Hilbert Series for Theories with Aharony Duals
Amihay Hanany, Chiung Hwang, Hyungchul Kim, Jaemo Park, Rak-Kyeong Seong
KKIAS-P14084IMPERIAL-TP-15-AH-01
Hilbert Series for Theories with Aharony Duals
Amihay Hanany, a Chiung Hwang, b Hyungchul Kim, b Jaemo Park, b,c andRak-Kyeong Seong da Theoretical Physics Group, Blackett Laboratory,Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom b Department of Physics, POSTECH,Pohang 790-784, Korea c Postech Center for Theoretical Physics (PCTP), POSTECH,Pohang 790-784, Korea d School of Physics, Korea Institute for Advanced Study,85 Hoegi-ro, Seoul 130-722, Korea
E-mail: [email protected] , c [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The algebraic structure of moduli spaces of 3d N = 2 supersymmetricgauge theories is studied by computing the Hilbert series which is a generating functionthat 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 bothmesonic and monopole operators. In order to compute the Hilbert series, recentlydeveloped techniques for Coulomb branch Hilbert series in 3d N = 4 are extended to3d N = 2. The Hilbert series computation leads to a general expression of the algebraicvariety which represents the moduli space of the U ( N c ) theory with N f flavors and itsAharony dual theory. A detailed analysis of the moduli space is given, including ananalysis of the various components of the moduli space. a r X i v : . [ h e p - t h ] M a y ontents U (1) 113.2.1 U (1) with 1 flavor: 3 identical components 113.2.2 U (1) with 2 flavors: 3 components with non-Abelian symmetry 143.3 Examples: U (2) and U (3) 173.3.1 U (2) with 3 flavors: 4 components (Higgs, Mixed and Coulomb) 173.3.2 U (3) with 4 flavors: 4 components (Higgs and Mixed) 203.4 General Result of the Moduli Space 24 N = 2 Superconformal Index 274.2 Limits of the N = 4 Superconformal Index 284.3 Generalized Limits for the N = 2 Superconformal Index 294.3.1 M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) and M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) and M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) Dualities between supersymmetric gauge theories have attracted much interest in thepast. In particular, dualities have shed light on understanding the strongly coupledregime of supersymmetric gauge theories. One way to identify dual supersymmetricgauge theories is to understand the structure of their vacuum moduli spaces. Recently,tools such as the Hilbert series [1–8] have been effectively used to obtain a betterunderstanding of vacuum moduli spaces of various supersymmetric gauge theories.Seiberg duality [9], proposed 20 years ago, is a quintessential example of an IR du-ality 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 3 d N = 2 analog of Seiberg dualitywas proposed in 1997 [10–12]. The duality which is now known as Aharony duality– 1 –elates a U ( N c ) theory with N f chiral fundamental and N f chiral anti-fundamentalmultiplets with a dual U ( N f − N c ) theory with N f chiral fundamentals and N f chiralanti-fundamentals. These Aharony dual theories have been studied extensively in thepast, with attempts to match the chiral rings of dual theories, in particular by comput-ing the corresponding superconformal indices [13–22]. In this work, we want to expressthe moduli space of Aharony dual theories as an affine algebraic variety by computingthe Hilbert series.Hilbert series are generating functions which count gauge invariant operators inthe chiral ring of the supersymmetric gauge theory. They have been used to extractinformation about the exact algebraic structure of vacuum moduli spaces [1–3]. For in-stance, Hilbert series for instanton moduli spaces [23–26] and vortex moduli spaces [27]have shed light on the algebraic structure of the corresponding moduli spaces. More-over, 4 d N = 1 theories represented by bipartite graphs on the torus known as branetilings [28, 29] have been studied with the help of Hilbert series. More recently, tech-niques have been developed for computing the Hilbert series for the Coulomb branchmoduli space of 3 d N = 4 theories in [8, 30, 31] and [26] which paved the way in fur-ther understanding among other things instanton moduli spaces as Coulomb branchesof extended Dynkin diagrams.In this work, we want to express the moduli space of 3 d N = 2 Aharony dualtheories as an algebraic variety. In order to compute the Hilbert series, recently de-veloped techniques for Coulomb branch Hilbert series in 3 d N = 4 [8] are extended to3 d N = 2. Given the Hilbert series, it is possible using plethystics [1, 3, 5] to extractinformation about the generators and first order relations amongst the generators ofthe moduli space.The moduli space for 3 d N = 2 supersymmetric gauge theories is the space ofdressed monopole operators. These operators are dressed with gauge invariant oper-ators which are invariant under a residual gauge symmetry left unbroken under themonopole background. Furthermore, the moduli space is partially lifted due to instan-ton effects [10–12, 32, 33]. As such, methods for the Coulomb branch Hilbert series for3 d N = 4 theories can be generalized for Aharony dual theories. In this work, we usea 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 theHilbert series of the moduli space. By doing so, we are able to express the moduli spaceas an algebraic variety for any U ( N c ) gauge theory with N f flavors and their Aharonydual theory.Our Hilbert series computation identifies the generators of the moduli space whichagree with previously known results [20]. Moreover, since the Hilbert series compu-tation gives the algebraic structure of the chiral ring, including relations amongst the– 2 –enerators, 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 Aharonydual theories. It is important to note that in order to know the entire algebraic struc-ture of the moduli space, it is crucial to compute the Hilbert series directly. Thesuperconformal index gives information on the moduli space only after one finds anappropriate limit to a Hilbert series.The work is structured as follows: section § d N = 2 supersym-metric gauge theories which are discussed in this paper. Section § U (3) are given and a gener-alization of the algebraic variety for the moduli space is presented. A further analysison the various components of the moduli space is presented. Section § Note added:
We acknowledge a future paper to appear in [34] that also discusses modulispaces of dressed monopole operators for 3 d N = 2 theories. U ( N c ) SU ( N f ) Q ai SU ( N f ) e Q ia Figure 1 . The quiver diagram for the 3d N = 2 theory with a U ( N c ) gauge group and N f flavors. This is up to numerical coefficients which can usually be absorbed into the elements of the chiralring. In this work, the numerical coefficients are not needed as the relations are homogeneous andthere is precisely one operator per relation. – 3 – he Theory.
We are interested in the moduli space of a 3d N = 2 U ( N c ) gaugetheory with N f flavors that has a global symmetry S ( U ( N f ) × U ( N f ) ). The vectormultiplet of the theory contains the adjoint real scalar σ and the gauge field A . Thescalar can be diagonalised to give σ = diag( σ , . . . , σ N c ). The theory also has chiralmultiplets containing chiral matter fields Q and ˜ Q which respectively transform in thefundamental and anti-fundamental representations of the gauge group U ( N c ). Thecorresponding 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 N c D3-branes are suspended between 2 NS5-branes andtheir positions along the x -direction are labelled by σ i , where i = 1 , . . . , N c . For eachof the flavour groups U ( N f ) and U ( N f ) , there is a stack of N f D5 branes attached tothe NS5 (cid:48) along the x -direction. Their positions along the x -direction are respectivelylabelled by the real masses m a and ˜ m b of Q and ˜ Q where a, b = 1 , . . . , N f . For thetheories considered here, the bare masses are set to zero.The moduli space of the 3 d N = 2 U ( N c ) theory receives quantum corrections. TheHiggs branch is parameterized by mesonic operators of the form M = Q ˜ Q which areinvariant under the gauge group U ( N c ). The remaining moduli space is parameterizedby chiral operators that are composed of supersymmetrized ’t Hooft monopole operators v m with magnetic charge m and mesonic operators of the form M m = Q ˜ Q which areinvariant under a residual subgroup H m ⊂ U ( N c ). In other words, there are chiralgauge invariant operators which are either bare monopole operators built out of v m , or dressed monopole operators which are built out of the mesonic operators M m and baremonopole operators v m .The ’t Hooft monopole operators are defined by introducing a Dirac monopolesingularity at an insertion point in the Euclidean path integral [36]. By Dirac quan-tization, the monopole operators are labelled by magnetic charges on a weight latticeΓ G ∨ of the GNO/Langlands dual group G ∨ [37–39]. For gauge group G = U ( N c ), themagnetic charge takes the form m = ( m , m , . . . , m N c ) , (2.1)where by fixing the action of the Weyl symmetry W G m ≥ m ≥ · · · ≥ m N c 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 [10–12, 32] lift most of the moduli space of the theory such thatmagnetic charges of the remaining monopole operators have m = · · · = m N c − = 0 . (2.2)– 4 – f D5 N f D5NS5 NS5
39 80 1 2 3 4 5 6 7 8 9D3 ⇥ ⇥ ⇥ · · · · · ⇥ · D5 ⇥ ⇥ ⇥ · · · ⇥ ⇥ · ⇥ NS5 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ · · · ·
NS5 ⇥ ⇥ ⇥ ⇥ · · ⇥ ⇥ · · D3 i j m a ˜ m b Figure 2 . The brane construction for the 3 d N = 2 U ( N c ) theory with N f flavors thathas a global symmetry S ( U ( N f ) × U ( N f ) ). There are N c D3-branes suspended between2 NS5-branes. The positions of the branes along the x -direction are given by the scalaradjoints σ i , where i = 1 , . . . , N c . There are also N f pairs of D5-branes which are attached tothe NS5 (cid:48) -branes along the x -direction. The position along the x -direction for the D5-branesare given by the real masses m a and ˜ m b for Q and ˜ Q respectively, where a, b = 1 , . . . , N f . The remaining GNO charges are m ≥ ≥ m N c . For convenience, the index for themagnetic charge variable m N c is relabelled to m such that the magnetic charges ofmonopole operators are of the form m ≥ ≥ m . (2.3)We introduce the following notation for bare monopole operators with magnetic charges( m , m ) = (+1 ,
0) : v m ≡ v + , ( m , m ) = (0 , −
1) : v m ≡ v − . (2.4)The bare and dressed monopole operators have magnetic charges m ≥ ≥ m .Non-zero magnetic charges m , m give effective masses | σ i − σ j | and | σ i | to the gauge– 5 –eld A and the matter fields Q, ˜ Q respectively. These massive fields are integrated outwith the gauge group G breaking into a residual subgroup H m ⊂ G . For our theorywith gauge group U ( N c ), the residual subgroup is one of the following: • m , m = 0: U ( N c ) • m (cid:54) = 0 , m = 0 or m = 0 , m (cid:54) = 0: U ( N c − × U (1) • m , m (cid:54) = 0: U ( N c − × U (1) × U (1)The above values for m , m can be thought of as 4 sublattices of the GNO lattice wherein each particular sublattice the gauge group U ( N c ) breaks into a particular residualsubgroup H m .The global symmetry of our theory is SU ( N f ) × SU ( N f ) × U (1) A × U (1) T × U (1) R ,where SU ( N f ) × SU ( N f ) 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 carriedby the bare monopole operators and matter fields are summarized in Table 1. U ( N c ) SU ( N c ) U (1) B SU ( N f ) SU ( N f ) U (1) A U (1) T U (1) R Q ai [1 , , . . . , z +1 [0 , . . . , , u r (cid:101) Q ia [0 , . . . , , z − , , . . . , ˜ u rv ± − N f ± − r ) N f − ( N c − Table 1 . The U ( N c ) theory with N f flavors (theory A). The table shows the fundamentaland anti-fundamental matter fields and bare monopole operators under gauge and globalsymmetries. 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 [10–12] maps theory A to a new theory for N f > N c . This dual theory is a N = 2 3 d theory with U ( N f − N c ) gauge symmetryand N f flavors and is called theory B . The dual theory has v ± and M ji of theory A asgauge singlets. In addition, there are fundamental q ai and anti-fundamental (cid:101) q ai underthe U ( N f − N c ) gauge symmetry as well as monopole operators V + , V − under the gaugegroup U ( N f − N c ). The monopole operators V ± respectively carry magnetic charges ofthe form ( ± , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) N f − N c − ) . (2.5)– 6 –he global symmetry of the theory B is the same as for theory A, SU ( N f ) × SU ( N f ) × U (1) A × U (1) T × U (1) R . The quiver diagrams of theories A and B are shown in Figure3. The matter fields and monopole operators under gauge and global symmetries oftheory B are summarized in Table 2. U ( N c ) SU ( N f ) Q ai SU ( N f ) e Q ia SU ( N f ) SU ( N f ) U ( N f N c ) q ai e q ia monopoles: v ± monopoles: V ± , v ± M ji theory Atheory B Figure 3 . The quiver diagrams of theories A and B under Aharony duality. U ( N f − N c ) SU ( N f − N c ) U (1) B SU ( N f ) SU ( N f ) U (1) A U (1) T U (1) R q ai [1 , , . . . , z +1 0 [0 , . . . , , ˜ u − − r (cid:101) q ia [0 , . . . , , z − , , . . . , u − − rV ± N f ± − (1 − r ) N f + ( N c + 1) v ± − N f ± − r ) N f − ( N c − M ji , . . . , , u [1 , , . . . , ˜ u r Table 2 . The gauge and global charges of the U ( N f − N c ) theory with N f flavors (theoryB). Theory B has a superpotential of the form W = (cid:101) q ia M ji q aj + v + V − + v − V + . (2.6)The superpotential above gives the following F-term relations, q (cid:101) q = 0 , V + = 0 , V − = 0 , (2.7)– 7 –hich imply that the singlets V ± do not contribute to the moduli space of theory B.Furthermore, there are F-terms (cid:101) qM = 0 , M q = 0 , (2.8)Overall, the relevant gauge invariant quantities that contribute to the moduli space ofthe dual theory are the singlets M ji and the monopoles v ± . These are precisely thegauge invariant mesonic operators and bare monopole operators of theory A and followfrom 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 chiralmultiplets with the superpotential W = − v + v − det M . (2.9)When N f = N c = 1, U (1) theory A and its dual XY Z theory B flow to the sameinteracting IR fixed point. On the other hand, when N c = N f >
1, it is known thatthe 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 -chargesof v ± and M ji are 1 − rN f and 2 r respectively. r is a parameter to be determined sothat 1 − rN f and 2 r give correct R -charges at the IR fixed point. These R -chargesare constrained by unitarity of the SCFT to be larger than 1/2 for interacting fields orto be equal to 1/2 for non-interacting fields. For N c = N f = 2, in order to meet theunitarity constraint one has 1 − r = 1 / r = 1 /
2, which in turn indicates that v ± and M ji are non-interacting. Therefore, the U (2) theory with two flavors, and itsdual 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 2 r cannot be larger than or equal to 1/2 simultaneously. This however doesn’tmean that the unitarity bound cannot be met. Instead, new U (1) symmetry emergein IR and the R -charges would get corrections from the new symmetry to meet theunitarity constraint for a IR fixed point. One can understand this better with theoryB and the reader is referred to [20, 41–43]. In general, this happens for cases which do not satisfy N f > N c − . Such theories have beenstudied in [40]. In this work, we only study cases for which the unitary bound is not broken. – 8 – owards the algebraic structure of the Moduli Spaces. The following sectionsfocus on theory A and refer to theory B via Aharony duality. The focus is to identify thealgebraic structure of the moduli spaces by computing the
Hilbert series [1–7] for the-ory A. The Hilbert series counts gauge invariant operators that characterizes the entirechiral ring. By direct generalization from the 3d N = 4 theories, the monopole opera-tors for 3d N = 2 theories are dressed by gauge invariant operators which are invariantunder the residual gauge symmetry left unbroken in the monopole background. Thefollowing section outlines the computation of the Hilbert series which counts dressedmonopole operators for 3d N = 2 theories. The Hilbert series counts gauge invariant operators on the moduli space of a supersym-metric gauge theory. By doing so, the Hilbert series identifies the algebraic structureof the moduli space of the theory. For the 3 d N = 2 theory with U ( N c ) gauge groupand N f flavors, the Hilbert series counts mesonic gauge invariant operators of the form M = Q ˜ Q on the Higgs branch and dressed monopole operators of the form M m v m onthe remaining moduli space of the theory. The aim of this section is to introduce thecomputation of the Hilbert series for the 3 d N = 2 theory. Conformal dimension of monopole operators.
For the 3 d N = 2 theory withgauge group U ( N c ) and N f flavors, the conformal dimension of a monopole operatorwith GNO charge m = ( m , . . . , m N c ) has the general form∆( m ) = N f (1 − r ) N c (cid:88) i | m i | − (cid:88) i 2, instanton effects [10–12, 32] lift most of the moduli space such that the remaining monopole operators carryonly magnetic charges m ≥ ≥ m . Accordingly, (3.1) simplifies for N c > m , m ) = ((1 − r ) N f − ( N c − m − m ) . (3.2)If N c = 1, the conformal dimension is∆( m ) = (1 − r ) N f | m | , (3.3)where m ∈ Z . – 9 – ilbert Series formula. The Hilbert series for the U ( N c ) theory with N f flavors isgiven by [8] g ( t , τ, a, u, ˜ u ; M U ( N c ) ,N f ) = ∞ (cid:88) m =0 0 (cid:88) m = −∞ τ J ( m ,m ) a K ( m ,m ) t ∆( m ,m ) P U ( N c ) ( m , m ; u, ˜ u, t ) , (3.4)where t counts the monopole operators according to their conformal dimension. J ( m , m ) = m + m and K ( m , m ) = − N f ( m − m ) are respectively the charges under the U (1) T topological and U (1) A axial symmetries. The respective fugacities are chosen to be τ and a . The above Hilbert series is further refined under the flavour symmetries SU ( N f ) and SU ( N f ) with the fugacities u and ˜ u 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) symmetrycan 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, t = t Nf − Nc +1) Nf +2 , (3.5)with r mapping to the value r (cid:55)→ r = ( N f − N c +1) N f +2 under the new U (1) symmetry. In thefollowing sections, fugacity t is used instead of t in the Hilbert series.The dressing of monopole operators comes from the classical factor P U ( N c ) ( m , m ; u, ˜ u, t )in (3.4). As discussed in section § 2, depending on the magnetic charge of the monopoleoperator, the gauge group U ( N c ) is broken to a residual subgroup H m ⊂ U ( N c ). Thedressing factor is a separate Hilbert series which counts mesonic operators of the form M m = Q ˜ Q which are invariant under the residual subgroup H m . It takes the form [44] P U ( N c ) ( m , m ; u, ˜ u, t ) = (cid:73) d µ H m ⊆ U ( N c ) PE (cid:104) [1 , , . . . , z [0 , . . . , , u wat / +[0 , . . . , , z [1 , , . . . , ˜ u w − at / (cid:105) , (3.6)where d µ H m is the Haar measure of H m and fugacities z and w correspond respectivelyto 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 aconcise form when one uses the highest weight generating function of characters of theflavour symmetry SU ( N f ) × SU ( N f ) . It is F N c ,N f = PE (cid:34) N c (cid:88) i =1 µ N f − i ν i a i t ir (cid:35) , (3.7)– 10 –here µ i , ν i count highest weights of SU ( N f ) representations. Monomials in µ i , ν i arereplaced by characters of SU ( N f ) N f − (cid:89) i =1 µ n i i (cid:55)→ [ n , . . . , n N f − ] SU ( N f ) u , N f − (cid:89) i =1 ν n i i (cid:55)→ [ n , . . . , n N f − ] SU ( N f ) ˜ u . (3.8) Plethystic Logarithm. The plethystic logarithm [1, 3, 5] of the Hilbert series g ( t ; M U ( N c ) ,N f )is defined as PL (cid:2) g ( t ; M U ( N c ) ,N f ) (cid:3) = ∞ (cid:88) k =1 µ ( k ) k log (cid:2) g ( t k ; M U ( N c ) ,N f ) (cid:3) , (3.9)where µ ( k ) is the M¨obius function. The plethystic logarithm has a series expansion in t . It extracts information from the Hilbert series about the algebraic structure of themoduli space. As an expansion in t , the initial positive terms refer to generators of themoduli space. The following negative terms refer to first order relations amongst thegenerators. 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 spaceis known to be a non-complete intersection where relations form higher order relationsknown 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) U (1) with 1 flavor: 3 identical components The Hilbert series is given by g ( t ; M U (1) , ) = ∞ (cid:88) m = −∞ t ∆( m ) P U (1) ( m ; t ) , (3.10)where fugacity t counts the bare monopole operators according to their conformaldimension. For a U (1) theory with N f = 1 the conformal dimension of the baremonopole operator is given by ∆( m ) = (1 − r ) | m | , (3.11)where r is the U (1) R-charge of the fundamental Q and anti-fundamental (cid:101) Q and m ∈ Z is the GNO magnetic flux. Under a refinement, fugacities τ and a , which respectively– 11 –orrespond to the topological symmetry U (1) T and axial symmetry U (1) A , can be addedto the formula in (3.10). The refined Hilbert series takes the form g ( t , τ, a ; M U (1) , ) = ∞ (cid:88) m = −∞ τ J ( m ) a K ( m ) t ∆( m ) P U (1) ( m ; t , y ) , (3.12)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 themonopole operators and mesonic operators are weighted equally. Under this new U (1)symmetry, r (cid:55)→ r = and the following fugacity map applies t = t , (3.13)where now t counts the number of v + , v − and M m = Q ˜ Q .In the Hilbert series formulas, the classical factor in its refined form is given by P U (1) ( m ; t, a ) = (cid:40) (cid:72) | w | =1 d ww PE (cid:2) wat / + w − at / (cid:3) = − a t m = 01 m (cid:54) = 0 , (3.14)where w is the U (1) gauge charge fugacity and a the U (1) A axial charge fugacity. C [ M ] C [ v + ] C [ v ] Figure 4 . The moduli space of the U (1) theory with N f = 1 is made of 3 1-dimensionalcones C which meet at the origin. The cones are indistinguishable and are each generated bythe mesonic generator and the monopole operators v + and v − . Summing up the refined series in (3.12) gives g ( t, τ, a ; M U (1) , ) = 11 − a t + 11 − τ a − t + 11 − τ − a − t − 2= 1 + a t + ( τ + τ − ) a − t + a t + ( τ + τ − ) a − t + . . . . (3.15) The U (1) R-charge was found to be r = in [11]. – 12 –he plethystic logarithm of the Hilbert series isPL (cid:2) g ( t, τ, a ; M U (1) , ) (cid:3) = a t + ( τ + τ − ) a − t − ( τ + τ − ) at − a − t + . . . . (3.16)The generators corresponding to the first positive terms can be identified from theabove plethystic logarithm as follows,PL term → generator a t → M = Q a ˜ Q a τ a − t → v + τ − a − t → v − , (3.17)the meson M and two monopoles v ± . The first order relations formed by the generatorsare as follows, PL term → relation − τ at → v + M = 0 − τ − at → v − M = 0 − a − t → v + v − = 0 . (3.18)The full moduli space of the U (1) theory with N f = 1 is an algebraic variety, M U (1) , = C [ M, v + , v − ] / (cid:104) v + M = 0 , v − M = 0 , v + v − = 0 (cid:105) . (3.19)The moduli space M U (1) , has the following components M U (1) , = C [ M ] , M + U (1) , = C [ v + ] , M − U (1) , = C [ v − ] , (3.20)where M U (1) , is the Higgs branch and M + U (1) , and M − U (1) , are the Coulomb branchesof the theory. The corresponding Hilbert series are g ( t, τ, a ; M U (1) , ) = − a t ,g ( t, τ, a ; M + U (1) , ) = − τa − t , g ( t, τ, a ; M − U (1) , ) = − τ − a − t . (3.21)The moduli space M U (1) , is made of 3 identical cones C which meet at the origin, asshown in Figure 4.The moduli space is the union of the 3 components. By removing contributionsfrom the intersections, the Hilbert series of the full moduli space can therefore beexpressed as g ( t, τ, a ; M U (1) , ) = g ( t, τ, a ; M U (1) , ) + g ( t, τ, a ; M + U (1) , ) + g ( t, τ, a ; M − U (1) , ) − , (3.22)where the intersections at the origin are taken care of by − .2.2 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 g ( t ; M U (1) , ) = ∞ (cid:88) m = −∞ t ∆( m ) P U (1) ( m ; t ) , (3.23)where t is the fugacity which counts bare monopole operators according to their con-formal dimension. For a U (1) theory with N f = 2 the conformal dimension of the baremonopole operator is given by ∆( m ) = 2(1 − r ) | m | , (3.24)where r is the U (1) R-charge of the fundamental Q i and anti-fundamental (cid:101) Q i and m ∈ Z is the GNO magnetic flux. The Hilbert series formula above can be refined withthe charges from the topological symmetry U (1) T and the axial symmetry U (1) A . Therespective fugacities are chosen to be τ and a . The refined Hilbert series is g ( t , τ, a, u, ˜ u ; M U (1) , ) = ∞ (cid:88) m = −∞ τ J ( m ) a K ( m ) t ∆( m ) P U (1) ( m ; t , a, u, ˜ u ) , (3.25)where J ( m ) = m and K ( m ) = − | m | are respectively the topological and axial chargesof 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, anew fugacity t can be introduced by mapping the value of r to r (cid:55)→ r = . As discussedin (3.5), the fugacity map is t = t .The classical factor of the Hilbert series formula is P U (1) ( m ; t, a, u, ˜ u ) and it isfurther refined under the flavour symmetries SU (2) × SU (2) . The fugacities u and˜ u respectively count charges under SU (2) and SU (2) . The refined classical factor isgiven by P U (1) ( m ; t, a, u, ˜ u ) = (cid:40) (cid:72) | w | =1 d ww PE (cid:2) w ( u + u − ) at / + w − (˜ u + ˜ u − ) at / (cid:3) = f m = 01 m (cid:54) = 0 , (3.26)where the integral gives f = (1 − a t )(1 − u ˜ ua t )(1 − u ˜ u − a t )(1 − u − ˜ ua t )(1 − u − ˜ u − a t ) . (3.27)– 14 – [ v + ] C [ v ] C [ M ji ] / h det M = 0 i Figure 5 . The moduli space of the U (1) theory with N f = 2 is made of 3 cones which meetat the origin. From the above Hilbert series corresponding to the classical component of the modulispace where the GNO magnetic flux is m = 0, one can identify the classical componentto be the conifold C . The 4 generators of the conifold are the mesonic operators M = Q ˜ Q which satisfy the quadratic relation det M = 0.Summing up the Hilbert series formula in (3.23) for the entire moduli space gives g ( t, τ, a, u, ˜ u ; M U (1) , ) = f + 11 − τ a − t + 11 − τ − a − t − . (3.28)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 3cones meet at the origin as shown in Figure 5.The Hilbert series has the following character expansion, g ( t, τ, a, u, ˜ u ; M U (1) , ) = ∞ (cid:88) n =0 (cid:104) [ n ] u [ n ] ˜ u a n + ( τ n + τ − n ) a − n (cid:105) t n − 2= 1 + [1] u [1] ˜ u a t + ( τ + τ − ) a − t + [2] u [2] ˜ u a t + ( τ + τ − ) a − t + . . . . (3.29)The plethystic logarithm of the refined Hilbert series of the full moduli space isPL (cid:2) g ( t, τ, a, u, ˜ u ; M U (1) , ) (cid:3) = [1] u [1] ˜ u a t + ( τ + τ − ) a − t − a t − a − t − [1] u [1] ˜ u ( τ + τ − ) t + . . . . (3.30)– 15 –rom the initial positive terms of the plethystic logarithm, one can identify the gener-ators of the moduli space, PL term → generator+[1] u [1] ˜ u a t → M ji = Q ai ˜ Q ja + τ a − t → v + + τ − a − t → v − . (3.31)The generators are the mesons and the bare monopoles. The first order relations formedamong the generators are identified as follows,PL term → relation − a t → det M = 0 − [1] u [1] ˜ u τ t → v + M ji = 0 − [1] u [1] ˜ u τ − t → v − M ji = 0 − a − t → v + v − = 0 . (3.32)The full moduli space of the U (1) theory with N f = 2 can be expressed as thefollowing algebraic variety, M U (1) , = C [ M ji , v + , v − ] / (cid:104) det M = 0 , v + M ji = 0 , v − M ji = 0 , v + v − = 0 (cid:105) . (3.33)The moduli space M U (1) , has the following components M U (1) , = C [ M ji ] / (cid:104) det M = 0 (cid:105) = C , M + U (1) , = C [ v + ] , M − U (1) , = C [ v − ] , (3.34)where the Higgs branch is given by M U (1) , and the Coulomb branch by M + U (1) , and M − U (1) , . The Higgs branch M U (1) , is the conifold C . The Hilbert series of the 3components are as follows, g ( t, τ, a, u, ˜ u ; M U (1) , ) = PE [[1] u [1] ˜ u a t − a t ] ,g ( t, τ, a, u, ˜ u ; M + U (1) , ) = − τa − t , g ( t, τ, a, u, ˜ u ; M − U (1) , ) = − τ − a − t . (3.35)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 contributionsfrom the intersections, the Hilbert series of M U (1) , therefore can be expressed as g ( t, τ, a, u, ˜ u ; M U (1) , ) = g ( t, τ, a, u, ˜ u ; M U (1) , ) + g ( t, τ, a, u, ˜ u ; M + U (1) , ) + g ( t, τ, a, u, ˜ u ; M − U (1) , ) − . (3.36)– 16 – .3 Examples: U (2) and U (3) U (2) with 3 flavors: 4 components (Higgs, Mixed and Coulomb) The Hilbert series for the U (2) theory with 3 flavors is given by g ( t ; M U (2) , ) = ∞ (cid:88) m =0 0 (cid:88) m = −∞ t ∆( m ,m ) P U (2) ( m , m ; t ) , (3.37)where fugacity t counts bare monopole operators according to their conformal dimen-sion. For the U (2) theory with N f = 3 the conformal dimension of the bare monopoleoperator is given by ∆( m , m ) = (2 − r )( m − m ) , (3.38)where m , m 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 g ( t , τ, a, u, ˜ u ; M U (2) , ) = ∞ (cid:88) m =0 0 (cid:88) m = −∞ τ J ( m ,m ) a K ( m ,m ) t ∆( m ,m ) P U (2) ( m , m ; t , a, u, ˜ u ) , (3.39)where J ( m , m ) = m + m and K ( m , m ) = − m − m ) are respectively thetopological and axial charges of a monopole operator with GNO charge m , m . Inaddition, the Hilbert series above is refined under the flavour symmetry SU (3) × SU (3) , where fugacities u and ˜ u count the charges of the respective symmetries assummarised in Table 1. By introducing a new U (1) symmetry that replaces U (1) R andweights 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 (cid:55)→ r = . Following(3.5), the fugacity map is t = t .The classical contribution comes from the factor P U (2) ( m , m ; t, a, u, ˜ u ). The GNOcharge lattice with m , m can be dividend into 4 sublattices under which monopoleoperators that contribute to the moduli space are charged. Depending on which GNOsublattice one is, the gauge symmetry is either broken or unbroken. Accordingly, the– 17 –lassical factor of the Hilbert series can be written as follows, P U (2) ( m , m ; t, a, u, ˜ u )= (cid:72) d µ SU (2) (cid:72) d µ U (1) PE (cid:2) [1] z w [0 , u at / + [1] z w − [1 , ˜ u at / (cid:3) = f m , m = 0 (cid:72) d µ U (1) PE (cid:2) w [0 , u at / + w − [1 , ˜ u at / (cid:3) = f (cid:26) m (cid:54) = 0 , m = 0 m = 0 , m (cid:54) = 01 m , m (cid:54) = 0 , (3.40)where the integrals above give f = PE (cid:2) [0 , u [1 , ˜ u a t − a t (cid:3) ,f = (1 − [0 , u [1 , ˜ u a t + [1 , u a t + [1 , ˜ u a t − [0 , u [1 , ˜ u a t + a t ) × PE (cid:2) [0 , u [1 , ˜ u a t (cid:3) . (3.41)Summing up the refined Hilbert series in (3.39) gives g ( t, τ, a, u, ˜ u ; M U (2) , ) = f + f (cid:20) − τ a − t + 11 − τ − a − t − (cid:21) + a − t (1 − τ a − t )(1 − τ − a − t ) . (3.42)The first few orders of the expansion of the Hilbert series is as follows, g ( t, τ, a, u, ˜ u ; M U (2) , ) = 1 + ([0 , u [1 , ˜ u a + ( τ + τ − ) a − ) t + (cid:0) ([0 , u [2 , ˜ u + [0 , u [1 , ˜ u ) a + [0 , u [1 , ˜ u ( τ + τ − ) a − + ( τ + τ − + 1) a − (cid:1) t + (cid:0) ([0 , u [3 , ˜ u + [1 , u [1 , ˜ u ) a + [0 , u [2 , ˜ u ( τ + τ − ) a +[1 , u [1 , ˜ u ( τ + τ − ) a − + ( τ + τ + τ − + τ − ) a − (cid:1) t + . . . . (3.43)The corresponding plethystic logarithm isPL (cid:2) g ( t, τ, a, u, ˜ u ; M U (2) , ) (cid:3) = [0 , u [1 , ˜ u a t + ( τ + τ − ) a − t − a t − [1 , u [0 , ˜ u ( τ + τ − ) at − [0 , u [1 , ˜ u a − t + . . . . (3.44)The plethystic logarithm encodes the generators and relations amongst generatorswhich define the moduli space. The generators of the moduli space correspond to theinitial positive terms of the plethystic logarithm. The generators are as followsPL term → generator+[0 , u [1 , ˜ u a t → M ji = Q ai ˜ Q ja + τ a − t → v + + τ − a − t → v − , (3.45)– 18 –here i, j = 1 , , 3. The corresponding first order relations between the generators areidentified as follows PL term → relation − a t → det M = 0 − [1 , u [0 , ˜ u τ at → v + R (2 , ij = 0 − [1 , u [0 , ˜ u τ − at → v − R (2 , ij = 0 − [0 , u [1 , ˜ u a − t → v + v − M ji = 0 , (3.46)where R (2 , ij = 12 (cid:15) jk k (cid:15) im m M k m M k m . (3.47)From the generators and first order relations, the moduli space can be expressedas the following algebraic variety, M U (2) , = C [ M ji , v ± ] / (cid:104) det M = 0 , v ± R (2 , ij = 0 , v + v − M ji = 0 (cid:105) . (3.48)Let us call the space of N × N matrices M ji with at most rank k as M k,N . Using thisspace, the components of the moduli space M U (2) , can be expressed as M U (2) , = M , M + U (2) , = M , × C [ v + ] , M − U (2) , = M , × C [ v − ] , M + − U (2) , = C [ v + , v − ] , (3.49)where M U (2) , and M + − U (2) , are identified as Higgs and Coulomb branches respectivelywhile M + U (2) , and M − U (2) , are mixed branches. The corresponding Hilbert series areas follows, g ( t, τ, a, u, ˜ u ; M U (2) , ) = f ,g ( t, τ, a, u, ˜ u ; M + U (2) , ) = f × − τ a − t ,g ( t, τ, a, u, ˜ u ; M − U (2) , ) = f × − τ − a − t ,g ( t, τ, a, u, ˜ u ; M + − U (2) , ) = 1(1 − τ a − t )(1 − τ − a − t ) , (3.50)where f and f correspond to the monopole dressing factors in (3.41). The 4 compo-nents intersect in various subspaces which are I = { } , I M = M , , I + = C [ v + ] , I − = C [ v − ] , (3.51)– 19 –here { } is the origin. The corresponding Hilbert series are g ( t, τ, a, u, ˜ u ; I M ) = f , g ( t, τ, a, u, ˜ u ; I ) = 1 ,g ( t, τ, a, u, ˜ u ; I + ) = − τa − t , g ( t, τ, a, u, ˜ u ; I − ) = − τ − a − t . (3.52)The union of the 4 components is the moduli space. By removing contributions fromthe intersections, the Hilbert series of the full moduli space M U (2) , can be expressedas g ( t, τ, a, u, ˜ u ; M U (2) , ) = g ( t, τ, a, u, ˜ u ; M U (2) , ) + g ( t, τ, a, u, ˜ u ; M + U (2) , )+ g ( t, τ, a, u, ˜ u ; M − U (2) , ) + g ( t, τ, a, u, ˜ u ; M + − U (2) , ) − g ( t, τ, a, u, ˜ u ; I M ) − g ( t, τ, a, u, ˜ u ; I + ) − g ( t, τ, a, u, ˜ u ; I − ) + g ( t, τ, a, u, ˜ u ; I ) . (3.53)This expression for the Hilbert series of the full moduli space is in agreement with theHilbert 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 g ( t ; M U (3) , ) = ∞ (cid:88) m =0 0 (cid:88) m = −∞ t ∆( m ,m ,m =0) P U (3) ( m , m ; t ) , (3.54)where t is the fugacity which counts bare monopole operators according to their con-formal dimension. The following is the conformal dimension of the bare monopoleoperator for the U (3) theory with N f = 4,∆( m , m ) = (2 − r )( m − m ) (3.55)where m , m are the GNO magnetic fluxes. Note that all monopole operators whichcontribute to the moduli spaces carry fluxes m ≥ , m ≤ m = 0, as discussedin section § U (1) T and axial U (1) A symmetry. The respectivefugacities are chosen to be τ and a . Accordingly, the refined Hilbert series takes theform g ( t , τ, a, u, ˜ u ; M U (3) , ) = ∞ (cid:88) m =0 0 (cid:88) m = −∞ τ J ( m ,m a K ( m ,m )2 t ∆( m ,m ) P U (3) ( m , m ; t , a, u, ˜ u ) , (3.56)– 20 –here J ( m , m ) = m + m and K ( m , m ) = − m − m ) are respectively thetopological and axial charges of a monopole operator with GNO charge m , m . TheHilbert series in (3.56) is further refined by the flavour symmetry SU (4) × SU (4) whose charges are counted respectively by fugacities u and ˜ u . A new U (1) symmetrythat replaces U (1) R can be introduced such that monopole operators v ± and mesonicoperators M m are counted equally by a new fugacity t . This new fugacity replaces t by mapping the value of r to r (cid:55)→ r = . The corresponding fugacity map is t = t .The classical contribution to the Hilbert series comes from the factor P U (3) ( m , m ; t, a, u, ˜ u ).The GNO charge lattice can be divided into 4 sublattices. Depending on which sub-lattice the GNO charge of a monopole operator is located, the gauge symmetry breaksunder the Higgs mechanism. The residual gauge symmetry determines the dressingof the monopole operator in the particular GNO charge sublattice. Accordingly, theclassical factor of the Hilbert series can be written as follows, P U (3) ( m , m ; t, a, u, ˜ u ) = (cid:72) d µ SU (3) (cid:72) d µ U (1) PE (cid:2) [1 , z w [0 , , u at / + [0 , z w − [1 , , ˜ u at / (cid:3) = f m , m = 0 (cid:72) d µ SU (2) (cid:72) d µ U (1) PE (cid:2) [1] z w [0 , , u at / + [1] z w − [1 , , ˜ u at / (cid:3) = f (cid:26) m (cid:54) = 0 , m = 0 m = 0 , m (cid:54) = 0 (cid:72) d µ U (1) PE (cid:2) z [0 , , u at / + z − [1 , , ˜ u at / (cid:3) = f m , m (cid:54) = 0 , (3.57)where the integrals above give f = PE (cid:2) [0 , , u [1 , , ˜ u a t − a t (cid:3) ,f = (1 − [0 , , u [1 , , ˜ u a t + [1 , , u a t + [1 , , ˜ u a t − [0 , , u [1 , , ˜ u a t + a t ) × PE (cid:2) [0 , , u [1 , , ˜ u a t (cid:3) ,f = (cid:0) − [0 , , u [0 , , ˜ u a t + ([1 , , u [1 , , ˜ u + [0 , , u [0 , , ˜ u ) a t − ([0 , , u + [1 , , u [1 , , ˜ u + [2 , , ˜ u ) a t + ([1 , , u [1 , , ˜ u +[0 , , u [2 , , ˜ u ) a t + ([2 , , u [0 , , ˜ u − [0 , , u [2 , , ˜ u ) a t − ([2 , , u [0 , , ˜ u + [1 , , u [1 , , ˜ u ) a t + ([2 , , u + [1 , , u [1 , , ˜ u +[0 , , ˜ u ) a t − ([1 , , u [1 , , ˜ u + [0 , , u [0 , , ˜ u ) a t +[0 , , u [0 , , ˜ u a t − a t (cid:1) × PE (cid:2) [0 , , u [1 , , ˜ u a t (cid:3) . (3.58)When one sums up the Hilbert series in (3.56), one obtains g ( t, τ, a, u, ˜ u ; M U (3) , ) = f + f (cid:20) − τ a − t + 11 − τ − a − t − (cid:21) + f a − t (1 − τ a − t )(1 − τ − a − t ) . (3.59)– 21 –he Hilbert series has the following expansion up to order t , g ( t, τ, a, u, ˜ u ; M U (3) , ) = 1 + (cid:0) [0 , , u [1 , , ˜ u a + ( τ + τ − ) a − (cid:1) t + (cid:0) ([0 , , u [2 , , ˜ u ) a + [0 , , u [1 , , ˜ u ( τ + τ − ) a − + ( τ + τ − + 1) a − (cid:1) t + (cid:0) ([0 , , u [3 , , ˜ u + [0 , , u [1 , , ˜ u + [1 , , u [0 , , ˜ u ) a + ([0 , , u [2 , , ˜ u +[0 , , u [0 , , ˜ u )( τ + τ − ) + [0 , , u [1 , , ˜ u ( τ + τ − ) a − +( τ + τ + τ − + τ − ) a − (cid:1) t + (cid:0) ([0 , , u [4 , , ˜ u + [0 , , u [2 , , ˜ u + [0 , , u [0 , , ˜ u + [1 , , u [1 , , ˜ u ) a +([0 , , u [3 , , ˜ u + [0 , , u [1 , , ˜ u )( τ + τ − ) a + ([0 , , u [2 , , ˜ u +[0 , , u [0 , , ˜ u )( τ + τ − ) a − + [0 , , u [1 , , ˜ u ( τ + τ − ) a − +( τ + τ + τ − + τ − + 1) a − (cid:1) t + . . . . (3.60)The plethystic logarithm of the Hilbert series isPL (cid:2) g ( t, τ, a, u, ˜ u ; M U (3) , ) (cid:3) = [0 , , u [1 , , ˜ u a t + ( τ + τ − ) a − t − a t − [1 , , u [0 , , ˜ u ( τ + τ − ) a t − [0 , , u [0 , , ˜ u a − t + . . . . (3.61)The generators of the moduli space are identified from the plethystic logarithm asfollows, PL term → generator+[0 , , u [1 , , ˜ u a t → M ji = Q ai ˜ Q ja + τ a − t → v + + τ − a − t → v − . (3.62)The generators form first order relations which arePL term → relation − a t → det M = 0 − [1 , , u [0 , , ˜ u τ a t → v + R (3 , ji = 0 − [1 , , u [0 , , ˜ u τ − a t → v − R (3 , ji = 0 − [0 , , u [0 , , ˜ u a − t → v + v − R (2 , j j i i = 0 , (3.63)where R (3 , ji = 16 (cid:15) ik k k (cid:15) jm m m M k m M k m M k m ,R (2 , j j i i = 12 (cid:15) i i k k (cid:15) j j m m M k m M k m . (3.64)– 22 –iven the generators and first order relations, the moduli space can be expressedas follows M U (3) , = C [ M ji , v ± ] / (cid:104) det M = 0 , v ± R (3 , ji = 0 , v + v − R (2 , j j i i = 0 (cid:105) . (3.65)Calling the space of N × N matrices with rank at most k as M k,N , the 4 componentsof the moduli space can be expressed as M U (3) , = M , , M + U (3) , = M , × C [ v + ] , M − U (3) , = M , × C [ v − ] , M + − U (3) , = M , × C [ v + , v − ] , (3.66)where M U (3) , is the Higgs branch and M + U (3) , , M − U (3) , and M + − U (3) , are mixed branches.There is no pure Coulomb branch for this theory. The corresponding Hilbert series are g ( t, τ, a, u, ˜ u ; M U (3) , ) = f ,g ( t, τ, a, u, ˜ u ; M + U (3) , ) = f × − τ a − t ,g ( t, τ, a, u, ˜ u ; M − U (3) , ) = f × − τ − a − t ,g ( t, τ, a, u, ˜ u ; M + − U (3) , ) = f × − τ a − t )(1 − τ − a − t ) , (3.67)where f , f and f are the dressing factors of the monopole operators in different GNOsublattices, as shown in (3.58). The 4 components intersect in various subspaces whichare I = M , , I M = M , , I + = M , × C [ v + ] , I − = M , × C [ v − ] . (3.68)The corresponding Hilbert series are g ( t, τ, a, u, ˜ u ; I M ) = f , g ( t, τ, a, u, ˜ u ; I ) = f ,g ( t, τ, a, u, ˜ u ; I + ) = f × − τa − t , g ( t, τ, a, u, ˜ u ; I − ) = f × − τ − a − t . (3.69)The union of the 4 components is the moduli space. By removing contributions fromthe intersections, the Hilbert series of the full moduli space M U (3) , can be expressedas g ( t, τ, a, u, ˜ u ; M U (3) , ) = g ( t, τ, a, u, ˜ u ; M U (3) , ) + g ( t, τ, a, u, ˜ u ; M + U (3) , )+ g ( t, τ, a, u, ˜ u ; M − U (3) , ) + g ( t, τ, a, u, ˜ u ; M + − U (3) , ) − g ( t, τ, a, u, ˜ u ; I M ) − g ( t, τ, a, u, ˜ u ; I + ) − g ( t, τ, a, u, ˜ u ; I − ) + g ( t, τ, a, u, ˜ u ; I ) . (3.70)– 23 –his expression for the Hilbert series of the full moduli space is in agreement with theHilbert series expression in (3.59). The Hilbert series g ( t, τ, a, u, ˜ u ; M U ( N c ) ,N f ) which has been computed in the abovesections all satisfy a general form. In order to present this general form, we make use ofthe highest weight generating function for the characters of irreducible representationsof the flavour symmetry SU ( N f ) × SU ( N f ) . The highest weight generating functionfor Hilbert series makes use of the map N c − (cid:89) i =1 µ n i i (cid:55)→ [ n , . . . , n N c − ] SU ( N f ) u , N c − (cid:89) i =1 ν n i i (cid:55)→ [ n , . . . , n N c − ] SU ( N f ) ˜ u , (3.71)where fugacities µ i and ν i count the highest weight of the irreducible representationsof SU ( N f ) × SU ( N f ) .Using the highest weight generating function of Hilbert series, one can for instanceexpress concisely the dressing factor for monopole operators as follows, F N c ,N f = PE (cid:34) N c (cid:88) i =1 µ N f − i ν i a i t i (cid:35) . (3.72)After the inclusion of the monopole operators the highest weight generating function is G ( t, τ, a, u, ˜ u ; M U ( N c ) ,N f ) = F N c ,N f + F N c − ,N f (cid:20) − τ a − N f t + 11 − τ − a − N f t − (cid:21) + F N c − ,N f a − N f t (1 − τ a − N f t )(1 − τ − a − N f t ) , (3.73)where t counts magnetic monopoles v ± and mesonic operators M m = Q ˜ Q and corre-sponds to U (1) symmetry which replaces U (1) R . By identifying the exponents of fu-gacities 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 func-tion is µ N f − ν a t + ( τ + τ − ) a − N f t − µ N f − ( N c +1) ν N c +1 a N c +1) t N c +1 − µ N f − N c ν N c ( τ + τ − ) a N c − N f t N c +1 − µ N f − ( N c − ν N c − a N c − − N f t N c +1 + . . . . (3.74)– 24 –he following product of mesonic operators is used in order to express relationsamongst moduli space generators, R ( N c ,N f ) j ...j Nf − Nc i ...i Nf − Nc = 1 N c ! (cid:15) i ...i Nf − Nc k ...k Nc (cid:15) j ...j Nf − Nc m ...m Nc M k m . . . M k Nc m Nc , (3.75)where R ( N c ,N c ) = det( M ) . (3.76)From the plethystic logarithm in (3.74), the general form of the generators can beidentified as PL term → generator µ N f − ν a t → M ji = Q ai ˜ Q ja τ ± a − N f t → v ± . (3.77)Furthermore, the general form of the first order relations formed amongst the generatorsare PL term → relation − µ N f − ( N c +1) ν N c +1 a N c +1) t N c +1 → R ( N c +1 ,N f ) j ...j Nf − Nc − i ...i Nf − Nc − = 0 − µ N f − N c ν N c ( τ + τ − ) a N c − N f t N c +1 → v ± R ( N c ,N f ) j ...j Nf − Nc i ...i Nf − Nc = 0 − µ N f − ( N c − ν N c − a N c − − N f t N c +1 → v + v − R ( N c − ,N f ) j ...j Nf − Nc +1 i ...i Nf − Nc +1 = 0 . (3.78)It is important to note that the terms in the plethystic logarithm in (3.74) whichcorrespond 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 contributionsfrom the monopole operators. One can show that the terms of the plethystic logarithmin (3.78) do not appear as operators in the Hilbert series expansion and that therelations 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, M U ( N c ) ,N f = C [ M ji , v ± ] / I , (3.79)where the quotienting ideal is I = (cid:104) R ( N c +1 ,N f ) = 0 , v ± R ( N c ,N f ) = 0 , v + v − R ( N c − ,N f ) = 0 (cid:105) . (3.80)Let us call M k,N the space of all N × N matrices M ji which at most have rank k . Interms of (3.75), one can write M k,N = C [ M ji ] / (cid:104) R ( k +1 ,N ) = 0 (cid:105) . Then using M k,N , the 4– 25 –omponents of the moduli space can be expressed as M U ( N c ) ,N f = M N c ,N f , M + U ( N c ) ,N f = M N c − ,N f × C [ v + ] , M − U ( N c ) ,N f = M N c − ,N f × C [ v − ] , M + − U ( N c ) ,N f = M N c − ,N f × C [ v + , v − ] , (3.81)where M U ( N c ) ,N f is the Higgs branch, M + U ( N c ) ,N f and M − U ( N c ) ,N f are mixed branches,and M + − U ( N c ) ,N f is a Coulomb branch when N c = 1 , N c > The corresponding highest weight generating functions for the Hilbert series are G ( t, τ, a, u, ˜ u ; M U ( N c ) ,N f ) = F N c ,N f , G ( t, τ, a, u, ˜ u ; M + U ( N c ) ,N f ) = F N c − ,N f × − τ a − t , G ( t, τ, a, u, ˜ u ; M − U ( N c ) ,N f ) = F N c − ,N f × − τ − a − t , G ( t, τ, a, u, ˜ u ; M + − U ( N c ) ,N f ) = F N c − ,N f × − τ a − t )(1 − τ − a − t ) , (3.82)where F N c ,N f , F N c − ,N f and F N c − ,N f are the dressing factors in (3.72) for the differentGNO sublattices. The 4 components of the moduli space intersect in the followingsubspaces, I M = M N c − ,N f , I = M N c − ,N f , I + = M N c − ,N f × C [ v + ] , I − = M N c − ,N f × C [ v − ] . (3.83)The corresponding highest weight generating functions for the Hilbert series are G ( t, τ, a, u, ˜ u ; I M ) = F N c − ,N f , G ( t, τ, a, u, ˜ u ; I ) = F N c − ,N f , G ( t, τ, a, u, ˜ u ; I + ) = F N c − ,N f × − τa − Nf t , G ( t, τ, a, u, ˜ u ; I − ) = F N c − ,N f × − τ − a − Nf t . (3.84)Taking into account all the intersections, the highest weight generating function for theHilbert series of the full moduli space M U ( N c ) ,N f can be expressed as G ( t, τ, a, u, ˜ u ; M U ( N c ) ,N f ) = G ( t, τ, a, u, ˜ u ; M U ( N c ) ,N f ) + G ( t, τ, a, u, ˜ u ; M + U ( N c ) ,N f )+ G ( t, τ, a, u, ˜ u ; M − U ( N c ) ,N f ) + G ( t, τ, a, u, ˜ u ; M + − U ( N c ) ,N f ) − G ( t, τ, a, u, ˜ u ; I M ) −G ( t, τ, a, u, ˜ u ; I + ) − G ( t, τ, a, u, ˜ u ; I − ) + G ( t, τ, a, u, ˜ u ; I ) . (3.85) Note that component M U ( N c ) ,N f is the dressing factor for components M + U ( N c +1) ,N f and M − U ( N c +1) ,N f and the dressing factor for component M + − U ( N c +2) ,N f . – 26 –his expression for the highest weight generating function for the Hilbert series of thefull moduli space is in agreement with the Hilbert series expression in (3.73). In this section, we examine the relation between the superconformal index and theHilbert series. The superconformal index by itself does not give information on themoduli space. Only by taking appropriate limits to a Hilbert series one can deriveinformation about the structure of the moduli space. The following section proposeslimits from the superconformal index which reproduce Hilbert series of certain sub-spaces of the moduli space of the 3d N = 2 theories. 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 , × SO (2)whose three Cartan elements are denoted by E, j and R . The superconformal index isdefined by [17] I ( x, u i ) = Tr( − F exp( − β (cid:48) { Q, S } ) x E + j (cid:32)(cid:89) i u F i i (cid:33) (4.1)where Q is a supercharge of quantum numbers E = , j = − and R = 1, and S = Q † . x is the fugacity for E + j and u i ’s are additional fugacities for global symmetries of thetheory. The trace is taken over the Hilbert space of the SCFT on R × S , or equivalentlyover the space of local gauge invariant operators on R . As usual, only the BPS states,which saturate the inequality { Q, S } = E − R − j ≥ , (4.2)contribute to the index.Using supersymmetric localization, the superconformal index can be exactly com-puted as follows, [15, 16] ( a ; q ) n is the q -Pochhammer symbol, defined by( a ; q ) n = n − (cid:89) k =0 (cid:0) − aq k (cid:1) . (4.3) – 27 – ( x, u, ˜ u, a, τ ) = (cid:88) m ∈ Z Nc /S Nc (cid:73) (cid:32) N c (cid:89) a =1 dz a πiz a (cid:33) |W m | τ (cid:80) a m a Z vector ( x, z, m ) Z chiral ( x, u, ˜ u, a, z, m ) , (4.4)where Z vector ( x, z, m ) = N c (cid:89) a,b =1( a (cid:54) = b ) x −| m a − m b | / (cid:0) − z a z − b x | m a − m b | (cid:1) ,Z chiral ( x, u, ˜ u, a, z, m ) = N c (cid:89) a =1 x (1 − r ) N f | m a | a − N f | m a | × N f (cid:89) i =1 (cid:0) z − a u − i a − x | m a | +2 − r ; x (cid:1) ∞ (cid:0) z a ˜ u − i a − x | m a | +2 − r ; x (cid:1) ∞ ( z a u i ax | m a | + r ; x ) ∞ ( z − a ˜ u i ax | m a | + r ; x ) ∞ . (4.5)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 , . . . , u N f ) , ˜ u =(˜ u , . . . , ˜ u N f ) , a and τ are the fugacities for the global symmetry SU ( N f ) × SU ( N f ) × U (1) A × U (1) T respectively. Note that (cid:81) N f i =1 u i = (cid:81) N f i =1 ˜ u i = 1. N = 4 Superconformal Index Let us review the proposal [45] for the relation between the superconformal index andthe Hilbert series of N = 4 theories [8, 23]. Let us denote by j H and j V the spins of thetwo SU (2) in the SO (4) R = SU (2) H × SU (2) V R -symmetry. x is the E + j fugacityand x (cid:48) is the j H − j V fugacity. The superconformal index for an N = 4 theory is I ( x, x (cid:48) ) = Tr (cid:48) ( − F x E + j x (cid:48) j H − j V = Tr (cid:48) ( − F t E − j V H t E − j H C (4.6)where we ignore other global symmetry fugacities and t H = xx (cid:48) , t C = xx (cid:48)− . Theprimed trace Tr (cid:48) 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 ofthe fermions in the N = 4 vector multiplet get the same quantum numbers as the– 28 –-terms and play the same role for the index as the F-terms for the Hilbert series. It isimportant to note that the index is unreliable when there are accidental IR correctionsto the R-symmetry.The proposed limits for getting the Hilbert series of the Higgs branch and theCoulomb branch from the superconformal index are Higgs branch: HS H ( t H ) = lim t C → I ( t H , t C ) , Coulomb branch: HS C ( t C ) = lim t H → I ( t H , t C ) , (4.7)where HS H and HS C are respectively the Hilbert series of the Higgs and Coulombbranches.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 indexin each limit captures the SU (2) V/H singlet scalar BPS states, which corresponds tothe Hilbert series of the Higgs/Coulomb branch of the N = 4 theory, respectively. N = 2 Superconformal Index N = 2 theories do not in general have distinct Higgs and Coulomb branches. Fur-thermore, there is only one U (1) R symmetry in the superconformal algebra for N = 2theories. 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 canchoose one of the N = 2 global U (1) symmetries and choose its charge to play the roleof j H − j V in N = 4. With these choices, it turns out that the resulting generalizedlimits of the N = 2 superconformal index give rise to Hilbert series of certain sub-spaces of the moduli space for the N = 2 theory. In addition, such generalized limitsof the N = 2 superconformal index are not unique because the N = 2 theories we areconsidering have several U (1) global symmetries.We will examine 4 limits of the superconformal index of the N = 2 U ( N c ) theorywith N f flavors. The BPS condition and certain constraints on the global U (1) sym-metry charges, which derive from the requirement that the limit is well-defined andnon-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 4limits corresponds to a Hilbert series of a certain subspace of the moduli space. 3 ofthem can be expressed in terms of the 4 main components of the moduli space whichare discussed in section § A crucial comment here is that the Higgs branch limit gives the Hilbert series only when a completeHiggsing of the gauge group occurs along the Higgs branch. – 29 – M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) = M U ( N c ) ,N f • M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) = M U ( N c ) ,N f ∪ M + U ( N c ) ,N f • M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) = M U ( N c ) ,N f ∪ M − U ( N c ) ,N f The 4th limit gives the Hilbert series of a subspace of the moduli space that cannot bedirectly expressed in terms of the 4 main components. It is a subspace of component M + − U ( N c ) ,N f as follows: • M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) ⊂ M + − U ( N c ) ,N f By considering all 4 limits, we are going to see that taking a limit of the superconformalindex cannot reproduce the Hilbert series of the whole component M + − U ( N c ) ,N f , and thusthat of the complete moduli space. The subsequent sections explain how we obtain theHilbert 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 stateis 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 astate whose energy is equal to the U (1) R charge R . Such a state then should be scalarBPS because of the N = 2 BPS condition E ≥ R + j . We cannot trace every scalarBPS state by taking a limit of the superconformal index because there are accidentalcancelations between the bosonic and the fermionic contributions to the index. Thissection explains which remaining states can be traced by taking an appropriate limitof the N = 2 superconformal index.For every factor U (1) k in the global symmetry of the theory, one can introduce acorresponding fugacity u k . In order to have a well-defined non-divergent limit of thesuperconformal index, we propose the condition that for a U (1) k factor in the globalsymmetry, the ratio of the U (1) k charge F k to the U (1) R charge R satisfies the followingbound F k R ≤ . (4.8)We have assumed for simplicity that F k is normalized such that the right hand sideis 1. The role of the above condition is going to become clearer when one revisits thegeneral form of the N = 2 index I ( x, u i ) = Tr (cid:48) ( − F x E + j (cid:32)(cid:89) i u F i i (cid:33) , – 30 –here we can make shifts of the E + j fugacity and the U (1) k fugacity, x → xy, u k → u k y − , such that in the limit y → y → I ( xy, u i ( (cid:54) = k ) , u k y − ) = lim y → Tr (cid:48) ( − F x E − R (cid:32)(cid:89) i u F i i (cid:33) y E − R )+ R − F k . (4.9)Again the primed trace Tr (cid:48) denotes that the trace is taken over the BPS states. Giventhe 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 → y . The remaining terms correspond to the contributions of BPS statessatisfying E = R = F k and j = 0. This is exactly the Hilbert series counting scalarBPS states of the theory, g ( x, u i ) = lim y → I ( xy, u i ( (cid:54) = k ) , u k y − ) , (4.10)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 BPScondition, and the requirement for having a well-defined non-divergent limit of the su-perconformal index all lead to precisely 4 limits of the N = 2 superconformal indexfor the theories we are considering. In the following sections, these limits are presentedand the resulting Hilbert series are identified with subspaces of the moduli space of the N = 2 theory. M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) and M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) We are considering N = 2 U ( N c ) theories with N f pairs of fundamental and anti-fundamental chiral multiplets which have a global symmetry of SU ( N f ) × SU ( N f ) × U (1) A × U (1) T . Let us consider here the U (1) A axial symmetry. Given the chargeassignments summarized in 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: − N f (1 − r ) N f − N c + 1 ≤ AR ≤ r (4.11)where r is the U (1) R charge of the fundamental and anti-fundamental chiral multiplets Q and ˜ Q . r is such that ∆ M = 2 r and ∆ V = (1 − r ) N f − N c +1 for mesonic and monopoleoperators respectively are larger than or equal to 1/2 due to unitarity. We can taketwo differently normalized versions of U (1) A such that each inequality in (4.11) takesthe form of (4.8). Then, as we have proposed, the Hilbert series of a subspace of themoduli space generated by generators saturating each inequality can be obtained from– 31 –he superconformal index. It turns out that the right inequality is saturated for themesonic operators M ji , which have the U (1) A charge 2 and the U (1) R charge 2 r , whereasthe 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 limitsof the superconformal index which give rise to the Hilbert series of two subspaces ofthe moduli space M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) and M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) . M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) is the same as component M U ( N c ) ,N f of the moduli space as discussed in section 3.4while M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) is only a subspace of component M + − U ( N c ) ,N f : M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) = M U ( N c ) ,N f , M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) = M + − U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) ⊂ M + − U ( N c ) ,N f . (4.12)Their Hilbert series are given by g ( x, τ, a, u, ˜ u, ; M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) ) = lim y → I ( xy, u, ˜ u, ay − r , τ ) , (4.13) g ( x, τ, a, u, ˜ u ; M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) ) = lim y → I ( xy, u, ˜ u, ay (1 − r ) − ( N c − /N f , τ ) . (4.14)Again x is the energy fugacity of the Hilbert series. u, ˜ u, a and τ are identified as thefugacities for SU ( N f ) × SU ( N f ) × U (1) A × U (1) T respectively. Computation. Using the limits, we claim that one can obtain the explicit formulaefor the Hilbert series of the two subspaces M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) and M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) from the superconformal index. Firstly, the Hilbert series of M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) is given by the limit (4.13). Since M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) is the same as component M U ( N c ) ,N f of the moduli space, g ( M U ( N c ) ,N f / (cid:104) v ± = 0 (cid:105) ) = g ( M U ( N c ) ,N f ) . (4.15)In this limit the monomial factor x (1 − r ) N f (cid:80) a | m a |− (cid:80) a
0. This is because the power of x , which is equalto ∆( m ) = (1 − r ) N f (cid:80) a | m a | − (cid:80) a
2. For a U (1)theory the vector multiplet does not contribute to the index. Only the contribution ofchiral multiplets is nontrivial, which becomes the monomial factor x (1 − r ) N f | m | a − N f | m | (4.17)under the limit. For the U (1) theory, M U (1) ,N f / (cid:10) M ji = 0 (cid:11) is nothing but component M + − U (1) ,N f of the moduli space. Therefore, g ( M U (1) ,N f / (cid:10) M ji = 0 (cid:11) ) = g ( M + − U (1) ,N f ) (4.18)and g ( x, τ, a ; M + − U (1) ,N f ) = lim y → I ( xy, u, ˜ u, ay − r , τ )= ∞ (cid:88) m = −∞ τ m a − N f | m | x (1 − r ) N f | m | = 1 − a − N f x − r ) N f (1 − τ a − N f x (1 − r ) N f ) (cid:0) − τ − a − N f x (1 − r ) N f (cid:1) . (4.19)For a U (1) theory, the nontrivial components of the moduli space are only component M U (1) ,N f and M + − U (1) ,N f because component M + U (1) ,N f and M − U (1) ,N f are included in M + − U (1) ,N f . The Hilbert series of component M U (1) ,N f is given by (4.16) and the Hilbertseries of component M + − U (1) ,N f is given by (4.19). Taking into account the fact that theirintersection is only the origin, for this special case of the U (1) theory, the completeHilbert series can be written as g ( x, τ, a, u, ˜ u, τ ; M U (1) ,N f ) = g ( x, a, u, ˜ u ; M U (1) ,N f ) + g ( x, τ, a ; M + − U (1) ,N f ) − g ( M U (1) ,N f ∩ M + − U (1) ,N f ) , (4.20)where we use g ( M U (1) ,N f ∩ M + − U (1) ,N f ) = 1 . (4.21)If we substitute x = t into (4.20), we recover the result in section 3.– 33 –ow let us consider a U ( N c ) theory with N c ≥ 2. In this case, the superconformalindex in the limit (4.14) is given by g ( x, τ, a ; M U ( N c ≥ ,N f / (cid:10) M ji = 0 (cid:11) ) = lim y → I ( xy, u, ˜ u, ay (1 − r ) − ( N c − /N f , τ )= ∞ (cid:88) m =0 0 (cid:88) m = −∞ τ m + m a − N f ( m − m ) x (1 − r ) N f ( m − m ) = 1(1 − τ a − N f x (1 − r ) N f − N c +1 ) (cid:0) − τ − a − N f x (1 − r ) N f − N c +1 (cid:1) . (4.22)The above Hilbert series shows that the chiral ring is freely generated by two monopoleoperators v ± . Note that especially for N c = 2, M U (2) ,N f / (cid:10) M ji = 0 (cid:11) is again component M + − U (2) ,N f . Therefore, g ( M U (2) ,N f / (cid:10) M ji = 0 (cid:11) ) = g ( M + − U (2) ,N f ) . (4.23) M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) and M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) Let us consider in this section the topological symmetry U (1) T . Given that onlymonopole operators are charged under U (1) T , we do not directly use the U (1) T sym-metry for formulating the N = 2 limit but use mixed symmetries U (1) + and U (1) − instead whose conserved currents are defined by J ± = rJ A ± ( N f − N c + 1) J T (4.24)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 asfollows, F + R ≤ , , (4.25) F − R ≤ , (4.26)where F ± are charges under U (1) ± .Recall that only the monopole operators v ± are charged under U (1) T with thecharges T = ± 1. Thus, the mesonic operators M ji just have the U (1) ± charges F ± = rA = 2 r and saturate both inequalities (4.25) and (4.26). On the other hand,– 34 – + F − v + (1 − r ) N f − N c + 1 − (1 + r ) N f + N c − v − − (1 + r ) N f + N c − − r ) N f − N c + 1 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 Ta-ble 3. As a result, v + saturates the bound (4.25) while v − saturates the bound (4.26).Furthermore, the inequality (4.25) is saturated at a subspace of the moduli space M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) while the inequality (4.26) is saturated at a subspace of the mod-uli space M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) . Each subspace can be expressed in terms of the maincomponents of the moduli space M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) = M U ( N c ) ,N f ∪ M + U ( N c ) ,N f , M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) = M U ( N c ) ,N f ∪ M − U ( N c ) ,N f . (4.27) Computation. In order to obtain the Hilbert series of M / (cid:104) v − = 0 (cid:105) , we propose thefollowing limit of the superconformal index, g ( x, u + ; M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) ) = lim y → I ( xy, u + y − )= lim y → Tr (cid:48) ( − F ( xy ) E + j ( u + y − ) rA +( N f − N c +1) T = lim y → Tr (cid:48) ( − F x E − R u rA +( N f − N c +1) T + y E − R − rA − ( N f − N c +1) T , (4.28)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 − hereis equivalent to the shifts of the U (1) A and U (1) T fugacities a → ay − r and τ → τ y − ( N f − N c +1) respectively. This is because u + and u − are written in terms of a, τ as– 35 – ± = a r τ ± Nf − Nc +1) . Therefore, (4.28) takes the form g ( x, τ, a, u, ˜ u, τ ; M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) )= lim y → I ( xy, u, ˜ u, ay − r , τ y − ( N f − N c +1) )= (cid:73) d µ U ( N c ) N c (cid:89) a =1 N f (cid:89) i =1 − z a u i ax r ) (1 − z − a ˜ u i ax r )+ ∞ (cid:88) m =1 τ m a − N f m x (1 − r ) N f m (cid:73) d µ U ( N c − N c − (cid:89) a =1 N f (cid:89) i =1 − z a u i ax r ) (1 − z − a ˜ u i ax r )= g ( x, a, u, ˜ u ; M U ( N c ) ,N f ) + τ a − N f x (1 − r ) N f − N c +1 − τ a − N f x (1 − r ) N f − N c +1 × g ( x, a, u, ˜ u ; M U ( N c − ,N f ) . (4.29)This is the same as the Hilbert series of the union of components M U ( N c ) ,N f and M + U ( N c ) ,N f , g ( M U ( N c ) ,N f / (cid:104) v − = 0 (cid:105) ) = g ( M U ( N c ) ,N f ) + g ( M + U ( N c ) ,N f ) − g ( M U ( N c ) ,N f ∩ M + U ( N c ) ,N f ) , (4.30)where M U ( N c ) ,N f ∩ M + U ( N c ) ,N f = M U ( N c − ,N f . (4.31)In the same way, the Hilbert series of M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) is obtained from thesuperconformal index as follows: g ( x, τ, a, u, ˜ u, τ ; M U ( N c ) ,N f / (cid:104) v + = 0 (cid:105) )= lim y → I ( xy, u, ˜ u, ay − r , τ y N f − N c +1 )= (cid:73) d µ U ( N c ) N c (cid:89) a =1 N f (cid:89) i =1 − z a u i ax r ) (1 − z − a ˜ u i ax r )+ − (cid:88) m = −∞ τ − m a − N f m x (1 − r ) N f m (cid:73) d µ U ( N c − N c − (cid:89) a =1 N f (cid:89) i =1 − z a u i ax r ) (1 − z − a ˜ u i ax r )= g ( x, a, u, ˜ u ; M U ( N c ) ,N f ) + τ − a − N f x (1 − r ) N f − N c +1 − τ − a − N f x (1 − r ) N f − N c +1 × g ( x, a, u, ˜ u ; M U ( N c − ,N f )= g ( M U ( N c ) ,N f ) + g ( M − U ( N c ) ,N f ) − g ( M U ( N c ) ,N f ∩ M − U ( N c ) ,N f ) , (4.32)– 36 –here M U ( N c ) ,N f ∩ M − U ( N c ) ,N f = M U ( N c − ,N f . (4.33)As we observed in section 4.3.1, M U ( N c ) ,N f / (cid:10) M ji = 0 (cid:11) is the same as component M + − U ( N c ) ,N f for a U (2) theory. Thus, for a U (2) theory, we can completely recover theHilbert series for each of the four components of the moduli space from those of thefour subspaces we have examined. Superconformal Index and Hilbert Series. In contrast to the N c = 1 , M + − U ( N c ) ,N f for N c ≥ U ( N c ) theory with N c ≥ M + − U ( N c ) ,N f could cancelwith the contribution of another fermionic operator. In that case any analytic manip-ulation of the superconformal index, for example taking a limit of the index, cannottrace the contribution of that chiral ring element. Let us consider an example. If weconsider the U (3) theory with five flavors, there is a chiral ring element of the form v + v − M ( j ( i M j i M j i M j i M j ) i ) , which has E + j = 6 and transforms in the representation[0 , , , × [5 , , , 0] of SU (5) × SU (5) whose dimension is given by 126 = 15876, andmost crucially has charges A = 0, T = 0. These charges make it easy to identify manynon-zero spin operators. Most of them are fermionic such that their contributions comewith a negative sign and could cancel the contributions of v + v − M ( j ( i M j i M j i M j i M j ) i ) .For example, the index contributions of v + v − M ( j ( i M j i M j i M j i M j ) i ) contain the follow-ing terms: . . . + u x + u u x + . . . . (4.34)On the other hand, the index contributions of the nonzero spin states contain . . . − u u x + . . . , (4.35)which comes from ( Q ψ † Q )( Q ψ † Q )( Q ψ † Q ). That contribution cancels out the term u u x in (4.34). On the other hand, the other term u x in (4.34) does not ap-pear in the contributions of the nonzero spin states. Therefore, the cancelation of u u x is accidental. In fact there are many cancelations between the contributions of v + v − M ( j ( i M j i M j i M j i M j ) i ) and those of the nonzero spin states. Because of these can-cellations, taking the limit of the index does not capture the presence of this operatorin the chiral ring. – 37 – cknowledgements A. H., J. P. and R.-K. S. gratefully acknowledge hospitality at the Simons Centerfor Geometry and Physics, Stony Brook University where some of the research forthis paper was performed. A. H. is grateful for the hospitality of the Korea Institutefor Advanced Study in Seoul and acknowledges private communication and invaluablediscussions with Stefano Cremonesi. He is also grateful for discussions with AlbertoZaffaroni and Noppadol Mekareeya. A. H. and R.-K. S. are grateful for the hospitalityof the ICMS in Edinburgh. H. 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