Hilbert Series for Theories with Aharony Duals

The algebraic structure of moduli spaces of 3d N=2 supersymmetric gauge theories is studied by computing the Hilbert series which is a generating function that counts gauge invariant operators in the chiral ring. These U(N_c) theories with N_f flavors have Aharony duals and their moduli spaces receive contributions from both mesonic and monopole operators. In order to compute the Hilbert series, recently developed techniques for Coulomb branch Hilbert series in 3d N=4 are extended to 3d N=2. The Hilbert series computation leads to a general expression of the algebraic variety which represents the moduli space of the U(N_c) theory with N_f flavors and its Aharony dual theory. A detailed analysis of the moduli space is given, including an analysis of the various components of the moduli space.


Introduction
Dualities between supersymmetric gauge theories have attracted much interest in the past. In particular, dualities have shed light on understanding the strongly coupled regime of supersymmetric gauge theories. One way to identify dual supersymmetric gauge theories is to understand the structure of their vacuum moduli spaces. Recently, tools such as the Hilbert series [1][2][3][4][5][6][7][8] have been effectively used to obtain a better understanding of vacuum moduli spaces of various supersymmetric gauge theories.
Seiberg duality [9], proposed 20 years ago, is a quintessential example of an IR duality that relates N = 1 SQCD theories with gauge group SU (N c ) and N f flavors with SU (N f − N c ) gauge theories with N f flavors. A 3d N = 2 analog of Seiberg duality was proposed in 1997 [10][11][12]. The duality which is now known as Aharony duality relates a U (N c ) theory with N f chiral fundamental and N f chiral anti-fundamental multiplets with a dual U (N f − N c ) theory with N f chiral fundamentals and N f chiral anti-fundamentals. These Aharony dual theories have been studied extensively in the past, with attempts to match the chiral rings of dual theories, in particular by computing the corresponding superconformal indices [13][14][15][16][17][18][19][20][21][22]. In this work, we want to express the moduli space of Aharony dual theories as an affine algebraic variety by computing the Hilbert series.
Hilbert series are generating functions which count gauge invariant operators in the chiral ring of the supersymmetric gauge theory. They have been used to extract information about the exact algebraic structure of vacuum moduli spaces [1][2][3]. For instance, Hilbert series for instanton moduli spaces [23][24][25][26] and vortex moduli spaces [27] have shed light on the algebraic structure of the corresponding moduli spaces. Moreover, 4d N = 1 theories represented by bipartite graphs on the torus known as brane tilings [28,29] have been studied with the help of Hilbert series. More recently, techniques have been developed for computing the Hilbert series for the Coulomb branch moduli space of 3d N = 4 theories in [8,30,31] and [26] which paved the way in further understanding among other things instanton moduli spaces as Coulomb branches of extended Dynkin diagrams.
In this work, we want to express the moduli space of 3d N = 2 Aharony dual theories as an algebraic variety. In order to compute the Hilbert series, recently developed techniques for Coulomb branch Hilbert series in 3d N = 4 [8] are extended to 3d N = 2. Given the Hilbert series, it is possible using plethystics [1,3,5] to extract information about the generators and first order relations amongst the generators of the moduli space.
The moduli space for 3d N = 2 supersymmetric gauge theories is the space of dressed monopole operators. These operators are dressed with gauge invariant operators which are invariant under a residual gauge symmetry left unbroken under the monopole background. Furthermore, the moduli space is partially lifted due to instanton effects [10-12, 32, 33]. As such, methods for the Coulomb branch Hilbert series for 3d N = 4 theories can be generalized for Aharony dual theories. In this work, we use a sum over a sublattice of GNO charges for the monopole operators which are dressed by suitable gauge invariant operators. The sum over the GNO sublattice generates the Hilbert series of the moduli space. By doing so, we are able to express the moduli space as an algebraic variety for any U (N c ) gauge theory with N f flavors and their Aharony dual theory.
Our Hilbert series computation identifies the generators of the moduli space which agree with previously known results [20]. Moreover, since the Hilbert series computation gives the algebraic structure of the chiral ring, including relations amongst the generators, 1 we are able to study in detail the structure of the vacuum moduli space, including the structure of its components.
This work compares the Hilbert series with the superconformal index for Aharony dual theories. It is important to note that in order to know the entire algebraic structure of the moduli space, it is crucial to compute the Hilbert series directly. The superconformal index gives information on the moduli space only after one finds an appropriate limit to a Hilbert series.
The work is structured as follows: section §2 introduces the 3d N = 2 supersymmetric gauge theories which are discussed in this paper. Section §3 introduces the Hilbert series and the method used to compute it for these theories. In particular, the section outlines the structure of the partially lifted GNO charge lattice and the summation of the dressed monopole operators which is necessary for the computation of the Hilbert series. Explicit examples up to gauge group U (3) are given and a generalization of the algebraic variety for the moduli space is presented. A further analysis on the various components of the moduli space is presented. Section §4 compares the superconformal index with the Hilbert series.
Note added: We acknowledge a future paper to appear in [34] that also discusses moduli spaces of dressed monopole operators for 3d N = 2 theories. Figure 1. The quiver diagram for the 3d N = 2 theory with a U (N c ) gauge group and N f flavors.

The Theory and Aharony Duality
The Theory. We are interested in the moduli space of a 3d N = 2 U (N c ) gauge theory with N f flavors that has a global symmetry S(U (N f ) 1 × U (N f ) 2 ). The vector multiplet of the theory contains the adjoint real scalar σ and the gauge field A. The scalar can be diagonalised to give σ = diag(σ 1 , . . . , σ Nc ). The theory also has chiral multiplets containing chiral matter fields Q andQ which respectively transform in the fundamental and anti-fundamental representations of the gauge group U (N c ). The corresponding quiver diagram of the theory is shown in Figure 1.
The theory can be realized with D3 branes in a D5 and NS5-brane background [35] as shown in Figure 2 The 't Hooft monopole operators are defined by introducing a Dirac monopole singularity at an insertion point in the Euclidean path integral [36]. By Dirac quantization, the monopole operators are labelled by magnetic charges on a weight lattice Γ G ∨ of the GNO/Langlands dual group G ∨ [37][38][39]. For gauge group G = U (N c ), the magnetic charge takes the form where by fixing the action of the Weyl symmetry W G m 1 ≥ m 2 ≥ · · · ≥ m Nc such that m ∈ Γ G ∨ /W G . Note that the magnetic charges m i can be considered conjugate to the σ i of the diagonalised scalar adjoint in the vector multiplet of the theory. Instanton effects [10][11][12]32] lift most of the moduli space of the theory such that magnetic charges of the remaining monopole operators have The remaining GNO charges are m 1 ≥ 0 ≥ m Nc . For convenience, the index for the magnetic charge variable m Nc is relabelled to m 2 such that the magnetic charges of monopole operators are of the form We introduce the following notation for bare monopole operators with magnetic charges The bare and dressed monopole operators have magnetic charges m 1 ≥ 0 ≥ m 2 . Non-zero magnetic charges m 1 , m 2 give effective masses |σ i − σ j | and |σ i | to the gauge field A and the matter fields Q,Q respectively. These massive fields are integrated out with the gauge group G breaking into a residual subgroup H m ⊂ G. For our theory with gauge group U (N c ), the residual subgroup is one of the following: The above values for m 1 , m 2 can be thought of as 4 sublattices of the GNO lattice where in each particular sublattice the gauge group U (N c ) breaks into a particular residual subgroup H m .
The global symmetry of our theory is SU 2 is the flavour symmetry, U (1) A is the axial symmetry, U (1) T is the topological symmetry and U (1) R is the R-symmetry. The global charges carried by the bare monopole operators and matter fields are summarized in Table 1. Aharony Duality. We call the 3d N = 2 theory with U (N c ) gauge group and N f flavors as the theory A. Aharony duality [10][11][12] maps theory A to a new theory for N f > N c . This dual theory is a N = 2 3d theory with U (N f − N c ) gauge symmetry and N f flavors and is called theory B. The dual theory has v ± and M j i of theory A as gauge singlets. In addition, there are fundamental q a i and anti-fundamental q ai under the U (N f − N c ) gauge symmetry as well as monopole operators V + , V − under the gauge group U (N f − N c ). The monopole operators V ± respectively carry magnetic charges of the form (±1, 0, . . . , 0 The global symmetry of the theory B is the same as for theory A, The quiver diagrams of theories A and B are shown in Figure  3. The matter fields and monopole operators under gauge and global symmetries of theory B are summarized in Table 2.  Theory B has a superpotential of the form The superpotential above gives the following F-term relations, which imply that the singlets V ± do not contribute to the moduli space of theory B. Furthermore, there are F-terms Overall, the relevant gauge invariant quantities that contribute to the moduli space of the dual theory are the singlets M j i and the monopoles v ± . These are precisely the gauge invariant mesonic operators and bare monopole operators of theory A and follow from the identifications made by Aharony duality. IR free theories. Let us comment on the case when N c = N f . The original U (N c ) theory with N f = N c flavors has a dual description, the theory of N f 2 + 2 chiral multiplets with the superpotential When N f = N c = 1, U (1) theory A and its dual XY Z theory B flow to the same interacting IR fixed point. On the other hand, when N c = N f > 1, it is known that the U (N c ) theory A and its dual theory B flow to a IR free theory. Firstly, let's consider the N c = N f = 2 case. As shown in Table 2, the R-charges of v ± and M j i are 1 − rN f and 2r respectively. r is a parameter to be determined so that 1 − rN f and 2r give correct R-charges at the IR fixed point. These R-charges are constrained by unitarity of the SCFT to be larger than 1/2 for interacting fields or to be equal to 1/2 for non-interacting fields. For N c = N f = 2, in order to meet the unitarity constraint one has 1 − 2r = 1/2 and 2r = 1/2, which in turn indicates that v ± and M j i are non-interacting. Therefore, the U (2) theory with two flavors, and its dual B theory, flow to a free theory in the IR.
For N c = N f > 2, the situation is more complicated because both R-charges 1−rN f and 2r cannot be larger than or equal to 1/2 simultaneously. 2 This however doesn't mean that the unitarity bound cannot be met. Instead, new U (1) symmetry emerge in IR and the R-charges would get corrections from the new symmetry to meet the unitarity constraint for a IR fixed point. One can understand this better with theory B and the reader is referred to [20,[41][42][43]. 2 In general, this happens for cases which do not satisfy N f > 4Nc−2 3 . Such theories have been studied in [40]. In this work, we only study cases for which the unitary bound is not broken.
Towards the algebraic structure of the Moduli Spaces. The following sections focus on theory A and refer to theory B via Aharony duality. The focus is to identify the algebraic structure of the moduli spaces by computing the Hilbert series [1][2][3][4][5][6][7] for theory A. The Hilbert series counts gauge invariant operators that characterizes the entire chiral ring. By direct generalization from the 3d N = 4 theories, the monopole operators for 3d N = 2 theories are dressed by gauge invariant operators which are invariant under the residual gauge symmetry left unbroken in the monopole background. The following section outlines the computation of the Hilbert series which counts dressed monopole operators for 3d N = 2 theories.
where r is the U (1) R charge of Q a i . As reviewed in section §2, instanton effects [10][11][12]32] lift most of the moduli space such that the remaining monopole operators carry only magnetic charges m 1 ≥ 0 ≥ m 2 . Accordingly, (3.1) simplifies for N c > 1 to where m ∈ Z.
Hilbert Series formula. The Hilbert series for the U (N c ) theory with N f flavors is given by [8] g(t, τ, a, u,ũ; and SU (N f ) 2 with the fugacities u andũ respectively. Instead of using fugacity t, one can identify a fugacity basis in terms of a new U (1) symmetry that weights the bare monopole operators v + and v − and mesonic operators M m equally. By doing so, a new fugacity t corresponding to this new U (1) symmetry can be introduced which counts degrees of chiral operators according to the number of v + , v − and M m . The fugacity map between t and t is as follows, under the new U (1) symmetry. In the following sections, fugacity t is used instead of t in the Hilbert series.
The dressing of monopole operators comes from the classical factor P U (Nc) (m 1 , m 2 ; u,ũ, t) in (3.4). As discussed in section §2, depending on the magnetic charge of the monopole operator, the gauge group U (N c ) is broken to a residual subgroup H m ⊂ U (N c ). The dressing factor is a separate Hilbert series which counts mesonic operators of the form M m = QQ which are invariant under the residual subgroup H m . It takes the form [44] where dµ Hm is the Haar measure of H m and fugacities z and w correspond respectively to the non-Abelian subgroup of H m and a U (1) factor of H m . The remaining U (1) factors in H m do not give charge to the matter fields. The dressing factor takes a concise form when one uses the highest weight generating function of characters of the flavour symmetry SU Plethystic Logarithm. The plethystic logarithm [1,3,5] of the Hilbert series g(t; where µ(k) is the Möbius function. The plethystic logarithm has a series expansion in t. It extracts information from the Hilbert series about the algebraic structure of the moduli space. As an expansion in t, the initial positive terms refer to generators of the moduli space. The following negative terms refer to first order relations amongst the generators. When the series terminates at this point, the moduli space is known to be a complete intersection moduli space. If the series does not terminate, the moduli space is known to be a non-complete intersection where relations form higher order relations known as syzygies [1,3,5]. We expect the moduli space of the U (N c ) theory with N f flavors to be in one of these two classes.

U (1) with 1 flavor: 3 identical components
The Hilbert series is given by where fugacity t counts the bare monopole operators according to their conformal dimension. For a U (1) theory with N f = 1 the conformal dimension of the bare monopole operator is given by where r is the U (1) R-charge of the fundamental Q and anti-fundamental Q and m ∈ Z is the GNO magnetic flux. Under a refinement, fugacities τ and a, which respectively correspond to the topological symmetry U (1) T and axial symmetry U (1) A , can be added to the formula in (3.10). The refined Hilbert series takes the form where J(m) = m and K(m) = −|m| are respectively the topological and axial U (1) charges of a monopole operator of GNO charge m. As discussed in (3.5), a new U (1) symmetry can be introduced under which the monopole operators and mesonic operators are weighted equally. Under this new U (1) symmetry, r → r 0 = 1 3 and the following fugacity map applies where now t counts the number of v + , v − and M m = QQ.
In the Hilbert series formulas, the classical factor in its refined form is given by where w is the U (1) gauge charge fugacity and a the U (1) A axial charge fugacity. 3 Summing up the refined series in (3.12) gives The plethystic logarithm of the Hilbert series is The generators corresponding to the first positive terms can be identified from the above plethystic logarithm as follows, the meson M and two monopoles v ± . The first order relations formed by the generators are as follows, The full moduli space of the U (1) theory with N f = 1 is an algebraic variety, The moduli space M U (1),1 has the following components where M 0 U (1),1 is the Higgs branch and M + U (1),1 and M − U (1),1 are the Coulomb branches of the theory. The corresponding Hilbert series are The moduli space M U (1),1 is made of 3 identical cones C which meet at the origin, as shown in Figure 4. The moduli space is the union of the 3 components. By removing contributions from the intersections, the Hilbert series of the full moduli space can therefore be expressed as 22) where the intersections at the origin are taken care of by −2.

U (1) with 2 flavors: 3 components with non-Abelian symmetry
The Hilbert series for the U (1) theory with 2 flavors is given by where t is the fugacity which counts bare monopole operators according to their conformal dimension. For a U (1) theory with N f = 2 the conformal dimension of the bare monopole operator is given by where r is the U (1) R-charge of the fundamental Q i and anti-fundamental Q i and m ∈ Z is the GNO magnetic flux. The Hilbert series formula above can be refined with the charges from the topological symmetry U (1) T and the axial symmetry U (1) A . The respective fugacities are chosen to be τ and a. The refined Hilbert series is where J(m) = m and K(m) = −2|m| are respectively the topological and axial charges of a monopole operator with GNO charge m as discussed in Table 1. Under a new U (1) symmetry that weights monopole operators v ± and mesonic operators M m equally, a new fugacity t can be introduced by mapping the value of r to r → r 0 = 1 2 . As discussed in (3.5), the fugacity map is t = t.
The classical factor of the Hilbert series formula is P U (1) (m; t, a, u,ũ) and it is further refined under the flavour symmetries SU (2) 1 × SU (2) 2 . The fugacities u and u respectively count charges under SU (2) 1 and SU (2) 2 . The refined classical factor is given by where the integral gives From the Hilbert series above one can observe that the moduli space is made of 3 cones, one being the conifold generated by the mesonic operators and the other two being two C, each generated by monopole operators of opposite topological U (1) T charge. The 3 cones meet at the origin as shown in Figure 5.
The Hilbert series has the following character expansion, The plethystic logarithm of the refined Hilbert series of the full moduli space is From the initial positive terms of the plethystic logarithm, one can identify the generators of the moduli space, The generators are the mesons and the bare monopoles. The first order relations formed among the generators are identified as follows, The full moduli space of the U (1) theory with N f = 2 can be expressed as the following algebraic variety, The moduli space M U (1),2 has the following components The 3 components of the moduli space intersect only at the origin. The moduli space is the union of the 3 components. By removing the contributions from the intersections, the Hilbert series of M U (1),2 therefore can be expressed as g(t, τ, a, u,ũ; M U (1),2 ) = g(t, τ, a, u,ũ; M 0 U (1),2 ) + g(t, τ, a, u,ũ; M + U (1),2 ) + g(t, τ, a, u,ũ; M − U (1),2 ) − 2 . The Hilbert series for the U (2) theory with 3 flavors is given by where fugacity t counts bare monopole operators according to their conformal dimension. For the U (2) theory with N f = 3 the conformal dimension of the bare monopole operator is given by where m 1 , m 2 are the GNO magnetic fluxes. The Hilbert series expression in (3.37) can be refined to include fugacities τ and a which respectively count charges of the topological U (1) T and axial U (1) A symmetries. The refined Hilbert series takes the following form where J(m 1 , m 2 ) = m 1 + m 2 and K(m 1 , m 2 ) = −3(m 1 − m 2 ) are respectively the topological and axial charges of a monopole operator with GNO charge m 1 , m 2 . In addition, the Hilbert series above is refined under the flavour symmetry SU (3) 1 × SU (3) 2 , where fugacities u andũ count the charges of the respective symmetries as summarised in Table 1. By introducing a new U (1) symmetry that replaces U (1) R and weights monopole operators v ± and mesonic operators M m equally, a new fugacity t can be introduced that replaces t by mapping the value of r to r → r 0 = 2 5 . Following (3.5), the fugacity map is t = t 4 5 . The classical contribution comes from the factor P U (2) (m 1 , m 2 ; t, a, u,ũ). The GNO charge lattice with m 1 , m 2 can be dividend into 4 sublattices under which monopole operators that contribute to the moduli space are charged. Depending on which GNO sublattice one is, the gauge symmetry is either broken or unbroken. Accordingly, the classical factor of the Hilbert series can be written as follows, where the integrals above give Summing up the refined Hilbert series in (3.39) gives .
The corresponding plethystic logarithm is PL g(t, τ, a, u,ũ; The plethystic logarithm encodes the generators and relations amongst generators which define the moduli space. The generators of the moduli space correspond to the initial positive terms of the plethystic logarithm. The generators are as follows where i, j = 1, 2, 3. The corresponding first order relations between the generators are identified as follows where From the generators and first order relations, the moduli space can be expressed as the following algebraic variety, Let us call the space of N × N matrices M j i with at most rank k as M k,N . Using this space, the components of the moduli space M U (2),3 can be expressed as where f 1 and f 2 correspond to the monopole dressing factors in (3.41). The 4 components intersect in various subspaces which are where {0} is the origin. The corresponding Hilbert series are g(t, τ, a, u,ũ; I M ) = f 2 , g(t, τ, a, u,ũ; I 0 ) = 1 , g(t, τ, a, u,ũ; I + ) = 1 1−τ a −3 t , g(t, τ, a, u,ũ; The union of the 4 components is the moduli space. By removing contributions from the intersections, the Hilbert series of the full moduli space M U (2),3 can be expressed as g(t, τ, a, u,ũ; M U (2),3 ) = g(t, τ, a, u,ũ; M 0 U (2),3 ) + g(t, τ, a, u,ũ; M + U (2),3 ) +g(t, τ, a, u,ũ; M − U (2),3 ) + g(t, τ, a, u,ũ; M +− U (2),3 ) − 2g(t, τ, a, u,ũ; I M ) −g(t, τ, a, u,ũ; I + ) − g(t, τ, a, u,ũ; I − ) + g(t, τ, a, u,ũ; I 0 ) . (3.53) This expression for the Hilbert series of the full moduli space is in agreement with the Hilbert series expression in (3.42).

U (3) with 4 flavors: 4 components (Higgs and Mixed)
The Hilbert series for the U (3) theory with 4 flavors is given by where J(m 1 , m 2 ) = m 1 + m 2 and K(m 1 , m 2 ) = −4(m 1 − m 2 ) are respectively the topological and axial charges of a monopole operator with GNO charge m 1 , m 2 . The Hilbert series in (3.56) is further refined by the flavour symmetry SU (4) 1 × SU (4) 2 whose charges are counted respectively by fugacities u andũ. A new U (1) symmetry that replaces U (1) R can be introduced such that monopole operators v ± and mesonic operators M m are counted equally by a new fugacity t. This new fugacity replaces t by mapping the value of r to r → r 0 = 1 3 . The corresponding fugacity map is t = t 2 3 . The classical contribution to the Hilbert series comes from the factor P U (3) (m 1 , m 2 ; t, a, u,ũ). The GNO charge lattice can be divided into 4 sublattices. Depending on which sublattice the GNO charge of a monopole operator is located, the gauge symmetry breaks under the Higgs mechanism. The residual gauge symmetry determines the dressing of the monopole operator in the particular GNO charge sublattice. Accordingly, the classical factor of the Hilbert series can be written as follows, . (3.59) The Hilbert series has the following expansion up to order t 4 , The plethystic logarithm of the Hilbert series is The generators of the moduli space are identified from the plethystic logarithm as follows, The generators form first order relations which are Given the generators and first order relations, the moduli space can be expressed as follows Calling the space of N × N matrices with rank at most k as M k,N , the 4 components of the moduli space can be expressed as The corresponding Hilbert series are g(t, τ, a, u,ũ; I M ) = f 2 , g(t, τ, a, u,ũ; I 0 ) = f 3 , g(t, τ, a, u,ũ;

General Result of the Moduli Space
The Hilbert series g(t, τ, a, u,ũ; M U (Nc),N f ) which has been computed in the above sections all satisfy a general form. In order to present this general form, we make use of the highest weight generating function for the characters of irreducible representations of the flavour symmetry SU (N f ) 1 × SU (N f ) 2 . The highest weight generating function for Hilbert series makes use of the map where fugacities µ i and ν i count the highest weight of the irreducible representations of Using the highest weight generating function of Hilbert series, one can for instance express concisely the dressing factor for monopole operators as follows, (3.72) After the inclusion of the monopole operators the highest weight generating function is , (3.73) where t counts magnetic monopoles v ± and mesonic operators M m = QQ and corresponds to U (1) symmetry which replaces U (1) R . By identifying the exponents of fugacities µ i and ν i in the expansion of the highest weight generating function in (3.73), one obtains the character expansion of the Hilbert series. The plethystic logarithm of the Hilbert series as a highest weight generating function is (3.74) The following product of mesonic operators is used in order to express relations amongst moduli space generators, From the plethystic logarithm in (3.74), the general form of the generators can be identified as (3.77) Furthermore, the general form of the first order relations formed amongst the generators are It is important to note that the terms in the plethystic logarithm in (3.74) which correspond to the above relations do not appear in the Hilbert series expansion itself. This can be seen when one expands the dressing factor in (3.72) with the contributions from the monopole operators. One can show that the terms of the plethystic logarithm in (3.78) do not appear as operators in the Hilbert series expansion and that the relations in (3.78) are satisfied.
From the above analysis of the plethystic logarithm, the moduli space of the U (N c ) theory with N f flavors can be expressed as the following algebraic variety, where the quotienting ideal is

The Superconformal Index and the Hilbert Series
In this section, we examine the relation between the superconformal index and the Hilbert series. The superconformal index by itself does not give information on the moduli space. Only by taking appropriate limits to a Hilbert series one can derive information about the structure of the moduli space. The following section proposes limits from the superconformal index which reproduce Hilbert series of certain subspaces of the moduli space of the 3d N = 2 theories.

The N = 2 Superconformal Index
Firstly, let us recall the definition of the superconformal index for 3d N = 2 theories. The bosonic subgroup of the 3d N = 2 superconformal group is SO(2, 3) × SO(2) whose three Cartan elements are denoted by E, j and R. The superconformal index is defined by [17] I (x, where Q is a supercharge of quantum numbers E = 1 2 , j = − 1 2 and R = 1, and S = Q † . x is the fugacity for E + j and u i 's are additional fugacities for global symmetries of the theory. The trace is taken over the Hilbert space of the SCFT on R×S 2 , or equivalently over the space of local gauge invariant operators on R 3 . As usual, only the BPS states, which saturate the inequality contribute to the index. Using supersymmetric localization, the superconformal index can be exactly computed as follows, [15,16] u,ũ, a, z, m) , Above, |W m | is the Weyl group order of the residual gauge group left unbroken by flux m. We have taken into account the gauge group U (N c ) and the matter content: the N f pairs of fundamental and anti-fundamental chiral multiplets. u = (u 1 , . . . , u N f ),ũ = (ũ 1 , . . . ,ũ N f ), a and τ are the fugacities for the global symmetry SU

Limits of the N = 4 Superconformal Index
Let us review the proposal [45] for the relation between the superconformal index and the Hilbert series of N = 4 theories [8,23]. Let us denote by j H and j V the spins of the two SU (2) in the SO(4) R = SU (2) H × SU (2) V R-symmetry. x is the E + j fugacity and x is the j H − j V fugacity. The superconformal index for an N = 4 theory is where we ignore other global symmetry fugacities and t H = xx , t C = xx −1 . The primed trace Tr denotes that the trace is taken over the BPS states. The BPS condition E = j H + j V + j [46] is used for the second equality. Under N = 2 twisting some of the fermions in the N = 4 vector multiplet get the same quantum numbers as the F-terms and play the same role for the index as the F-terms for the Hilbert series. It is important to note that the index is unreliable when there are accidental IR corrections to the R-symmetry. The proposed limits for getting the Hilbert series of the Higgs branch and the Coulomb branch from the superconformal index are 6 Higgs branch: where HS H and HS C are respectively the Hilbert series of the Higgs and Coulomb branches.
Note that the BPS condition E = j H + j V + j implies inequalities E ≥ j H and E ≥ j V . Using (4.6) the first limit in (4.7) restricts to BPS states with E = j H implying j V = j = 0. Similar arguments apply for the second limit in (4.7). Therefore, the index in each limit captures the SU (2) V /H singlet scalar BPS states, which corresponds to the Hilbert series of the Higgs/Coulomb branch of the N = 4 theory, respectively.

Generalized Limits for the N = 2 Superconformal Index
N = 2 theories do not in general have distinct Higgs and Coulomb branches. Furthermore, there is only one U (1) R symmetry in the superconformal algebra for N = 2 theories. Nevertheless, one may try to generalize the limits in (4.7) for N = 2 theories. The N = 2 U (1) R charge plays the role of j H + j V in N = 4. In addition, one can choose one of the N = 2 global U (1) symmetries and choose its charge to play the role of j H − j V in N = 4. With these choices, it turns out that the resulting generalized limits of the N = 2 superconformal index give rise to Hilbert series of certain subspaces of the moduli space for the N = 2 theory. In addition, such generalized limits of the N = 2 superconformal index are not unique because the N = 2 theories we are considering have several U (1) global symmetries.
We will examine 4 limits of the superconformal index of the N = 2 U (N c ) theory with N f flavors. The BPS condition and certain constraints on the global U (1) symmetry charges, which derive from the requirement that the limit is well-defined and non-divergent, can be used to show that there are just 4 relevant limits to consider. This is further elaborated in the following section. Here it is noted that each of the 4 limits corresponds to a Hilbert series of a certain subspace of the moduli space. 3 of them can be expressed in terms of the 4 main components of the moduli space which are discussed in section §3.4. These 3 subspaces are as follows: The 4th limit gives the Hilbert series of a subspace of the moduli space that cannot be directly expressed in terms of the 4 main components. It is a subspace of component M +− U (Nc),N f as follows: By considering all 4 limits, we are going to see that taking a limit of the superconformal index cannot reproduce the Hilbert series of the whole component M +− U (Nc),N f , and thus that of the complete moduli space. The subsequent sections explain how we obtain the Hilbert series of each subspace from the superconformal index.
Recall why the limits in (4.7) capture scalar BPS states: if energy E of a BPS state is equal to the R-charge j H/V , the state is scalar BPS due to the N = 4 BPS condition E ≥ j H + j V + j [46]. The idea for N = 2 theories is the same. We try to identify a state whose energy is equal to the U (1) R charge R. Such a state then should be scalar BPS because of the N = 2 BPS condition E ≥ R + j. We cannot trace every scalar BPS state by taking a limit of the superconformal index because there are accidental cancelations between the bosonic and the fermionic contributions to the index. This section explains which remaining states can be traced by taking an appropriate limit of the N = 2 superconformal index.
For every factor U (1) k in the global symmetry of the theory, one can introduce a corresponding fugacity u k . In order to have a well-defined non-divergent limit of the superconformal index, we propose the condition that for a U (1) k factor in the global symmetry, the ratio of the U (1) k charge F k to the U (1) R charge R satisfies the following bound We have assumed for simplicity that F k is normalized such that the right hand side is 1. The role of the above condition is going to become clearer when one revisits the general form of the N = 2 index where we can make shifts of the E + j fugacity and the U (1) k fugacity, x → xy, u k → u k y −1 , such that in the limit y → 0 one has Again the primed trace Tr denotes that the trace is taken over the BPS states. Given the BPS condition E ≥ R + j and the condition R ≥ F k from (4.8), the power of y for each term is non-negative. Therefore, the limit y → 0 only leaves terms which are independent of y. The remaining terms correspond to the contributions of BPS states satisfying E = R = F k and j = 0. This is exactly the Hilbert series counting scalar BPS states of the theory, where u i 's are the global symmetry fugacities and x is the energy fugacity. The choice of global U (1) symmetries, the constraints set by the N = 2 BPS condition, and the requirement for having a well-defined non-divergent limit of the superconformal index all lead to precisely 4 limits of the N = 2 superconformal index for the theories we are considering. In the following sections, these limits are presented and the resulting Hilbert series are identified with subspaces of the moduli space of the N = 2 theory.  Table 1, one can identify bounds for the ratio of the U (1) A charge A to the U (1) R charge R for a BPS state as follows: where r is the U (1) R charge of the fundamental and anti-fundamental chiral multiplets Q andQ. r is such that ∆ M = 2r and ∆ V = (1−r)N f −N c +1 for mesonic and monopole operators respectively are larger than or equal to 1/2 due to unitarity. We can take two differently normalized versions of U (1) A such that each inequality in (4.11) takes the form of (4.8). Then, as we have proposed, the Hilbert series of a subspace of the moduli space generated by generators saturating each inequality can be obtained from the superconformal index. It turns out that the right inequality is saturated for the mesonic operators M j i , which have the U (1) A charge 2 and the U (1) R charge 2r, whereas the left inequality is saturated for the monopole operators v ± , which have the U (1) A charge −N f and the U (1) R charge (1 − r)N f − N c + 1. Therefore, we propose two limits of the superconformal index which give rise to the Hilbert series of two subspaces of the moduli space M U (Nc),N f / v ± = 0 and M U (Nc),   Again x is the energy fugacity of the Hilbert series. u,ũ, a and τ are identified as the fugacities for SU Computation. Using the limits, we claim that one can obtain the explicit formulae for the Hilbert series of the two subspaces M U (Nc),N f / v ± = 0 and M U (Nc),N f / M j i = 0 from the superconformal index. Firstly, the Hilbert series of M U (Nc),N f / v ± = 0 is given by the limit (4.13). Since M U (Nc),N f / v ± = 0 is the same as component where dµ U (Nc) is the Haar measure for U (N c ). The formula in (4.16), which is obtained from the index formula (4.4), is equivalent to the classical contribution of the mesonic operators in (3.6) if we substitute x = t. Next the limit (4.14) gives the Hilbert series of M U (Nc),N f / M j i = 0 . We consider the U (1) case first and then consider general U (N c ) cases with N c ≥ 2. For a U (1) theory the vector multiplet does not contribute to the index. Only the contribution of chiral multiplets is nontrivial, which becomes the monomial factor  U (1),N f is given by (4.19). Taking into account the fact that their intersection is only the origin, for this special case of the U (1) theory, the complete Hilbert series can be written as where we use If we substitute x = t into (4.20), we recover the result in section 3. Now let us consider a U (N c ) theory with N c ≥ 2. In this case, the superconformal index in the limit (4.14) is given by (4.22) The above Hilbert series shows that the chiral ring is freely generated by two monopole operators v ± . Note that especially for N c = 2, Let us consider in this section the topological symmetry U (1) T . Given that only monopole operators are charged under U (1) T , we do not directly use the U (1) T symmetry for formulating the N = 2 limit but use mixed symmetries U (1) + and U (1) − instead whose conserved currents are defined by where J A and J T are the conserved currents of U (1) A and U (1) T . Following Table 1, one can show that the ratios of U (1) ± to the R-charge are bounded from above as follows, where F ± are charges under U (1) ± . Recall that only the monopole operators v ± are charged under U (1) T with the charges T = ±1. Thus, the mesonic operators M j i just have the U (1) ± charges F ± = rA = 2r and saturate both inequalities (4.25) and (4.26). On the other hand, Table 3. Saturated U (1) + and U (1) − charges for the monopole operators v ± . the two monopole operators have different U (1) ± charges and are summarized in Table 3. As a result, v + saturates the bound (4.25) while v − saturates the bound (4.26).  where u + is the fugacity of U (1) + and the other global symmetry fugacities are omitted. This tells us that only the contributions satisfying E = R = rA + (N f − N c + 1)T and j = 0 remain under the limit. One can check that the shift u + → u + y −1 here is equivalent to the shifts of the U (1) A and U (1) T fugacities a → ay −r and τ → τ y −(N f −Nc+1) respectively. This is because u + and u − are written in terms of a, τ as U (Nc),N f for a U (2) theory. Thus, for a U (2) theory, we can completely recover the Hilbert series for each of the four components of the moduli space from those of the four subspaces we have examined.
Superconformal Index and Hilbert Series. In contrast to the N c = 1, 2 cases, the Hilbert series of component M +− U (Nc),N f for N c ≥ 3 cannot be reproduced as a limit of the superconformal index. Because of this reason, one cannot obtain the exact Hilbert series of a U (N c ) theory with N c ≥ 3 by taking a limit of the superconformal index. The index contribution of a chiral ring element in component M +− U (Nc),N f could cancel with the contribution of another fermionic operator. In that case any analytic manipulation of the superconformal index, for example taking a limit of the index, cannot trace the contribution of that chiral ring element. Let us consider an example. If we consider the U (3) theory with five flavors, there is a chiral ring element of the form i 5 ) , which has E + j = 6 and transforms in the representation [0, 0, 0, 5]×[5, 0, 0, 0] of SU (5) 1 ×SU (5) 2 whose dimension is given by 126 2 = 15876, and most crucially has charges A = 0, T = 0. These charges make it easy to identify many non-zero spin operators. Most of them are fermionic such that their contributions come with a negative sign and could cancel the contributions of v + v − M i 5 ) and those of the nonzero spin states. Because of these cancellations, taking the limit of the index does not capture the presence of this operator in the chiral ring.