T^{\sigma}_{\rho}(G) Theories and Their Hilbert Series

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d N=4 superconformal gauge theories T^{\sigma}_{\rho}(G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, \sigma is a partition of G and \rho a partition of the dual group G^\vee. In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of N=4 superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G=SU(N) but some interesting results are also given for orthogonal and symplectic groups.

1 Introduction An efficient way of encoding the information on the chiral ring of a supersymmetric theory is given by the Hilbert series of the moduli space of supersymmetric vacua, which is the generating function for the gauge invariant chiral operators. There has been recent progress in the analysis of the chiral ring and moduli space of an N = 4 superconformal gauge theory in 2 + 1 dimensions. We can now compute the Hilbert series for both the Higgs and Coulomb branch and use it to test dualities, most notably mirror symmetry [1]. The Hilbert series for the Higgs branch, which is classical, can be computed in a conventional way from the Lagrangian using Molien-Weyl integrals. The Coulomb branch is more complicated, but in spite of the complex structure of the chiral ring and the quantum corrections, it is still possible to write the Hilbert series by counting monopole operators dressed with scalar fields [2]. We refer to the Hilbert series for the Coulomb branch also as monopole formula [2]. The formula can be applied to any 3d N = 4 supersymmetric gauge theory that has a Lagrangian description and that is "good" or "ugly" in the sense of [3]. 1 In this paper we discuss the general properties of the Hilbert series for the threedimensional superconformal field theories known as T σ ρ (G) [3]. These are linear quiver theories associated with a partition σ of G and a partition ρ of the GNO (or Langlands) dual group G ∨ [5]. They are defined in terms of a general set of D3 branes ending on NS5 and D5-branes [6], possibly in the presence of O3 planes [3,7]. By construction, the mirror of T σ ρ (G) is T ρ σ (G ∨ ). These theories serve as basic building blocks for constructing a large class of more complicated theories. In [8,9] we already analyzed the special case of the Coulomb branch of T ρ (G), corresponding to σ = (1, · · · , 1), and we proposed a general formula for the Coulomb branch Hilbert series in terms of Hall-Littlewood polynomials. In this paper we define generalized Hall-Littlewood functions that give a general expression for the Coulomb branch Hilbert series of T σ ρ (G), or equivalently the Higgs branch Hilbert series of T ρ σ (G ∨ ), with background charges. The relevant formulae are (4.2) and (6.7).
The Hall-Littlewood polynomials are a class of symmetric function that have appeared in related context in both the mathematical and physical literature. In physics, they appeared as blocks in the computation of a limit of the superconformal index [10,11] for class S theories in four dimensions [12]. The relation with our results for T ρ (G) theories can be seen after compactification and mirror symmetry and was discussed in details in [9]. In mathematics, the Hall-Littlewood polynomials have appeared as characters of the cotangent bundle of flag varieties, which can be computed by localization. 2 The relation with our work comes from the fact that the moduli space of T σ ρ (G) can be expressed in terms of nilpotent orbits of G ∨ which have the cotangent bundles of flag varieties as smooth resolutions. We took inspiration from these mathematical results to derive our formula for the Hilbert series of T σ ρ (G). A similar approach has been successfully applied to the computation of the Hilbert series of instanton moduli spaces [15] or the Hilbert series of non-compact Calabi-Yau's [16].
In this paper we mainly focus on the case G = SU (N ) where all derivations are neat and we can make very general statements. The case of other classical groups, where there are complications with the algebraic description of nilpotent orbits and issues with discrete groups, is briefly discussed at the end of the paper. A regular and interesting pattern seems to emerge also in other types, but we leave the general analysis for future work.
The paper is organized as follows. In section 2 we present the quivers for the T σ ρ (G) theory for a general classical group and generic partitions σ and ρ. We have been able to find in the literature only particular examples and we state here the general result which follows from the constructions in [3,7]. In section 3 we discuss the brane construction of the T σ ρ (SU (N )) theories and we review the general expressions for the Coulomb branch Hilbert series (based on the monopole formula [2]) and the Higgs branch Hilbert series (based on the Molien-Weyl formula). We allow for background magnetic charges for flavor symmetries in the Coulomb branch [8,17] and baryonic charges in the Higgs branch [18][19][20]. The two set of parameters are exchanged by mirror symmetry and we provide a precise map in section 3.4. Section 4 contains the derivation of our main formula (4.2) for unitary groups. We first derive the baryonic Higgs branch Hilbert series of T σ (SU (N )) by a direct evaluation of the the Molien-Weyl integral. We complement the result with a second derivation based on a localization formula for the character of the Higgs branch moduli space, which can be interpreted as a nilpotent orbit of SU (N ). The localization formula apply to the standard resolution of the orbit as a cotangent bundle of a flag variety and can be expressed in terms of generalized Hall-Littlewood functions. In section 4.2.3 we show that the Higgs branch Hilbert series of T ρ σ (SU (N )) can be obtained from that of the theory T σ (SU (N )) by taking residues with respect to the flavor fugacities. A mirror statement holds for the Coulomb branch: the Coulomb branch Hilbert series of T σ ρ (SU (N )) can be obtained from that of T σ (SU (N )) by taking residues with respect to the fugacities for the topological symmetry. Section 5 contains several explicit examples for unitary groups. Finally, section 6 contains the generalization of our results to orthogonal and symplectic groups. After reviewing some general facts about partitions and orbit resolutions for orthogonal and symplectic groups, we present a generalised Hall-Littlewood formula (6.7) for a generic group G. We present a series of results for U Sp(4) and SO (5) and discuss in details subtleties related to the choice of SO/O gauge groups in the quiver. Other useful results, including explicit examples for other groups of low rank, are given in a series of appendices.
2 Quiver diagrams for T σ ρ (G) with G a classical group The theories T σ ρ (G) are a class of 3d N = 4 superconformal field theories arising as infrared limits of linear quivers with unitary or alternating orthogonal-symplectic gauge groups [3]. G is a classical group and σ and ρ are partitions of G and G ∨ , respectively, as defined below. The theories T σ ρ (G) can be defined in terms of configurations of D3 branes suspended between NS and D5-branes [6], possibly in the presence of an orientifold O3 plane [3,7]. G is determined by the type of orientifold and the two partitions σ and ρ specify how the D3 branes end on the D5-branes and the NS5-branes, respectively. An example for G = SU (N ) is depicted in figure  1. By construction, the mirror of T σ ρ (G) is T ρ σ (G ∨ ). 3 The quiver for T σ ρ (G) can be extracted from the brane construction using standard brane moves [6] and paying attention to the presence of orientifolds [3,7].
We could not find in the literature the quiver for the T σ ρ (G) theory for a general classical group and generic partitions σ and ρ and we present it here. We also discuss the T σ ρ (U Sp (2N )) theories. Note that the quivers for T ρ (SO(N )) have been explicitly written in [21].
Partitions of G are defined as follows. A partition of G = SU (N ) is a nonincreasing sequence of integer numbers (parts) corresponding to a partition of N . Partitions for other classical groups are required to satisfy some constraints. A partition of G = SO(N ) is a partition of N where any even part appears an even number of times. The partition is called a B-or a D-partition if N is odd or even, respectively. A partition of G = U Sp(2N ) is a partition of 2N where any odd part appears an even number of times. Such a partition is called a C-partition. With these definitions, partitions are in one-to-one correspondence with the nilpotent orbits of the group G and also with the homomorphisms Lie(SU (2)) → Lie(G) [22]. The interpretation of these constraints in terms of D3 branes ending on D5-branes in the presence of an O3 plane is given in [3,7].
2.2 T σ ρ (SO(2N )) These theories can be realised on the worldvolume of N D3 branes parallel to an orientifold O3 − plane and ending on systems of half D5 branes and of half NS5 branes. The partitions σ and ρ determine how the D3 branes end on the half D5 branes and on the half NS5 branes respectively. In this case both σ and ρ are D-partitions of SO(2N ), of lengths and .
The quiver diagram for T σ ρ (SO(2N )) consists of alternating (S)O/U Sp groups depicted in (2.6), where each grey nodes with a label N denotes an O(N ) or SO(N ) group and each black node with a label N denotes a U Sp(N ) group.
The length of the quiver (2.6) is L = 2 /2 − 1, unless N L = M L = 0, in which case we remove such nodes from the quiver and the length reduces to L − 1.
The labels M j , with 1 ≤ j ≤ L, for the flavor symmetries are determined by σ T as in (2.3). On the other hand, the labels N j , with 1 ≤ j ≤ L, for the gauge symmetries are given by where [n] +(resp. −) denotes the smallest (resp. largest) even integer ≥ n (resp. ≤ n).
2.3 T σ ρ (SO(2N + 1)) These theories can be realised on the worldvolume of N D3 branes parallel to an orientifold O3 − plane and ending on systems of half D5 branes and of half NS5 branes. The partitions σ and ρ determine how the D3 branes end on the half D5 branes and on the half NS5 branes respectively: σ is a B-partition of SO(2N + 1), of length , and ρ is a C-partition of U Sp(2N ), of length .
The quiver diagram for T σ ρ (SO(2N +1)) consists of alternating (S)O/U Sp groups depicted in (2.8), where each grey nodes with a label N denotes an O(N ) or SO(N ) group and each black node with a label N denotes a U Sp(N ) group.
where the length of the quiver is given by L = 2 /2 .
The labels M j , with 1 ≤ j ≤ L, for the flavor symmetries are determined by σ T as in (2.3). On the other hand, the labels N j , with 1 ≤ j ≤ L, for the gauge symmetries are given by where [n] + is the smallest odd integer ≥ n.
Here and in the following, the distinction between SO(N ) and O(N ) groups is important. Theories with SO(N ) gauge groups have typically more BPS gauge invariant operators compared with the same theory with gauge group O(N ) and we have different theories according to the choice of O/SO factors. Examples are discussed in section 6.
2.4 T σ ρ (U Sp(2N )) These theories can be realised on the worldvolume of N D3 branes parallel to an orientifold O3 + plane and ending on systems of half D5 branes and of half NS5 branes. The partitions σ and ρ determine how the D3 branes end on the half D5 branes and on the half NS5 branes respectively: σ is a C-partition of U Sp(2N ), of length , and ρ a B-partition of SO(2N + 1), of length .
The quiver diagram for T σ ρ (U Sp(2N )) consists of alternating (S)O/U Sp groups depicted in (2.10), where each grey nodes with a label N denotes an O(N ) or SO(N ) group and each black node with a label N denotes a U Sp(N ) group.
where L = 2 /2 and if N L and M L are both zero, we remove such nodes from the quiver, in which case the length of quiver (2.10) is L − 1.
The labels M j , with 1 ≤ j ≤ L, for the flavor symmetries are determined by σ T as in (2.3). On the other hand, the labels N j , with 1 ≤ j ≤ L, for the gauge symmetries are given by 2N )) These theories can be realised on the worldvolume of N D3 branes parallel to an orientifold O3 + plane and ending on systems of half D5 branes and of half NS5 branes. The partitions σ and ρ determine how the D3 branes end on the half D5 branes and on the half NS5 branes respectively. In this case both σ and ρ are C-partitions of U Sp(2N ), of lengths and respectively.
The quiver diagram for T σ ρ (U Sp (2N )) consists of alternating (S)O/U Sp groups depicted in (2.12), where each grey nodes with a label N denotes an O(N ) or SO(N ) group and each black node with a label N denotes a U Sp(N ) group.
and if both N L and M L are zero, the nodes are removed from the quiver and the length of the quiver (2.12) is L − 1.
The labels M j , with 1 ≤ j ≤ L, for the flavor symmetries are determined by σ T as in (2.3). On the other hand, the labels N j , with 1 ≤ j ≤ L, for the gauge symmetries are given by (2.14) 3 The Hilbert series of T σ ρ (SU (N )) In this section we state the general formulae for the Hilbert series of the Coulomb and Higgs branch of T σ ρ (SU (N )) theories. The Hilbert series for the Coulomb branch can be written using the monopole formula [2], while the Hilbert series for the Higgs branch can be written as a Molien-Weyl integral. We introduce background magnetic fluxes for the flavor symmetries in the Coulomb branch [8] and baryonic charges in the Higgs branch [18][19][20]. We explain how fugacities, fluxes and charges are related by mirror symmetry. We also provide a useful brane description of the theory.

Brane configurations
The theory T σ ρ (SU (N )) can be realized with N D3 branes suspended between NS5-branes and D5-branes, where and are the length of the partitions ρ and σ respectively. We order the branes in such a way that all the D5-branes are on one side of the NS5 branes (on the left in figure 1). The parts of ρ = (ρ 1 , . . . , ρ ) are the net number of D3 branes ending on the NS5 branes going from the interior to the exterior of the configuration and the parts σ = (σ 1 , . . . , σ ) are the net number of D3 branes ending on the D5-branes again going from the interior to the exterior. Since the partitions are ordered as in (2.1), the smallest part of ρ and σ are associated with the most external NS5 and D5-branes, respectively, and they increase going into the interior. The configuration must satisfy the s-rule requiring that no more than one D3 brane can end on the same pair of NS5 and D5-branes [6]. The quiver can be read after splitting the D3 branes among the NS5 branes and moving the D5-branes inside the NS5 intervals as in the example in Figure 1 and Figure 2. The result is the quiver in (2.2).
Unless otherwise stated we always use the following convention in reading the quiver from the brane configuration. When talking about order we always refer to the brane configuration where all the D5 are on one side of the NS5, as in figure 1. The gauge groups are numbered by following the NS5 intervals from the interior to the exterior of the configuration, or, equivalently, in the direction which goes from the D5 to the NS5. The first gauge group U (N 1 ) in (2.2) is the one living on the NS5 interval closer to the D5-branes. Figure 1. A brane construction for T (3,2,2) (2,2,2,1) (SU (7)). The partition σ = (3, 2, 2) gives the net number of D3 branes ending on each D5-brane from the interior to the exterior. The partition ρ = (2, 2, 2, 1) gives the net number of D3 branes ending on each NS5-brane from the interior to the exterior. Here x i are the fugacities associated with each NS5 brane and n j are the background monopole charges associated with each D5-brane.
x 1 (2,2,2,1) (SU (7)) after the D5-branes are moved inside the NS5-brane intervals. Right: the linear quiver is read off from the brane configuration. We adopt the convention that the i-th gauge group corresponds to the D3-brane interval between x i and x i+1 : hence U (1), U (2), U (1) from left to right are regarded as the first, second and third gauge groups respectively, and similarly U (2) and U (1) are regarded as the second and third flavor groups respectively. (3,2,2) (SU (7)), obtained by exchanging D5branes and NS5-branes in Figure 1. Bottom left: the D5-branes are moved inside the NS5-brane intervals. Bottom right: the quiver diagram read off from the bottom left brane configuration. We adopt the convention that the i-th gauge group corresponds to the D3brane interval between n i and n i+1 : hence U (1) and U (2) are regarded as the first and the second gauge groups respectively, and similarly U (1) and U (3) are regarded as the first and the second flavor groups respectively.

3.2
The monopole formula for the Coulomb branch of T σ ρ (SU (N )) It is convenient to associate fugacities and fluxes to the NS5 and D5-branes, respectively. We assign fugacities x = (x 1 , . . . , x ) to each NS5 brane and we order them from the interior to the exterior of the branes configuration as in Figure 1. We also assign fluxes n = (n 1 , · · · n ) to the D5-branes and we order them again from the interior to the exterior of the branes configuration as in Figure 1.
The monopole formula [2] for quiver (2.2) is given by 4 (3.1) where m j = (m j,1 , . . . , m j,N j ) with 1 ≤ j ≤ L are dynamical magnetic charges associated with gauge group U (N j ), n j = ( n j,1 , . . . , n j,M j ) are background magnetic charges associated with the flavor group U (M j ), and is the number of gauge groups. Here z j and y j are fugacities for the topological U (1) symmetries associated with the gauge groups U (N j ) and flavor groups U (M j ) respectively. Since the flavor symmetry is actually ( L j=1 U (M j ))/U (1), these fugacities are not independent. Rather they satisfy the constraint which ensures that a shift of the magnetic charges corresponding to the removed U (1) does not affect the monopole formula (3.1). We will refer to this as a shift symmetry in the following. We now need to translate the topological fugacities z and y and the background magnetic fluxes n in terms of the previously defined variables x and n.
The fugacities z and y are related to x as Due to monopole operators, the topological symmetry U (1) −1 is enhanced to where ρ T = ( ρ 1 , . . . , ρ ), with ρ 1 ≥ . . . ≥ ρ > 0 is the transpose partition of ρ [3]. ρ i − ρ i+1 is the number of parts of ρ equal to i. As expected by mirror symmetry, (3.6) is the flavor symmetry of the mirror theory T ρ σ (SU (N )). The x become fugacities for the non-abelian symmetry (3.6). We can split the x into pieces where, by definition, x i is the set of x k with ρ k = i. The x i are fugacities for the group U ( ρ i − ρ i+1 ). The constraint (3.5) ensures that the overall U (1) is removed in (3.6). Notice that the splitting (3.7) reverses the order of the x i . The x i are constructed by collecting together all the fugacities of the NS5 associated with the parts of ρ equal to i and the index i increases going in the direction which goes from the NS5 to the D5, from the exterior to the interior, while the x i are ordered in the opposite direction. For the flavour symmetry the background monopole fluxes n j are related to the n = (n 1 , · · · , n ) in a similar manner. n j is the set of fluxes n k with σ k = j. The n i are fugacities for the group U ( σ i − σ i+1 ). Once we move the D5 inside the NS5 intervals, the fluxes in n j are those associated with the D5-branes living in the j-th interval, with the intervals ordered going from the D5 to the NS5 branes, according to our general convention. Notice that, in this case also, the splitting of the fluxes into the n i pieces reverses the original order of the n i . Let us discuss some examples. In Figure 1, we have n 1 = ∅, n 2 = (n 3 , n 2 ), n 3 = (n 1 ), and x 1 = (x 4 ), x 2 = (x 3 , x 2 , x 1 ). Notice that the order of the x i and n i has been reversed. The splitting of n i , corresponding to the flavour symmetry, is manifest in Figure 2. On the other hand, the splitting of the topological fugacities x i is not manifest in Figure 2, but this becomes apparent in the mirror quiver depicted in Figure 3.

3.3
The baryonic generating function for the Higgs branch of T σ ρ (SU (N )) The baryonic Hilbert series for quiver (2.2) is given by the Molien-Weyl integral [19,20] g[T σ ρ (SU (N ))](t; w 1 , . . . , w ; B 1 , . . . , where L = − 1 as before. w j = (w j,1 , . . . , w j,M j ) with 1 ≤ j ≤ are fugacities for the flavor symmetry 10) and the integration variables s i,k with 1 ≤ k ≤ N i parameterise the Cartan of the gauge groups U (N i ). The integration over the U (1) center of each U (N i ) factor selects the operators of baryonic charge B i for the leftover SU (N i ) gauge groups. 1 ≤ i ≤ L with the understanding that terms with occurrences of s L+1,p should not be included in the product. The numerator in (3.9) contains the Haar measure and the contribution of the F-term relations while the denominator receives contributions from the fundamental and bifundamental fields in the quiver.

Mapping of parameters under mirror symmetry
Under mirror symmetry T σ ρ (SU (N )) is exchanged with T ρ σ (SU (N )). The Coulomb branch of the former is identified with the Higgs branch of the latter and, at the level of Hilbert series, we have H mon [T σ ρ (SU (N ))](t; x; n 1 , · · · , n ) = x s(n) g[T ρ σ (SU (N ))](t; x 1 , . . . , x ; B 1 , . . . , B −1 ) , (3.11) where the relation between fugacities and charges in the two sides of the equation can be determined by comparing global symmetries and following the brane configuration under S-duality. The result is the following. The x i are defined in terms of x as in (3.7). The x i with i = 1, . . . are fugacities for the global symmetry S ( i U ( ρ i − ρ i+1 )) which is the topological symmetry of T σ ρ (SU (N )) and the flavor symmetry of T ρ σ (SU (N )). The x are associated with the NS5-branes in the Coulomb picture as in Figure 1 and with the D5-branes after S-duality, consistently with the identification made above.
The baryonic charges B i , which can also be viewed as magnetic charges for the topological symmetry, are instead given by where the n i and n i are related as discussed at the end of section 3.2. Recall that the n i are associated with the D5-branes and ordered in the direction which goes from the NS5-branes to the D5. After an S-duality the n i are associated with the NS5branes and ordered in the direction which goes from the D5-branes to the NS5 of the final configuration. The baryonic charge of the group living in the i-th NS5 interval is given by the difference between the fluxes on the two NS5 branes delimiting the interval. We follow the convention that the gauge groups are ordered in the direction which goes from the D5-branes to the NS5 even after S-duality.
The prefactor x s(n) is determined by the brane configuration of T ρ σ (SU (N )) as follows. First of all, write down the brane configuration of T ρ σ (SU (N )) as obtained from mirror symmetry, labelling each NS5-brane by n 1 , n 2 , . . . , n from the interior to the exterior, and each D5-branes by x 1 , x 2 , . . . , x from the interior to the exterior as in Figure 3. The relevant contributions to x s(n) come from D3-branes that stretch between an NS5-brane and a D5-brane and not from those split between NS5-brane intervals. In particular, any D3-brane that stretches between a D5-brane labelled by x i and an NS5-brane labelled by n j contributes the monomial x n j −n 1 i to the prefactor. Multiplying all such contributions, the prefactor x s(n) is then given by (3.13) The rationale for this prefactor comes from the residue computation presented in Appendix. To illustrate the above procedure, we provide an example of T (2,2,1,1) (3,2,1) (SU (6)) in (5.30). The dotted horizontal lines indicated in blue and red indicate the D3brane segments that contribute non-trivially to the prefactor x s(n) . In this example, We have explicitly tested the relation (3.11) in several different cases. Notice that there is an ambiguity in associating the n i corresponding to the same block n j to the NS5 branes after S-duality. However, the Coulomb branch formula is manifestly invariant under permutations of fluxes belonging to the same flavor symmetry U (M i ). One can check that also the Higgs branch formula is the same for set of fluxes B i obtained by permuting the entries in the various blocks n j .

The generalised Hall-Littlewood formula for T σ ρ (SU (N ))
In this section we provide a closed formula for the Hilbert series of the Coulomb branch of T σ ρ (SU (N )), or equivalently the Hilbert series of the Higgs branch of T ρ σ (SU (N )). The Higgs branch part of the computation can be reinterpreted in the language of localization and generalizes a known connection between Hall-Littlewood polynomials and Hilbert series of cotangent bundles of flag varieties [29,30]. Subtleties and complications arising for other classical groups are discussed in section 6.
To state the formula we first need to repackage the magnetic fluxes in yet another form. We construct a string of N integers by repeating σ i times each flux n i n σ = (n σ 1 1 , n σ 2 2 , · · · , n σ ) , where n a means n repeated a times.

4.1
The formula for T σ ρ (SU (N )) The Coulomb branch formula for T σ ρ (SU (N )) can be written as and it is valid when the fluxes are fully ordered n 1 ≥ n 2 ≥ · · · ≥ n . The notations are defined as follows.
1. Q n σ σ is a generalised Hall-Littlewood function for the group SU (N ), given by where • ∆ + is the set of positive roots of SU (N ), which can be written in standard notation as α = e i − e j (with 1 ≤ i < j ≤ N ).
• ∆ σ is the set of positive roots in the diagonal blocks associated with σ: • the sum over w is over the Weyl group of SU (N ).
• n σ determines a point in the weight lattice of U (N ). It is a dominant weight left invariant by the elements of the Weyl group i=1 S σ i that only permutes indices within the blocks associated with σ.
• The factor indicated in blue enters in the definition of the standard Hall-Littlewood polynomial. The factor indicated in purple is a modification appearing for non-trivial partitions σ = (1 N ).
2. The power p σ (n σ ) is given by where for the positive root α = e i −e j (with 1 ≤ i < j ≤ N ), α(n σ ) = n σ i −n σ j and d σ (e i − e j ) depends only on the index i: it is equal to the size of block in the decomposition given by σ to which e i belongs.
which we shall henceforth abbreviate as a, is determined by the following decomposition of the fundamental representation of SU (N ) to G ρ × ρ(SU (2)): where G ρ k = U (r k ) denotes a subgroup of G ρ corresponding to the part k of the partition ρ that appears r k times and the x k are defined as in (3.7). Formula (4.5) determines a as a function of t and { x k } as required. Of course, there are many possible choices for a; choices related by outer automorphisms of SU (N ) are equivalent.
4. The prefactor K ρ (x; t) can be determined as follows. The embedding specified by ρ induces the decomposition where a on the left hand side is the same a as in (4.5). Each term in the previous formula gives rise to a plethystic exponential, giving 4.2 Derivation of the Hall-Littlewood formula for T σ ρ (SU (N )) We first consider the Coulomb branch formula (3.1) for the theory T σ (SU (N )), where omitted partitions are understood to be the trivial one (1 N ). By mirror symmetry we can equivalently compute the baryonic Higgs branch Hilbert series for T σ (SU (N )) using equation (3.11).

T σ (SU (N )): computing residues for the gauge fugacities
In the case of T σ (SU (N )) the quiver is where (n) and [n] indicate a U (n) gauge and flavor group respectively. By defining N 0 = N and s 0,k = x k , we can rewrite the Molien-Weyl formula as follows where N i = k=i+1 σ k with i = 0, · · · , − 1. We need to identify the poles that contributes to the integral (4.9). We choose to evaluate the integral adding (minus) the contributions from all the poles outside the unit circle. For positive baryonic charges B i ≥ 0 there are no poles at s i,k = ∞. Assuming |t| < 1, the poles for the fugacities of the gauge group U (N i ) are of the form . Most of these poles give the same contribution to the integral due to the permutation symmetry of the fugacities s i,k for each i and this contribution is compensated by the factors N i !. Let us consider the contribution of the pole for the gauge group U (N i ). The residue of the i-th term in the product in (4.9) is (4.12) Combining the contributions of all the groups and observing that, by iteration, s i,k = x k /t i and s i,k /s i,p = x k /x p we obtain the contribution where P runs over all the entries (k, p) of the upper triangular part of an N × N matrix with diagonal blocks of sizes (σ , · · · , σ 1 ) removed. Here σ corresponds to the block on the top of the matrix. All other poles in (4.10) give contributions that are obtained by permuting the x i . Permutations that exchange only indices belonging to same blocks can be reabsorbed by a permutation of the s i,k and do not lead to new contributions. We can rewrite the result in a more compact form in terms of roots. Using the conventions where the positive roots of SU (N ) (e i − e j with i < j) corresponds to the entries (i, j) of the hermitian matrix in the Lie algebra Lie(SU (N )), we find t pσ(n σ ) where ∆ + is the set of positive roots of SU (N ), ∆ σ is the set of positive roots in the diagonal blocks of size σ i and W L(σ) is the subgroup of the Weyl group of SU (N ) which just permutes roots inside the various blocks. 5 p σ (n σ ) is defined in (4.4) and n σ in (4.1). It is convenient to write the expression (4.14) as follows where Q n σ σ (x; t) and K ρ have been defined in (4.3) and (4.7) respectively. We can extend the sum to the entire Weyl group since the fluxes are equal inside the blocks. We have thus recovered the expression (4.2).
The computation is valid for B i ≥ 0 which, using (3.12) and (4.1), correspond to fully ordered fluxes n 1 ≥ n 2 ≥ · · · ≥ n . For other values of B i , extra poles at infinity might affect the result and give a more complicated expression.

The localisation formula
We can reinterpret the previous computation in terms of localisation. A similar approach has been successfully used for the computation of the Hilbert series of noncompact Calabi-Yaus [16] and the Hilbert series of instanton moduli spaces [15]. We use localisation in the following form. Suppose that we have a line bundle L over a smooth manifold X with a holomorphic action of a torus µ : T → X with isolated fixed points. The Lefschetz fixed point formula states that [31,32] where P are the fixed points of the torus action, m j P , j = 1, · · · , dimX are the weights of the linearization of the torus action µ on the tangent space of X at the point P and m L P is the weight of the action of µ on the fiber of the line bundle at P . q denotes a set of fugacities for the action of T . Whenever the higher cohomology groups H (0,i) (X, L) , i ≥ 1 vanish the left hand side of (4.16) computes the Hilbert series counting holomorphic sections of the line bundle L.
In order to use formula (4.16) we need to find an algebraic description of the Higgs branch of T σ (SU (N )), a smooth resolution of it, and the conditions under which the higher cohomology groups vanish.
It is known that, as an algebraic variety, the Higgs branch of T σ (SU (N )) is the closure of the nilpotent orbit of Jordan type σ T [23,33,34] Higgs(T σ (SU (N ))) = O σ T . (4.17) Recall that a partition λ = (λ 1 · · · , λ l ) of N naturally identifies a nilpotent matrix N λ in Lie(SU (N )) with Jordan blocks of dimension λ i . The nilpotent orbit O λ of type λ is, by definition, the orbit of N λ under the adjoint action of SU (N ). Notice that the transpose of σ enters in (4.17).
It is also well known that the singular variety O σ T has a smooth resolution, called the Springer resolution, in terms of the cotangent bundle of a flag variety. P here is a parabolic subgroup of SU (N ) consisting of the upper triangular block matrices with blocks of dimensions σ i . The non-zero entries in P are those belonging to diagonal blocks of dimensions σ i ×σ i in addition to all the entries above the diagonal. The homogeneous space SU (N )/P parametrizes all the possible flags of type σ in C N , i.e. the set of vector subspaces We can also give a different characterization of T * (SU (N )/P ) which is sometime useful. The elements in P belonging to the diagonal blocks form a subgroup S( i U (σ i )) of P , called the Levi subgroup and denoted by L(σ). Accordingly, the Lie algebra of P decomposes as where the nil-radical n(P ) consists of nilpotent matrices. The cotangent bundle T * (SU (N )/P ) can be written as SU (N ) × P n(P ), which is the quotient of SU (N ) × n(P ) by the equivalence relation The resolution in (4.19) is just given by µ : (g, n) → gng −1 .
We can now use the localisation formula (4.16). We can apply the formula to X = T * (SU (N )/P ) since it has the same holomorphic functions as O σ . The torus action is induced by the Cartan subgroup of SU (N ) and by the scaling symmetry, with associated fugacities x and t. The Cartan subgroup of SU (N ) acts in the obvious way on the cosets in SU (N )/P and its action is naturally extended to the cotangent bundle. The scaling symmetry acts on the cotangent fiber. This torus action has isolated fixed points. A coset gP in SU (N )/P is fixed by the action of the Cartan torus T ⊂ SU (N ) if and only if T ⊂ gP g −1 and this selects g ∈ W SU (N ) /W L(σ) where W SU (N ) is the Weyl group of SU (N ) and W L(σ) the Weyl group of the Levi subgroup of P . The fiber at the fixed points must be zero because of the scaling symmetry. In order to use (4.16) we need to linearize the torus action around the fixed points. The tangent space to T * (SU (N )/P ) at a fixed point can be written as where the first factor is the tangent space to the flag manifolfd SU (N )/P and the second to the cotangent fiber. The torus action on an element of the root space α in Lie(SU (N ))/Lie(P ) is x α while on the corresponding element in the dual space Lie(SU (N )) * /Lie(P ) * is t 2 x −α . We also consider a line bundle L associated with the fluxes n, which give the weight of the representation of the Cartan subgroup of SU (N ) on the fiber. The right hand side of (4.16) then reads where ∆ + is the set of positive roots of SU (N ), while ∆ σ is the set of positive roots in L(σ). The product in (4.23) covers all the roots in Lie(SU (N ))/Lie(P ) which correspond to the entries in the lower triangular part of the matrix with the exclusion of those living in the diagonal blocks. When σ = (1, · · · , 1) the sum in (4.23) runs over all the positive roots and the expression in (4.23) becomes a (dual) Hall-Littlewood polynomial [29]. The expression (4.23) is the baryonic Hilbert series of the Higgs branch of T σ (SU (N )) provided the higher cohomology group of the line bundle L vanish. Sufficient conditions for the vanishing have been discussed in [35] (see Proposition 3.7) and require that n σ is a dominant weight and it is fixed by the action of W L(σ) . This requires that all the entries in n σ are ordered and equal in the blocks corresponding to the partition σ where n a means n repeated a times. The rest of the computation is the same as in section 4.2.1. We can manipulate expression (4.23) and obtain again the final formula (4.15). The sum is extended to the entire Weyl group assuming the condition (4.24) on the fluxes. In this approach the overall prefactor t pσ(n σ ) is found by an explicit comparison with the monopole formula.

Computing residues in the flavor fugacities
We state the following general observation: The Higgs branch Hilbert series of T ρ σ (SU (N )) can be obtained from that of the theory T (1 N ) σ (SU (N )) by taking residues with respect to the flavor fugacities.
The Higgs branch Hilbert series (4.9) for T ρ σ (SU (N )) has poles corresponding to a particular limit of the fugacities. The residue at this pole is the Higgs branch Hilbert series for T ρ σ (SU (N )), where ρ is obtained from ρ by moving the last box in the partition ρ to a previous column. For example, we can go from the trivial partition ρ = (1 N ) to ρ = (2, 1 N −1 ) as follows: where w 1 = (x N −1 , · · · , x 2 ), w 2 = (x 1 ) and the first line receives the contribution from the residue We give a proof and a generalization of this formula to partitions ρ and ρ of SU (N ) which are related by moving a single box in Appendix C. Any partition ρ can be obtained from the trivial partition (1 N ) by an iteration of the previous move. Therefore by repeated residue computations we may extract the Higgs branch Hilbert series of any T ρ σ (SU (N )) theory from that of T (1 N ) σ (SU (N )). We can do a completely analogous computation in the Coulomb branch. The mirror of the previous observation is: The Coulomb branch Hilbert series of T σ ρ (SU (N )) can be obtained from that of T σ (1 N ) (SU (N )) by taking residues with respect to the topological fugacities.
As discussed in section 6 of [8], the monopole formula for T σ ρ (SU (N )) has poles corresponding to a particular limit of the fugacities. The residue at this pole gives the monopole formula for T σ ρ (SU (N )), where ρ is obtained from ρ by moving the last box in the partition ρ to a previous column. This was proven in [8] for the case σ = (1 N ) but it can straightforwardly extended to the case of a general σ.
For example, we can go from the trivial partition ρ = (1 N ) to ρ = (2, 1 N −1 ) by computing By repeated residue computations we may extract the Coulomb branch Hilbert series of any T σ ρ (SU (N )) theory from that of T σ (1 N ) (SU (N )). By carefully mapping the fugacities under mirror symmetry, we see that the two previous observations are consistent with (3.11). Notice that in taking residues with respect to the flavor symmetries we obtain a prefactor with powers of x i in the Higgs branch computation but not in the Coulomb branch one. This is consistent with and explains the prefactor (3.13) in (3.11).
The observations can be now used to conclude our proof of (4.2). The Higgs branch Hilbert series of T ρ σ (SU (N )) can be obtained from (4.15) by taking residues with respect to the flavor fugacities. Notice that the poles in formula (4.15) come only from the factor K (1 N ) (x, t). The partition ρ can be obtained from (1 N ) by a set of moves like those in (C.1). It is not difficult to see that this set of moves has the effect of replacing x with a ρ (t, x) given in (4.5). The multiplicative factors in (4.25) and (C.7) cancel some terms in the denominator of K (1 N ) (x, t) and transform it into K ρ (x, t). They also introduce a prefactor which coincides with (3.13). In this way we obtain the general expression for the Higgs branch Hilbert series T ρ σ (SU (N )). Removing the prefactor according to (3.11), we obtain precisely the general expression for the Coulomb branch Hilbert series of the mirror T σ ρ (SU (N )) given in (4.2). Geometrically, the structure of the factor K ρ (x, t) is related to the fact that, as an algebraic variety, the Coulomb branch of T σ ρ (SU (N )), equivalently the Higgs branch of T ρ σ (SU (N )), is the intersection of the nilpotent orbit of type σ T with the Slodowy slice of type ρ [3,23], (4.28) The Slodowy slice is defined as follows. The partition ρ identifies a homomorphism ρ : Lie(SU (2)) → Lie(SU (N )) by saying that the image of J + = J 1 + iJ 2 , where J i are the standard generators of SU (2), is a nilpotent matrix of Jordan type ρ. A theorem by Jacobson and Morozov guarantees that the map between partitions and homomorphisms is one-to-one [22]. The Lie algebra of SU (N ) decomposes under the homomorphism ρ into a set of irreducible G ρ × SU (2) representations [R j ; 2j] as in (4.6). Let t j be the SU (2) lowest weight in each representation [R j ; 2j]. The Slodowy slice associated with the partition ρ is the subspace of Lie(SU (N )) consisting of the elements of the form The various terms entering (4.7) schematically correspond to such description of the slice.

Applications of the Hall-Littlewood formula for T σ ρ (SU (N ))
In this section we demonstrate the previous results for unitary groups of small rank.
Let us map the results obtained from the right-hand side of (4.2) to those obtained from the monopole formula.
We know that H mon and H coincide for fully ordered fluxes. We see that in certain specific cases this constraint can be relaxed. In general, whenever there are two background fluxes present in the theory, it is always possible to find an ordering of such fluxes in the generalised Hall-Littlewood formula to match the result obtained from the monopole formula. The reason is the symmetry of the monopole formula under permutation of the fluxes belonging to the same flavor group and under change of sign of all the background fluxes together with a reflection x → x −1 of the fugacities. We present another example in the next subsection. Note that when there are three or more background fluxes, this is not always possible; we comment on this below (5.33).
Let us compare this result to the baryonic generating function of the mirror T (3,1) (SU (4)) :
This can be equated to the monopole formula as follows: where the prefactor x n 2 −n 1 1 is due to the D3-brane indicated by the dotted red horizontal line in the diagram below.

Orthogonal and symplectic groups
In this section we discuss the generalisation of our results to other classical groups SO(N ) and U Sp(2N ). We consider the case of U Sp (2N ) in Appendix E. Life is much harder in other types. Complications with orthogonal and symplectic nilpotent orbits and issues with discrete groups make it difficult to state general results. We present few examples for the case of orthogonal and symplectic groups with low rank, mostly for the Coulomb branch of T σ (G) and the Higgs branch of T σ (G) at zero external fluxes, leaving the general analysis for future work. We also provide a generalised Hall-Littlewood formula and discuss its condition of validity. A regular and interesting pattern seems to emerge, which would be interesting to study in more detail. A non-exhaustive list of differences with the unitary case is the following.
• Many of the T σ ρ (G) theories with orthogonal and symplectic groups are bad in the language of [3], meaning that the dimension of some monopole operator computed using the ultraviolet R-symmetry violates the unitary bound. The monopole formula written in terms of the Lagrangian data is ill-defined and divergent. Complications arise also in the Higgs branch where typically there is no complete Higgsing. Such theories are supposed to flow to an interacting superconformal point in the IR but, unfortunately, we have no general description of it.
• Recall that, for T σ ρ (G), σ and ρ are partitions of G and G ∨ , respectively. Partitions of G, as defined in section 2, are in one-to-one correspondence with the nilpotent orbits of the group G and also with the homomorphisms Lie(SU (2)) → Lie(G) [22]. The Coulomb branch of T σ ρ (G), equivalently the Higgs branch of T ρ σ (G ∨ ), as an algebraic variety, can be still written as an intersection of a nilpotent orbit with a Slodowy slice, but this time of the group G ∨ [3] O σ ∨ ∩ S ρ . (6.1) ρ is indeed a partition of G ∨ and determines the Slodowy slice through the homomorphism ρ : Lie(SU (2)) → Lie(G ∨ ). To determine the orbit itself we need a map ∨ : σ → σ ∨ from partitions of G to partitions of G ∨ . Such a map is well known in the mathematical literature [36,37] and we discuss it below. It has also explicitly appeared in the physical literature in the context of the (2, 0) theory compactified on Riemann surfaces with punctures [38,39].
• The quivers T σ ρ (G) contains orthogonal gauge groups as nodes. The distinction between an SO(N ) and an O(N ) gauge group is important. Theories with SO(N ) gauge groups have typically more BPS gauge invariant operators compared with the same theory with gauge group O(N ). We often have different interesting quivers which we can write under the name of T σ ρ (G) and that differ in the choice of O/SO factors. Their Coulomb branch is typically a covering of (6.1). We discuss examples in section (6.3).
• There is a Springer map T * (G/P ) → O σ ∨ , where the parabolic group P is related to σ ∨ by yet another nontrivial map that we discuss below. In other types, the Springer map is not necessarily one-to-one. We can always write a generalisation of the Hall-Littewood function (4.2) for generic classical group G, which computes the Hilbert series of some covering of the moduli space (6.1). 6 The Hall-Littlewood formula computes the Coulomb branch Hilbert series of the quiver T σ ρ (G) for a specific choice of SO/O factors.
All these features are discussed in the explicit examples which are discussed below. We first discuss some general properties of partitions of a classical group G, we write a generalised Hall-Littlewood function and we present the results for U Sp(4) and SO (5). Other groups of low rank are discussed in Appendix D. We provide mirror pairs and we test mirror symmetry by evaluating the monopole formula in the Coulomb branch of T σ (G) theories and the Molien-Weyl integral in the Higgs branch of T σ (G). Whenever the SO/O factors in the theory T σ ρ (G) can be chosen in physically inequivalent ways, we put subscripts in order to differentiate the theories: we adopt the convention that the subscript (I) refers to the theory which has moduli space (6.1); other subscripts correspond to various coverings of (6.1).

Properties of partitions of a classical group G
As discussed above, for any partition σ of G we need to define two auxiliary objects. One is a partition σ ∨ of the dual group G ∨ . The Coulomb branch of T σ ρ (G) is expressed as an algebraic variety in terms of σ ∨ as in (6.1). The other is the Levi type σ L of the parabolic group corresponding to σ ∨ , which is needed to write a resolution of the moduli space.
Recall that a partition σ of a classical group identifies the Jordan type of a nilpotent element of Lie(G) up to conjugacy. The Jordan types of matrices in the Lie algebra of a classical group are restricted as follows [22]. A partition of type A is just a non-increasing sequence of integers. A partition of type B and D is a non-increasing sequence of integers where all the even parts appear an even number of times. A partition of type C is a nonincreasing sequence of integers where all the odd parts appear an even number of times.
For each non-increasing sequence of integers σ we can define a B-, C-and D-collapse as the maximal partition τ ≤ σ of type B, C and D, respectively.
• For G = U Sp(2N ), σ ∨ is obtained by transposing σ, adding a new part equal to 1 to the transpose partition and B-collapsing.
∨ is an inclusion-reversing map between the orbits of G and G ∨ which becomes one-to-one when restricted to the so-called special orbits [36,37]. Consider now the nilpotent orbit associated with σ ∨ . We are interested in maps where P is a parabolic subgroup of G with dim O σ ∨ = 2 dim G/P . Such P is called a polarization of σ ∨ . For G = SU (N ), polarizations exist and are unique. As we already discussed in section 4.2.2, we have σ ∨ = σ T and, up to conjugation, the associated parabolic group P is the group of upper triangular block matrices with blocks of size σ i . The algebra of P decomposes as LieP = LieL(σ) ⊕ n(P ) (6.4) where n(P ) is the nil-radical of LieP and L(σ) is the Levi subgroup. Here LieL(σ) = i U (σ i ) consists of block diagonal matrices of sizes σ i . Notice that the original partition σ determines the structure of blocks in P while the transpose partition σ T determines the Jordan type of the nilpotent orbit.
The conditions for the existence of a polarization of σ ∨ for other classical groups have been discussed in [40][41][42]. The structure of parabolic groups is more complicated than for SU (N ), see for example section 2 of [42]. The Levi subgroup L(σ) is now of the form L(σ) = g × n i=1 U (l i ) 2 where g is a classical group of the same type (B, C or D) as G and each factor U (l i ) appears an even number of times. We denote with σ L the set of numbers (l 1 , l 1 , · · · , l n , l n , p), where p is the dimension of the block corresponding to g. Notice that σ L is not strictly a partition of G. There is yet another map from the Levi type of a parabolic group G to a partition of G [40] S : σ L → { partitions of G } (6.5) defined as follows. Let π be the set obtained by ordering the parts of σ L in non-increasing order. Define the set of indices I(π) = {j ∈ N| j = n (mod 2), π j even for SO and odd for USp, π j ≥ π j+1 + 2} (6.6) where n is the dimension of the matrix giving the classical representation of the group G.
The map S is defined by a series of moves. For all the indices j belonging to I(π) we simultaneosly decrease by one unit the parts π j and increase by one unit the corresponding part π j−1 . S(σ L ) is then a partition of G (see for example, Theorem 2.7 in [42]). Moreover, σ L is the Levi type of a polarization of σ ∨ if and only if S(σ L ) = σ ∨ . All the polarizations of classical groups of small rank are explicitly tabulated in [40], including all the cases considered in this paper.
In contrast with G = SU (N ), polarizations are not unique for other classical groups. Moreover the map (6.3) is not necessarily one-to-one and, therefore, it is not necessarily a resolution. The degree of the map (6.3) can be explicitly computed from (6.6): it is 2 |I(π)| except for the special case of σ L of SO groups with no special part p and all other parts odd where it is given by 2 |I(π)|−1 (see Theorem 8 in [42]). When the degree is one the map (6.3) is a Springer resolution of O σ ∨ .

The generalised Hall-Littlewood formula for a classical group G
The generalised Hall-Littlewood formula for a classical group is expressed in terms of geometric data of the dual group. It is then convenient to write the generalised Hall-Littlewood formula for the dual group G ∨ .
The Coulomb branch Hilbert series for T σ ρ (G ∨ ) is where the notations are defined as follows.
1. σ is a partition of G ∨ and ρ is a partition of G.

2.
Here Q n σ is the modified dual Hall-Littlewood polynomial associated to a Lie group G, given by • L(σ) denote the Levi subgroup associated with the partition σ ∨ . L(σ) can be computed as described above and is explicitly tabulated in [40] for all cases considered in this paper.
• ∆ σ is the set of positive roots in the diagonal blocks associated with L(σ).
• W G denotes the Weyl group of G.
• W L(σ) denotes the Weyl group of the Levi subgroup L(σ).
• n = r i=1 n i e i , with {e 1 , . . . , e r } the standard basis of the weight lattice and r the rank of G. The Hall-Littlewood formula applies when n is a dominant weight of G invariant under the action of W L(σ) .
3. The power p σ (n) is a linear function of n that generalizes the expression (4.4).
Examples are given in Table 10.
4. The argument a ρ (t, x), which we shall henceforth abbreviate as a, 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. Formula (4.5) determines a as a function of t and {x k } as required. Of course, there are many possible choices for a; choices that are related to each other by outer automorphisms of G are equivalent.
5. The prefactor K ρ (x; t) is independent of n and can be determined as follows. The embedding specified by ρ induces the decomposition where a on the left hand side is the same a as in (4.5). Each product in the previous formula gives rise to a term in the plethystic exponential 6.3 T σ (SO(5)) and T σ (U Sp (4)) We now consider the case of the theories T σ 1 (SO(5)) and T σ 2 (U Sp(4)) where many of the differences with the unitary case are manifest. Here σ 1 is a B 2 -partition and σ 2 is a C 2 -partition. The possible partitions and the mirror pairs are summarized in Table 2. Important data associated with each partition and the Hilbert series for the Coulomb branch of the T σ (G) theory are given in Table 3. (5)) T σ (U Sp(4)) C 2 -part. σ Quiver of T σ (U Sp(4)) T σ (SO (5)) Table 2. Quiver diagrams for T σ (SO(5)) and their mirror theories. The asterisk * indicates the gauge group that renders the quiver a 'bad' theory.
For U Sp(4), the positive roots are e i − e j , e i + e j and 2e i , with 1 ≤ i < j ≤ 2. The C 2partition σ ∨ 1 can be obtained from σ 1 by transposing, deleting a box in the last tuple and C-collapsing. The B 2 -partition σ ∨ 2 can be obtained from σ 2 by adding a box, transposing and B-collapsing. The subscript below σ indicates that we are using the theory T σ (G) II ; we are correspondingly using a polarization where the the degree of map (6.3), as computed using (6.6), is 2. The HS given above are computed from the Hall-Littlewood formula. For comparison, the HS for the Higgs branch of T (1,1,1,1) (SO(5)) and T (1,1,1,1,1) (U Sp(4)) were explicitly computed in (4.34), (4.36) of [8] and the HS for the Higgs branch of the quiver [U Sp(4)] − SO(2) for the theory T (3,1,1) (U Sp(4)) (II) was computed in (D.4) of [8].
We make some general observations about these tables.
• We expect that the T σ ρ (G) theories for isomorphic groups should be equivalent with an appropriate mapping between the partitions, even if the quivers are different. We verify this explicitly at the level of Hilbert series for SO(5) ∼ U Sp(4). In the tables we report in parallel the results for SO(5) and U Sp(4) and the correspondence between partitions.
• In Table 2 we have taken some of orthogonal groups to be SO. The distinction between an SO(N ) and an O(N ) gauge group is important. Theories with SO(N ) gauge groups have typically more BPS gauge invariant operators compared with the same theory with gauge group O(N ). In the Higgs branch, an SO(N ) gauge symmetry allows for baryonic operators and extra mesonic operators which are odd under parity. In the Coulomb branch, the magnetic lattice of SO(N ) is different from that of O(N ). As a general rule, we have only considered quivers without baryonic operators. Even with this restriction, we have often different interesting quivers which we can write under the name of T σ ρ (G) and differ in the choice of O/SO factors. We use the notation T σ ρ (G) (I) and T σ ρ (G) (II) to differentiate these theories. In our conventions, T σ ρ (G) (I) is a quiver with Coulomb branch moduli space equal to (6.1). The Coulomb branch of T σ ρ (G) (II) is instead a double cover of (6.1).
• In Table 3 we present the Hilbert series for the Coulomb branch of the T σ (G) theory based on the Hall-Littlewood formula (6.7). The last two rows in Table 3 contain the two non-trivial partitions σ of SO(5) or U Sp (4). We see that they both correspond to the same σ ∨ . The Hilbert series in the last two rows in Table 3 correspond to two different polarizations of σ ∨ , one of degree one and one of degree two. We have chosen the O/SO factors in the quivers Table 2 in order to match the two different Hilbert series. This involves choosing the theory T σ (G) (II) in some cases.
• We have explicitly computed the Hilbert series for the Higgs branch (using the Molien-Weyl integral) and the Coulomb branch (using the monopole formula) of all the quivers given in the tables whenever they are well defined. The result obviously coincides with that given in Table 3 based on the Hall-Littlewood formula. The monopole formula fails when the quiver is bad. In particular, we can only compute the monopole formula the Coulomb branch Hilbert series of T σ (U Sp(4)), since in general the T σ (SO(5)) theories are bad. Recall that, in general, a linear quiver theory is 'bad' if it contains one of the following items: In the case of a bad quiver, there is no complete Higgsing along the Higgs branch and the F-flat moduli space is not a complete intersection. As a result the Hilbert series needs to be computed using other techniques, for example using Macaulay2 [43]. 7 To fully appreciate the differences between the quivers and the subtlelties about O/SO factors we need a longer discussion which is given in the next subsections.

Relations between quivers with O and SO gauge groups: the Higgs branch of T σ (G)
The Higgs branch of the theories T σ (SO(N )) when all orthogonal gauge groups are of O type (and not SO) was explicitly shown to be the nilpotent orbit O σ ∨ in [44]. The argument can be generalized to T σ (U Sp(N )) [38]. One can see with methods similar to those in Appendix B.3 that the presence of SO groups in Table 2, Table 5 and Table 4 does not introduce extra baryonic operators in the chiral ring and in most of the cases does not affect the Higgs branch. For some particular theories, ungauging the parity in a group O might introduce extra mesonic operators. Consider for example the theory T (3,1,1) (U Sp(4)). We have two choices for the corresponding quiver, [U Sp(4)] − O(2), which we call T (3,1,1) (U Sp (4) (2) is obviously a two-fold covering of the nilpotent orbit (2,2). This is an example of the theories that we call T σ (G) (I) and T σ (G) (II) , with T σ (G) (I) giving the hyperKähler quotient description of a nilpotent orbit and T σ (G) (II) a covering of it.
Let us also notice that hyperKähler quotient constructions for all the nilpotent orbits of all classical groups have been given in the mathematical literature a long time ago [34]. The corresponding hyperKähler quotient is sometimes different from ours, allowing for U Sp groups with odd number of half-hypermultiplets. The quiver corresponding to the T σ (G) theories have always an even number of half-hypermultiplets for any U Sp gauge group in order to cancel parity anomalies and provide a somehow non-minimal (in terms of groups in the quiver) hyperKähler quotient construction of the nilpotent orbits of G. One can use the result in Appendix B.3 to show that the various different formulations for the hyperKähler quotient construction of the same nilpotent orbit are equivalent.

Relations between quivers with O and SO gauge groups: the Coulomb branch of T σ (U Sp(4))
In this section we focus for simplicity on the theories T σ (U Sp(4)). A parallel analysis can be done for T σ (SO (5)).
Part of the story about the theories that we have called T σ (G) (I) and T σ (G) (II) is related to the fact the map ∨ is not injective. The C 2 -partitions σ = (2, 1, 1) and σ = (2, 2) correspond both to the orbit σ ∨ = (3, 1, 1). We should expect that the theories T (2,1,1) (U Sp(4)) and T (2,2) (U Sp(4)) describe the same physics although they have different quivers. We now discuss in what sense this is true. To understand the following discussion, it is important to notice that σ ∨ has two different polarizations, of Levi type σ L = (1 2 , 3) and (2 2 , 1) corresponding to maps (6.3) of degree one and two respectively.
Let us check explicitly that the Coulomb branch of T (2,2) (U Sp(4)) (II) is a double cover of the Coulomb branch of T (2,1,1) (U Sp (4)) at the level of Hilbert series with vanishing background fluxes. This can be seen by gauging the parity in an SO(2) factor in the quiver T (2,2) (U Sp(4)) (II) . The result is the quiver T (2,2) (U Sp(4)) (I) given in Table 4, which is different from the quiver of T (2,1,1) (U Sp(4)) but it is has the same Coulomb branch. The two quivers indeed differ by replacing SO(3) gauge group with O(2) together with shifts in flavor symmetries. We argue now that this move does not change the monopole formula. The reason is the following. As we discuss in appendix A the weight lattice and the classical P factors for O(2) are the same as those for SO (3). Moreover, the dimension of the monopole operator, as a function of the dynamical magnetic charges (but for vanishing background magnetic charges), does not change because the shift in flavors compensates the contribution of the vector multiplet that has been changed. In this way, the monopole formula for the two theories is the same.
To illustrate this, we give the formulae for the dimension of the monopole operators in the two quivers below: where u, a, b are topological charges for SO(2), U Sp(2) and SO(3) gauge groups, and n is the background monopole flux for the global symmetry U Sp (2).
where u, a, b are topological charges for SO(2), U Sp (2) and O(2) gauge groups, and n is the background monopole flux for the global symmetry SO (2). Observe that when n = 0, the two blue terms in (6.12) cancel with each other and the equality between (6.12) and (6.13) can be established. It would be interesting to understand better the role of background fluxes in these theories.
A similar analysis applies to SO(5) and the relevant quivers are given in Table 5. In all other examples considered in Appendix D, whenever two partitions σ 1 and σ 2 correspond to the same σ ∨ , we have two different quivers. One has moduli space (6.1). The other comes in two versions, related by ungauging the parity in one of the O(N ) gauge groups, with moduli space (6.1) or a covering of it, respectively. It is interesting to notice that, in the mathematical literature, the map ∨ comes equipped with a local system, typically a set of discrete symmetries, which is non-trivial precisely when ∨ is not injective [45,46]. It would be interesting to see if there is a relation with our results. A Monopole formula for orthogonal and special orthogonal gauge groups We state the following general observation The P -factor and the GNO lattice of magnetic charges (in the summation of the monopole formula) of an O(2k) group are the equal to those of SO(2k + 1) group. 8 We demonstrate this with an example. Let us compare the following data for SO(4), O(4) and SO(5) groups.
• For SO(4) group, the magnetic fluxes is (m 1 , m 2 ) with m 1 ≥ |m 2 | ≥ 0 and −∞ < m 2 < ∞. The residual gauge symmetries in the presence of various magnetic charges are presented in Table 6.
Monopole fluxes Residual gauge symmetry P -factor (0, 0) SO(4) Table 6. Data for SO(4) group 8 On the other hand, the weights and the roots of O(2k) gauge group appearing in the dimension formula are the same as those of SO(2k) gauge group.
Monopole fluxes Residual gauge symmetry P -factor (0, 0) O(4)  (4), the invariants involving epsilon tensors are projected out by the parity and the Casimir invariant at order 4 becomes another independent one; hence we have .

B Different theories with the same Higgs branch
In this section we analyze the Higgs branch of various N = 4 theories with single gauge group in three dimensions, focussing on pairs of theories that have the same Higgs branch. The results for orthogonal groups are useful to understand the Higgs branch of the T σ (G) theories, the choice of SO/O factors and the equivalence of Higgs branches of different theories.
FI terms and it reduces to the suborbit (2 Nc−k , 1 k ). 10 In fact, this is a special case of the general isomorphism between the Higgs branches of U (N c ) gauge theory with N f flavors and U (N f − N c ) gauge theory with N f flavors at non-vanishing FI parameter [48], which follows from the Grassmannian duality Gr (N c Let us mention certain features of the Higgs branch corresponding to the orbit (2 Nc−k/2 ), where k is even. As discussed above, the rank of the meson is r = N c − k/2. According to (2.14) of [47], this corresponds to a submanifold with enhanced gauge group U (k/2).

B.2 Symplectic gauge groups
We now consider the Higgs branches of U Sp(2N c ) and U Sp(2N c − 2k) gauge theories with 2N c + 1 − k flavors.
• The Higgs branch of U Sp(2N c ) gauge theory with SO(4N c + 2 − 2k) flavor symmetry has a meson M as the generator. It is a matrix of size (4N c + 2 − 2k) × (4N c + 2 − 2k) satisfying M 2 = 0. The maximal rank of M can be 2N c + 1 − k. Keeping the constraints on D-partitions into account, we conclude that the Higgs branch corresponds to the orbit (2 2Nc+1−k ) for odd k and (2 2Nc−k , 1 2 ) for even k.
These orbits correspond to the dual partitions ρ ∨ which can be obtained by transposing and D-collapsing of the following partitions ρ:

(B.4)
Using the rule given in section 2, we conclude that • The Higgs branch of U Sp(2N c ) gauge theory with SO(4N c + 2 − 2k) flavor symmetry is equal to that of T (2Nc+1−k,2Nc+1−k) [SO(4N c +2−2k)], even though the quiver of the latter is not the same as that of the former, having gauge group U Sp(2(N c − [k/2])).
• The theory U Sp(2N c − 2k) with SO(4N c + 2 − 2k) flavor symmetry is identical to Since the orbit (2 2Nc−2k , 1 2k+2 ) is a suborbit of (2 2Nc+1−k ) and of (2 2Nc−k , 1 2 ), we reach the conclusion that the Higgs branch of the U Sp(2N c − 2k) gauge theory is a subvariety of the U Sp(2N c ) gauge theory. In some cases, these orbits correspond to the dual partitions ρ ∨ which can be obtained by transposing and C-collapsing of the following partitions ρ:

B.3 Orthogonal gauge groups
Using the rule given in section 2, we conclude that • The Higgs branch of O(N c ) gauge theory with U Sp(2N c − 2k) flavor symmetry is equal to that of T (Nc−k,Nc−k,1) [U Sp(2N c − 2k)], even though the quiver of the latter is not the same as that of the former.
Observe that the F -flat space is a 4 complex dimensional non-complete-intersection space.
Integrating over the SO(3) gauge group, we obtain the Hilbert series of C 2 /Z 2 .
The generator of this space contains only the meson, which does not involve a contraction with the epsilon tensor (i.e. no baryon). This moduli space may as well be viewed as the Higgs branch of O(3) gauge theory with one flavor, since the parity symmetry does not project out any gauge invariant quantity from the SO(3) counterpart.
It is worth pointing out that the gauge symmetry is not completely broken at a generic point on the hypermultiplet moduli space; rather SO(3) or O(3) gauge symmetry is broken to SO(2) so that the space is 3 − (3 − 1) = 1 quaternionic dimensional.
As argued above, the O(1) gauge theory with 1 flavor has the same Higgs branch. The Higgs branch of O(1) with 1 flavor is the reduced moduli space of 1 U Sp(2) instanton on C 2 ; this space is C 2 /Z 2 , in agreement with the above computation.

C Computing residues in the Higgs branch
In this appendix we derive the baryonic generating function of T ρ σ (SU (N )) from that of T ρ σ (SU (N )), where ρ is obtained from ρ by moving the last box to a previous column, by computing residues at certain poles of the latter. Since all partitions can be obtained by the partition (1, 1 . . . , 1) by repeatedly moving a single box, it suffices to consider Let us suppose that H > 1 and return to the special case of H = 1 later. The quiver diagram of T ρ σ (SU (N )) is as follows: The quiver diagram of T ρ σ (SU (N )) is Let us start with the baryonic generating function of T ρ σ (SU (N )) given by (3.9). Taking fugacities w 1,h for the last Cartan U (1) in the flavor symmetry U (h) and w H,1 for the flavor symmetry U (1) to be as follows  Note that the prefactor w − H i=1 B i becomes 1/x s(n) given by (3.13) of the new quiver T ρ σ (SU (N )) after substituting B i = n i − n i+1 , w = w H+1,M H+1 +1 . For the case of H = 1, the quiver diagrams of T ρ σ (SU (N )) and T ρ σ (SU (N )) are respectively as follows:

D More examples for orthogonal and symplectic groups
We present another set of examples for U Sp(6) and SO (8). The relevant information are contained in the following series of tables. For partitions σ with the same σ ∨ we have two different quivers; one of the two comes in two versions, (I) and (II), corresponding to a moduli space that is a nilpotent orbit or its double covering. The same subscripts are used to distinguish an orbit and its covering in Table 10 and Table 13. We list in the tables only the quivers whose Hilbert series can be obtained in terms of the Hall-Littlewood formula.
D.1 T σ (U Sp (6)) We present the quiver diagrams of T σ (U Sp (6)) and their mirror duals T σ (SO (7)) for various C 3 -partitions σ in Table 9. Information about the associated nilpotent orbits are provided in Table 10. All statements of equality of Coulomb/Higgs branches between different theories hold for vanishing background charges.
C 3 -part. σ Quiver of T σ (U Sp (6)) Quiver with the same Quiver of Tσ(SO (7)) Quiver with the same Coulomb branch as T σ (U Sp (6)) Higgs branch of Tσ(SO(7)) Table 9. Quiver diagrams for T σ (U Sp(6)) and their mirror theories. The asterisk * indicates the gauge group that renders the quiver a 'bad' theory. Each black node labeled by M denotes a U Sp(M ) group, each gray node labeled by N denotes an SO(N ) group and for an orthogonal group O(N ) is spelt out explicitly. Whenever a gauge group O(N ) is indicated by the asterisk, such a gauge group can be taken as O(N ) or SO(N ) without changing the Higgs branch; see Appendix B.3.

D.2 T σ (SO(8))
We present the quiver diagrams of T σ (SO(8)) and their mirror duals T σ (SO(8)) for various D 4 -partitions σ in Table 12. Information about the associated nilpotent orbits is provided in Table 13.  (8)) and their mirror theories. The asterisk * indicates the gauge group that renders the quiver a 'bad' theory.    Table 13. For SO (8), the positive roots are e i − e j , e i + e j , with 1 ≤ i < j ≤ 4. The D 4 -partition σ ∨ can be obtained from σ by adding a box, transposing and D-collapsing.
E More on T ρ σ (U Sp (2N )) theories In this appendix we provide more details on T ρ σ (U Sp (2N )) theories, which are realised on the worldvolume of N D3 branes parallel to an O3 + plane and ending on systems of half D5 branes and of half NS5 branes. σ and ρ, which determine how the D3 branes end on the D5 and on the NS5 branes, are both C-partitions of U Sp (2N ).
Some examples of the T σ ρ (U Sp (2N )) theory were given in sections 7 and 9 of [7]. In section (2.5) we provided a prescription to write down the quiver diagram for general C-partitions σ and ρ. Let us present some examples here: • If σ = ρ = (1 2N ), we refer to the theory as T (U Sp (2N )). The quiver diagram is