Quiver Theories and Formulae for Slodowy Slices of Classical Algebras

We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra $\mathfrak g$. 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 $\mathcal S\cap \mathcal N$ of $\mathfrak{g}$. We calculate refined Hilbert series for Classical algebras up to rank $4$ (and $A_5$), 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 $\mathcal S\cap \mathcal N$; 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 $T_\sigma^\rho(G)$ 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.


Contents
List of Tables   1  Sub-regular Slodowy Slices of Classical Groups  10  2 Hilbert Series for Slodowy Slices of A 1 , A 2 , A 3 and A 4 . 20 3 Hilbert Series for Slodowy Slices of A 5 . 21 4 Generators for Slodowy Slice to Ar101s. 23 5 Generators for Slodowy Slice to Ar1111s. 25  Quivers for A 1 to A 4 Slodowy Slices. 13 3 Quivers for A 5 Slodowy Slices. 14 4 BCD Series Linear and Balanced Quiver Types 30 5 Quivers for B 1 to B 3 Slodowy Slices. 34 6 Quivers for B 4 Slodowy Slices. 35 7 Quivers for C 1 to C 3 Slodowy Slices. 36 8 Quivers for C 4 Slodowy Slices. 37 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 g were 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 3d N " 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 O ρ of a Lie algebra g of a Classical group G has a transverse slice, or Slodowy slice S ρ , lying within the algebra g. 1 There is a variety defined by the intersection between the Slodowy slice and the nilpotent cone N of the algebra: S N ,ρ " N X S ρ . In this paper, we deal almost entirely with these intersections S N ,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 g that commutes with (or stabilises) the sup2q. In the case of the sub-regular nilpotent orbit S N ,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 relevance of nilpotent orbits and Slodowy slices to 3d N " 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 1 ρ identifies the embedding of sup2q into g that defines the nilpotent orbit. 2 For general background on nilpotent orbits the reader is referred to [6].
theory. Brane systems of the type of [11] are relevant for the A series and systems with three dimensional orientifold planes [12] for the BCD series 3 .
In the course of these latter papers, a class of superconformal field theories ("SCFT") was proposed, with moduli spaces defined by the intersections between Slodowy slices and nilpotent orbits. These are termed T ρ σ pGq theories, where G is 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 S N ,ρ , and a relationship between S-duality and dualities arising from the 3d mirror symmetry [17] of Classical quiver theories was conjectured 4 .
For example, in the case of an A series nilpotent orbit O ρ , where ρ describes the embedding of sup2q into supnq that defines the nilpotent orbit, and ρ " p1 N q corresponds to the trivial nilpotent orbit, these dualities entail that the Higgs branch of a linear quiver based on a partition ρ T , yields the closure of the nilpotent orbitŌ ρ , while the Coulomb branch of a linear quiver based on the partition ρ gives its Slodowy slice S N ,ρ . The application of 3d mirror symmetry to this pair of linear quivers yields a pair of "balanced" quivers, with the Coulomb branch of the former yieldingŌ ρ and the Higgs branch of the latter yielding S N ,ρ . 5 More recently, in [19] and [21], nilpotent orbits and Slodowy slices have been used in the study of 6d N " p2, 0q 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 f of g has a ring of holomorphic functions transforming in irreps of the sub-group F of 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][25][26].
Importantly, following a result in [3], the Hilbert series of Slodowy slices S N ,ρ are always complete intersections, i.e. quotients of geometric series. It was shown in [27] 3 Note 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]. 4 In the case of nilpotent orbits of C and D type, the precise match between quivers and orbits was given in [18]. Subsequently, [19] described the relation for B type and unified all classical cases via the introduction of the Barbasch-Vogan map [20]. 5 The 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.
how the HS of the Slodowy slices of A series and certain BCD series algebras can be calculated from the Coulomb branches of linear quivers (or from the Higgs branches of their 3d mirror duals). [27] also identified a relationship between Slodowy slices and the (modified) Hall Littlewood polynomials of g, under the mapping g Ñ sup2q b f.
Methods of calculating Hilbert series for T ρ σ pGq theories with multi-flavoured quivers of unitary or alternating O{U Sp 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{U Sp type requires close attention to the distinction between SO and O groups.
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 2 we summarise some facts about nilpotent orbits and review the relationship between a Slodowy slice S N ,ρ and the homomorphism ρ defining the embedding of sup2q into g (and thus of the mapping of irreps of G into the irreps of SU p2q) associated with a nilpotent orbit O ρ . We give some simple representation theoretic formulae for calculating the dimensions and Hilbert series of a Slodowy slice, given a homomorphism ρ.
In Section 3 we treat A series Slodowy slices, summarising the relevant Higgs branch and Coulomb branch formulae, describing the quivers upon which they act, and tabulating the commutant 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 T ρ σ pSU pN qq theories to summarise the known exact A series dualities between quiver theories for Slodowy slices and nilpotent orbits. We find matrix formulations for the generators of A series Slodowy slices and their relations, which explicate the mechanism of symmetry breaking and the residual symmetries of the parent group.
In Section 4 we 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{U Sp quivers with appropriately balanced gauge nodes. As in the case of BCD nilpotent orbits [1], calculation of these Higgs branches requires taking Z 2 averages over the SO and O´forms of O group characters. We also identify a limited set of Higgs branch constructions based on D series Dynkin diagrams. We summarise the restricted set of Coulomb branch monopole constructions that are available for S N ,ρ , which are based on alternating SO{U Sp linear quivers. We highlight apparent restrictions on 3d mirror symmetry between Higgs and Coulomb branches of BCD quiver theories; these include the requirements that the nilpotent orbit O ρ should be special, and that the O{U Sp 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.
Taken together with other recent studies [1,29], this analysis of Hilbert series is relevant for the understanding of T ρ σ pGq theories of type BCD. It highlights the difference between orthogonal Opnq and special orthogonal SOpnq 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.

Relationship to Nilpotent Orbits
Each nilpotent orbit O ρ of a Lie algebra g is defined by the conjugacy class g X of nilpotent elements X P g under the group action [6]. Each nilpotent element X forms part of a standard sup2q triple tX, Y, Hu and, following the Jacobson Morozov theorem, the conjugacy classes are in one to one correspondence with the embeddings of sup2q into g, described by some homomorphism ρ. The closure of each orbitŌ ρ , can, as discussed in [1,2], be described as a moduli space, by a refined Hilibert series of representations of G, graded according to the degree of symmetrisation of the underlying nilpotent element.
The closuresŌ ρ of the nilpotent orbits of g form a poset, ordered according to their inclusion relations 6 . The closure of the maximal (also termed principal or regular ) nilpotent orbit is called the nilpotent cone N ; it contains all the orbits O ρ and 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: All nilpotent orbits have an even (complex) dimension and are HyperKähler cones. The closure of the minimal nilpotent orbit of g corresponds 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: where t is a counting fugacity, χ G adjoint is the character of the adjoint representation and td 1 , . . . , d r u are the degrees of the symmetric Casimirs of G, which is of rank r.
Slodowy slices are defined as slices S ρ Ď g that are transverse in the sense of [3] to the orbit O ρ . The varieties S N ,ρ that concern the present study are slices inside the nilpotent cone N . They can be constructed as: Naturally, the slice S N ,ρ transverse to the trivial nilpotent orbit is the entire nilpotent cone N and the slice S N ,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: (2.4)

Dimensions and Symmetry Groups
The dimensions of a Slodowy slice S N ,ρ plus those of the nilpotent orbit O ρ combine to the dimensions of the nilpotent cone N : The elements of the Slodowy slice S N ,ρ lie in a subalgebra f, which is the centraliser of the nilpotent element X P g, so that rX, fs " 0, and f is often termed the commutant of sup2q in g. The structure of f and the dimensions of S N ,ρ and O ρ can be determined by analysing the embedding of sup2q Ñ g. Following [5], a homomorphism ρ can be described by a root space map from g to sup2q, and this is conveniently encoded in a Characteristic set of Dynkin labels. 7 The Characteristic rq 1 . . . q r s provides a map from the simple root fugacities tz 1 , . . . , z r u of g to the simple root fugacity tzu of sup2q: where the labels q i P t0, 1, 2u. This induces corresponding weight space maps under which each representation of G of dimension N branches to representations rns of SU p2q 7 A Characteristic Gr. . .s is distinct from highest weight Dynkin labels r. . . , . . .s G .
at some multiplicity m n . This branching is conveniently described using partition notation, p|rN´1s| m N´1 , . . . , |rns| mn , . . . , 1 m 0 q, which lists the dimensions of the SU p2q irreps, using exponents 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 ρ with Characteristic r202s, which generates the 10 dimensional nilpotent orbit of of A 3 , induces the following maps: (2.7) These SU p2q partitions are invariant under the symmetry group F Ď G of the Slodowy slice and hence the multiplicities encode representations of F . Under the branching, the adjoint representation of G decomposes to representations of the product group SU p2q b F with branching coefficients a nm : (2.8) Other than for the trivial nilpotent orbit (in which the adjoint of G branches to itself times an SU p2q singlet), the adjoint of SU p2q and the adjoint (if any) of F each appear separately in the decomposition, so that rankrGs ě rankrF s ě 0. Along with the requirement that any multiplicities m n appearing in a partition must be dimensions of representations of F , this often makes it possible to determine the Lie algebra f of the Slodowy slice directly from the partition data. In the example 2.7 the presence of a single SU p2q singlet in the partition of the adjoint of A 3 entails that the symmetry group of the Slodowy slice to the r202s orbit is simply U p1q.
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 F irreps: (2.9)

Hilbert Series
The branching of the adjoint representation of G determines 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 mapρ of the adjoint that is obtained from 2.8 by the replacement of SU p2q irreps by their highest weight fugacities χ SU p2q rns Ñ t n , taking the particular choice of t from 2.2 as the counting variable: (2.10) When the adjoint map 2.10 is applied to the generators of the nilpotent cone 2.2, the replacement of the SU p2q representations rns by the counting fugacity t n entails that the resulting Hilbert series only transforms in the symmetry group of the Slodowy slice. Thus, g HS , or, written more explicitly: (2.11) The expression 2.11 gives the refined Hilbert series of the Slodowy slice in terms of its generators, 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 F replaced by their dimensions, m n " ř m a nm |χ F rms |, can be calculated directly from the adjoint partition under ρ, without knowledge of the precise details of the embedding 2.8: Finally, the freely generated Hilbert Series for the canonical Slodowy slices S ρ are related to those of their nilpotent intersections S N ,ρ simply by the exclusion of the Casimir relations: In Sections 3 and 4 we 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 S N ,ρ .

Sub-Regular Singularities
As shown in [3,4], the Slodowy slices of sub-regular orbits S N ,subregular take the form of ADE type singularities, C 2 {Γ, where Γ is a finite group of type ADE. Under the nilpotent orbit grading by t 2 used herein, these take the forms in table 1. The intersection S N ,subregular 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].

Group
Singularity Dimension Hilbert Series The dicyclic group of order 4k is denoted as Dic k . 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.

Quiver Types
The constructions for the Slodowy slices of A series nilpotent orbits draw upon the same two quiver types as the constructions for the closures of the nilpotent orbits. These are shown in figure 1: 1. Linear quivers based on partitions. These quivers L A pρq consist of a SU pN 0 q flavour node connected to a linear chain of U pN i q gauge nodes, where the decrements between nodes, ρ i " N i´1´Ni , constitute an ordered partition of N 0 , ρ " tρ 1 , . . . , ρ k u, where ρ i ě ρ i`1 and ř k i"1 ρ i " N 0 .
2. Balanced quivers based on Dynkin diagrams. These quivers B A pN f q consist of a linear chain of U pN i q gauge nodes (in the form of an A series Dynkin diagram), with each gauge node connected to a flavour node of rank 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 Spb i U N f i q 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. For theories with A series gauge nodes, the requirement of balance 9 is that, for each gauge node, the sum of the ranks of adjacent gauge nodes plus the number of attached flavours should equal twice its rank: This condition of balance, B " tBalancepiqu " 0 can be rearranged as: where the flavour and gauge nodes have been written as vectors N f " pN f 1 , . . . , N f k q and N " pN 1 , . . . , N k q, and A is the Cartan matrix of A k . In the case of nilpotent orbits, A series 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 A series orbits and can be related by 3d mirror symmetry [17].
For Slodowy slices, the roles of these quiver types are reversed: the linear A series quivers provide a complete set of Coulomb branch constructions, while the balanced A series 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 pN 0 q 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 discussed in section 2.2. For the A series, it is convenient to write the SU p2q partition of the fundamental representation under ρ as: so that the multiplicities of partition elements, which may be zero, are mapped to the flavour vector N f . The linear quiver L A pρq can be extracted simply by writing ρrf und.s in long form. The ranks N of the gauge nodes of the balanced quiver B A pN f pρqq can be found from N f by inverting 3.2. Alternatively, the balanced quivers B A pN f pρqq can be obtained by applying 3d mirror symmetry transformations to the linear quivers L A pρq, and vice versa. We can use the notation above to express the key relationships and dualities involving A series quivers for the Slodowy slices of nilpotent orbits: 4) or, taking the transpose of ρ: The quivers for A series 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 A series orbits, is known as the Lusztig-Spaltenstein map [31].  These linear or balanced quiver types correspond to the limiting cases of T ρ σ pSU pN qq theories [10,28], where one of the partitions σ or ρ is taken as the trivial partition: Those quivers, whose Higgs or Coulomb branches yield Slodowy slices of A series groups up to rank 5, are tabulated in figures 2 and 3, labelled by the nilpotent orbit, giving the partition ρ of the fundamental, the dimensions of the Slodowy slice, and the residual symmetry group. The balanced quivers used in the Higgs branch construction always have gauge nodes equal in number to the rank of G " SU pN q, while the linear quivers used in the Coulomb branch constructions always have a number of flavours equal to the fundamental dimension of G " SU pN q. The quivers L A pp1 N qq and B A pN f p1 N qq for the Higgs and Coulomb branch constructions of the Slodowy slice to the trivial nilpotent orbit are identical.

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 A series nilpotent orbits, as elaborated in [1]. Pairs of bi-fundamental fields (and their complex conjugates) 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 Weyl 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 A series Slodowy slices is: g where dµ is the Haar measure for the U pN 1 q b . . . U pN k q product group. Note that the bifundamental fields are symmetrised with the fugacity t, while the HyperKähler quotient ("HKQ") is symmetrised with t 2 .
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 A series Slodowy slice, and this can be simplified further by the balance condition 3.2:ˇˇg For further details of the calculation methodology the reader is referred to the Plethystics Program Literature. The same Hilbert series can in principle also be obtained algebraically by working with matrix generators and relations, as in section 3.5.

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 " 4 superconformal 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 configurations 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 p2q R global symmetry) of a monopole flux is given by applying the following general schema [10] to the quiver diagram: 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 q over a gauge node contains a number of different charges.
The calculation of Hilbert series for Coulomb branches of A type quivers draws on the unitary monopole formula, which follows from specialising 3.9 to unitary gauge groups. Each U pN i q gauge node carries a monopole flux q i " pq i,1 , . . . , q i,N i q comprising one or more monopole charges q i,m . The fluxes are assigned the collective coordinate q " pq 1 , . . . , q r q. Each flavour node carries N f i flavours of zero charge. 10 With these specialisations, the conformal dimension ∆pqq associated with a flux q yields the formula: where (i) the summations are taken over all the monopole charges within the flux q and (ii) the linking pattern between nodes is defined by the A ij off-diagonal A r Cartan matrix terms, which are only non-zero for linked nodes. 11 With conformal dimension defined as above, the unitary monopole formula for a Coulomb branch HS is given by the schema [32]: where: 1. The limits of summation for the monopole charges are 8 ě q i,1 ě . . . q i,m ě . . . q i,N i ě´8 for i " 1, . . . r.
2. The monopole flux over the gauge nodes is counted by the fugacity z " pz 1 , . . . , z r q, where the z i are fugacities for the simple roots of A r .
3. The monomial z q combines the monopole fluxes q i into total charges for each z i and is expanded as z q " 10 Flavour nodes may also carry non-zero charges, although these are not required by the Slodowy slice (or nilpotent orbit) constructions. 11 For theories with simply laced quivers of ADE type, A ij " 0 or´1, for i ‰ j.

The term P
U pN q q encodes the degrees d i,j of the Casimirs of the residual U pN q symmetries that remain at the gauge nodes under a monopole flux q: Recalling that a U pN q group has Casimirs of degrees 1 through N , the residual symmetries can be determined as in [32]. 12 Alternatively, the residual symmetries for a flux q i can be fixed from the sub-group of U pN i q identified by the Dynkin diagram formed by those monopole charges that equal their successors tq i,m : q i,m " q i,m`1 u, (or equivalently, correspond to zero Dynkin labels).
The exact calculation of a Coulomb branch HS can be carried out by evaluating 3.11 as a geometric series over each sub-lattice of monopole charges q, for which both conformal dimension ∆pqq and the symmetry factors P U pN q q are 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.

Hilbert Series
The Hilbert series of the Slodowy slices of algebras A 1 to A 4 , calculated as above, are summarised in table 2. 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 P LrHSs, using character notation rn 1 , . . . , n r s to label A r irreps. Symmetrisation of these generators using the P E 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 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 S N ,p1 N q has the same Hilbert series as the nilpotent cone, (ii) the slice to the sub-regular orbit S N ,pN´1,1q has the Hilbert series of a Kleinian singularity of typeÂ N´1 , and (iii) the slice to the maximal nilpotent orbit S N ,pN q is trivial.
2. As expected, the Slodowy slices S N ,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 F into G.
4. Complex representations always appear combined with their conjugates to give real representations.
5. The adjoint maps often contain singlet generators at even powers of t up 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, although these also raise several new intricacies, as will be seen in section 4.

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.

Fundamental Decomposition
From 3.3, it follows that the character of the fundamental representation of G decomposes into fundamental representations of a unitary product group: where rns ρ are irreps of the SU p2q associated with the nilpotent orbit embedding ρ, and the U p1q charges q i on the flavour nodes satisfy the overall gauge condition 13 Once this decomposition has been identified, the mapping of the adjoint of G into matrix generators follows, by taking the product of the fundamental and antifundamental characters, and eliminating a singlet. This can be checked against the adjoint partition ρ : χ G adjoint .

Orbit
Dimension Unrefined HS pnq q denotes the character of the D 1 " SOp2q reducible representation q n`q´n of U p1q.
pnq q denotes the character of the D 1 " SOp2q reducible representation q n`q´n of U p1q.

Generators from Quiver Paths
Alternatively the operators can be read from a quiver of type B A pN f pρqq, following the prescription: 1. Draw the chiral multiplets explicitly as arrows in the quiver: 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 P ij paq, defined as below.
4. The operator P ij paq transforms under the fundamental representation of U pN f i q and the antifundamental representation of U pN f j q and sits on an irrep of SU p2q R with spin s " A{2, where A is the number of arrows in the path that defines P ij paq. 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 t A .
5. Therefore, there is a one to one correspondence between operators P ij paq and irreducible representations in the decomposition of the adjoint representation of A k in 2.10.
Definition P ij paq: Let P ij paq be an operator P ij paq with i, j P t1, 2, . . . , ku and a P t1, 2, . . . , minpi, jqu. P ij p1q is defined as the operator formed by products of operators represented by arrows in the shortest possible path that starts at node N f i and ends at node N f j (note that i and j could be equal). P ij p2q represents a path that differs from P ij p1q only in that it has been extended to incorporate the arrows between the gauge nodes N minpi,jq and N minpi,jq´1 . P ij p3q differs from P ij p2q in that it also includes arrows between the gauge nodes N minpi,jq´1 and N minpi,jq´2 . In this way P ij paq is defined recursively as an extension of P ij pa´1q.
Example 1. Let us start with the balanced A 3 quiver based on the fundamental partition ρ " p2, 1 2 q, whose Higgs branch is the the Slodowy slice S N ,p2,1 2 q to the nilpotent orbit Ar101s. The quiver is: From table 2, the Hilbert series is: g To obtain this using the prescription in section 3.5.1, we first identify the fugacity map for the group decomposition using 3.13: (3.17) Next the irreps rns ρ of SU p2q ρ are mapped to the fugacity t n`2 , giving the generators: Subtracting the relations´t 4´t6´t8 , corresponding to Casimirs of A 3 , we obtain:
The generators in 3.19 can be understood as operators from paths in the quiver 3.15: P ij paq Quiver Path Generator Table 4. Generators for Slodowy Slice to Ar101s.
The irrep of each generator corresponds with the flavor nodes where the path starts and ends. The U p1q fugacity q " q 1 {q 2 . The exponent of the fugacity t corresponds to the length of the path.
Example 2. Now consider the balanced quiver based on the A 4 partition p3, 2q, whose Higgs branch is the the Slodowy slice S N ,p3,2q to the nilpotent orbit Ar1111s: The group decomposition is: The paths in the quiver can be used to predict the generators in table 5. Subtracting relations´ř 5 i"1 t 2i , corresponding to the special condition in 3.21, which eliminates one of the U p1q symmetries, and the Casimirs of A 4 , and substituting q for q 2 {q 3 gives the expected P LrHSs:

P Lrg
HiggsrB A pN f p3,2qqs HS in accordance with table 2.

Matrices and Relations
In this section we offer a reinterpretation of the previous results for Slodowy slices S N ,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 S N ,p1 N q " N and that of the Kleinian singularity S N ,pN´1,1q " C 2 {Z N . First, let us remember that the Kleinian singularity S N ,pN´1,1q " C 2 {Z N can be defined as the set of points parametrized by three complex variables x, y, z P C, subject to one relation: x N " yz. Secondly, the nilpotent cone S N ,p1 N q " N can be defined as a set of complex variables arranged in a NˆN matrix M P C NˆN , subject to the following relations: trpM p q " 0 @p " 1, 2, . . . , N. (3.24) We want to show that a Slodowy slice S N ,ρ can be viewed as an intermediate case between these two descriptions. In order to do this we build examples of varieties P ij paq Quiver Path Generator Table 5. Generators for Slodowy Slice to Ar1111s.
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 P ij paq found in the balanced quivers B A pN f pρqq. For example, let us study the balanced quiver whose Higgs branch is the Slodowy slice S N ,p2,1 3 q : (3.25) One can assemble the generators P ij paq into three different complex matrices M , A and B of dimensions 3ˆ3, 3ˆ1 and 1ˆ3 respectively. Let us show how this can be done explicitly. There are six paths of the form P ij paq: P 11 p1q, P 22 p1q, P 22 p2q, P 12 p1q, P 21 p1q. Out of these six sets of operators P 22 p1q can be removed by the relation´t 2 that removes the center of mass and P 22 p2q by the first Casimir invariant relation´t 4 . This means that there is a remaining set of generators transforming in the following irreps: P 11 p1q Ñ pr1, 1s`r0, 0sqt 2 , P 12 p1q Ñ pr1, 0sqqt 3 , (3.26) One can now assemble these generators in three complex matrices that transform in the usual way under the global symmetry U p3q: The chiral ring is then parametrized by the set of all matrices tM, A, Bu, subject to one relation at order t 6 , another relation at order t 8 and a final relation at order t 10 . These relations are invariant under the global U p3q symmetry. One can choose the following set of relations: 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 " p1´t 6 qp1´t 8 qp1´t 10 q p1´t 2 q 9 p1´t 3 q 6 .
(3.31) This is the same Hilbert series as that of the variety S N ,p2,1 3 q computed in table 2.
In tables 6 and 7 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 S N ,ρ . Note that we rewrite the Kleinian singularity in terms of 1ˆ1 matrices, to clarify the connection with the algebraic description of the other Slodowy slices.

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 L B{C{D pσq consist of a B, C or D series flavour node of vector irrep dimension N 0 connected to an alternating linear chain of pSqO{U SppN i q gauge nodes of non-increasing vector dimension. For a subset of these linear quivers, the decrements, σ i " N i´1´Ni , between nodes constitute an ordered partition of N 0 , σ " tσ 1 , . . . , σ k u, where σ i ě σ i`1 and ř k i"1 σ i " N 0 . More generally, however, the σ i form a sequence of non-negative integers, subject to ř k i"1 σ i " N 0 , and to selection rules, such that U Sp nodes of odd dimension do not arise. Recall, the nilpotent orbits of a BCD algebra correspond to a subset of the partitions ρ of N , once these have been subjected to selection rules, 14 and linear quivers L B{C{D pρ T q provide a complete set of Higgs branch constructions. Also, balanced quivers B B{C{D pN f q provide Coulomb branch constructions, using the O{U Sp 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 3d mirror symmetry, as discussed further in section 5. Many of these linear quivers have "Higgs equivalent" quivers, L B{C{D pσq, with a different choice of orthogonal gauge node dimensions, but the same Higgs branches; these are generally described by sequences σ rather than partitions ρ T : a U Sp´O´U Sp subchain with the sub-partition p. . . , n, n, . . .q has the Higgs equivalent sequence p. . . , σ i , σ i`1 , . . .q " p. . . , n´1, n`1, . . .q, in which the vector dimension of the central O node is increased by 1 [1]. Returning to Slodowy slices, the roles of these quiver types are essentially reversed: balanced quivers B B{C{D provide a complete set of Higgs branch refined Hilbert series constructions, while linear quivers L B{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 figure 4. . BCD linear and balanced quiver types. In the linear quivers L BC , L CD and L DC , the ranks and fundmental dimensions of the gauge nodes (blue) are in non-increasing order L to R and the quivers are in the form of alternating B´C or D´C chains. In the balanced quivers, B B{C{D , the gauge nodes (blue) inherit their balance, taking account of attached gauge and flavour nodes (red), from a quiver for the nilpotent cone. Nodes labelled C r represent the group U Spp2rq. Nodes labelled B r and D r represent SO{Op2r`1q and SO{Op2rq respectively. Nodes labelled BC, BD or DC indicate a group of one of the two types, subject to the alternation rule and to balance.
We refer to the quivers of type L BC , L CD or L DC , which contain pure B´C, C´D or D´C chains, 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 B and D series between O and SO groups. However, it is not easy to identify the global symmetry of the Coulomb branch of a O{U Sp 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 G under the homomorphism ρ.
The balanced quivers B B{C{D pN f pρqq are found via a modification of the A series method explained in section 3.1. Firstly, the SU p2q partition of a BCD series vector representation under ρ can be used to define a vector N f pρq of alternating O{U Sp flavour nodes, similarly to 3.3: (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.  While balance is zero for these A and B series quivers, it alternates as +/-2 on the C and D series quivers. Defining the vector B " pBalancep1q, . . . , Balancepkqq, these canonical quivers obey a generalisation of 3.2: where N is constructed from the dimensions of gauge node vector irreps.
The gauge node balance condition 4.2 can be extended from N to general Slodowy slices S N ,ρ , permitting the calculation of the gauge node vector N from the flavour node vector N f . In effect, the quivers B B{C{D pN f pρqq descend from the canonical linear quivers for N , through a series of transitions that leave the balance vector B invariant. These balanced quivers provide Higgs branch constructions for BCD Slodowy slices. They are tabulated in figures 5 to 10, along with the partitions of the fundamental, the dimensions of the Slodowy slices, and their residual symmetry groups. 15 On the other hand, the identification of linear quivers L B{C{D pσq for Coulomb branch constructions of BCD series Slodowy slices poses a number of complications.
1. There is no bijection between partitions of N and nilpotent orbits of OpN q or U SppN q. So the quiver L B{C{D pρq is valid only for partitions ρ of special nilpotent orbits; in the other cases L B{C{D pρq (unlike L B{C{D pρ T q) would contain U SppN q vectors of odd dimension N .
2. In the case of Coulomb branch constructions, GNO duality [34] is relevant. This indicates that, since the non-simply laced B and C groups are GNO dual to each other, partitions of B type will be necessary to produce quivers whose Coulomb branches generate Slodowy slices of C algebras, and vice versa.

3.
A quiver L B{C{D pρ T q may have several Higgs equivalent quivers L B{C{D pσq, 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.

The Coulomb branches of quivers with
O gauge 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.
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-Vogan map 16 d BV pρq [20], which provides a bijection between the partitions of real vector representations associated with B series special nilpotent orbits and those of pseudoreal vector representations associated with C series special nilpotent orbits. By making use of Higgs equivalences, to select canonical linear quivers of type L BC , L CD or L DC , which can be done for all special nilpotent orbits, the d BV pρq 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.
The specific transformations from the partitions ρ T to the sequences σ are summarised in table 9. Within these; ρ T indicates the transpose of a partition; ρ N ÑN˘1 Coulomb rL DC pσqs Table 9. Coulomb Branch Quiver Candidates for Slodowy Slices indicates incrementing(decrementing) the first(last) term of a partition by 1; ρ B , ρ C , or ρ D indicates collapsing a partition to a lower partition that is a valid B, C, or D partition [6]; | BC or | CD indicates shifting D or B nodes in a linear quiver to a 'Higgs equivalent' quiver that consists purely of B´C or of C´D pairs of nodes. The transformations can be written more concisely as σ " d BV pρq Tˇc anonical . The resulting linear quivers, L CD pσq, L BC pσq and L DC pσq, whose Coulomb branches are candidates for Slodowy slices of BCD groups up to rank 4, are included in figures 5 through 10. The quivers L DC pp1 N qq and B D pN f p1 N qq for the Higgs and Coulomb branch constructions of the Slodowy slice to the trivial nilpotent orbit are the same. These tables also include identified quivers of type D G pN f prd BV pρqsqq, whose Higgs branch Hilbert series match those of B B{C{D pN f pρqq.      Type IIB String Theory Brane Systems Note that all the resulting quivers, presented in figures 5 through 10 represent 3d N " 4 gauge theories that admit an embedding in Type IIB string theory. 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 T ρ σ pGq was introduced to denote the SCFT at the superconformal fixed point. These brane systems and 3d quivers 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.

Higgs Branch Constructions
In the case of the balanced unitary quivers D D pN f q, based on D series Dynkin diagrams, the calculation of Higgs branch Hilbert series proceeds similarly to the A algebras. This leads 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 B B{C{D pN f q, modifications to the A series Higgs branch formula are required. The O{U Sp alternating chains are taken to comprise bifundamental (half) hypermultiplet fields that transform in vector representations r1, 0, . . . , 0s B{D b r1, 0, . . . , 0s C . Also, it is necessary to average the integrations over the disconnected SO and O´components of the O gauge 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]. 17 In other respects, the calculation of the Higgs branch of a balanced BCD quiver follows a similar Weyl integration to the A series. The general Higgs branch formula for BCD series Slodowy slices is: 17 The effect of non-connected O group components is also discussed in [36].
(4.3) In 4.3, G n alternates between OpN q and U SppN q, dµ is the Haar measure for the G 1 pN 1 q b . . . G k pN k q product group, HKQ rG n pN n q, ts is the HyperKahler quotient for a gauge node, and the summation indicates that the calculation is averaged over the non-connected SO and O´components of O gauge groups [1]. The HKQ is given by HKQ rG n pN n q, ts " P E rradjoints Gn , t 2 s, where for the Op2rq´component of an Op2rq group, radjoints Op2rq´" Λ 2 " rvectors Op2rq´‰ . 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 O and U Sp vectors), and each gauge group offsets the generators by HS relations numbering twice the dimension of the gauge group (once for the Weyl integration and once for the HKQ). This gives a rule for the dimensions of a Slodowy slice calculated from a balanced B B{C{D pN f pρqq quiver:ˇˇg where K n " "`1 if G n " B{D 1 if G n " C and 4.2 is used to calculate Npρq.

Coulomb Branch Constructions
While the O{U Sp 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:  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 " pq 1 , . . . , q r q for unitary nodes, s " ps 1 , . . . , s r q for symplectic nodes and o " po 1 , . . . , o r q for orthogonal nodes, the relationships between monopole and integer weight space lattices can be summarised as in table 10.

Group
Monopole Lattice Basis Transformations Dynkin Labels rn 1 , . . . , n r s     Gauge Group ∆pNodeq U prq´ř 1ďiăjďr |q i´qj | B r´ř r i"1 |o i |´ř 1ďiăjďr |o i˘oj | C r´2 ř r i"1 |s i |´ř 1ďiăjďr |s i˘sj | D r´ř 1ďiăjďr |o i˘oj | Table 13. Gauge Node Conformal Dimensions Gauge Groups ∆pBifundamentalq U pr 1 q´U pr 2 q 1 2 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{U Sp{U gauge group identified by the Dynkin diagram formed by those monopole charges that equal their successors (or, equivalently, 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.
O vs SO gauge nodes. Both the characters of vector irreps and symmetry factors depend on whether a D series gauge node is taken as SO or as O. As noted in [28], the Casimirs of an Op2nq symmetry group are the same as those of SOp2n`1q, due to the absence of a Pfaffian in Op2nq (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 QuotientĀpOq 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 Z 2 action. The brane configurations [10,12,15] for linear quivers L B{C{D pσq do not present this effect, and therefore all gauge nodes are SO.
6. Fugacities. In the unitary monopole formula, z in 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{U Sp monopole formula due to the non-unitary gauge groups involved. Thus, while it can be helpful to include fugacities f " pf 1 , . . . , f r q 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 @f i : f i Ñ 1.
In order for a Coulomb branch Hilbert series not to lead to divergences when the fugacities f are set to unity, it is necessary that no sub-lattice of the monopole lattice (other than the origin) should have a conformal dimension of zero (to ensure that the fugacities f only appear as generators when coupled with t k , where k ą 0). A necessary (albeit not always sufficient) condition on O{U Sp 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 (o 1 , or s 1 ) 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 D 2´C1´D1 is assigned a unit conformal dimension of 1 and is a "good" node. Quivers with zero conformal dimension are identified as such in figures 5 through 10. Their Hilbert series clearly do not match those of the Higgs branch constructions for Slodowy slices, and are not tabulated here.
Providing (i) a nilpotent orbit Oρ is special (so that the Barbasch-Vogan map can be uniquely applied), and (ii) that the quiver L BC{CD{DC pσpρqq does not suffer from zero conformal dimension, the O{U Sp monopole formula 4.5 can be used to calculate unrefined Hilbert series for Slodowy slices; these match those calculated on the Higgs branch of B B{C{D pN f pρqq using 3.7.

Hilbert Series
The Hilbert series of the Slodowy slices of algebras B 1 to B 4 , C 1 to C 4 and D 2 to D 4 , calculated 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 B B{C{D pN f pρqq.
Whenever the flavour symmetry groups are from the B or the D series, a choice has to be made between the characters of SOpN q or OpN q´. In the tables, B{D flavour nodes have been taken as SO type, with the exception of B 0 where the Op1q fugacity k i "˘1 has been used (with indices dropped where no ambiguity arises) 18 .
The Hilbert series are presented in terms of their generators, or P LrHSs, using character notation rn 1 , . . . , n r s G to label irreps. Symmetrisation of these generators using the P E 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 S N ,p1 N q 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 typeÂ 2r´1 for the B series, D r`1 for the C series, andD r for the D series, and (iii) the slice to the maximal nilpotent orbit is trivial.
2. The Slodowy slices S N ,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 t up to (twice) the degree of the highest Casimir of G; these generators may be cancelled by one or more Casimir relations.

The global symmetry groups of the Slodowy slice generators include mixed BCD
Lie groups (or A series isomorphisms), as well as finite groups of type B 0 , 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 C n nilpotent orbits with vector partitions of type p2n´k, kq, it was identified in [22] that these isomorphisms with D n`1 extend further down the Hasse diagram: S N ,Cp2n´k,kq " S N ,Dp2n´k`1,k`1q . This occurs due to matching chains of Kraft-Procesi transitions [13] within such slices.
6. We have not attempted an exhaustive analysis of Z 2 factors associated with the choice of SO vs O flavour groups and the ensuing subtleties. Whilst Higgs branch constructions based on the balanced quivers of type B B{C{D pN f pρqq are available for all Slodowy slices, Coulomb branch constructions based on L BC{CD{DC quivers or Higgs branch constructions based on the quivers of type D G pN f q 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 BCD Slodowy slices do not have Coulomb branch constructions as L BC{CD{DC quivers, either because their underlying nilpotent orbits are not special, or due to zero conformal dimension problems under the O{U Sp monopole formula. While the issue of zero conformal dimension (∆ " 0) is less prevalent for low dimension Slodowy slices, the problem is inherent in maximal B r´Cr´Br´1 sub-chains, and so affects many C series Slodowy slices; certain other quivers are also problematic.     Unrefined HS   10   14

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. where rns ρ are bosonic (odd dimension) or fermionic (even dimension) irreps of the SU p2q associated with the nilpotent orbit embedding ρ. The requirement that the partition ρ obeys the BCD selection rules ensures that the U Sp irreps are all of even dimension. Once this decomposition has been identified, the mapping of the adjoint of G into matrix generators 2.8 follows, either by symmetrising the U Sp vector, or by antisymmetrising the O vector. This can be checked against the adjoint partition ρ : χ G adjoint . Note that a choice can be made whether to use the SO form of orthogonal group characters or the O´form.

Generators from Quiver Paths
For orthosymplectic quivers, the method in section 3.5.2 can be applied, with a few changes. An operator P ij paq formed from a path in the quiver is defined identically. However, for orthosymplectic quivers, P ij paq " P ji paq T , and a path yields only one generator when i ‰ j. Other differences follow from the irreducible representations of the operators P ij paq and the gauge group invariants. There are two cases: The set of operators P ij paq gives us all the generators of the chiral ring. The relations are inherited from those of the nilpotent cone N , and for S N ,ρ are always the Casimir invariants of G. Now, an OpN f i q flavour node (of rank ą 0) always contributes (at least) a path P ii p1q of length 2 that starts at OpN f i q, goes to the gauge node U SppN i q and comes back to OpN f i q. Since the gauge node in the middle of the path is U Sp, the operator transforms in the second antisymmetrization Λ 2 rf und.s O " radjoints O . Similarly, a U SppN f i q flavour node always contributes (at least) a path P ii p1q of length 2 that starts at U SppN f i q, goes to the gauge node OpN i q and comes back to U SppN f i q. Since the gauge node in the middle of the path is O, the operator transforms in the second symmetrization Sym 2 rf und.s U Sp " radjoints U Sp . Consequently, the adjoint of every flavour group appears as a generator at path length 2.
Example Consider the balanced quiver based on the partition p2 2 , 1 4 q, whose Higgs branch is the the Slodowy slice S N ,p4,2q to the nilpotent orbit Dr0100s: The decomposition of G to SU p2q b F is: The Hilbert series of the chiral ring of operators in the Higgs branch has generators P ij paq given by the quiver paths in table 21. For D 4 the Casimirs give relations, t 4´2 t 8´t12 , therefore, the PL[HS] read directly from the quiver is:

Matrices and Relations
Finally, in tables 22 to 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 S N ,ρ of B 1 to B 3 nilpotent orbits. The analysis can in principle be continued to higher rank.
p5, 1q 0 -- Table 24. D 2 and D 3 varieties, generated by complex matrices M , N , and A and their relations, which have Hilbert series matching Slodowy slices S N ,ρ . The matrix M "´M T is antisymmetric, N " N T is symmetric and Ω represents a square matrix that is antisymmetric and invariant under the action of U Spp2nq. pf pq denotes the Pfaffian.

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 A 1 to A 5 plus BCD algebras up to rank 4. There are essentially two families of quivers, the balanced unitary type tB A " D A , D D u and the canonically balanced orthosymplectic type tB B{C{D u. 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 A series slices and D series slices of low dimension. The orthosymplectic quivers yield constructions of all BCD Slodowy slices. The global symmetry F of 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 Z 2 symmetries (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 G representations to SU p2q b F follow, via the Higgs branch formula, from the SU p2q homomorphisms into G of 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 A series, the existence of a bijection between partitions and their transposes (the Luztig-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 L A , which are not generally balanced.
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 tL BC , L CD , L DC u 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 directly from the mapping of the adjoint representation of G into SU p2q b F , using 2.11. This mapping follows from the decomposition of the fundamental/vector of G Ñ SU p2q b F under 3.13 or 4.6.
Dualities and 3d Mirror Symmetry The A series findings verify the known 3d mirror symmetry relations 3.4 and 3.5. Under these, linear or balanced quivers based on partitions ρ can be used either for Higgs branch or Coulomb branch constructions; one combination yields a Slodowy slice and the other combination yields a (generally different) dual nilpotent orbit under the Lusztig-Spaltenstein map ρ T , as illustrated in figure 11.  Figure 11. A Series 3d Mirror Symmetry. All constructions give refined Hilbert series for a partition ρ and its dual ρ T under the Lusztig-Spaltenstein map.
The analysis of BCD series 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: and, for D series Dynkin type quivers of Characteristic height 2: O ρ " Coulomb rD D prρsqs , S N ,d BV pρq " Higgs rD D prρsqs , where d BV pρq is the dual partition to ρ under the D series 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 figure 12.
Note that even for these cases there is a further obstruction: the difference between SO and O nodes in the quiver [28,29]. For the A series, 3d mirror 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 figure 12 a different choice needs to be made depending on the branch of the quiver. This is not quite the same as 3d mirror symmetry. On the other hand, there is a pair of SCFTs, T ρ σ pGq and T σ ρ pG _ q [10,18,19], which are predicted to have precisely the two different gauge algebras depicted in one of the diagrams of figure 12: if T ρ σ pGq corresponds to quiver L BC{CD{DC pρ T q, then T σ ρ pG _ q has the quiver B B{C{D pN f pd BV pρqqq, along with the Higgs and Coulomb branches depicted in the same diagram. However, the present results, together with [1,28,29], show that this cannot be the case, since there are factors of Z 2 in the gauge group of the quiver for T ρ σ pGq that differ depending on the branch being computed. This is a very intriguing point that needs to be addressed in future 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, S ρ 1 ,ρ 2 "Ō ρ 1 X S ρ 2 , which include nilpotent orbits S ρ,trivial and Slodowy slices S N ,ρ as limiting cases. 19 . The indications are that Higgs branch methods should provide a fruitful means of analysing such spaces.
Further Work Besides a study of quivers for S ρ 1 ,ρ 2 moduli 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 C r gauge node have been computed. A case that also appears in our study is the quiver rD 2r s´pC r q, where the expected Slodowy slices are formed in quite a surprising way 20 . It remains a challenge to develop such techniques to obtain Coulomb branch calculations for the Slodowy slices of the other quivers with ∆ " 0 in 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. j . We label field (or R-charge) counting variables with t, adding subscripts if necessary. Under the conventions in this paper, the fugacity t corresponds to an Rcharge of 1/2 and t 2 corresponds to an R-charge of 1. We may refer to series, such as 1`f`f 2`. . ., by their generating functions 1{ p1´f q. 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.

Generating Function Notation Definition
Refined HS (Weight coordinates) g G HS px, tq 8 ř n"0 a n pxqt n Refined HS (Simple root coordinates) g G HS pz, tq 8 ř n"0 a n pzqt n Unrefined HS g G HS ptq 8 ř n"0 a n t n " 8 ř n"0 a n p1qt n Table 25. Types of Generating Function