Coulomb branch Hilbert series and Hall-Littlewood polynomials

There has been a recent progress in understanding the chiral ring of 3d $\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_\rho(G)$ theories in terms of Hall-Littlewood polynomials. Here $G$ is a classical group and $\rho$ is a certain partition related to the dual group of $G$. The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_\rho(G)$ theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.

) 45 C.2 T (4,4) (SO (8)) and the very even partition (4,4) 47 C.3 T (3,3,1,1) (SO (8)) and T (3,2,2,1) (SO(8)) 49 C.4 T (3,1,1) (U Sp(4)) and T (2,2) (SO(5)) 52 C.5 T (2,2,1) (U Sp(4)) and T (2,1,1) (SO (5) Identifying the chiral ring and moduli space on the Coulomb branch of an N = 4 supersymmetric gauge theory in 2 + 1 dimensions has been a long standing problem. On a generic point of the Coulomb branch, the triplet of scalars in the N = 4 vector multiplets acquires a vacuum expectation value, and the gauge fields that remain massless are abelian and can be dualized to scalar fields. Semiclassically, the Coulomb branch is parametrised by the vacuum expectation values of these four scalars. This classical description, however, receives quantum corrections. The chiral ring associated with the Coulomb branch, in fact, has a complicated structure involving monopole operators in addition to the classical fields in the Lagrangian. In spite of the complicated structure of the chiral ring and quantum corrections on the Coulomb branch, it is still possible to enumerate in an exact way the gauge invariant BPS operators that have a non-zero expectation value along the Coulomb branch [1]. The idea is that the chiral ring of the quantum Coulomb branch can be described in terms of monopole operators dressed with scalar fields from the vector multiplet. The generating function that enumerates such BPS operators according to their quantum numbers is called the Coulomb branch Hilbert series. This function can be computed for any 3d N = 4 supersymmetric gauge theory that has a Lagrangian description and that are good or ugly in the sense of [2]. We review the method proposed in [1], henceforth called the monopole formula for Coulomb branch Hilbert series, in section 2.
Another way to compute the Coulomb branch Hilbert series of a given theory is to use mirror symmetry as a working assumption (see e.g. [3]). Mirror symmetry exchanges the Coulomb branch of the theory in question with the Higgs branch of another theory [4], where the latter does not receive quantum corrections. The Higgs branch Hilbert series can be computed in a conventional way from the Lagrangian of the mirror theory using Molien integrals. This method has certain limitations, for example, when the Lagrangian of the mirror theory is not available. Even when the latter is known, if the theory contains gauge groups of large ranks or large number of hypermultiplets, the computation of the Molien integrals can become very cumbersome in practice.
One of the aims of this paper (and its companion [5]) is to develop a machinery for efficiently computing Coulomb branch Hilbert series for several classes of N = 4 gauge theories. We can obtain the Hilbert series of the theories in question by 'gluing' together the Hilbert series of building blocks. A similar method has been applied successfully to the computation of Higgs branch Hilbert series. The gluing procedure consists in gauging a common global flavor symmetry of the building blocks. For the gluing machinery to work, we need to define and compute Coulomb branch Hilbert series in the presence of background magnetic fluxes associated to monopole operators for the global symmetry. The gluing is performed by summing over the background monopole fluxes with an appropriate weight, as discussed in detail in section 2.
In this paper we discuss the general properties of the Hilbert series with background fluxes and we provide computations for a class of simple theories, the three-dimensional superconformal field theories known as T ρ (G) [2]. The latter are linear quiver theories with non-decreasing ranks associated with a partition ρ and a flavor symmetry G and were defined in terms of boundary conditions for 4d N = 4 SYM with gauge group G [2]. We are able to give a closed analytic expression for the Coulomb branch Hilbert series of T ρ (G). These expressions serve as basic building blocks for constructing a large class of more complicated theories. In a companion paper [5] we discuss the particularly interesting case of the mirror of Sicilian theories arising from twisted compactification of the 6d (2, 0) theory on a circle times a Riemann surface with punctures, which can be obtained by gluing copies of T ρ (G) theories [6].
One of the main results of this paper is an intriguing relation between the Coulomb branch Hilbert series of the T ρ (G) and a class of symmetric functions, as discussed in sections 3 and 4. We conjecture that, given a classical group G and a corresponding partition ρ of the dual group, the Coulomb branch Hilbert series of T ρ (G) with background monopole fluxes for the flavor symmetry G can be written in terms of Hall-Littlewood polynomials (see, e.g. [7] and Appendix B). We give several pieces of evidence in support of our conjecture and others are given in the companion paper [5]. Our general formula is (4.9), and for the special case of G = SU (N ) the formula is given by (3.9). We shall henceforth refer to this form of the Coulomb branch Hilbert series as the Hall-Littlewood formula. 1 Hall-Littlewood polynomials have also appeared in the recent literature in the context of the superconformal index of four dimensional N = 2 Sicilian theories [9]. As we will see in [5], this is not a coincidence. Our conjecture is actually inspired by the results in [9]. Turning off the background monopole fluxes for the flavor group in (4.9), we obtain a simple expression for the Hilbert series of the Coulomb branch of T ρ (G) for any classical G, formula (4.17), which shows that Coulomb branches of T ρ (G) theories are complete intersections.
In the rest of the paper we examine the structure of the Coulomb branch Hilbert series of the theory and the physical meaning of the background monopole fluxes.
In section 5 we study the action of mirror symmetry on the background monopole charges under the flavor symmetry of a theory, in order to shed light on their physical meaning. These charges are mapped to baryonic charges on the Higgs branch of the mirror theory. Indeed, in many examples we consider, we compute the generating function of the Coulomb branch Hilbert series and match it with the baryonic generating function [10] on the Higgs branch of the mirror theory. This relation is given by (5.1).
We continue and examine the analytic properties of the Coulomb branch Hilbert series of T ρ (SU (N )) in section 6. Similarly to the observation of [11] in the context of superconformal indices, we find that the Coulomb branch Hilbert series of T ρ (SU (N )) has a pole whose residue corresponds to that of a new theory T ρ (SU (N )), where the Young diagram of ρ can be obtained from that of ρ by moving one box to a different position. The analytic structure further substantiates our conjecture that the Hall-Littlewood formula computes the Coulomb branch Hilbert series of T ρ (G) theories.
Let us summarize the key results of this paper below.
• The Coulomb branch Hilbert series for theories arising from the 'gluing' precedure is given by (2.6).
• The Hall-Littlewood formula for a general T ρ (G) is given by (4.9), and by (3.9) in the special case of G = SU (N ).
• The relations between the generating function of Coulomb branch Hilbert series and the baryonic generating function of the mirror theory is given by (5.1).
In the next section we review the monopole formula for the Coulomb branch Hilbert series [1].
2 Coulomb branch Hilbert series of a 3d N = 4 gauge theory We are interested in the Coulomb branch of three-dimensional N = 4 superconformal field theories which have a Lagrangian ultraviolet description as gauge theories of vector multiplets and hypermultiplets. The branch is parameterized by the vacuum expectation value of the triplet of scalars in the N = 4 vector multiplets and by the vacuum expectation value of the dual photons, at a generic point where the gauge group is spontaneously broken to its maximal torus. This results in a singular HyperKähler cone of quaternionic dimension equal to the rank of the gauge group G. The Coulomb branch is not protected against quantum corrections and the associated chiral ring has a complicated structure involving monopole operators in addition to the classical fields in the Lagrangian.

The monopole formula for the Coulomb branch Hilbert series
In [1] a general formula for the Hilbert series of the Coulomb branch of an N = 4 theory was proposed, which we now review. We will work in the N = 2 formulation, where the N = 4 vector multiplet decomposes into an N = 2 vector multiplet and a chiral multiplet Φ transforming in the adjoint representation of the gauge group. The Hilbert series is the generating function of the chiral ring, which enumerates gauge invariant BPS operators which have a non-zero expectation value along the Coulomb branch modulo holomorphic relations. The N = 2 vector multiplets are replaced in the description of the chiral ring by monopole operators, which are subject to relations that arise at the quantum level. The magnetic charges of the monopoles m are labeled by the weight lattice of the GNO dual gauge group G ∨ [12]. The monopoles can be dressed with the scalar components φ of the chiral multiplet Φ that preserve some supersymmetry. As can be seen from the supersymmetry transformations of an N = 4 theory [1], the components φ that are BPS live in the Lie algebra of the group H m which is left unbroken by the monopole flux. The residual gauge symmetry which is left in the monopole background consists of a continuous part H m and of a discrete part corresponding to the Weyl group W G ∨ of G ∨ , which acts on both the monopole flux m and on the φ. Due to the action of the Weyl group, the gauge invariant operators can be labeled by a flux belonging to a Weyl Chamber of the weight lattice Γ G ∨ . They will be dressed by all possible products of φ invariant under the action of the residual group H m .
The final formula counts all gauge invariant BPS operators according to their dimension and reads: The sum is over a Weyl Chamber of the weight lattice Γ G ∨ of the GNO dual group [12]. P G (t; m) is a factor which counts the gauge invariants of the residual gauge group H m made with the adjoint φ, according to their dimension. It is given by where d i (m), i = 1, . . . , rank H m are the degrees of the independent Casimir invariants of H m , also known as exponents of H m . t ∆(m) is the quantum dimension of the monopole operator, which is given by [2,[13][14][15] where the first sum over positive roots α ∈ ∆ + (G) of G is the contribution of N = 4 vector multiplets and the second sum over the weights of the matter field representation R i under the gauge group is the contribution of the N = 4 hypermultiplets H i , i = 1, . . . , n. Half-hypermultiplets contribute to ∆(m) with a factor of 1 4 instead of 1 2 . If the gauge group G is not simply connected there is a nontrivial topological symmetry group under which the monopole operators may be charged, the center Z(G ∨ ) of G ∨ . Let z be a fugacity valued in the topological symmetry group and J(m) the topological charge of a monopole operator of GNO charges m. The Hilbert series of the Coulomb branch (2.1) can then be refined to bound ∆ ≥ 1/2. This ensures that the Hilbert series (2.1) is a Taylor series of the form 1 + O(t 1/2 ) at t → 0.
The formula bypasses previous techniques for determining the Coulomb moduli space, which were based on compactification of 4d N = 2 theories, the computation of the quantum corrections to the metric of the moduli space or the use of mirror symmetry. The latter method is useful only when the mirror gauge group is sufficiently small. We demonstrated the utility of the Coloumb branch formula with many explicit examples in [1]. On the other hand, even the Coloumb branch formula is difficult to evaluate when the gauge group becomes large. The main problems are the number of independent sums, which is equal to the rank of the gauge group, and the presence of absolute values in (2.4). So we need to find alternative tools for evaluating the formula. A quite efficient way is to look for an analytic formula for the Coulomb branch Hilbert series in the presence of background magnetic fluxes for the flavor symmetry group. Such Coulomb branch Hilbert series with background magnetic fluxes can then be used in the gluing technique to derive the Coulomb branch Hilbert series of more general theories.

The Hilbert series with background magnetic fluxes
Hilbert series with background monopole fluxes are defined as follows. If a theory has gauge group G and a global flavor symmetry G F acting on the matter fields we can define a Hilbert series as in (2.1) but in the presence of background monopole fluxes for the global symmetry group G F : In this formula m F is a weight of the dual group G ∨ F . By using the full global symmetry we can restrict its value to a Weyl chamber of G ∨ F and take m F ∈ Γ * The sum in (2.5) is only over the magnetic fluxes m of the gauge group G. The background fluxes m F enter explicitly in the dimension formula (2.3) through all the hypermultiplets that are charged under the global symmetry G F , which acts on the Higgs branch of our theory.
A general way of constructing complicated theories is to start with a collection of theories and gauge some common global symmetry G F that they share. The Hilbert series of the final theory is given by summing over the monopoles of G F and including the contribution to the dimension formula of the N = 4 dynamical vector multiplets associated with G F : where α F are the positive roots of G F and the product with the index i runs over the Hilbert series of the i-th theory that is taken into the gluing procedure. If we have explicit analytic formulae for the Hilbert series H (i) G,G F (t, m F ) with background fluxes of the original theories, that resum the RHS of (2.5), the evaluation of H(t) requires to perform a sum without any absolute value, since we can always make α F (m F ) positive by choosing m F in the main Weyl chamber.
The formulae (2.5) and (2.6) can be immediately generalized to include fugacities for the topological symmetries acting on the Coulomb branch.
The Hilbert series with background magnetic fluxes are interesting objects per se. We may ask what happens to the background fluxes under mirror symmetry. As we will see in section 5, the magnetic fluxes m F are mapped to baryonic charges in the mirror theory and the Hilbert series with background magnetic charges are mapped to baryonic generating functions [10,16,17] for the mirror theory with certain U (1) gauge factors removed. We will provide many explicit examples in Section 5.
In the next sections we will provide explicit and general formulae for an interesting class of 3d N = 4 superconformal theories which may serve as building blocks for the construction of more general theories. In a companion paper [5] we will apply the result to the mirrors of M5-brane theories compactified on a circle times a Riemann surface with punctures.

Hilbert series and Hall-Littlewood polynomials
In this and the following section we discuss the Coulomb branch Hilbert series of a certain class of N = 4 supersymmetric gauge theories in three dimensions, the theories called T ρ (G) [2], with G a classical group and ρ a partition of a certain number discussed in detail below. Such theories can be naturally realized using brane configurations as in [18]. In this section we discuss the Coulomb branch Hilbert series of T ρ (SU (N )). Other classical groups G are discussed in section 4.2.
We shall discuss how the Coulomb branch Hilbert series of T ρ (SU (N )) in the presence of background magnetic fluxes for SU (N ) can be expressed in terms of certain symmetric functions known as Hall-Littlewood polynomials (see e.g. [7]). The main formula of this section is the Coulomb branch Hilbert series (3.9) for the basic building block T ρ (SU (N )). We summarize key information about Hall-Littlewood polynomials in Appendix B.

The theory T ρ (SU (N ))
The quiver diagram for T ρ (SU (N )) is 2 where the partition ρ of N is given by with the restriction that ρ is a non-increasing sequence: This condition ensures that the T ρ (SU (N )) is a 'good' theory [2]. The brane configuration of T ρ (SU (N )) is depicted in Figure 1. ρ corresponds to a collection of the linking numbers of each NS5-brane. The theory associated with the partition ρ = (1, · · · , 1) is usually called T (SU (N )) without any further specification. Figure 1. The brane configuration of T ρ (SU (N )). The numbers in red indicates the number of D3-branes in each intervals. The labels in blue denote the fugacities x i for NS5-branes and the background fluxes n j for D5-branes.
The theory has a manifest U (1) d topological symmetry associated to the U (1) d center of the gauge group. The topological symmetry is enhanced by quantum corrections to a non-abelian global symmetry which is determined as follows [2]. We refer to ρ i as the parts of the partition ρ. Let r k be the number of times that part k appears in the partition ρ. The Coulomb branch global symmetry associated with the theory T ρ (SU (N )) is where S denotes the removal of the overall U (1). For example, the flavor symmetry associated with ρ = (5, 5, 4, 4, 4, 2, 1, 1) is S(U (2) × U (3) × U (1) × U (2)). From (3.2), the number of gauge groups d is related to r k in (3.4) by this is the rank of the global symmetry G ρ acting on the Coulomb branch. The maximal torus U (1) d of G ρ , which is manifest as a topological symmetry in the quiver, enhances to a non-abelian G ρ due to hidden symmetry generators whose associated conserved currents are monopole operators. There is also a manifest SU (N ) flavor symmetry acting on the Higgs branch. In the following we provide two formulae to compute the Coulomb branch Hilbert series of T ρ (SU (N )) with background fluxes. One is the first principle formula given in (2.5), that we present in (3.6) and refer to as the monopole formula; the other is the formula involving the Hall-Littlewood polynomial, that we present in (3.9) and refer to as the Hall-Littlewood formula. We conjecture the equivalence of the two formulae, which we have analytically checked for small values of N and tested perturbatively at very large order in t in many different cases. As shown in section 6, the equivalence of the monopole formula (2.5) with the Hall-Littlewood formula (3.9) for the case of the maximal partition ρ = (1, 1, · · · , 1) implies the equivalence of the two formulae for a generic partition ρ, due to their common analytic structure.

Monopole formula for the Coulomb branch Hilbert series
As discussed in section 2, we can define a Coulomb branch Hilbert series of the theory depending on magnetic background fluxes n i for the SU (N ) flavor symmetry and refined by fugacities z i for the U (1) d topological symmetry.
The monopole formula (2.5) for the Coulomb branch Hilbert series of T ρ (SU (N )) with background fluxes reads H[T ρ (SU (N ))](t; z 0 , z 1 , . . . , z d ; n 1 , . . . , n N ) where the dimension of bare monopole operators is given by (2.3) In (3.6), the integers (m 1,N i , · · · , m N i ,N i ) are the GNO magnetic fluxes for the group U (N i ) and the sum is restricted to the fundamental Weyl chamber by restricting to ordered N i -tuples m 1,N i ≥ · · · , ≥ m N i ,N i . The integers (n 1 , · · · , n N ) are instead the background magnetic fluxes for the flavor symmetry group SU (N ). The three sets of sums in the dimension ∆ take into account the contribution of the fundamental hypermultiplets, the bi-fundamental hypermultiplets and the vector multiplets respectively. Notice that the background fluxes n i enter explicitly in the dimension ∆ through the contribution of the N fundamental hypermultiplets. Finally, the classical factors P U (N k ) are defined in (2.2) and more details are given in Appendix A of [1]. In the same paper the reader can find many simple examples of the use of the monopole formula.
In (3.6), we also use • the fugacities z k to keep track of the topological charges N k j=1 m j,N k of U (1) ⊂ U (N k ) gauge group, with k = 1, . . . , d.
The flavor symmetry G F is in fact SU (N ) rather than U (N ), therefore there is no associated topological symmetry even when G F is (weakly) gauged. We fix the extra unwanted topological U (1) by imposing on these fugacities the constraint With this convenient choice the Hilbert series (3.6) is invariant under a common shift of the fluxes n i . The formula is also invariant under permutations of the n i , the Weyl group of SU (N ). Combining the two invariances, we can always restrict the values of the fluxes to n 1 ≥ n 2 ≥ · · · ≥ n N ≥ 0, which will allow to compare with the Hall-Littlewood formula (3.9). Using the shift symmetry we could further set n N = 0.

Hall-Littlewood formula for the Coulomb branch Hilbert series
We claim that the Coulomb branch Hilbert series of this theory (3.6) can be written in terms of HL polynomials as where we explain the notations below: 1. n 1 , . . . , n N are the background GNO charges for U (N ) group, with n 1 ≥ n 2 ≥ · · · ≥ n N ≥ 0 . (3.10) 2. The Hall-Littlewood polynomial associated with the group U (N ) is given by . (3.11) 3. The notation δ U (N ) denotes the sum over positive roots of the group U (N ) acting on the background charges n i : Note that this is minus the contribution of the background vector multiplet in the monopole dimension formula.
4. The fugacities x 1 , . . . , x d+1 are naturally associated to the NS5-branes in the brane construction as depicted in Figure 1. They are related to the fugacities z 0 , . . . , z d for the manifest topological symmetry group U (1) d /U (1) by the fugacity map The symmetry U (1) d /U (1) enhances to the non-abelian G ρ due to monopole operators. x 1 , . . . , x d+1 are subject to the constraint which fixes the overall U (1). Using the map (3.13), the constraint (3.8) is rewritten as (3.14) 5. w r denotes the weights of the SU (2) representation of dimension r: Hence the notation t    6. The prefactor K U (N ) ρ (x; t) is given by where ρ T denotes the transpose of the partition ρ and we associate the factors to each box in the Young tableau. The powers of t inside a i j and a i k are positive by construction. The graphical illustration of a i j is given by Fig. 3 of [9], where j labels the column from left to right and i labels the rows from the bottom to the top. For instance: (a) For the full puncture 3 ρ = (1 N ), we have ρ T = (N ) and so The representation theoretic explanations for xt 1 2 wρ and the prefactor K U (N ) ρ (x; t) are presented in Sec. 4.1 of [19]. We summarize this in Sec. 4.3 of this paper.
We have explicitly checked in a large number of examples that the HL formula (3.9) coincides with the monopole formula (3.6). For instance, we can consider the case N = 3. There are two relevant partitions, (1, 1, 1) and (2, 1). The T (1,1,1) (SU (3)) theory, also known simply as T (SU (3)), has SU (3) global symmetry acting on the Coulomb branch, as it follows from (3.4). The monopole formula (3.6) reads (3.22) 3 We often write 'puncture' in analogy to the literature on M5 branes on a Riemann surface, but take it to mean 'partition' in this paper.
where m i,k , with i = 1, . . . , k, denote the GNO charges for the U (k) gauge group, x 1 x 2 x 3 = 1, and ∆ (1,1,1) is twice the dimension of monopole operators in the [3]−(2)−(1) quiver: We used the fugacity map (3.13) and we imposed the constraint (3.14). One can check that (3.22) reproduces the Hall-Littlewood formula (3.24) The T (2,1) (SU (3)) theory has Coulomb branch symmetry U (1) and the monopole formula reads where m denotes the GNO charge for the U (1) gauge group, x 2 1 x 2 = 1, and ∆ (2,1) is twice the dimension of monopole operators in the [3] − (1) quiver: (3.26) We again used the fugacity map (3.13) and we imposed the constraint (3.14). Again one can check that (3.25) reproduces the Hall-Littlewood formula (3.27) We will further demonstrate the HL formula (3.9) in a number of examples in a companion paper [5], where we will successfully compare our formula for the Hilbert series of the Coulomb branch of mirrors of genus 0 3d Sicilian theories with the Hilbert series of the Higgs branch of the Sicilian theories themselves, computed as the Hall-Littlewood limit of the superconformal index of the 4d theories in [9,20].

The Coulomb branch of T ρ (SU (N )) is a complete intersection
Coulomb branches of T ρ (SU (N )) theories for various partitions ρ were studied in [3] using mirror symmetry; it was found that many of these algebraic varieties are complete intersections. In this section, we provide a direct argument that for any partition ρ, the Coulomb branch of T ρ (SU (N )) is a complete intersection.
Setting n 1 = n 2 = . . . = n N = 0 in (3.9) and using the identity we obtain where we have used (3.18) in the second equality. It follows from the remark below (3.19) that the powers of t appearing inside the PE are strictly positive. The form of (3.29) shows that the Coulomb branch, denoted by C[T ρ (SU (N ))], is a complete intersection. Its complex dimension is given by where the negative contribution comes from the product of N factors in (3.29) and the positive contribution comes from the terms inside the PE. This is in agreement with (2.3) and (2.4) of [21]. Cancelling a common factor of (1 − t) between numerator and denominator of (3.29), we see that there are length( In this section we generalize the results on T ρ (G ∨ ) to other classical groups beyond G = SU (N ), namely SO(N ) and U Sp(2N ). One of the key results in this section is the formula (4.9) for the Coulomb branch Hilbert series, involving the Hall-Littlewood polynomial. We demonstrate in many examples below that this formula is uniform for all classical gauge groups. We conjecture that it can be used also for large classes of 'bad' theories, where the monopole formula is not working, as we shall discuss in several examples below and in the Appendix. 4 In the following construction, we will also need the GNO, or Langlands, dual of G, denoted as G ∨ [12]. Recall that the Lie algebras A and D are self-dual, while B and C are exchanged by GNO duality.
As pointed out in [2] and [21], T ρ (G ∨ ) is constructed as a boundary theory of 4d N = 4 super Yang-Mills on a half-space, with the half-BPS boundary condition specified by a homomorphism ρ : Lie(SU (2)) → Lie(G). The homomorphisms ρ can be classified, up to conjugation, by the nilpotent orbits of the Lie algebra of G. This classification puts certain restrictions on the partition ρ which we discuss in section 4.1. The quiver diagrams of the corresponding T ρ (G ∨ ) theories are presented in section 4.2. We compute their Coulomb branch Hilbert series in section 4.3.
Due to the Jacobson-Morozov theorem, such embedding can be classified up to a conjugacy by the nilpotent orbit of G. As discussed in [22] and section 2.1 of [21], the possible cases are as follows: • For G = SO(N ), the partition ρ of N satisfies the condition that any even part in ρ must appear an even number of times. The partition ρ is called a B-or a D-partition if N is odd or even, respectively. For instance, the B-partitions for SO (3) are (3) and (1, 1, 1); the D-partitions for SO(4) are (4), (3, 1), (2,2) and (1, 1, 1, 1). Given a partition ρ satisfying this condition, there is a unique nilpotent orbit associated to it, except for the case when all the parts ρ i are even and each even integer appears even times. Such a partition is referred to as a very even partition, whose distinct nilpotent orbits are exchanged by the outer automorphism of SO(N ). An example of a very even partition is ρ = (4, 4) for G = SO(8).
Below we present quiver diagrams for T ρ (G ∨ ) for G = SO(N ) and U Sp(2N ). Such quivers already appeared as subquivers in the work of [23].
where is even and with [n] +(resp.−) the smallest (resp. largest) even integer ≥ n (resp. ≤ n) and the node U Sp(0) being removed.
T ρ (U Sp(2N )) theory. In this case we consider the Langlands dual of where is odd and with [n] ± defined as above and the node U Sp(0) being removed.

Hall-Littlewood formula for T ρ (G ∨ ) with a classical group G
In this section we generalize the Hall-Littlewood formula (3.9) to a more general group G. We follow closely a similar discussion in section 4.1 of [19]; see also [21] for a comprehensive presentation. Several explicit examples are presented in subsequent subsections and in Appendix C.
As discussed earlier, the partition ρ defines a homomorphism ρ : Lie(SU (2)) → Lie(G) such that The global symmetry G ρ associated to the puncture and acting on the Coulomb branch is given by the commutant of ρ(SU(2)) in G. Explicitly, for a given group G = U (N ), SO(N ) or U Sp(2N ) and a puncture ρ = [ρ i ] with r k the number of times that part k appears in the partition ρ, we have . . be fugacities for the global symmetry G ρ , and r(G) the rank of G. We conjecture that the Coulomb branch Hilbert series is given by the HL formula where we explain the notations below.
1. The Hall-Littlewood polynomial Ψ n G (a(t, x); t) associated with a classical group G is given in Appendix B.
2. The power δ G ∨ (n) is the sum over positive roots α ∈ ∆ + (G ∨ ) of the flavor group G ∨ acting on the background monopole charges n: Explicitly, for classical groups G, these are given by (4.11) 3. The argument a(t, x), which we shall henceforth abbreviate as a, of the HL polynomial is determined by the following decomposition of the fundamental representation of G to G ρ × ρ(SU (2)): where G ρ k denotes a subgroup of G ρ corresponding to the part k of the partition ρ that appears r k times. Formula (4.12) determines a as a function of t and {x k } as required. Of course, there are many possible choices for a; the choices that are related to each other by outer automorphisms of G are equivalent.
is a prefactor independent of n, determined as follows. The embedding associated with ρ induces the decomposition of the adjoint representation of G where a on the left hand side is the same a as in (4.12). Note that j R j gives the decomposition of the Slodowy slice [21]. Each component in the slice gives rise to a plethystic exponential, giving We provide a number of examples below; see for example (C.35) for ρ = (3, 3, 1, 1) of G = SO (8) and (C.37) for ρ = (3, 2, 2, 1) of G = SO(8).

The Coulomb branch of T ρ (G ∨ ) is a complete intersection
In this subsection we generalize the computation in Section 3.2 for any classical group G. We will show that formula (4.9) implies that the Coulomb branch of T ρ (G ∨ ) is a complete intersection for any classical group G and any partition ρ.
Turning off the background monopole fluxes n 1 = n 2 = . . . = n r(G) = 0, the HL polynomials reduce to where d i (G), with i = 1, . . . , r(G), are the exponents of G: (4.16) Thus we obtain where the representations R j of the group G ρ are given by (4.13). This shows that the Coulomb branch, denoted by C(T ρ (G ∨ )), of T ρ (G ∨ ) is indeed a complete intersection: there are j dim Gρ (R j ) generators subject to r(G) relations, one per independent Casimir invariant of G. The complex dimension of the Coulomb branch is given by where dim Gρ (R j ) denotes the dimension of representation R j of the group G ρ . According to Theorem 6.1.3 of [22], the first term in (4.18) can be related to the partition as where r k is the number of times that k appears in the partition ρ. Thus, from (4.18) and (4.19), we have the dimension formula in accordance with (2.3) of [21]: (4.20) In this subsection we focus on ρ = (1, 1, . . . , 1). The case of non maximal punctures is discussed in Appendix C. From now on, we shall abbreviate T (1,...,1) (G) as T (G). The quiver diagrams for T (SO(N )) and T (U Sp(2N )) are given in Fig. 54 of [2]; we summarize them below.
Note that edges connecting O and U Sp represent bifundamental half -hypermultiplets. Remark also that the quiver in (4.23) is a 'bad theory'; for example, the number of flavors under the U Sp(2) gauge group is 2 (because there are 4 half-hypers charged under this gauge group), which is smaller than 2(1) + 1 = 3 (see (5.9) of [2]). Mirror symmetry exchanges G with the Langlands or GNO dual G ∨ , therefore T (D N ) is self-dual, whereas T (C N ) and T (B N ) form a mirror pair [2].
From many examples presented in the following subsections, we deduce that the Hilbert series of Coulomb branch for each of such theories is given by where n 1 , . . . , n N are the background charges, δ G ∨ (n) is given by (4.11), and Since the quiver (4.23) for T (B N ) is a bad quiver, we cannot compute the Coulomb branch Hilbert series from the monopole formula, which diverges. On the other hand one may expect that the quiver flows to an interacting conformal fixed point in the infrared. Since the formula (4.24) involving the Hall-Littlewood polynomial is well defined, we conjecture that it computes the Hilbert series of the Coulomb branch of the infrared SCFT. Below we demonstrate this by computing the Coulomb branch Hilbert series of T (SO(5)), which is a bad theory, using the Hall-Littlewood formula and find a matching (4.38) with that of T (U Sp(4)), which is a good theory.
When comparing with the monopole formula, we will face the problem of matching fugacities. Unlike in the case of unitary groups, in the Coulomb branch of the T (G ∨ ) theories with orthogonal and symplectic groups the Cartan subalgebra of the global symmetry is not fully manifest. Some Cartan generators do not correspond to topological symmetry of the Coulomb branch, but arise instead as monopole operators [2]. It is not clear to us how to introduce fugacities for these generators in the monopole formula.
Let us demonstrate these formulae in the examples below.

T (SO(4))
The quiver of T (SO (4)) is with a Coulomb branch symmetry SO(4). In addition there is a flavor symmetry SO(4) acting on the Higgs branch. The conjectured HL formula for the Coulomb branch Hilbert series of T (SO (4)) is where explicit expression for small values of n 1 and n 2 are given in Appendix B. For n 1 = n 2 = 0, we have in terms of the Hilbert series of C 2 /Z 2 Hence the Coulomb branch of T (SO(4)) is (C 2 /Z 2 ) 2 ; this is identical to the Higgs branch of the same theory, in agreement with the self-mirror property.
Comparison with monopole formula. We can compare the conjectured formula (4.27) with the monopole formula for Coulomb branch Hilbert series where m is the topological charge for SO(2) gauge group with x 1 the corresponding fugacity, and k is the monopole charges in U Sp(2) gauge group. Here (4.32) It can be checked that Note that in the monopole formula (4.31) we can refine only one fugacity x 1 of SO(4), whereas in the conjectured Hall-Littlewood formula (4.27) for the Hilbert series both fugacities x 1 and x 2 appear. This requires some explanation. Although the Coulomb branch symmetry is SO(4), only a U (1) subgroup is manifest. The reason is that the only manifest symmetries in the Coulomb branch are the topological symmetries and the quiver (SO(2)) − (U Sp(2)) − [SO(4)] has a single abelian factor. The remaining SO(4) generators, including the other generator of the Cartan subalgebra, correspond to monopole operators [2]. In particular the other generator of the Cartan subalgebra is provided by the monopole operator with m = 0, k = 1 along with n i = 0. It would be interesting to understand how to include fugacities for the full Cartan subalgebra in the monopole formula for the Coulomb branch Hilbert series. We will encounter this phenomenon many times in the following. (4)) and T (SO (5)) The HL formula for the Coulomb branch of T (SO(5)) is given by

T (U Sp
where the character of the adjoint representation [2, 0] of U Sp(4) is Since the Lie algebra of U Sp(4) is isomorphic to that of SO(5), we also have where the character of adjoint representation [0, 2] of SO (5) is These two expressions can be matched as follows: The matching reflects the translation between representations of U Sp(4) and SO (5): where (n 1 , n 2 ) denotes the representation of B 2 = SO(5) in the standard e-basis, 5 where n 1 ≥ n 2 ≥ 0 with n 1 , n 2 all integers or all half-integers. One can explicitly check that (4.36) with y 2 = 1 can also be obtained from the monopole formula (2.1) for the Coulomb branch Hilbert series. Again, only one fugacity corresponding to the topological charge of the SO(2) group is manifest.

Background magnetic charges and baryonic charges
Having computed the Coulomb branch Hilbert series of many theories with background magnetic charges turned on, it is natural to ask which quantities on the Higgs branch are such background charges mapped to under mirror symmetry. In this section we show that the answer is the baryonic charges in the mirror of the theory in question. So, in a sense, as expected from mirror symmetry which acts as S-duality on brane configurations in type IIB string theory, the magnetic background fluxes are mapped to background electric fluxes.
The relation between the Hilbert series with background fluxes of a theory and its mirror can be made precise as following. If the gauge group G M of the mirror theory contains a U (1) factor, it also has a U (1) topological symmetry acting on its Coulomb branch. In the original theory, this U (1) will be acting on the Higgs branch. Ungauging the U (1) gauge symmetry in the mirror is equivalent to gauging the corresponding global U (1) symmetry in the original theory. We thus have the following equality where we explain the notation as follows. Even though the correspondence (5.1) holds separately for each U (1) group, for simplicity we will mostly present examples where we gauge (resp. ungauge) the maximal torus of the symmetry group acting on the Coulomb branch (resp. Higgs branch of the mirror theory). Given a theory T , we use the notations T B for the theory obtained by replacing all U (N i ) gauge groups in T by SU (N i ) gauge groups, and T J for the theory obtained by gauging the maximal torus of the flavor symmetry. B and J label the baryonic and topological symmetries gained in the two processes. If theories T and T M are mirror theories, so are T B and T M J . The equivalence of ungauging a U (1) on the Coulomb branch of a theory and gauging a U (1) on the Higgs branch of the mirror theory is straightforward to see at the level of Hilbert series [1]. The equivalence of gauging a U (1) on the Coulomb branch of a theory and ungauging a U (1) on the Higgs branch of the mirror theory is much less trivial to see at the level of Hilbert series. We show it in a number of examples in the rest of the section. Let b be the fugacity associated with the baryonic charges. The baryonic generating function is then given by where x is the fugacity for SU (2). After a rescaling t → t 2 needed to compare with 4d quantities, the function is indeed the F -flat Hilbert series [24,25] of (1)− [2] quiver and the baryonic generating function of C 2 /Z 2 ; see (4.6) of [10]. The baryonic generating function can be related to the Coulomb branch Hilbert series of T (SU (2)) with the background flux turned on. Indeed, we find that where H[T (SU (2))] is given by (3.9) with ρ = (1, 1), which extends to the Weyl chamber n < 0 by replacing n → −n, or equivalently by the monopole formula (3.6): The summation in (5.3) has an interpretation of generating function for the Coulomb branch Hilbert series with background fluxes. Equation (5.3) relates this generating function to the baryonic generating function on the Higgs branch of the theory. For reference, we provide the expressions for such a function in the regions n ≥ 0 and n < 0: .

T (SU (3)) theory
As before, the baryonic generating function of (1)−(2)− [3] quiver is equal to the Higgs branch Hilbert series of the same theory with the two unitary gauge groups replaced by special unitary groups: where • b 1 and b 2 are baryonic fugacities; they can also be viewed as electric fugacities for the U (1) node and for the U (1) center of the U (2) node respectively, • x 1 , x 2 , x 3 , with x 1 x 2 x 3 = 1, are fugacities of the SU (3) flavor symmetry, • the Haar measure of SU (2) is Coulomb branch Hilbert series and baryonic generating function We are interested in relating (5.6) to the Coulomb branch Hilbert series of T (SU (3)).

Hall-Littlewood formula
Alternatively to the monopole formula (5.8), the Coulomb branch Hilbert series is also given by the HL formula (3.9): where we emphasize that this formula is valid only if n 1 ≥ n 2 ≥ 0 and we take x 1 x 2 x 3 = 1. The relation with (5.6) can therefore be separated into 6 regions, as in Appendix A.2 of [17]. Explicitly, we obtain where x 1 x 2 x 3 = 1. Each summand indicates the baryonic generating function, in terms of the Coulomb branch Hilbert series, in one of the 6 Weyl chambers of SU (3).

T (SU (N )) theory
It is straightforward to derive a similar relation to (5.3) and (5.10) for a general T (SU (N )) theory. Using the same notation as before, we obtain , (5.13) where the power ± is + for N even and − for N odd. (5.13) shows that the Hilbert series of the Higgs branch of T (SU (N )) B and the Coulomb branch of T (SU (N )) J coincide. The latter is described by a quiver where the flavor group is replaced by N U (1) nodes joined to the U (N − 1) node, with the overall U (1) gauge group factored out. The corresponding quiver diagram is depicted in Figure 2.

T (2,1) (SU (3)) theory: (1) − [3] quiver
The mirror theory of T (2,1) (SU (3)) is The baryonic generating function of the mirror of T (2,1) (SU (3)) is given by This again confirms the relation between the generating function of the Coulomb branch Hilbert series and the baryonic generating function of the mirror theory.

T (3,1,1) (U Sp(4)) theory: (SO(2)) − [U Sp(4)] quiver
We now consider the T (3,1,1) (U Sp (4)) theory, which corresponds to SO(2) gauge theory with 4 flavors of half-hypermultiplets in the two-dimensional vector representation. The aim of this section is to present certain subtleties that do not appear in T ρ (SU (N )) theory. The Coulomb branch Hilbert series of this theory is discussed in appendix C.4. In the following we shall compute the baryonic generating of the Higgs branch of the mirror theory and establish some relations with such a Coulomb branch Hilbert series.
The Coulomb branch Hilbert series of T (3,1,1) (U Sp (4)) The monopole formula for the Coulomb branch Hilbert series of T (3,1,1) (U Sp (4) where z 2 keeps track of the topological charge of the gauge group SO(2), and If we set n 1 = n 2 = 0, the Coulomb branch Hilbert series simplifies to Note that this is the Hilbert series of C 2 /Z 4 , the Coulomb branch of the U (1) gauge theory with 4 flavors of hypermultiplets of charge 1, which is the same as the SO(2) gauge theory with 4 flavors of half-hypermultiplets in the vector representation of SO(2).

Mirror of T (3,1,1) (U Sp(4)) and the baryonic generating function
The mirror theory of T (3,1,1) (U Sp (4)) theory was discussed in section 6.2 of [23]. Its quiver is given by (4)): This is a 'bad' theory, since the number of flavors of each U Sp(2) gauge node is 2, which is less than 2(2) + 1 = 5. In fact, this theory flows to the mirror theory of U (1) gauge theory with 4 flavors, as we shall demonstrate below (5.23). Here we are interested in the Higgs branch, which is protected from quantum corrections, so we can use the 'bad' quiver to compute it. The computation of the F -flat Hilbert series and the baryonic generating function of this mirror theory, obtained by ungauging the SO(2) group, is rather techical, and we relegate it to Appendix D.2. We present only the end result of the baryonic generating function, which is given by (D.23): Note that (5.21) is, in fact, the Hilbert series of (C 2 /Z 2 ) 2 , where b 1 and b 2 are fugacities for the SU (2) isometry associated with each copy of C 2 /Z 2 . The Higgs branch Hilbert series of quiver (5.20) is given by (D.24): (4))](t; x 1/2 y −1/2 ; 0, 0) ; (5.23) this is equal to the Coulomb branch Hilbert series (5.17) with vanishing background fluxes n 1 = n 2 = 0, as predicted by mirror symmetry. Indeed, this indicates that the 'bad' theory (5.21) flows to the mirror theory of U (1) gauge theory with 4 flavors in the infra-red.

Relation between (5.17) and (5.21)
We find that (4))](t; z; n, n)b 2n , (5.24) where G Z 2 (t; z, z −1 ; b) is a Z 2 projection of G(t; z, z −1 ; b) defined as follows: Note that this Z 2 projection is not to be identified with the parity of the orthogonal gauge group O(2), cf. (D.8), which does not commute with the Cartan elements of SO (2). It is rather the Z 2 subgroup of SO(2) that consists of gauge rotations by multiples of π. As such, it projects out the odd powers of b in G(t; z, z −1 ; b). This example displays several subtleties that are not present for T ρ (SU (N )). Let us comment and list certain open questions as follows: 1. There are two background fluxes n 1 and n 2 for the flavor symmetry U Sp(4) of the theory T (3,1,1) (U Sp (4)) but only one baryonic charge b (and correspondingly only one manifest topological SO(2) symmetry) in the mirror theory. This mismatch is very much like the situation discussed in (4.33). This is related to the absence of Cartan generators of the symmetry U Sp(4) on the Coulomb branch of the mirror of T (3,1,1) (U Sp(4)), where only SO(2) is manifest. The remaining Cartan generators of U Sp(4) correspond to monopole operators.
2. The coefficients of odd powers of b in (5.21) (i.e. those with odd baryonic charges) cannot be matched with the Coulomb branch Hilbert series (5.17) for any background fluxes. Only those of even powers of b can be matched with the Coulomb branch Hilbert series (5.17); this happens when n 1 is set to be equal to n 2 . The Z 2 projection is there to get rid of the odd powers of b in (5.21) and hence the matching can be done as in (5.24). We also observe that by gauging the U (1)'s associated to n 1 and n 2 and ungauging the U (1) associated to m on the Coulomb branch side, one can reproduce the full G in (5.21).

Analytic structure of the Coulomb branch Hilbert series
In this section we examine the analytic structure of the Coulomb branch Hilbert series of T ρ (SU (N )), from the complementary perspectives of the Hall-Littlewood formula and of the monopole formula. The study of the analytic structure further substantiates our conjecture that the Hall-Littlewood formula computes the Coulomb branch Hilbert series of T ρ (SU (N )) theories. The analysis of the HL formula is very similar to that of HL index of 4d Sicilian theories in [11]. The prefactor K U (N ) ρ (x; t) in the HL formula (3.9) has a pole corresponding to a particular box corresponding to the rightmost part of the partition ρ. Taking the residue of the Hilbert series at this pole isolates the Coulomb branch Hilbert series of a new theory T ρ (SU (N )), where ρ is obtained from ρ by moving such a box to a previous column.
We can identify the same simple pole in the monopole formula for the Coulomb branch Hilbert series with background fluxes of T ρ (SU (N )). Computing the residue yields the monopole formula for T ρ (SU (N )), in complete analogy with the analysis of the HL formula. It thus follows that the equivalence of the monopole formula (2.5) with the Hall-Littlewood formula (3.9) in the case the maximal partition ρ = (1, 1, · · · , 1) implies the equivalence of the two formulae for a generic partition ρ.
Let us demonstrate the idea in the following example. The general case is discussed in Section 6.2 and Appendix E. Another example of the application of the same technique is discussed in Appendix F. The relation between 4d superconformal indices of Sicilian theories including the former or the latter puncture was studied in [11]. In this section, we perform a similar computation to find the relation between the Coulomb branch Hilbert series for the two T ρ (SU (3)) theories. From the HL formula (3.9), we obtain the expressions (3.24) and (3.27) for the Coulomb branch Hilbert series of T (1,1,1) (SU (3)) and T (2,1) (SU (3)) respectively.
Formula (3.24) has simple poles at x i x −1 j → t for i = j, due to the plethystic exponential factor in the second line (the HL polynomials are regular). In view of the permutation symmetry, we can focus on the pole at x 1 x −1 3 → t. We set and consider the limit where z → 1. Similarly to the discussion in section 2.2 of [11] for the HL limit of superconformal indices, the residue of (3.24) as z → 1 in (6.2) reproduces the Hilbert series (3.27) with a certain prefactor that is easily determined: (3))](t; y 1 , y 2 ; n 1 , n 2 ) .

(6.3)
Analytic structure of the monopole formula The aim of this section is to show that the residue formula (6.3) can also be derived by means of the monopole formula of the Hilbert series (3.22). The idea is that when evaluating this expression at the values of x i corresponding to a pole, one introduces extra powers of t depending on the monopole fluxes which make the series diverge along certain directions in the GNO lattice. Taking the residue, some gauge groups effectively disappear, reproducing the quiver corresponding to a different partition.
Example: pole at x 1 x −1 3 → t in (3.22). The monopole formula (3.22) has a simple pole when the fugacities x are as in (6.2) and z → 1, because there is a region of the summation in (3.22) where the series diverges. To see this, we consider the region where m 1,1 , m 2,2 m 1,2 , n j . The singularity depends on the summation over m 2,2 , which depends only on z: where L is a lower cutoff larger than m 1,2 and n j . Because P U (2) (t; m 1,2 , m 2,2 ) becomes 1/(1 − t) 2 for m 2,2 m 1,2 , we find that (3))](t; y 1 t 1/2 z, y 2 , y 1 t −1/2 z −1 ; n 1 , n 2 ) The factor in the first square brackets in (6.9) is the Coulomb branch Hilbert series of U (1) with one flavor, which is mirror to a twisted hypermultiplet and evaluates to The factor in the second square brackets is the Coulomb branch Hilbert series for the quiver [3] − (1) with background fluxes n, corresponding to the partition ρ = (2, 1). Therefore we conclude from the monopole formula that as deduced previously from the HL formula in (6.3).

Moving the last box: from partition
The previous argument can be generalized to the following theories: where the total numbers of gauge groups are respectively d and d − 1, and This corresponds to moving the last box in a partition ending with 1 as follows: .
Note that any partition σ of N can be obtained from (1 N ) by an iteration of the previous move. Therefore by repeated residue computations we may extract the Coulomb branch Hilbert series of any T σ (SU (N )) theory from that of T (SU (N )).
Let us denote the fugacities corresponding to each column of ρ and ρ by ρ : where the fugacities are coloured in accordance with the boxes in (6.14). They are subject to the constraints The corresponding brane configurations are depicted in Figure 3. Figure 3. The top and bottom brane configurations correspond to the partitions ρ to ρ , respectively. The notation is as indicated in Figure 1.
Analogously to the analysis of the HL limit of superconformal indices of 4d Sicilian theories in [11], we find that the HL formula (3.9) for the Coulomb branch Hilbert series of T ρ (SU (N )) has a simple pole at x d−h+1 x −1 d+1 = t 1 2 (1+ ) . We parametrize (6.18) in agreement with (6.17), and compute the residue at the pole z → 1. The residue of the prefactor (3.18) is whereas the HL polynomial with arguments determined by the partition ρ in (3.9) tends to the HL polynomial with arguments determined by ρ . Therefore we find that This result can be reproduced from the monopole formula for the Coulomb branch Hilbert series along the lines of section 6.1. The simple pole corresponds to a noncompact flat direction in the Coulomb branch where the D3-branes depicted in purple in Figure 3 are sent to infinity. The identification of the simple pole and the computation of the residue is tedious but straightforward. We leave it to Appendix E.
This agreement of analytic structures is a nontrivial consistency check of the conjectured equivalence of the monopole and the Hall-Littlewood formula for the Coulomb branch Hilbert series. It guarantees the equivalence of the two formulae for any T ρ (SU (N )) theories once this is established for T (SU (N )).

Conclusions
This paper is a first step towards the computation of the Coulomb branch Hilbert series for wide classes of N = 4 gauge theories. The monopole formula introduced in [1] allows to determine the Hilbert series of any good or ugly N = 4 theory as an infinite sum over magnetic fluxes. Due to the presence of many sums and absolute values in the dimension formula for the monopoles, it is sometimes difficult to obtain explicit analytic expressions for the Hilbert series, especially when the gauge group becomes large. The gluing techniques that we have introduced in section 2 allows to solve this problem in many cases. Knowing analytic formulae for the Coulomb branch Hilbert series in the presence of background fluxes for a general class of building blocks, we can determine the Hilbert series of more general theories by gluing. The mechanism requires a sum over the background monopole fluxes and involves no absolute values.
We have given a closed analytic expression for the Coulomb branch Hilbert series with background fluxes for the class of theories T ρ (G) which can serve as building blocks for constructing a wide classes of N = 4 gauge theories. In particular, all the mirrors of Sicilian theories, obtained from M5-branes compactified on S 1 times a Riemann sphere with punctures {ρ i }, can be obtained by gluing the corresponding T ρ (G) theories. In the companion paper [5] we will compute the Hilbert series for the Coulomb branch of the mirror of Sicilian theories of type A and D. By mirror symmetry, this is equal to the Higgs branch Hilbert series of the Sicilian theory, which can can be evaluated by the Hall-Littlewood (HL) limit of the superconformal index [9,20] when the genus of the Riemann surface is zero. We will find perfect agreement with the results in [9,20], which also involve Hall-Littlewood polynomials and were obtained in a completely different manner. The agreement with [9] further substantiates our conjecture that the Hall-Littlewood formula computes the Coulomb branch Hilbert series of T ρ (G) theories. In [5] we will also compute the Coulomb branch Hilbert series of mirrors of Sicilian theories with genus greater than one, for which there is no other available method.
It would be interesting to extend our analysis to cover the more general class of theories T µ ρ (G) defined in [2], where µ and ρ are partitions related to G and the dual G ∨ respectively. The Coulomb and Higgs branches of these theories are not generally complete intersections. Computing their Coulomb branch Hilbert series with background fluxes would allow us to obtain the Hilbert series of an even wider class of N = 4 gauge theories.
• The plethystic exponential of a multivariate function f (t 1 , . . . , t n ) that vanishes at the origin, f (0, . . . , 0) = 0, is defined as • An irreducible representation of a simple group G can be denoted by its highest weight vector.
-With respect to a basis consisting of the fundamental weights (also known as the ω-basis), we write the highest vector as [a 1 , . . . , a r ] with r = rank G. This is the Dynkin label.
-With respect to a basis of the dual Cartan subalgebra (also known as the ebasis or the standard basis), we denote the the highest vector by (λ 1 , . . . , λ r ).

The Weyl character formulae
In the following, we present the character formulae that we use throughout the paper. Our convention for characters is different from that of LiE 6 .

B Hall-Littlewood polynomials
The Hall-Littlewood (HL) polynomial associated to a group G and a representation λ is a polynomial labelled by the highest weight vector λ = r i=1 λ i e i , with {e 1 , . . . , e r } the standard basis of the weight lattice and r the rank of G, defined as (see e.g. Note that, for G = U (N ), we define the Hall-Littlewood polynomial without a normalisaton factor N (t) in comparison with (5.17) of [9]. This coincides with R λ in Section III.1 of [7].
where W G denotes the Weyl group of G and ∆ + (G) the set of positive roots of G. Explicit details for classical groups G can be listed as follows: • For G = U (N ), W G = S N is a group of permutations of the elements in {e 1 , . . . , e N }.
The partition λ is subject to λ 1 ≥ λ 2 ≥ . . . ≥ λ N ≥ 0 with λ i ∈ Z. The representation λ corresponds to the Dynkin label The corresponding HL polynomial is where the positive roots are we set x 1 · · · x N = 1 and restrict λ N = 0.
The partition λ is subject to λ 1 ≥ λ 2 ≥ . . . ≥ |λ N | ≥ 0. The Dynkin label of the representation λ is a = (a 1 , . . . , a N ), with The corresponding HL polynomial is We collect explicit expressions for HL polynomials for groups with small ranks below.

C Examples of non-maximal B and C punctures
In this appendix, we focus on non-trivial partitions ρ providing many examples for SO and U Sp groups. For groups with low ranks, there are certain isomorphism between their Lie algebras, e.g. SO(5) and U Sp(4), SO(6) and SU (4). We use such isomorphisms as a tool to check the Hall-Littlewood formula (4.9) in many examples; these checks are similar in spirit to what we performed in section 4.4.3.
Furthermore, in many examples below, we use the Hilbert series as a tool to understand the relationships between the nilpotent orbit and the Higgs/Coulomb branches of the theories. Let us summarize some highlights below: • In appendix C.2 we consider the "very even" partition ρ = (4, 4) of SO (8), which corresponds to two distinct nilpotent orbits of SO(8) [21,22,26]. We study how the Coulomb branches of the theories corresponding to these two orbits are related.

(C.13)
C.2 T (4,4) (SO (8)) and the very even partition (4,4) In this appendix, we consider the partition (4, 4) of SO (8). This partition is "very even", therefore it corresponds to two different nilpotent orbits of SO(8) (see, e.g. Recipe 5.2.6 of [22] and [21,26]). These two types of puncture (4,4) are related by an outer automorphism of SO(8) that interchanges the two spinor representations [0, 0, 0, 1] and [0, 0, 1, 0]; we distinguish these punctures by subscripts I and II. Even though the distinction of these two types is not apparent in the quiver diagram, the Hall-Littlewood formulae for these two partitions are not equal, even though they can be related.

Quiver and mirror theory
For both types of the (4, 4) puncture, the quiver diagram of T (4,4) (SO (8) This theory is a 'bad theory', since the number of flavors is 4, less than 2(2) + 1 = 5. The Coulomb branch Hilbert series cannot be computed from the monopole formula, but we expect that the Hall-Littlewood formula gives the correct result.
The mirror theory of this quiver can be determined using brane configurations as in Figure 13 of [23]; the quiver diagram is given by The Coulomb branch Hilbert series The global symmetry group of the (4, 4) puncture is U Sp (2). Two types of punctures corresponds to different embeddings of U Sp(2) in SO (8). From (4.12), we consider the following decomposition: There are two inequivalent choices of the fugacity maps corresponding to the two types of (4,4) puncture: 8 (I) : : For the two types of (4,4) puncture, the HL formula for the Coulomb branch Hilbert series of (C.14) is given by (8) (4,4) (t; x)× Ψ (n 1 ,n 2 ,n 3 ,n 4 ) D (t 3/2 x, t 1/2 x, t 1/2 x, t −3/2 x; t) , (C.20) Note that (C.20) is consistent with (C.17), namely (t Note that the space is 4 complex dimensional as required. For SO (8) In this appendix we consider T (3,3,1,1) (SO(8)) and T (3,2,2,1) (SO(8)) theories, which are mirrors to the second and third example on pp. 24-25 of [21]. Their quiver diagrams are given below: Note that the quiver for T (3,3,1,1) (SO (8)) is a 'good theory', whereas the quiver for T (3,2,2,1) (SO (8)) is a 'bad theory' since the number of flavors under the SO(4) gauge group is 2, smaller than 4 − 1 = 3. The partitions ρ = (3, 3, 1, 1) and ρ = (3, 2, 2, 1) define the Nahm poles of the above theories. The Hitchin pole of each theory can be obtained by a procedure consisting of a transposition and a series of D-collapses [21,26], whereby the box in the bottom row of the left most column is moved to the next right column. The D-collapsing is to be repeated until the number of columns that contain even boxes is even. 9 The resulting D-partition ρ defines the Hitchin pole of the theory.
Below we show step-by-step the procedure to obtain the Hitchin poles for both theories. Let us start from ρ = (3, 3, 1, 1).
In this appendix, we show that the Hilbert series of the Coulomb branch of these quivers, which are both 2 quaternionic dimensional, are equal. As discussed in (4.8), the Coulomb branch global symmetry G ρ corresponding to both partitions is U Sp (2).
The contribution from the positive parity of O(2). This is the Higgs branch of (SO (2)) − [U Sp(4)], whose Hilbert series is 2) The first few terms in the U Sp(4) character expansion of g + (τ ; x) are and the unrefined Hilbert series is It should be observed that the U Sp(4) representations appearing in (D.3) embed into SU (4) representations, namely g + (τ ; y) = where a fugacity map between SU (4) and U Sp(4) is 12 Note that (D.5) is in fact the Hilbert series of the reduced moduli space of 1 SU (4) instanton on C 2 , or of the Higgs branch of U (1) gauge theory with 4 flavors.
The contribution from the negative parity of O(2). This is given by where 1 2 denotes a two-by-two identity matrix and Here we take χ Combining g + and g − . The Higgs branch Hilbert series of (O(2)) − [U Sp (4)] is therefore where we abbreviate [a 1 , a 2 ] x = χ U Sp (4) [a 1 ,a 2 ] (x). The unrefined Hilbert series is (D.10)

D.1.2 The Kibble branch of (SO(4)) − [U Sp(4)]
Let us denote the half-hypermultiplets in the theory by Q i a , where a = 1, . . . , 4 is an SO(4) gauge index and i = 1, . . . , 4 is a U Sp(4) flavor index. The superpotential is W = Q i a Q j b ϕ ab J ij , where J is a symplectic matrix and ϕ is an adjoint field under SO(4). Using Macaulay2 [29], we find the space of F -term solutions (i.e. the F -flat space) defined by is a 11 complex dimensional space; not 16 − 1 2 (4 × 3) = 10 complex dimensional if the gauge group SO(4) is completely broken. Hence we conclude that at a generic point of the hypermultiplet moduli space (also known as the Kibble branch [30]), the gauge symmetry SO(4) is broken to SO(2). The remaining unbroken symmetry on the Kibble branch is indeed as expected, since (SO(4)) − [U Sp(4)] is a 'bad' theory.
The Hilbert series of the F -flat space can be computed using Macaulay2: where we abbreviate [a 1 , The unrefined Hilbert series of the F -flat space is Note that the F -flat space is indeed 11 complex dimensional.
Integrating over the Haar measure of SO(4), we obtain the Kibble branch Hilbert series of (SO (4) (4)) and the baryonic generating function In this section of the appendix, we discuss the computation of the Higgs branch Hilbert series and the baryonic generating function of the mirror theory of T (3,1,1) (U Sp (4)) theory, whose quiver is given by (5.20).
Let us now compute the baryonic generating on the Higgs branch of this mirror theory. Let us denote the chiral fields in the quiver as follows: Since the mirror theory is a 'bad' theory, we expect that the space of F -terms solutions (i.e. the F -flat space) 19) has many branches. In order to determine the relevant branch (9 complex dimensional space), we perform the primary decomposition, which can be done using mathematical packages such as STRINGVACUA [31] or SINGULAR [32]. After such a decomposition, we find using Macaulay2 [29] the Hilbert series of the relevant branch of the F -flat space. Since the result is too long to be reported here, we present the first few terms in the series expansion in τ : F (t; x, y; z 1 , b, z 2 ) where x, y are the fugacities of the two SO(2) flavor symmetries and z 1 , b, z 2 are respectively the fugacitites of the gauge groups U Sp(2), SO(2), U Sp (2). We have used the shorthand notation a ±n = a n + a −n . The unrefined F -flat Hilbert series is F (τ ; 1, 1; 1, 1, 1) = (1 + τ )(1 + 3τ ) 2 (1 − τ ) 9 . (D.21) Let us next integrate over the two U Sp(2) gauge groups, but not the SO(2) gauge group: The baryonic generating function for this theory is obtained by ungauging the SO(2) group; this amounts to multiplying the above function by (1 − τ 2 ) −1 to remove the contribution of the F -term from this SO(2) group: where the dimension formula ∆(ρ 1 , · · · , ρ d−h ) corresponds to the quiver tail [N ] − · · · − (h + + ρ d−h ) − (h + ) including the contribution of the U (h + ) vector multiplet, and ∆( , 1 h ) corresponds to the tail [h + ] − (h) − · · · − (2) − (1) without the contribution of the U (h + ) vector multiplet. Explicitly, the latter is given by .

(E.2)
We will show that the monopole formula (E.1) for the Coulomb branch Hilbert series has a simple pole at x d−h+1 x −1 d+1 = t 1 2 (1+ ) , and compute the residue at z → 1 when the fugacities associated to the partition ρ satisfy (6.18), to reproduce (6.21).
The pole is due to the region in which m 1,1 , m 2,2 , . . . , m h,h are much larger than other m i,j and all n i . In the brane picture at the top of Figure 3, this limit corresponds to considering the h rightmost pairs of adjacent NS5-branes, and moving one D3-brane attached to each such pair far out on the Coulomb branch. This limit leaves the brane picture at the bottom of Figure 3   Next, we focus on the changes in the topological factors in the second line of (E.1). The factor that involves the fugacities x d−h+1 and x d+1 associated to the blue boxes becomes, using (6.18) and (E. 3), (E.7) The factor that involves the fugacities associated to the h − 1 pink boxes becomes where the quantity in the curly bracket {· · · } becomes parts of the new topological factors after the box is moved. The topological factors associated to the grey boxes are not affected.
Finally we consider the fate of the classical factors of the U (j) gauge groups, for j = 1, . . . , h, in the limit where m 1,1 , m 2,2 , . . . , m h,h are much larger than other m i,j and all n i . The limit breaks U (j) → U (j − 1) × U (1) along the Coulomb branch, and Combining the relevant terms together, we summarize each factor below: • The power of t 1/2 . We combine (E.4) and (E.7) and yield • The power of z. This comes from (E.7), • The power of y d−h+1 . This comes from (E.7), • The factors containing x's in the (1 h−1 ) block. These come from (E.8), (E.13) • The classical factors. These come from (E.9), (E.14) Finally, we perform the summations over the large magnetic charges {m j,j } h j=1 , or equivalently over m h,h and q 1 , . . . , q h−1 : The remaining factors are finite as z → 1, that we set in the following.

The summations over
Combining (E.15) and (E.16) with the classical factor PE[ht] of U (1) h in (E.14), we recover P ρρ in (6.20). The remaining factors and summations combine with the factors and summations of the spectator part of the quiver to reconstruct the monopole formula for the Coulomb branch Hilbert series of T ρ (SU (N )). Hence we have reproduced formula (6.21) from the monopole formulae.