)),
¯
Oρ=HiggsLB/C/D(ρ T),(5.1)
and, for Dseries Dynkin type quivers of Characteristic height 2:

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 353
Tab l e 24
D2and D3varieties, generated by complex matrices M, N, and Aand their relations, which have Hilbert series matching
Slodowy slices SN,ρ . The matrix M=−MTis antisymmetric, N=NTis symmetric and represents a square matrix
that is antisymmetric and invariant under the action of USp(2n). pf () denotes the Pfaffian.
Orbit Partition Dim. Generators; Degree Relations
D[00] (14)4M4×4;2tr(M2)=0
pf ( M ) =0
D[20] (22)2N2×2;2tr ((N )2)=0
D[02] (22)2N2×2;2tr ((N )2)=0
D[22] (4)0– –
D[000] (16)12 M6×6;2
tr(M2)=0
tr(M4)=0
pf ( M ) =0
D[011] (22,12)6
M2×2;2
N2×2;2
A2×2;3
tr(AAT) =0
tr(M4)+tr((N )4)=tr(AATM+ANAT)
D[200] (3,13)4M3×3;2
A3×1;4
ij k Mij Ak=0
tr(M4)=ATA
D[022] (32)2M2×2;2
N2×2;4
tr(N) =0
tr(M4)=tr(N2)
D[222] (5,1)0– –
Fig. 11. A Series 3d Mirror Symmetry. All constructions give refined Hilbert series for a partition ρand its dual ρT
under the Lusztig–Spaltenstein map.
¯
Oρ=Coulomb[DD([ρ])],
SN,dBV (ρ) =Higgs[DD([ρ])],(5.2)
where dBV (ρ) is the dual partition to ρunder the Dseries Barbasch–Vogan map.
If we restrict ourselves (i) to special nilpotent orbits, (ii) to quivers with positive conformal
dimension, and (iii) to unrefined Hilbert series, then we can summarise the more limited 3d
mirror symmetry for the BCD series as in Fig. 12.
Note that even for these cases there is a further obstruction: the difference between SO and
Onodes in the quiver [28,29]. For the A series, 3dmirror symmetry involves a pair of quivers
for which the Coulomb branch and Higgs branch are swapped. In the BCD series however, once
the gauge algebra of the quiver is specified there is still the question of whether the gauge groups
are orthogonal or special orthogonal. As shown in Fig. 12 a different choice needs to be made
depending on the branch of the quiver. This is not quite the same as 3dmirror symmetry.
On the other hand, there is a pair of SCFTs, Tρ
σ(G) and Tσ
ρ(G∨)[10,18,19], which are
predicted to have precisely the two different gauge algebras depicted in one of the diagrams
of Fig. 12: if Tρ
σ(G) corresponds to quiver LBC/CD/DC(ρT), then Tσ
ρ(G∨)has the quiver
BB/C/D(Nf(dBV (ρ))), along with the Higgs and Coulomb branches depicted in the same di-

354 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Fig. 12. BCD Series 3d Mirror Symmetry. Solid arrows indicate Higgs branches which give refined Hilbert series for
a partition ρ. Dashed arrows indicate Higgs branches which give refined Hilbert series for the Barbasch–Vogan dual
partition dBV (ρ) of a special nilpotent orbit. Dotted arrows indicate Coulomb branches which give unrefined Hilbert
series for those special nilpotent orbits whose quivers have positive conformal dimension.
agram. However, the present results, together with [1,28,29], show that this cannot be the case,
since there are factors of Z2in the gauge group of the quiver for Tρ
σ(G) that differ depending
on the branch being computed. This is a very intriguing point that needs to be addressed in fu-
ture studies, especially since it has consequences for the way effective gauge theories can be
employed to understand the dynamics of Dp-branes in the presence of Op-planes.
Thus, it is the Higgs branch that provides the means to conduct a refined analysis of the HS
of BCD series nilpotent orbits and Slodowy slices. These represent only a subset of the BCD
series moduli spaces, Sρ1,ρ2≡¯
Oρ1∩Sρ2, which include nilpotent orbits Sρ,trivial and Slodowy
slices SN,ρ as limiting cases.21 The indications are that Higgs branch methods should provide a
fruitful means of analysing such spaces.
Further work Besides a study of quivers for Sρ1,ρ2moduli spaces, it would be interesting to
extend this analysis to the Slodowy slices of Exceptional groups. While Higgs branch quiver
constructions are not available for nilpotent orbits of Exceptional groups, a limited number of
Coulomb branch quiver constructions are known. For Slodowy slices, where the situation is
somewhat reversed by dualities, some Higgs branch constructions should be available, based, for
example, on Dynkin diagrams of the E series.
With respect to the Coulomb branch, it would be interesting to understand (i) whether some
non-linear fugacity map can be developed for the orthosymplectic monopole formula in order to
obtain refined Hilbert series, and (ii) whether a modified monopole formula can be found that
avoids the zero conformal dimension problem associated with many orthosymplectic quivers.
A recent advance has been made on this front in [38], where Coulomb branches of bad quivers
with a single Crgauge node have been computed. A case that also appears in our study is the
quiver [D2r] −(Cr), where the expected Slodowy slices are formed in quite a surprising way.22
It remains a challenge to develop such techniques to obtain Coulomb branch calculations for the
Slodowy slices of the other quivers with  =0in our tables.
More generally, the family of transverse spaces and symmetry breaking associated with
Slodowy slices provides a rich basis set of quivers that can be extended or used as building
blocks to understand the relationships between a wide array of quiver theories and their Higgs
and/or Coulomb branches.
21 Such BCD series moduli spaces Sρ1,ρ2generalise naturally to any pair of nilpotent orbits (unlike Tρ
σ(O/ U Sp)
theories, which are restricted to special orbits).
22 [38] computes that there are two most singular points in this Coulomb branch, related by a Z2action. Crucially, at
each point, an SCFT denoted TUSp(2r),2rhas a Coulomb branch identical to the expected Slodowy slice (identified in
[38]as the Higgs branch of the corresponding DG(Nf)quiver).

S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357 355
Acknowledgements
We would like to thank Stefano Cremonesi and Benjamin Assel for helpful conversations
during the development of this project. S.C. is supported by an EPSRC DTP studentship
EP/M507878/1. A.H. is supported by STFC Consolidated Grant ST/J0003533/1, and EPSRC
Programme Grant EP/K034456/1.
Appendix A. Notation and terminology
We refer to Slodowy slices and nilpotent orbits either by their Lie algebras g, or by the Lie
groups Gin which they transform. While such references are relatively interchangeable for USp
groups, with Lie algebras of Ctype, it can be important to distinguish between Oand SO
forms of orthogonal groups, which may share the same Bor Dtype Lie algebra, but whose
representations have different characters. We have sought to highlight those areas where this
distinction is important in the text.
We freely use the terminology and concepts of the Plethystics Program, including the Plethys-
tic Exponential (“PE”), its inverse, the Plethystic Logarithm (“PL”), the Fermionic Plethystic
Exponential (“PEF”) and, its inverse, the Fermionic Plethystic Logarithm (“PFL”). For our pur-
poses:
PEd

i=1
Ai,t≡
d

i=1
1
(1−Ait),
PE−
d

i=1
Ai,t≡
d

i=1
(1−Ait),
PEd

i=1
Ai,−t≡
d

i=1
1
(1+Ait),
PE−
d

i=1
Ai,−t≡PEF d

i=1
Ai,t≡
d

i=1
(1+Ait),
(A.1)
where Aiare monomials in weight or root coordinates or fugacities. The reader is referred to
[24]or [26]for further detail.
We present the characters of a group Geither in the generic form XG(xi), or as [irrep]G,
or using Dynkin labels as [n1,...,n
r]G, where ris the rank of G. We often represent sin-
glet irreps implicitly via their character 1. We typically label unimodular Cartan subalgebra
(“CSA”) coordinates for weights within characters by x≡(x1...x
r)and simple root coor-
dinates by z≡(z1...z
r), dropping subscripts if no ambiguities arise. The Cartan matrix Aij
mediates the canonical relationship between simple root and CSA coordinates as zi=
j
xAij
j
and xi=
j
zA−1ij
j.
We label field (or R-charge) counting variables with t, adding subscripts if necessary. Under
the conventions in this paper, the fugacity tcorresponds to an R-charge of 1/2 and t2corresponds
to an R-charge of 1. We may refer to series, such as 1 +f+f2+..., by their generating functions

356 S. Cabrera et al. / Nuclear Physics B 939 (2019) 308–357
Tab l e 25
Types of generating function.
Generating function Notation Definition
Refined HS (Weight coordinates) gG
HS(x, t )
∞

n=0
an(x)t n
Refined HS (Simple root coordinates) gG
HS(z, t )
∞

n=0
an(z)tn
Unrefined HS gG
HS (t)
∞

n=0
antn≡
∞

n=0
an(1)tn
1/
(1−f). Different types of generating function are indicated in Table 25; amongst these, the
refined HS faithfully encode the group theoretic information about a moduli space.
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