Coulomb branches for rank 2 gauge groups in 3dN=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} gauge theories

The Coulomb branch of 3-dimensional N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} gauge theories is the space of bare and dressed BPS monopole operators. We utilise the conformal dimension to define a fan which, upon intersection with the weight lattice of a GNO-dual group, gives rise to a collection of semi-groups. It turns out that the unique Hilbert bases of these semi-groups are a sufficient, finite set of monopole operators which generate the entire chiral ring. Moreover, the knowledge of the properties of the minimal generators is enough to compute the Hilbert series explicitly. The techniques of this paper allow an efficient evaluation of the Hilbert series for general rank gauge groups. As an application, we provide various examples for all rank two gauge groups to demonstrate the novel interpretation.


Introduction
The moduli spaces of supersymmetric gauge theories with 8 supercharges have generically two branches: the Higgs and the Coulomb branch. In this paper we focus on 3-dimensional N = 4 gauge theories, for which both branches are hyper-Kähler spaces. Despite this fact, the branches are fundamentally different. The Higgs branch M H is understood as hyper-Kähler quotient in which the vanishing locus of the N = 4 F-terms is quotient by the complexified gauge group. The F-term equations play the role of complex hyper-Kähler moment maps, while the transition to the complexified gauge group eliminates the necessity to impose the Dterm constraints. Moreover, this classical description is sufficient as the Higgs branch is protected from quantum corrections. The explicit quotient construction can be supplemented by the study of the Hilbert series, which allows to gain further understanding of M H as a complex space. Classically, the Coulomb branch M C is the hyper-Kähler space where W G is the Weyl group of G and rk(G) denotes the rank of G. However, the geometry and topology of M C are affected by quantum corrections. Recently, the understanding of the Coulomb branch has been subject of active research from various viewpoints: the authors of [1] aim to provide a description for the quantum-corrected Coulomb branch of any 3d N = 4 gauge theory, with particular emphasis on the full Poisson algebra of JHEP08(2016)016 the chiral ring C[M C ]. In contrast, a rigorous mathematical definition of the Coulomb branch itself lies at the heart of the attempts presented in [2][3][4]. In this paper, we take the perspective centred around the monopole formula proposed in [5]; that is, the computation of the Hilbert series for the Coulomb branch allows to gain information on M C as a complex space. Let us briefly recall the set-up. Select an N = 2 subalgebra in the N = 4 algebra, which implies a decomposition of the N = 4 vector multiplet into an N = 2 vector multiplet (containing a gauge field A and a real adjoint scalar σ) and an N = 2 chiral multiplet (containing a complex adjoint scalar Φ) which transforms in the adjoint representation of the gauge group G. In addition, the selection of an N = 2 subalgebra is equivalent to the choice of a complex structure on M C and M H , which is the reason why one studies the branches only as complex and not as hyper-Kähler spaces.
The description of the Coulomb branch relies on 't Hooft monopole operators [6], which are local disorder operators [7] defined by specifying a Dirac monopole singularity for the gauge field, where m ∈ g = Lie(G) and (θ, ϕ) are coordinates on the 2-sphere around the insertion point. An important consequence is that the generalised Dirac quantisation condition [8] exp (2πim) = 1 G (1.4) has to hold. As proven in [9], the set of solutions to (1.4) equals the weight lattice Λ w ( G) of the GNO (or Langlands) dual group G, which is uniquely associated to the gauge group G. For Coulomb branches of supersymmetric gauge theories, the monopole operators need to be supersymmetric as well, see for instance [10]. In a pure N = 2 theory, the supersymmetry condition amounts to the singular boundary condition σ ∼ m 2r for r → ∞ , (1.5) for the real adjoint scalar in the N = 2 vector multiplet. Moreover, an N = 4 theory also allows for a non-vanishing vacuum expectation value of the complex adjoint scalar Φ of the adjoint-valued chiral multiplet. Compatibility with supersymmetry requires Φ to take values in the stabiliser H m of the "magnetic weight" m in G. This phenomenon gives rise to dressed monopole operators. Dressed monopole operators and G-invariant functions of Φ are believed to generate the entire chiral ring C[M C ]. The corresponding Hilbert series allows for two points of view: seen via the monopole formula, each operator is precisely counted once in the Hilbert series -no over-counting appears. Evaluating the Hilbert series as rational function, however, provides an over-complete set of generators that, in general, satisfies relations. In order to count polynomials in the chiral ring, a notion of degree or dimension is required. Fortunately, in a CFT one employs the conformal dimension ∆, which for BPS states agrees with the SU(2) R highest weight. Following [10][11][12][13], the conformal dimension of a BPS bare where R i denotes the set of all weights ρ of the G-representation in which the i-th flavour of N = 4 hypermultiplets transform. Moreover, Φ + denotes the set of positive roots α of the Lie algebra g and provides the contribution of the N = 4 vector multiplet. Bearing in mind the proposed classification of 3d N = 4 theories by [11], we restrict ourselves to "good" theories (i.e. ∆ > 1 2 for all BPS monopoles). If the centre Z( G) is non-trivial, then the monopole operators can be charged under this topological symmetry group and one can refine the counting on the chiral ring.
Putting all the pieces together, the by now well-established monopole formula of [5] reads Here, the fugacity t counts the SU(2) R -spin, while the (multi-)fugacity z counts the quantum numbers J(m) of the topological symmetry Z( G). This paper serves three purposes: firstly, we provide a geometric derivation of a sufficient set of monopole operators, called the Hilbert basis, that generates the entire chiral ring. Secondly, employing the Hilbert basis allows an explicit summation of (1.7), which we demonstrate for rk(G) = 2 explicitly. Thirdly, we provide various examples for all rank two gauge groups and display how the knowledge of the Hilbert basis completely determines the Hilbert series.
The remainder of this paper is organised as follows: section 2 is devoted to the exposition of our main points: after recapitulating basics on (root and weight) lattices and rational polyhedral cones in subsection 2.1, we explain in subsection 2.2 how the conformal dimension decomposes the Weyl chamber of G into a fan. Intersecting the fan with the weight lattice Λ w ( G) introduces affine semi-groups, which are finitely generated by a unique set of irreducible elements -called the Hilbert basis. Moving on to subsection 2.3, we collect mathematical results that interpret the dressing factors P G (t, m) as Poincaré series for the set of H m -invariant polynomials on the Lie algebra h m . Finally, we explicitly sum the unrefined Hilbert series in subsection 2.4 and the refined Hilbert series in 2.5 utilising the knowledge about the Hilbert basis. After establishing the generic results, we provide a comprehensive collection of examples for all rank two gauge groups in section 3-8. Lastly, section 9 concludes.
Before proceeding to the details, we present our main result (2.35) already at this stage: the refined Hilbert series for any rank two gauge group G.
The form of (1.8) is chosen to emphasis that the terms within the curly bracket represent the numerator of the Hilbert series as rational function, i.e. the curly bracket is a proper polynomial in t without poles. On the other hand, the first fraction represents the denominator of the rational function, which is again a proper polynomial by construction.

Preliminaries
Let us recall some basic properties of Lie algebras, cf. [14], and combine them with the description of strongly convex rational polyhedral cones and affine semi-groups, cf. [15]. Moreover, we recapitulate the definition and properties of the GNO-dual group, which can be found in [9,16].
Root and weight lattices of g. Let G be a Lie group with semi-simple Lie algebra g and rk(G) = r. Moreover, G is the universal covering group of G, i.e. the unique simply connected Lie group with Lie algebra g. Choose a maximal torus T ⊂ G and the corresponding Cartan subalgebra t ⊂ g. Denote by Φ the set of all roots α ∈ t * . By the choice of a hyperplane, one divides the root space into positive Φ + and negative roots Φ − . In the half-space of positive roots one introduces the simple positive roots as irreducible basis elements and denotes their set by Φ s . The roots span a lattice Λ r (g) ⊂ t * , the root lattice, with basis Φ s .
Besides roots, one can always choose a basis in the complexified Lie algebra that gives rise to the notion of coroots α ∨ ∈ t which satisfy α (β ∨ ) ∈ Z for any α, β ∈ Φ. Define α ∨ to be a simple coroot if and only if α is a simple root. Then the coroots span a lattice Λ ∨ r (g) in t -called the coroot lattice of g.

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The dual lattice Λ w (g) of the coroot lattice is the set of points µ ∈ t * for which µ(α ∨ ) ∈ Z for all α ∈ Φ. This lattice is called weight lattice of g. Choosing a basis B of simple coroots (2.1) one readily defines a basis for the dual space via The basis elements λ α are precisely the fundamental weights of g (or G) and they are a basis for the weight lattice. Analogous, the dual lattice Λ mw (g) ⊂ t of the root lattice is the set of points m ∈ t such that α(m) ∈ Z for all α ∈ Φ. In particular, the coroot lattice is a sublattice of Λ mw (g).
As a remark, the lattices defined so far solely depend on the Lie algebra g, or equivalently on G, but not on G. Because any group defined via G/Γ for Γ ⊂ Z(G) has the same Lie algebra.
Weight and coweight lattice of G. The weight lattice of the group G is the lattice of the infinitesimal characters, i.e. a character χ : T → U(1) is a homomorphism, which is then uniquely determined by the derivative at the identity. Let X ∈ t then χ(exp (X)) = exp (iµ(X)), wherein µ ∈ t * is an infinitesimal character or weight of G. The weights form then a lattice Λ w (G) ⊂ t * , because the exponential map translates the multiplicative structure of the character group into an additive structure. Most importantly, the following inclusion of lattices holds: Note that the weight lattice Λ w of g equals the weight lattice of the universal cover G. As before, the dual lattice for Λ w (G) in t is readily defined Λ * w (G) := Hom (Λ w (G), Z) = ker t → T X → exp(2πiX) . (2.4) As we see, the coweight lattice Λ * w (G) is precisely the set of solutions to the generalised Dirac quantisation condition (1.4) for G. In addition, an inclusion of lattices holds which follows from dualising (2.3).
GNO-dual group and algebra. Following [9,16], a Lie algebra g is the magnetic dual of g if its roots coincide with the coroots of g. Hence, the Weyl groups of g and g agree. The magnetic dual group G is, by definition, the unique Lie group with Lie algebra g and weight lattice Λ w ( G) equal to Λ * w (G). In physics, G is called the GNO-dual group; while in mathematics, it is known under Langlands dual group.

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Polyhedral cones. A rational convex polyhedral cone in t is a set σ B of the form where B ⊆ Λ ∨ r , the basis of simple coroots, is finite. Moreover, we note that σ B is a strongly convex cone, i.e. {0} is a face of the cone, and of maximal dimension, i.e. dim(σ B ) = r. Following [15], such cones σ B are generated by the ray generators of their edges, where the ray generators in this case are precisely the simple coroots of g.
For a polyhedral cone σ B ⊆ t one naturally defines the dual cone (2.7) One can prove that σ ∨ B equals the rational convex polyhedral cone generated by B * , i.e.
which is well-known under the name (closed) principal Weyl chamber. By the very same arguments as above, the cone σ B * is generated by its ray generators, which are the fundamental weights of g. If d = 0 then H m,0 is hyperplane through the origin, sometimes denoted as central affine hyperplane. A theorem [17] then states: a cone σ ⊂ R n is finitely generated if and only if it is the finite intersection of closed linear half spaces. This result allows to make contact with the usual definition of the Weyl chamber. Since we know that σ B * is finitely generated by the fundamental weights {λ α } and the dual basis is {α ∨ }, one arrives at σ B * = ∩ α∈Φs H + α ∨ ,0 ; thus, the dominant Weyl chamber is obtained by cutting the root space along the hyperplanes orthogonal to some root and selecting the cone which has only positive entries.

Effect of conformal dimension
Next, while considering the conformal dimension ∆(m) as map between two Weyl chambers we will stumble across the notion of affine semi-groups, which are known to constitute the combinatorial background for toric varieties [15].

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Conformal dimensions -revisited. Recalling the conformal dimension ∆ to be interpreted as the highest weight under SU(2) R , it can be understood as the following map Where σ G B * is the cone spanned by the fundamental weights of g, i.e. the dual basis of the simple roots Φ s of g. Likewise, σ SU(2) B * is the Weyl chamber for SU(2) R . Upon continuation, ∆ becomes a map between the dominant Weyl chamber of G and SU(2) R ∆ : By definition, the conformal dimension (1.6) has two types of contributions: firstly, a positive contribution |ρ(m)| for a weight ρ ∈ Λ w (G) ⊂ t * and a magnetic weight m ∈ Λ w ( G) ⊂ t * . By definition Λ w ( G) = Λ * w (G); thus, m is a coweight of G and ρ(m) is the duality paring. Secondly, a negative contribution −|α(m)| for a positive root α ∈ Φ + of g. By the same arguments, α(m) is the duality pairing of weights and coweights. The paring is also well-defined on the entire the cone.
Fan generated by conformal dimension. The individual absolute values in ∆ allow for another interpretation; we use them to associate a collection of affine central hyperplanes and closed linear half-spaces Here, µ ranges over all weights ρ and all positive roots α appearing in the theory. If two weights µ 1 , µ 2 are (integer) multiples of each other, then H µ 1 ,0 = H µ 2 ,0 and we can reduce the number of relevant weights. From now on, denote by Γ the set of weights ρ and positive roots α which are not multiples of one another. Then the conformal dimension contains Q := |Γ| ∈ N distinct hyperplanes such that there exist 2 Q different finitely generates cones (2.13) By construction, each cone σ 1 , 2 ,..., Q is a strongly convex rational polyhedral cone of dimension r, for non-trivial cones, or 0, for trivial intersections. Consequently, each cone is generated by its ray generators and these can be chosen to be lattice points of Λ w ( G). Moreover, the restriction of ∆ to any σ 1 , 2 ,..., Q yields a linear function, because we effectively resolved the absolute values by defining these cones. It is, however, sufficient to restrict the considerations to the Weyl chamber of G; hence, we simply intersect the cones with the hyperplanes defining σ G B * , i.e. (2.14)

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Naturally, we would like to know for which µ ∈ Λ w (G) the hyperplane H µ,0 intersects the Weyl chamber σ G B * non-trivially, i.e. not only in the origin. Let us emphasis the differences of the Weyl chamber (and their dual cones) of G and G: It is possible to prove the following statements: and µ = 0, i.e. µ = α∈Φs g α α where at least one g α = 0, then H µ,0 intersects σ G B * at one of its boundary faces.
i.e. µ = α∈Φs g α α with at least one g α > 0 and at least one Consequently, a weight µ ∈ Λ w (G) appearing in ∆ leads to a hyperplane intersecting the Weyl chamber of G non-trivially if and only if neither µ nor −µ lies in the rational cone spanned by the simple roots Φ s of G.
Therefore, the contributions −|α(m)|, for α ∈ Φ + , of the vector multiplet never yield a relevant hyperplane. From now on, assume that trivial cones C p are omitted in the index set I for p. The appropriate geometric object to consider is then the fan F ∆ ⊂ t defined by the family F ∆ = {C p , p ∈ I} in t. A fan F is a family of non-empty polyhedral cones such that (i) every non-empty face of a cone in F is a cone in F and (ii) the intersection of any two cones in F is a face of both. In addition, the fan F ∆ defined above is a pointed fan, because {0} is a cone in F ∆ (called the trivial cone).
Semi-groups. Although we already know the cone generators for the fan F ∆ , we have to distinguish them from the generators of F ∆ ∩ Λ w ( G), i.e. we need to restrict to the weight lattice of G. The first observation is that are semi-groups, i.e. sets with an associative binary operation. This is because the addition of elements is commutative, but there is no inverse defined as "subtraction" would lead out of the cone. Moreover, the S p satisfy further properties, which we now simply collect, see for instance [17]. Firstly, the S p are affine semi-groups, which are semi-groups that can be embedded in Z n for some n. Secondly, every S p possesses an identity element, here m = 0, and such semi-groups are called monoids. Thirdly, the S p are positive because the only invertible element is m = 0.

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Now, according to Gordan's Lemma [15,17], we know that every S p is finitely generated, because all C p 's are finitely generated, rational polyhedral cones. Even more is true, since the division into the C p is realised via affine hyperplanes H µ i ,0 passing through the origin, the C p are strongly convex rational cones of maximal dimension. Then [15,Prop. 1.2.22.] holds and we know that there exist a unique minimal generating set for S p , which is called Hilbert basis.
The Hilbert basis H(S p ) is defined via where an element is called irreducible if and only if m = x + y for x, y ∈ S p implies x = 0 or y = 0. The importance of the Hilbert basis is that it is a unique, finite, minimal set of irreducible elements that generate S p . Moreover, H(S p ) always contains the ray generators of the edges of C p . The elements of H(S p ) are sometimes called minimal generators. As a remark, there exist various algorithms for computing the Hilbert basis, which are, for example, discussed in [18,19]. For the computations presented in this paper, we used the Sage module Toric varieties programmed by A. Novoseltsev and V. Braun as well as the Macaulay2 package Polyhedra written by René Birkner.
After the exposition of the idea to employ the conformal dimension to define a fan in the Weyl chamber of G, for which the intersection with the weight lattice leads to affine semi-groups, we now state the main consequence: The collection {H(S p ) , p ∈ I} of all Hilbert bases is the set of necessary (bare) monopole operators for a theory with conformal dimension ∆.
At this stage we did not include the Casimir invariance described by the dressing factors P G (t, m). For a generic situation, the bare and dressed monopole operators for a GNO-charge m ∈ H(S p ) for some p are all necessary generators for the chiral ring C[M C ]. However, there will be scenarios for which there exists a further reduction of the number of generators. For those cases, we will comment and explain the cancellations.

Dressing of monopole operators
One crucial ingredient of the monopole formula of [5] are the dressing factors P G (t, m) and this section provides an algebraic understanding. We refer to [14,20,21] for the exposition of the mathematical details used here.
It is known that in N = 4 the N = 2 BPS-monopole operator V m is compatible with a constant background of the N = 2 adjoint complex scalar Φ, provided Φ takes values on the Lie algebra h m of the residual gauge group H m ⊂ G, i.e. the stabiliser of m in G. Consequently, each bare monopole operator V m is compatible with any H minvariant polynomial on h m . We will now argue that the dressing factors P G (t, m) are to be understood as Hilbert (or Poincaré) series for this so-called Casimir-invariance.
Chevalley-Restriction Theorem. Let G be a Lie group of rank l with a semi-simple Lie algebra g over C and G acts via the adjoint representation on g. Denote by P(g) the algebra of all polynomial functions on g. The action of G extends to P(g) and I(g) G denotes JHEP08(2016)016 the set of G-invariant polynomials in P(g). In addition, denote by P(h) the algebra of all polynomial functions on h. The Weyl group W G , which acts naturally on h, acts also on P(h) and I(h) W G denotes the Weyl-invariant polynomials on h. The Chevalley-Restriction Theorem now states where the isomorphism is given by the restriction map p → p| h for p ∈ I(g) G . Therefore, the study of H m -invariant polynomials on h m is reduced to W Hm -invariant polynomials on a Cartan subalgebra t m ⊂ h m .
Finite reflection groups. It is due to a theorem by Chevalley [22], in the context of finite reflection groups, that there exist l algebraically independent homogeneous elements p 1 , . . . , p l of positive degrees d i , for i = 1, . . . , l, such that (2. 19) In addition, the degrees d i satisfy The degrees d i are unique [21] and tabulated for all Weyl groups, see for instance [21, section 3.7]. However, the generators p i are themselves not uniquely determined.
Poincaré or Molien series. On the one hand, the Poincaré series for the I(h) W G is simply given by On the other hand, since h is a l-dimensional complex vector space and W G a finite group, the generating function for the invariant polynomials is known as Molien series [23] . (2.22) Therefore, the dressing factors P G (t, m) in the Hilbert series (1.7) for the Coulomb branch are the Poincaré series for graded algebra of H m -invariant polynomials on h m .
Harish-Chandra isomorphism. In [5], the construction of the P G (t, m) is based on Casimir invariants of G and H m ; hence, we need to make contact with that idea. Casimir invariants live in the centre Z(U(g)) of the universal enveloping algebra U(g) of g. Fortunately, the Harish-Candra isomorphism [24] provides us with Consequently, Z(U(g)) is a polynomial algebra with l algebraically independent homogeneous elements that have the same positive degrees d i as the generators of I(h) W G . It is known that for semi-simple groups G these generators can be chosen to be the rk(G) Casimir invariants; i.e. the space of Casimir-invariants is freely generated by l generators (together with the unity). Figure 1. A representative fan, which is spanned by the 2-dim. cones C

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(2) p for p = 1, . . . , L, is displayed in 1a. In addition, 1b contains a 2-dim. cone with a Hilbert basis of the two ray generators (black) and two additional minimal generators (blue). The ray generators span the fundamental parallelotope (red region).
Conclusions. So far, G (and H m ) had been restricted to be semi-simple. However, in most cases H m is a direct product group of semi-simple Lie groups and U(1)-factors. We proceed in two steps: firstly, U(1) acts trivially on its Lie-algebra ∼ = R, thus all polynomials are invariant and we obtain Secondly, each factor G i of a direct product G 1 ×· · ·×G M acts via the adjoint representation on on its own Lie algebra g i and trivially on all other g j for j = i. Hence, the space of G 1 × · · · × G M -invariant polynomials on g 1 ⊕ · · · ⊕ g M factorises into the product of the I(g i ) G i such that For abelian groups G, the Hilbert series for the Coulomb branch factorises in the Poincaré series G-invariant polynomials on g times the contribution of the (bare) monopole operators. In contrast, the Hilbert series does not factorise for non-abelian groups G as the stabiliser H m ⊂ G depends on m.

Consequences for unrefined Hilbert series
The aforementioned dissection of the Weyl chamber σ G B * into a fan, induced by the conformal dimension ∆, and the subsequent collection of semi-groups in Λ w ( G)/W G provides an immediate consequence for the unrefined Hilbert series. For simplicity, we illustrate the consequences for a rank two example. Assume that the Weyl chamber is divided into a fan generated the 2-dimensional cones C The Hilbert basis H(S (2) p ) for S (2) p := C (2) p ∩ Λ G w contains the ray generators {x p−1 , x p }, such that H(S (1) p ) = {x p }, and potentially other minimal generators u p κ for κ in some finite index set. Although any element s ∈ S (2) p can be generated by {x p−1 , x p , {u p κ } κ }, the representation s = a 0 x p−1 + a 1 x p + κ b κ u p κ is not unique. Therefore, great care needs to be taken if one would like to sum over all elements in S (2) p . A possible realisation employs the fundamental parallelotope see also figure 1b. The number of points contained in P(C However, as known from solid state physics, the discriminant counts each of the four boundary lattice points by 1 4 ; thus, there are d(C (2) p ) − 1 points in the interior. Remarkably, each point s ∈ Int(P(C (2) p )) is given by positive integer combinations of the {u p κ } κ alone. A translation of P(C (2) p ) by non-negative integer combinations of the ray-generators {x p−1 , x p } fills the entire semi-group S (2) p and each point is only realised once. Now, we employ this fact to evaluate the un-refined Hilbert series explicitly. .

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Next, we utilise that the classical dressing factors, for rank two examples, only have three different values: in the (2-dim.) interior of the Weyl chamber W , the residual gauge group is the maximal torus T and P G (t, IntW ) ≡ P 2 (t) = 2 i=1 1 (1−t) . Along the 1-dimensional boundaries, the residual gauge group is a non-abelian subgroup H such that T ⊂ H ⊂ G and the P G (t, , for the two degree b i Casimir invariants of H. At the (0-dim.) boundary of the boundary, the group is unbroken and P G (t, 0) ≡ P 0 (t) = 2 i=1 1 (1−t d i ) contains the Casimir invariants of G of degree d i . Thus, there are a few observations to be addressed.
1. The numerator of (2.28), which is everything in the curly brackets {. . .}, starts with a one and is a polynomial with integer coefficients, which is required for consistency.
2. The denominator of (2.28) is given by P G (t, 0)/ L p=0 (1 − t ∆(xp) ) and describes the poles due to the Casimir invariants of G and the bare monopole (x p , ∆(x p )) which originate from ray generators x p .
3. The numerator has contributions ∼ t ∆(xp) for the ray generators with pre-factors P 1 (t) P 0 (t) − 1 for the two outermost rays p = 0, p = L and pre-factors P 2 (t) P 0 (t) − 1 for the remaining ray generators. None of the two pre-factors has a constant term as P i (t → 0) = 1 for each i = 0, 1, 2. Also deg(1/P 0 (t)) ≥ deg(1/P 1 (t)) ≥ deg(1/ P 2 (t)) = 2 and is a polynomial for any rank two group. For the examples considered here, we also obtain for some k 1 , k 2 ∈ N. In summary, ( P G (t,xp) P G (t,0) − 1)t ∆(xp) describes the dressed monopole operators corresponding to the ray generators x p . 4. The finite sums s∈Int(P(C (2) p )) t ∆(s) are entirely determined by the conformal dimensions of the minimal generators u p κ .
5. The first contributions for the minimal generators u p κ are of the form which then comprise the bare and the dressed monopole operators simultaneously.
6. If C p )) in (2.28) is zero, as the interior is empty. Also indicated by d(C (2) p ) = 1.

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In conclusion, the Hilbert series (2.28) suggests that ray generators are to be expected in the denominator, while other minimal generators are manifest in the numerator. Moreover, the entire Hilbert series is determined by a finite set of numbers: the conformal dimensions of the minimal generators {∆(x p ) | p = 0, 1, . . . , L} and {{∆(u . . , L} as well as the classical dressing factors. Moreover, the dressing behaviour, i.e. number and degree, of a minimal generator m is described by the quotient P G (t, m)/P G (t, 0). Consolidating evidence for this statement comes from the analysis of the plethystic logarithm, which we present in appendix A. Together, the Hilbert series and the plethystic logarithm allow a better understanding of the chiral ring.
We illustrate the formula (2.28) for the two simplest cases in order to hint on the differences that arise if d(C (2) p ) > 1 for cones within the fan.
Example: one simplicial cone Adapting the result (2.28) to one cone C . Example: one non-simplicial cone Adapting the result (2.28) to one cone C 1 with Hilbert basis {x 0 , x 1 , {u κ }}, fundamental parallelotope P, and discriminant d > 1, we find

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where we assumed each component J i (m) to be a linear function in m. By the very same arguments as in (2.28), one can evaluate the refined Hilbert series explicitly and obtains .
The interpretation of the refined Hilbert series (2.35) remains the same as before: the minimal generators, i.e. their GNO-charge, SU(2) R -spin, topological charges J, and their dressing factors, completely determine the Hilbert series. In principle, this data makes the (sometimes cumbersome) explicit summation of (1.7) obsolete.
3 Case: U(1) × U(1) In this section we analyse the abelian product U(1) × U(1). By construction, the Hilbert series simplifies as the dressing factors are constant throughout the lattice of magnetic weights. Consequently, abelian theories do not exhibit dressed monopole operators.

Set-up
The weight lattice of the GNO-dual of U(1) is simply Z and no Weyl-group exists due the abelian character; thus, Λ w ( U(1) × U(1)) = Z 2 . Moreover, since U(1) × U(1) is abelian the classical dressing factors are the same for any magnetic weight (m 1 , m 2 ), i.e.
which reflects the two degree one Casimir invariants.

Two types of hypermultiplets
Set-up. To consider a rank 2 abelian gauge group of the form U(1) × U(1) requires a delicate choice of matter content. If one considers N 1 hypermultiplets with charges (a 1 , b 1 ) ∈ N 2 under U(1) × U(1), then the conformal dimension reads

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However, there exists an infinite number of points {m 1 = b 1 k, m 2 = −a 1 k, k ∈ Z} with zero conformal dimension, i.e. the Hilbert series does not converge due to a decoupled U(1). Fixing this symmetry would reduce the rank to one. Fortunately, we can circumvent this problem by introducing a second set of N 2 hypermultiplets with charges (a 2 , b 2 ) ∈ N 2 , such that the matrix has maximal rank. The relevant conformal dimension then reads Nevertheless, this set-up would introduce four charges and the summation of the Hilbert series becomes tricky. We evade the difficulties by the choice a 2 = b 1 and b 2 = −a 1 . Dealing with such a scenario leads to summation bounds such as Having the summation variable within a floor or ceiling function seems to be an elaborate task with Mathematica. Therefore, we simplify the setting by assuming ∃ k ∈ N such that b 1 = ka 1 . Then we arrive at For this conformal dimension, there exists exactly one point (m 1 , m 2 ) with zero conformal dimension -the trivial solution. Further, by a redefinition of N 1 and N 2 we can consider a 1 = 1.
Hilbert basis. Consider the conformal dimension (3.2f) for a 1 = 1. By resolving the absolute values, we divide Z 2 into four semi-groups which all descend from 2-dimensional rational polyhedral cones. The situation is depicted in figure 2. Next, one needs to compute the Hilbert basis H(S) for each semi-group S. In this example, it follows from the drawing that H(S 1 , red circled points complete the basis for S 2 . Green circles correspond to the remaining minimal generators of S  For a fixed k ≥ 1 we obtain 4(k + 1) basis elements.
Hilbert series. We then compute the following Hilbert series for which we obtain The minimal generators which are ray generators or poles of the Hilbert series.
The minimal generators, labelled by l = 1, 2, . . . , k − 1, which are not ray generators. while the numerator R(t, z 1 , z 2 ) is too long to be displayed, as it contains 1936 monomials. Nonetheless, one can explicitly verify a few properties of the Hilbert series. For example, the Hilbert series (3.6) has a pole of order 4 at t → 1, because R(1, z 1 , z 2 ) = 0 and the derivatives d n dt n R(t, z 1 , z 2 )| t=1 = 0 for n = 1, 2, . . . 9 (at least for z 1 = z 2 = 1). Moreover, the degrees of numerator and denominator depend on the relations between N 1 , N 2 , and k; however, one can show that the difference in degrees is precisely 2, i.e. it matches the quaternionic dimension of the moduli space.
Discussion. Analysing the plethystic logarithm and the Hilbert series, the monopole operators corresponding to the Hilbert basis can be identified as follows: Eight poles of the Hilbert series (3.6) can be identified with monopole generators as shown in table 1a. Studying the plethystic logarithm clearly displays the remaining set, which is displayed in table 1b.

Remark.
A rather special case of (3.2c) is a 2 = 0 = b 1 , for which the theory becomes the product of two U(1)-theories with N 1 or N 2 electrons of charge a or b, respectively. In detail, the conformal dimension is simply JHEP08(2016)016 U(1) Figure 3. Quiver gauge theory whose Coulomb branch is the reduced moduli space of one SO(5)instanton.
such that the Hilbert series becomes For the unrefined Hilbert series, that is z 1 = 1 = z 2 , the rational function HS a U(1) (t, N ) equals the Hilbert series of the (abelian) ADE-orbifold C 2 /Z a·N , see for instance [25]. Thus, the U(1)×U(1) Coulomb branch is the product of two A-type singularities.
Quite intuitively, taking the corresponding limit k → 0 in (3.6) yields the product which are U(1) theories with N 1 and N 2 electrons of unit charge. The unrefined rational functions are the Hilbert series of Z N 1 and Z N 2 singularities in the ADE-classification. From figure 2 one observes that in the limit k → 0 the relevant rational cones coincide with the four quadrants of R 2 and the Hilbert basis reduces to the cone generators.

Reduced moduli space of one SO(5)-instanton
Consider the Coulomb branch of the quiver gauge theory depicted in figure 3 with conformal dimension given by Instead of associating (3.10) with the quiver of figure 3, one could equally well understand it as a special case of a U(1) 2 theory with two different hypermultiplets (3.2c).
Hilbert basis. Similar to the previous case, the conformal dimensions induces a fan which, in this case, is generated by four 2-dimensional cones 3 . Blue circled lattice points complete the bases for S The intersection with the Z 2 lattice defines the semi-groups S (2) p := C (2) p ∩ Z 2 for which we need to compute the Hilbert bases. Figure 4 illustrates the situation and we obtain Figure 5. Quiver gauge theory whose Coulomb branch is the reduced moduli space of one SU(3)instanton.
Symmetry enhancement. The information conveyed by the Hilbert basis (3.12), the Hilbert series (3.13), and the plethystic logarithm (3.14) is that there are eight minimal generators of conformal dimension one which, together with the two Casimir invariants, span the adjoint representation of SO (5). It is known [25,26] that (3.13) is the Hilbert series for the reduced moduli space of one SO(5)-instanton over C 2 .

Reduced moduli space of one SU(3)-instanton
The quiver gauge theories associated to the affine Dynkin diagramÂ n have been studied in [5]. Here, we consider the Coulomb branch of theÂ 2 quiver gauge theory as depicted in figure (5) and with conformal dimension given by The intersection with the Z 2 lattice defines the semi-groups S p ∩ Z 2 for which we need to compute the Hilbert bases. Figure 6 illustrates the situation. We compute the Hilbert bases to read Hilbert series. for j = 1, . . . , 6. The black circled points denote the ray generators, which coincide with the minimal generators.

PL(HS
Symmetry enhancement. The information conveyed by the Hilbert basis (3.17), the Hilbert series (3.18), and the plethystic logarithm (3.19) is that there are six minimal generators of conformal dimension one which, together with the two Casimir invariants, span the adjoint representation of SU (3). As proved in [5], the Hilbert series (3.18) can be resumed as In this section we aim to consider two classes of U(2) gauge theories wherein U(2) ∼ = SU(2)×U(1), i.e. this is effectively an SU(2) theory with varying U(1)-charge. As a unitary group, U(2) is self-dual under GNO-duality.

Set-up
To start with, let consider the two view points and elucidate the relation between them.
Moreover, the Weyl-group is S 2 and acts via permuting the two Cartan generators; consequently, (2), we need to find the weight lattice of the GNO-dual, i.e. find all solutions to the Dirac quantisation condition, see for instance [9]. Since we consider the product, the exponential in (1.4) factorises in exp(2πi n T U(1) ) and exp(2πi m T SU(2) ), where the T 's are the Cartan generators. Besides the solution (n, m) corresponding to the weight lattice of U(1) × SO (3), there exists also the solution for which both factors are equal to −1. The action of the Weyl-group S 2 restricts then to non-negative m i.e.
Relation between both. To identify both views with one another, we select the U(1) as diagonally embedded, i.e. identify the charges as follows: The two classes of U(2)-representations under consideration in this section are for a ∈ N 0 . Following (4.2), we define the fugacities q := √ y 1 y 2 for U(1) and x := y 1 y 2 for SU(2), (4.4) and consequently observe , (4.5a) , (4.5b)
Dressing factors. Lastly, the calculation employs the classical dressing function as presented in [5]. (Note that we rescaled t to be t 2 for later convenience.) Following the discussion of appendix A, monopoles with m = 0 have precisely one dressing by a U(1) In contrast, there are no dressed monopole operators for m = 0.

N hypermultiplets in the fundamental representation of SU(2)
The conformal dimension for a U(2) theory with N hypermultiplets transforming in [1, a] is given as such that the Hilbert series is computed via where the ranges of n, m have been specified above. Here we use the fugacity t 2 instead of t to avoid half-integer powers.
Hilbert basis. The conformal dimension (4.7) divides Λ w (U(2))/S 2 into semi-groups via the absolute values |m|, |(2a + 1)n + m|, and |(2a + 1)n − m|. Thus, there are three semi-groups Figure 7. The Weyl-chamber for the example a = 4. The black circled lattice points are the ray generators. The blue circled lattice points complete the Hilbert basis (together with two ray generators) for S Hilbert series. Computing the Hilbert series yields The Hilbert series (4.11) has a pole of order 4 at t → 1, because R(t = 1, z) = 0 and d n dt n R(t, z)| t=1 = 0 for n = 1, 2, 3. Hence, the moduli space is of (complex) dimension 4. As a comment, the additional (1 − t 2 )-term in the denominator can be cancelled with a corresponding term in the numerator either explicitly for each a = fixed or for any a, but the resulting expressions are not particularly insightful.
Discussion. The four poles of the Hilbert series (4.11), which are graded as z ±2 and z ±1 , can be identified with the four ray generators (0, ±1) and (a + 1 2 , ± 1 2 ), i.e. they correspond to bare monopole operators. In addition, the bare monopole operator for the minimal generator (1, 0) is present in the denominator (4.11b), too.
In contrast, the family of monopoles {(l + 1 2 , ± 1 2 ) , l = 0, 1, . . . , a − 1} is not directly visible in the Hilbert series, but can be deduced unambiguously from the plethystic logarithm. These monopole operators correspond the minimal generators of S (2) ± which are not ray generators. Table 2 provides as summary of the monopole generators and their properties. As a remark, the family of monopole operators (l + 1 2 , ± 1 2 ) is not always completely present in the plethystic logarithm. We observe that l-th bare operator is a generator if N ≥ 2(a − l + 1), while the dressing of the l-th object is a generator if N > 2(a − l + 1). The reason for the disappearance lies in a relation at degree ∆(1, 0) + ∆(a + 1 2 , ± 1 2 ) + 2, which coincides with ∆(l + 1 2 , ± 1 2 ) for N − 1 = 2(a − l + 1), such that the terms cancel in the PL. (See also appendix A.) Thus, for large N all above listed objects are generators.

Case: a = 0, complete intersection
For the choice a = 1, we obtain the Hilbert series for the 2-dimensional fundamental representation [1,0] which agrees with the results of [5]. Let us comment on the reduction of generators compared to the Hilbert basis (4.10). The minimal generators have conformal dimensions 2∆( 1 2 , ± 1 2 ) = N −2, 2∆(1, 0) = 2N −4, and 2∆(0, ±1) = 2N . Thus, (1, 0) is generated by ( 1 2 , ± 1 2 ) and (0, ±1) are generated by utilising the dressed monopoles of ( 1 2 , ± 1 2 ) and suitable elements in their Weyl-orbits. Figure 8. The Weyl-chamber for odd a, here with the example a = 3. The black circled lattice points correspond to the ray generators originating from the fan. The blue/red circled points are the remaining minimal generators for S 2,± , respectively. Similarly, the orange/green circled point are the generators that complete the Hilbert basis for S

N hypermultiplets in the adjoint representation of SU(2)
The conformal dimension for a U(2)-theory with N hypermultiplets transforming in the adjoint representation of SU(2) and arbitrary even U(1)-charge is given by Already at this stage, one can define the four semi-groups induced by the conformal dimension, which originate from 2-dimensional cones It turns out that the precise form of the Hilbert basis depends on the divisibility of a by 2; thus, we split the considerations in two cases: a = 2k − 1 and a = 2k.

Case: a = 1 mod 2
Hilbert basis. The collection of semi-groups (4.14) is depicted in figure 8. As before, we compute the Hilbert basis H for each semi-group of the minimal generators.

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Hilbert series. The computation of the Hilbert series yields Inspection of the Hilbert series (4.16) reveals that it has a pole of order 4 as t → 1 because one explicitly verifies Discussion. The denominator of the Hilbert series (4.16) displays poles for the five bare monopole operators (0, ±1), (2k, ±1), and (1, 0), which are ray generators and charged under U(1) J as ±2, ±2, and 0, respectively. The remaining operators, corresponding to the minimal generators which are not ray generators, are apparent in the analysis of the plethystic logarithm. The relevant bare and dressed monopole operators are summarised in table 3.
The plethystic logarithm, moreover, displays that not always all monopoles of the family (j + 1 2 , ± 1 2 ) are generators (in the sense of the PL). The observation is:  Figure 9. The Weyl-chamber for a = 0 mod 2, here with the example a = 4. The black circled lattice points correspond to the ray generators originating from the fan. The blue/red circled points are the remaining minimal generators for S 2,± , respectively.
then the j-th operator (bare as well as dressed) is truely a generator in the PL. The reason behind lies in a relation at degree ∆(k − 1 2 , ± 1 2 )+∆(1, 0), which coincides with ∆(j + 1 2 , ± 1 2 ) for k − j = N . (See also appendix A.) Hence, for large enough N all above listed operators are generators.

Case: a = 0 mod 2
Hilbert basis. The diagram for the minimal generators is provided in figure 9. Again, the appearing (bare) monopoles correspond to the Hilbert basis of the semi-groups. Table 4. Summary of the monopole operators for even a.
Hilbert series. The computation of the Hilbert series for this case yields (4.18c) The Hilbert series (4.18) has a pole of order 4 as t → 1 because one can explicitly verify that R(t = 1, z, N ) = 0, d dt R(t, z, N )| t=1 = 0, and d n dt n R(t, z, N )| t=1,z=1 = 0 for n = 2, 3.
Discussion. The five monopoles corresponding to the ray generators, i.e. (0, ±1), (k + 1 2 , ± 1 2 ), and (1, 0), appear as poles in the Hilbert series (4.18) and are charged under U(1) J as ±2, ±1, and 0, respectively. The remaining minimal generator can be deduced by inspecting the plethystic logarithm. We summarise the monopole generators in table 4. Similarly to the case of odd a, the plethystic logarithm displays that not always all monopoles of the family (j + 1 2 , ± 1 2 ) are generators. The observation is: if k − j + 1 ≥ N then the j-th bare operator is a generator in the PL, while for k − j + 2 ≥ N then also the dressing of the j-th monopole is a generator. The reason behind lies, again, in a relation at degree ∆(k − 1 2 , ± 1 2 ) + ∆(1, 0) + 2, which coincides with ∆(j + 1 2 , ± 1 2 ) for k − j = N . (See also appendix A.) Hence, for large enough N all above listed operators are generators.

Direct product of SU(2) and U(1)
A rather simple example is obtained by considering the non-interacting product of an SU(2) and a U(1) theory. Nonetheless, it illustrates how the rank two Coulomb branches contain the product of rank one Coulomb branches as subclasses.
As first example, take N 1 fundamentals of SU (2) and N 2 hypermultiplets charged under U(1) with charges a ∈ N. The conformal dimension is given by for m ∈ N and n ∈ Z (4. 19) and the dressing factor splits as such that the Hilbert series factorises The rank one Hilbert series have been presented in [5]. Moreover, HS a U(1) (t, N 2 ) equals the A a·N 2 −1 singularity C 2 /Z a·N 2 ; whereas HS [1] SU(2) (t, N 1 ) is precisely the D N 1 singularity. The second, follow-up example is simply a theory comprise of N 1 hypermultiplets in the adjoint representation of SU (2) and N 2 hypermultiplets charged under U(1) as above. The conformal dimension is modified to and Hilbert series is obtained as Applying the results of [5], HS [2] SU(2) (t, N 1 ) is the Hilbert series of the D 2N 1 -singularity on C 2 . Summarising, the direct product of these SU(2)-theories with U(1)-theories results in moduli spaces that are products of A and D type singularities, which are complete intersections. Moreover, any non-trivial interactions between these two gauge groups, as discussed in subsection 4.2 and 4.3, leads to a very elaborate expression for the Hilbert series as rational functions. Also, the Hilbert basis becomes an important concept for understanding the moduli space.
This section concerns all Lie groups with Lie algebra D 2 , which allows to study products of the rank one gauge groups SO(3) and SU (2), but also the proper rank two group SO(4).

Set-up
Let us consider the Lie algebra D 2 ∼ = A 1 × A 1 . Following [9], there are five different groups with this Lie algebra. The reason is that the universal covering group SO(4) of SO(4) has a non-trivial centre Z( SO(4)) = Z 2 × Z 2 of order 4. The quotient of SO(4) by any of the five JHEP08(2016)016 different subgroups Z( SO (4)) yields a Lie group with the same Lie algebra. Fortunately, working with SO(4) allows to use the isomorphism SO(4) = Spin(4) ∼ = SU(2) × SU (2). We can summarise the setting as displayed in table 5. Here, we employed SU(2) = SO (3) and that for semi-simple groups G 1 , G 2 holds [9]. Moreover, the GNO-charges are defined via the following sublattices of the weight lattice of Spin(4) (see also figure 10) The important consequence of this set-up is that the fan defined by the conformal dimension will be the same for a given representation in each of the five quotients, but the semi-groups will differ due to the different lattices Λ w ( G) used in the intersection. Hence, we will find different Hilbert basis in each quotient group. Nevertheless, we are forced to consider representations on the root lattice as we otherwise cannot compare all quotients.
Dressings. In addition, we have chosen to parametrise the principal Weyl chamber via m 1 ≥ |m 2 | such that the classical dressing factors are given by [5] , Weyl chamber m 1 ≥ |m 2 | Regardless of the quotient SO(4)/Γ, the space of Casimir invariance is 2-dimensional. We choose a basis such that the two degree 2 Casimir invariants stem either from SU (2) or Next, we can clarify all relevant bare and dressed monopole operators for an (m 1 , m 2 ) that is a minimal generator. There are two cases: on the one hand, for m 2 = ±m 1 , i.e. at the boundary of the Weyl chamber, the residual gauge group is either U(1) i × SU(2) j or U(1) i × SO(3) j (for i, j = 1, 2 and i = j), depending on the quotient under consideration. Thus, only the degree 1 Casimir invariant of the U(1) i can be employed for a dressing, as the Casimir invariant of SU(2) j or SO(3) j belongs to the quotient SO(4)/Γ itself. Hence, we get Alternatively, we can apply the results of appendix A and deduce the dressing behaviour at the boundary of the Weyl chamber to be i.e. only one dressed monopole arises. On the other hand, for m 1 > |m 2 | ≥ 0, i.e. in the interior of the Weyl chamber, the residual gauge group is U(1) 2 . From the resulting two degree 1 Casimir invariants one constructs the following monopole operators: (5.5b) 1 In a different basis, the Casimir invariants for SO(4) are the quadratic Casimir and the Pfaffian. Figure 11. The semi-group S (2) and its ray-generators (black circled points) for the quotient Spin(4) and the representation [2, 0].
Using appendix A, we obtain that monopole operator with GNO-charge in the interior of the Weyl chamber exhibit the following dressings P A 1 ×A 1 (t, m 1 , m 2 )/P A 1 ×A 1 (t, 0, 0) = 1 + 2t + t 2 , which agrees with our discussion above.

Representation [2, 0]
The conformal dimension for this case reads Following the ideas outlined earlier, the conformal dimension (5.6) defines a fan in the dominant Weyl chamber. In this example, ∆ is already a linear function on the entire dominant Weyl chamber; thus, we generate a fan which just consists of one 2-dimensional rational cone (4) Hilbert basis. Starting from the fan (5.7) with the cone C (2) , the Hilbert basis for the semi-group S (2) := C (2) ∩ K [0] is simply given by the ray generators Hilbert series. We compute the Hilbert series to

Quotient Spin
which is a complete intersection with 6 generators and 2 relations. The generators are given in table 6. Remark. The Hilbert series (5.9) can be compared to the case of SU(2) with n fundamentals and n a adjoints such that 2N = n + 2n a , cf. [5]. One derives at HS [2,0] Spin(4) (t, N ) = HS

Quotient SO(4)
The centre of the GNO-dual SO(4) is a Z 2 , which we choose to count if (m 1 , m 2 ) belongs to K [0] or K [2] . A realisation is given by In other words, z is a Z 2 -fugacity.
Hilbert basis. The semi-group S (2) := C (2) ∩ K [0] ∪ K [2]. has a Hilbert basis as displayed in figure 12 or explicitly Hilbert series. The Hilbert series for SO(4) is given by which is a rational function with a palindromic polynomial of degree 4N − 2 as numerator, while the denominator is of degree 4N . Hence, the difference in degrees is 2, i.e. the quaternionic dimension of the moduli space. In addition, the denominator (5.13) has a pole of order 4 at t → 1, which equals the complex dimension of the moduli space.
Plethystic logarithm. Analysing the PL yields for N ≥ 3 PL(HS Figure 12. The semi-group S (2) and its ray-generators (black circled points) for the quotient SO(4) and the representation [2,0]. The red circled lattice point completes the Hilbert basis for S (2) .
such that we have generators as summarised in table 7.
Gauging a Z 2 . Although the Hilbert series (5.13) is not a complete intersection, the gauging of the topological Z 2 reproduces the Spin(4) result (5.9), that is The dual group is SU(2) × SO(3) and the summation extends over (m 1 , m 2 ) ∈ K [0] ∪ K [1] . The non-trivial centre Z 2 × {1} gives rise to a Z 2 -action, which we choose to distinguish the two lattices K [0] and K [1] as follows: for (m 1 , m 2 ) ∈ K [1] . Figure 13. The semi-group S (2) for the quotient SO(3) × SU(2) and the representation [2,0]. The black circled points are the ray generators. Hilbert basis. The semi-group S (2) := C (2) ∩ K [0] ∪ K [1] has a Hilbert basis comprised of the ray generators. We refer to figure 13 and provide the minimal generators for completeness: Hilbert series. Computing the Hilbert series and using explicitly the Z 2 -properties of Remark. Comparing to the case of SU(2) with n a adjoints and SO(3) with n fundamentals presented in [5], we can re-express the Hilbert series (5.19) as the product  Figure 14. The semi-group S (2) for the quotient SU(2) × SO(3) and the representation [2,2]. The black circled points are the ray generators.
where the z 1 -grading belongs to SO(3) with N fundamentals. The minimal generator ( 1 2 , 1 2 ) is the minimal generator for SO (3) with N fundamentals, while (1, −1) is the minimal generator for SU (2) with N adjoints.
Hilbert series. Similar to the previous case, employing the Z 2 -properties of z 2 we obtain the following Hilbert series: 23) which is a complete intersection with 6 generators and 2 relations. We summarise the generators in table 9. lattice Remark. Also, the equivalence

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holds, which then also implies Thus, the moduli space is a product of two complete intersections.

Quotient PSO(4)
Taking the quotient with respect to the entire centre of SO(4) yields the projective group PSO(4), which has as GNO-dual Spin(4) ∼ = SU(2) × SU (2). Consequently, the summation extends over the whole weight lattice [3] and there is an action of Z 2 × Z 2 on this lattice, which is chosen as displayed in table 10. [3] has a Hilbert basis that is determined by the ray generators. Figure 15 depicts the situation and the Hilbert basis reads Hilbert series. An evaluation of the Hilbert series yields 27) which is a complete intersection with 6 generators and 2 relations. Table 11 summarises the generators with their properties. Figure 15. The semi-group S (2) and its ray-generators (black circled points) for the quotient PSO(4) and the representation [2,0]. Gauging a Z 2 . Now, we utilise the Z 2 × Z 2 global symmetry to recover the Hilbert series for all five quotients solely from the PSO(4) result. Firstly, to obtain the SO(4) result, we need to average out the contributions of K [1] and K [3] , which is achieved for z 1 → ±1 (we also relabel z 2 for consistence), see also PSO(4) (t, z 1 =1, z 2 =z, N ) + HS [2,0] PSO(4) (t, z 1 = − 1, z 2 =z, N ) . (5.28a) Secondly, a subsequent gauging leads to the Spin(4) result as demonstrated in (5.16), because one averages the K [2] contributions out. Thirdly, one can gauge the other Z 2factor corresponding to z 2 → ±1, which then eliminates the contributions of K [2] and K [3]  (5.28b) Lastly, for obtaining the SU(2) × SO(3) Hilbert series one needs to eliminate the K [1] and K [2] contributions. For that, we have to redefine the Z 2 -fugacities conveniently. One choice is Remark. As for most of the cases in this section, the Hilbert series (5.27) can be written as a product of two complete intersections. Employing the results of [5] for SO(3) with n fundamentals, we obtain HS [2,0] PSO(4) (t, z 1 , z 2 , N ) = HS [1] SO(3) (t, z 1 , n = N ) × HS [1] SO(3) (t, z 1 z 2 , n = N ) .

Representation [2, 2]
Let us use the representation [2,2] to further compare the results for the five different quotient groups. The conformal dimension reads As described in the introduction, the conformal dimension (5.30) defines a fan in the dominant Weyl chamber, which is spanned by two 2-dimensional rational cones The numerator of (5.33) is a palindromic polynomial of degree 14N − 4; while the denominator is a polynomial of degree 14N − 2. Hence, the difference in degree is two, which equals the quaternionic dimension of the moduli space. In addition, denominator of (5.33) has a pole of order four at t = 1, which equals the complex dimension of the moduli space.

Quotient SO(4)
Hilbert basis. The semi-groups S (2) [2] have Hilbert bases which again equal (the now different) ray generators. The situation is depicted in figure 17 and the Hilbert bases are as follows: Hilbert series. The Hilbert series reads HS [2,2] SO(4) (t, z, N ) = The numerator of (5.36) is a palindromic polynomial of degree 10N − 2 (neglecting the dependence on z); while the denominator is a polynomial of degree 10N . Hence, the difference in degree is two equals the quaternionic dimension of the moduli space. Moreover, the denominator has a pole of order four at t = 1, which equals the complex dimension of the moduli space.
± := C (2) [1] have Hilbert bases that go beyond the set of ray generators. We refer to figure 18    which matches the quaternionic dimension of the moduli space. Also, the denominator has a pole of order four at t = 1, which equals the complex dimension of the moduli space.

Quotient SU(2) × SO(3)
Hilbert basis. The semi-groups S (2) [3] have Hilbert bases that go beyond the set of ray generators. Figure 19 depicts the situation and the Hilbert bases are computed to be H(S We observe that the bases (5.39) and (5.42) are related by reflection along the m 2 = 0 axis, which in turn corresponds to the interchange of K [1] and K [3] .
Hilbert series. The Hilbert series reads The numerator of (5.43) is palindromic polynomial of degree 14N − 4; while the denominator is a polynomial of degree 14N − 2. Hence, the difference in degree is two, which equals the quaternionic dimension of the moduli space. In addition, the denominator has a pole of order four at t = 1, which matches the complex dimension of the moduli space. As before, comparing the quotients SO(3) × SU(2) and SU(2) × SO(3) as well as the symmetry of (5.30), it is natural to expect the relationship HS [2,2] SO(3)×SU(2) (t, z 1 , N ) 2 ) (1 + 2t + t 2 ) + t ∆(2,0) (1 + 2t + t 2 ) + . . .  where we can summarise the monopole generators as in table 13. Note the change in GNO-charges in accordance with the use of K [3] instead of K [1] .

Quotient PSO(4)
Hilbert basis. The semi-groups S (2) [3] have Hilbert bases that are determined by the ray generators. Figure The numerator of (5.47) is palindromic polynomial of degree 10N − 2; while the denominator is a polynomial of degree 10N . Hence, the difference in degree is two, which corresponds JHEP08(2016)016 Table 14. The generators for the chiral ring of a PSO(4) gauge theory with matter in [2,2].
to the quaternionic dimension of the moduli space. Similarly to the previous cases, the denominator of (5.47) has a pole of order four at t = 1, which equals the complex dimension of the moduli space.

Representation [4, 2]
The conformal dimension for this case reads As before, the conformal dimension (5.50) defines a fan in the dominant Weyl chamber of, which is spanned by three 2-dimensional cones The numerator of (5.53) is an anti-palindromic polynomial of degree 64N − 10, while the denominator is of degree 64N − 8. Consequently, the difference in degree is two. Moreover, the rational function (5.53) has a pole of order four as t → 1 because R(t=1, N ) = 0, but d dt R(t, N )| t=1 = 0. Plethystic logarithm. Inspecting the PL yields for N ≥ 3 PL(HS [4,2] Spin (4)

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leads to an identification of generators as in table 15. We observe that (2, 0) has only 2 dressings, although we would expect 3. We know from other examples that there should be a relation at 2∆(1, 1) + 2 = 20N − 2 which is precisely the dimension of the second dressing of (2, 0).

Quotient SO(4)
Hilbert basis. The semi-groups S [2] have Hilbert bases as shown in figure The numerator (5.56b) is a palindromic polynomial of degree 64N − 10, while the denominator is of degree 64N − 8. Consequently, the difference of the degree is two. Also, the Hilbert series (5.56) has a pole of order four as t → 1, because R(t=1, z, N ) = 0 and d dt R(t, z, N )| t=1 = 0, but d 2 dt 2 R(t, z, N )| t=1 = 0.
The numerator of (5.60) is an anti-palindromic polynomial of degree 64N − 10, while the denominator is of degree 64N − 8. Thus, the difference in degrees is again 2. In addition, the Hilbert series (5.60) has a pole of order 4 as t → 1, because R(t=1, z 1 , N ) = 0, but d dt R(t, z 1 , N )| t=1 = 0. Plethystic logarithm. Analysing the PL yields

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verfies the set of generators as presented in table 17. The coloured term indicates that we suspect a cancellation between one dressing of ( 5 2 , 1 2 ) and one relation because ∆(  Figure 24. The semi-groups for the quotient SU(2) × SO(3) and the representation [4,2]. The black circled points are the ray generators.
As before, we can try to compare the quotients SO(3)×SU(2) and SU(2)×SO(3). However, due to the asymmetry in m 1 , m 2 or the asymmetry of the fan in the Weyl chamber, the Hilbert series for the two quotients are not related by an exchange of z 1 and z 2 .

Comparison to O(4)
In this subsection we explore the orthogonal group O(4), related to SO(4) by Z 2 . To begin with, we summarise the set-up as presented in [28, appendix A]. The dressing factor P O(4) (t) and the GNO lattice of O(4) equal those of SO (5). Moreover, the dominant Weyl chamber is parametrised by (m 1 , m 2 ) subject to m 1 ≥ m 2 ≥ 0. Graphically, the Weyl chamber is the upper half of the yellow-shaded region in figure 10 with the lattices K [0] ∪ K [2] present. Consequently, the dressing function is given as It is apparent that O(4) has a different Casimir invariant as SO (4), which comes about as the Levi-Civita tensor ε is not an invariant tensor under O(4). In other words, the Pfaffian of SO (4) is not an invariant of O(4). Now, we provide the Hilbert series for the three different representations studied above.

Representation [2, 0]
The conformal dimension is the same as in (5.6) and the rational cone of the Weyl chamber is simply C (2) = Cone ((1, 0), (1, 1)) , (5.71) such that the cone generators and the Hilbert basis for S (2) The upper half-space of figure 12 depicts the situation. The Hilbert series is then computed to read which clearly displays the palindromic numerator, the order four pole for t → 1, and the order two pole for t → ∞, i.e. the difference in degrees of denominator and numerator is two. By inspection of (5.72) and use of the plethystic logarithm PL(HS (5.74)

Representation [2, 2]
The conformal dimension is the same as in (5.30) and the rational cone of the Weyl chamber is still C (2) = Cone ((1, 0), (1, 1)) , (5.75) such that the cone generators and the Hilbert basis for S (2) := C (2) ∩ K [0] ∪ K [2] coincide. The upper half-space of figure 17 depicts the situation. We note that the Weyl chamber for SO(4) is already divided into a fan by two rational cones, while the Weyl chamber for O(4) is not. The computation of the Hilbert series then yields HS [2,2] O(4) (t, N ) = Again, the rational function clearly displays a palindromic numerator, an order four pole for t → 1, and an order two pole for t → ∞, i.e. the difference in degrees of denominator and numerator is two. By inspection of (5.76) and use of the plethystic logarithm PL(HS for N ≥ 2, we can summarise the generators as in table 21. The dressings behave as discussed earlier.

Representation [4, 2]
The conformal dimension is given in (5.50) and the Weyl chamber is split into a fan generated by two rational cones The computation of the Hilbert series then yields HS [4,2] O(4) (t, N ) = , As before, the rational function (5.80) clearly displays a palindromic numerator, an order four pole for t → 1, and an order two pole for t → ∞, i.e. the difference in degrees of denominator and numerator is two. By inspection of (5.80) and use of the plethystic logarithm PL(HS [4,2] O(4) ) = for N ≥ 2, we can summarise the generators as in table 22. The dressing behaviour of (1, 0), (1, 1) is as discussed earlier; however, we need to describe the dressings of (2, 1) and

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(3, 1) as it differs from the SO(4) counterparts. Again, we compute the quotient of the dressing factor of the maximal torus divided by the trivial one, i.e.

Case: USp(4)
This section is devoted to the study of the compact symplectic group USp(4) with corresponding Lie algebra C 2 . GNO-duality relates them with the special orthogonal group SO(5) and the Lie algebra B 2 .

Set-up
For studying the non-abelian group USp(4), we start by providing the contributions of N a,b hypermultiplets in various representations [a, b] of USp(4) to the conformal dimensions wherein i, j = 1, 2, and the contribution of the vector multiplet is given by Such that we will consider the following conformal dimension

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and we can vary the representation content via The Hilbert series is computed as usual where the summation for m 1 , m 2 has been restricted to the principal Weyl chamber of the GNO-dual group SO(5), whose Weyl group is S 2 (Z 2 ) 2 . Thus, we use the reflections to restrict to non-negative m i ≥ 0 and the permutations to restrict to a ordering m 1 ≥ m 2 . The classical dressing factor takes the following form [5]: , m 1 = m 2 = 0 . (6.4)

Hilbert basis
The conformal dimension (6.2a) divides the dominant Weyl chamber of SO(5) into a fan. The intersection with the corresponding weight lattice Λ w (SO(5)) introduces semi-groups S p , which are sketched in figure 26. As displayed, the set of semi-groups (and rational cones that constitute the fan) differ if N 3 = 0. The Hilbert bases for both case are readily computed, because they coincide with the set of ray generators.

Dressings
Before evaluating the Hilbert series, let us analyse the classical dressing factors for the minimal generators (6.5) or (6.6). Firstly, the classical Lie group USp(4) has two Casimir invariants of degree 2 and 4 and can they can be written as Tr 4 , respectively. Again, we employ the diagonal form of the adjoint valued scalar field Φ.
The number and the degrees of dressed monopole operators of charge (2, 1) are consistent with the quotient P USp(4) (t, m 1 > m 2 > 0)/P USp(4) (t, 0, 0) = 1 + 2t + 2t 2 + 2t 3 + t 4 of the dressing factors. For "generic" values of N 1 , N 2 and N 3 the Coulomb branch will be generated by the two Casimir invariants together with the bare and dressed monopole operators corresponding to the minimal generators of the Hilbert bases. However, we will encounter choices of the three parameters such that the set of monopole generators can be further reduced; for example, in the case of complete intersections.

Generic case
The computation for arbitrary N 1 , N 2 , and N 3 yields The numerator (6.10c) is an anti-palindromic polynomial of degree 6N 1 + 8N 2 + 26N 3 − 16; while the denominator is of degree 6N 1 + 8N 2 + 26N 3 − 14. The difference in degrees is 2, which equals the quaternionic dimension of the moduli space. In addition, the pole of (6.10) at t → 1 is of order 4, which matches the complex dimension of the moduli space. For that, one verifies explicitly R(t = 1, N 1 , N 2 , N 3 ) = 0, but d dt R(t, N 1 , N 2 , N 3 )| t=1 = 0. Consequently, the above interpretation of bare and dressed monopoles from the Hilbert series (6.10) is correct for "generic" choices of N 1 , N 2 , and N 3 . In particular, N 3 = 0 for this arguments to hold. Moreover, we will now exemplify the effects of the Casimir invariance in various special case of (6.10) explicitly. There are cases for which the inclusion of the Casimir invariance, i.e. dressed monopole operators, leads to a reduction of basis of monopole generators.
The moduli space is then generated by the Casimir invariants and the bare and dressed monopole operators corresponding to (1, 0), but this is to be understood as a rather "nongeneric" situation. JHEP08(2016)016

Representation [0, 1]
This choice is realised for N 2 = N , and N 1 = N 3 = 0 and the Hilbert series simplifies to 14) The Hilbert series (6.14) has a pole of order 4 at t = 1 as well as a palindromic polynomial as numerator. Moreover, the result (6.14) reflects the expected basis of monopole operators as given in the Hilbert basis (6.6).

Representation [2, 0]
This choice is realised for N 1 = 2N , N 2 = N , and N 3 = 0 and the Hilbert series reduces to Also, the rational function (6.15) has a pole of order 4 for t → 1 and a palindromic numerator. Evaluating the plethystic logarithm yields for all N > 1 PL(HS This proves that bare monopole operators, corresponding to the the minimal generators of (6.6), together with their dressing generate all other monopole operators.
We see, employing the previous results for N > 4, that the bare monopole (2, 1) and the last relation at t 62 coincide. Hence, the term ∼ t 62 disappears from the PL.
• For N = 3 PL(HS [1,1] USp (4) We see, employing again the previous results for N > 4, that the some monopole contributions of (2, 1) and the some of the relations coincide, cf. the coloured terms. Hence, there are, presumably, cancellations between generators and relations.
Summarising, the Hilbert series (6.19) and its plethystic logarithm display that the minimal generators of (6.5) are indeed the basis for the bare monopole operators, and the corresponding dressings generate the remaining operators.
We see that, presumably, one generator and one relation cancel at t 48 .
• For N = 1 PL(HS [3,0] USp (4) Again, we confirm that the minimal generators of the Hilbert basis (6.5) are the relevant generators (together with their dressings) for the moduli space.
7 Case: G 2 Here, we study the Coulomb branch for the only exceptional simple Lie group of rank two.

Set-up
The group G 2 has irreducible representations labelled by two Dynkin labels and the dimension formula reads dim[a, b] = 1 120 (a + 1)(b + 1)(a + b + 2)(a + 2b + 3)(a + 3b + 4)(2a + 3b + 5) . (7.1) In the following, we study the representations given in table 23. The three categories defined are due to the similar form of the conformal dimensions. The Weyl group of G 2 is D 6 and the GNO-dual group is another G 2 . Any element in the Cartan subalgebra h = span(H 1 , H 2 ) can be written as H = n 1 H 1 + n 2 H 2 . Restriction to the principal Weyl chamber is realised via n 1 , n 2 ≥ 0.
We will now exemplify the three different representations.
Evaluating the Hilbert series for N > 3 yields  We observe that the numerator of (7.8) is a palindromic polynomial of degree 4N −5; while, the denominator has degree 4N − 3. Hence, the difference in degree between denominator and numerator is 2, which equals the quaternionic dimension of moduli space. In addition, the Hilbert series (7.8) has a pole of order 4 as t → 1, which matches the complex dimension of the moduli space. As discussed in [5], the plethystic logarithm has the following behaviour: Hilbert basis. According to [5], the monopole corresponding to GNO-charge (1, 0), which has ∆(1, 0) = 4N −10, can be generated. Again, this is due to the specific form (7.7).
The calculation for the Hilbert series is analogous to the previous cases and we obtain (7.15) One readily observes, the numerator of (7.15) is a palindromic polynomial of degree 34N − 10 and the denominator is of degree 34N − 8. Hence, the difference in degree between denominator and numerator is 2, which is precisely the quaternionic dimension of moduli space. Also, the Hilbert series has a pole of order 4 as t → 1, which equals the complex dimension of the moduli space. Having in mind the minimal generators (7.5), the appearing objects in (7.15) can be summarised as in table 25.

Category 2
Hilbert basis. The representations [1,1], [0, 2], and [3,0] have schematically conformal dimensions of the form for a j , b j ∈ N and A j , B 1 , B 2 , C ∈ Z. The novelty of this conformal dimension, compared to (7.3), is the difference |n 1 − n 2 |, i.e. a hyperplane that intersects the Weyl chamber non-trivially. As a consequence, there is a fan generated by two 2-dimensional rational polyhedral cones of φ i and the next G 2 -Casimir C 6 is by four higher in degree and has a complicated structure as well.
We will now exemplify the three different representations.
Computing the Hilbert series provides the following expression The numerator (7.24b) is a anti-palindromic polynomial of degree 198N − 26; whereas the denominator is of degree 198N − 24. Hence, the difference in degree between denominator and numerator is 2, which coincides with the quaternionic dimension of moduli space.
Plethystic logarithm. Although the bare monopole V dress,0 is generically a necessary generator due to its origin as an ray generators of (7.21), not all dressings V dress (1,1) might be independent.
Interpreting the appearing operators leads to a list of chiral ring generators as presented in table 27. The behaviour of the Hilbert series is absolutely identical to the case [1,1], because the conformal dimension is structurally identical. Therefore, we do not provide further details.

Case: SU(3)
The last rank two example we would like to cover is SU(3), for which the computation takes a detour over the corresponding U(3) theory, similar to [5]. The advantage is that we can simultaneously investigate the rank three example U(3) and demonstrate that the method of Hilbert bases for semi-groups works equally well in higher rank cases.

Set-up
In the following, we systematically study a number of SU(3) representation, where we understand a SU(3)-representation [a, b] as an U(3)-representation with a fixed U(1)-charge.
Preliminaries for U (3). The GNO-dual group of U(3), which is again a U(3), has a weight lattice characterised by m 1 , m 2 , m 3 ∈ Z and the dominant Weyl chamber is given JHEP08(2016)016 by the restriction m 1 ≥ m 2 ≥ m 3 , cf. [5]. The classical dressing factors associated to the interior and boundaries of the dominant Weyl chamber are the following: , , m 1 = m 2 = m 3 .
The contributions of N (a,b) hypermultiplets transforming in [a, b] to the conformal dimension are as follows: where i, j = 1, 2, 3. In addition, the contribution of the vector-multiplets reads as Consequently, one can study a pretty wild matter content if one considers the conformal dimension to be of the form

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Preliminaries for SU (3). As noted in [5], the reduction from U(3) to SU(3) (with the same matter content) is realised by averaging over U(1) J , for the purpose of setting m 1 + m 2 + m 3 = 0, and multiplying by (1 − t 2 ), such that Tr(Φ) = 0 for the adjoint scalar Φ. In other words As a consequence, the conformal dimension for SU (3) , m 1 = m 2 = 0 . Following the ideas outline previously, Λ w ( U(3))/W U(3) can be described as a collection of semi-groups that originate from a fan. Since this is our first 3-dimensional example, we provide a detail description on how to obtain the fan. Consider the absolute values |am 1 + bm 2 + cm 2 | in (8.7) as Hesse normal form for the hyperplanes which pass through the origin. Take all normal vectors n j , define the matrices M i,j = ( n i , n j ) T (for i < j) and compute the null spaces (or kernel) K i,j := ker(M i,j ). Linear algebra tell us that dim(K i,j ) ≥ 1, but by the specific form 2 of ∆ we have the stronger condition rk(M i,j ) = 2 for all i < j; thus, we always have dim(K i,j ) = 1. Next, we select a basis vector e i,j of K i,j and check if e i,j or −e i,j intersect the Weyl-chamber. If it does, then it is going to be an edge for the fan and, more importantly, will turn out to be a ray generator (provided one defines e i,j via the intersection with the corresponding weight lattice). Now, one has to define all 3-dimensional cones, merge them into a fan, and, lastly, compute the Hilbert bases. The programs Macaulay2 and Sage are convenient tools for such tasks.
As two examples, we consider the conformal dimension (8.7) for N R = 0 and N R = 0 and preform the entire procedure. That is: firstly, compute the edges of the fan; secondly, define the all 3-dimensional cones and; thirdly, compute the Hilbert bases.

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Case N R = 0: in this circumstance, we deduce the following edges (8.10) All these vectors are on the boundaries of the Weyl chamber. The set of 3-dimensional cones that generate the corresponding fan is given by A computation shows that all four cones are strictly convex, smooth, and simplicial. The Hilbert bases for the resulting semi-groups comprise solely the ray generators From the above, we expect 6 bare monopole operators plus their dressings for a generic theory with N R = 0. Since all ray generators lie at the boundary of the Weyl chamber, the residual gauge groups are U(3) for ±(1, 1, 1) and U(2) × U(1) for the other four GNOcharges.
Case N R = 0: here, we compute the following edges: , Now, we need to proceed and define all 3-dimensional cones that constitute the fan and, in turn, will lead to the semi-groups that we wish to study.
All of the cones are strictly convex and simplicial, but only the cones C p for p = 1, 2, 3, 6, . . . , 13, 16 are smooth. Now, we compute the Hilbert bases for semi-groups S We observe that there are four semi-groups S p for p = 4, 5, 14, 15 for which the Hilbert bases exceeds the set of ray generators by an additional element. Consequently, we expect 16 bare monopoles plus their dressings for a generic theory with N R = 0. However, the dressings exhibit a much richer structure compared to N R = 0, because some minimal generators lie in the interior of the Weyl chamber.

Fan and cones for SU(3)
The conformal dimension (8.7) divides the Weyl chamber of the GNO-dual into two different fans, depending on N R = 0 or N R = 0.
Case N R = 0: for this situation, which is depicted in figure 30a, there are three rays ∼ |m 1 |, |m 1 − m 2 |, |m 1 + 2m 2 | present that intersect the Weyl chamber non-trivially. The corresponding fan is generated by two 2-dimensional cones Case N R = 0: for this circumstance, which is depicted in figure 30b, there are two additional rays ∼ |m 1 −2m 2 |, |m 1 +3m 2 | present, compared to N R = 0, that intersect the Weyl chamber non-trivially. The corresponding fan is now generated by four 2-dimensional cones  Following the description of dressed monopole operators as in [5], we diagonalise the adjointvalued scalar Φ along the moduli space, i.e. diagΦ = (φ 1 , φ 2 , φ 3 ) . Moreover, the Casimir invariants of U(3) can then be written as C j = Tr(Φ j ) = 3 l=1 (φ l ) j for j = 1, 2, 3. We will now elaborate on the possible dressed monopole operators by means of the insights gained in section 2.3 and appendix A.

Dressings for SU(3)
To determine the dressings, we take the adjoint scalar Φ and diagonalise it, keeping in mind that it now belongs to SU(3), that is While keeping in mind that each φ i has dimension one, we can write down the dressings (in the dominant Weyl chamber): (1, 0) can be dressed by two independent U(1)-Casimir invariants, i.e. directly by φ 1 and φ 2 V dress,(0,0) (1,0) such that the dressings have conformal dimension ∆(1, 0) + 1. Next, out of the three degree 2 combinations of φ i , only two of them are independent and we choose them to be having dimension ∆(1, 0)+3. Alternatively, we utilise appendix A and compute the number and degrees of the dressed monopole operators of magnetic charge (1, 0) via the quotient P SU(3) (t 2 , 1, 0)/P SU(3) (t 2 , 0, 0) = 1 + 2t 2 + 2t 4 + t 6 . For the two monopoles of GNO-charge (1, 1) and (2, −1), the residual gauge group is SU(2) × U(1), i.e. the monopoles can be dressed by a degree one Casmir invariant of the U(1) and by a degree two Casimir invariant of the SU(2). These increase the dimensions by one and two, respectively. Consequently, we obtain and similarly

Category
The Hilbert series is then readily computed One can check that R(N F , N A , t = 1, z) = 0 and d n dt n R(N F , N A , t, z)| t=1,z=1 = 0 for n = 1, 2. Thus, the Hilbert series (8.33) has a pole of order 6, which matches the dimension of the moduli space. Moreover, one computes the degree of the numerator (8.33c) to be 12N F + 16N A − 10 and the degree of the denominator (8.33b) to be 12N F + 16N A − 4, such that their difference equals the dimension of the moduli space. The interpretation follows the results (8.12) obtained from the Hilbert bases and we summarise the minimal generators in table 31.
Reduction to SU(3). Following the prescription (8.6), we derive the following Hilbert series: , (8.34a) An inspection yields that the numerator ( [1,0] representation Considering N hypermultiplets in the fundamental representation is on extreme case of (8.4), as N A = 0 = N R . We recall the results of [5] and discuss them in the context of Hilbert bases for semi-groups.
Intermediate step at U(3). The Hilbert series has been computed to read HS [1,0] U (3) (N, t, z . Although the form of the Hilbert series (8.37) is suggestive: it has a pole of order 4 for t → 1 and the numerator is palindromic, there is one drawback: no monopole operator of conformal dimension (2N − 6) exists. Therefore, we provide a equivalent rational function to emphasis the minimal generators:   We see that numerator of (8.43) is a palindromic polynomial of degree 20N − 14; while the degree of the denominator is 20N − 10. Hence, the difference in the degrees is 4, which coincides with the complex dimension of the moduli space. The same holds for the order of the pole of (8.43) at t → 1. The interpretation of the appearing monopole operators, and their dressings, is completely analogous to (8.34) and reproduces the picture concluded from the Hilbert bases (8.12). To be specific, 2∆(1, 0) = 8N − 8 and 2∆(1, 1) = 2∆(2, −1) = 12N − 12.
The Hilbert series (8.45) has a pole of order 6 as t → 1, because R(N, t = 1, z) = 0 and d n dt n R(N, t, z)| t=1 = 0 for n = 1, 2. Therefore, the moduli space is 6-dimensional. Also, the degree of (8.45c) is 52N − 10, while the degree of (8.45b) us 52N − 4; thus, the difference in degrees equals the dimension of the moduli space. As this example is merely a special case of (8.33), we just summarise the minimal generators in table 32. It is apparent that the numerator of (8.46) is a palindromic polynomial of degree 38N − 14; while the degree of the denominator is 38N − 10; hence, the difference in the degrees is 4, which equals the complex dimension of the moduli space. The structure of (8.46) is merely a special case of (8.34), and the conformal dimensions of the minimal generators are 2∆(1, 0) = 14N − 8 and 2∆(1, 1) = 2∆(2, −1) = 24N − 12.
• N R = 1: here, (2, 1) and (3, −1) are independent, but not all of their dressings, as we see  and the numerator R(N, t, z) is with 13492 monomials too long to be displayed. Nevertheless, we checked explicitly that R(N, t = 1, z) = 0 and d n dt n R(N, t, z)| t=1,z=1 = 0 for all n = 1, 2 . . . , 10. Therefore, the Hilbert series (8.54) has a pole of order 6 at t = 1, which equals the dimension of the moduli space. In addition, the degree of R(N, t, z) is 296N −62, while the denominator (8.54b) is of degree 296N − 56; therefore, the difference in degrees is again equal to the dimension of the moduli space.
The Hilbert series (8.54) appears as special case of (8.48) and as such the appearing monopole operators are the same. For completeness, we provide in table 34 the conformal dimensions of all minimal (bare) generators (8.15). The GNO-charge (3, 2, 1) is not apparent in the Hilbert series, but we know it to be present due to the analysis of the Hilbert bases (8.15).
From the examples of section 3-8, we see that (A.10) is at most satisfied for scenarios with just a few generators, but not for elaborate cases. Nevertheless, there are some observations we summarise as follows: • The bare and dressed monopole operators associated to the GNO-charge m are described by P G (t,m) P G (t,0) t ∆(m) . In particular, we emphasis that the quotient of dressing factors provides information on the number and degrees of the dressed monopole operators.
(A.11) where the last equality holds because the order of the Weyl group equals the product of the degrees of the Casimir invariants. Since W Hm ⊂ W G is a subgroup of the finite group W G , Lagrange's theorem implies that |W G | |W Hm | ∈ N holds. The situation becomes obvious whenever m belongs to the interior of the Weyl chamber, because H m = T and thus # dressed monopoles +1 bare monopole • The significance of the PL is limited, as, for instance, a positive contribution ∼ t ∆(X 1 ) can coincide with a negative contribution ∼ t ∆(X 2 )+∆(X 3 ) , but this does not necessarily imply that the object of degree ∆(X 1 ) can be generated by others. The situation becomes clearer if there exists an additional global symmetry Z( G) on the moduli space. The truncated PL for (2.35) is obtained from (A.9) by the replacement t ∆(X) → z J(X) t ∆(X) . (A.13) Then the "syzygy" z J(X 2 +X 3 ) t ∆(X 2 )+∆(X 3 ) can cancel the "generator" z J(X 1 ) t ∆(X 1 ) only if the symmetry charges agree z J(X 1 ) = z J(X 2 +X 3 ) , in addition to the SU(2) R iso-spin.
Lastly, we illustrate the truncation with the two simplest examples:
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