JHEP02(2017)037 Springer )] February 1, 2017 February 7, 2017 N : : November 28, 2016 : [SU( )] theory. We compute 10.1007/JHEP02(2017)037 Accepted Published N T Received doi: [SU( T = 4 Published for SISSA by N d a,b [email protected] , . 3 and Hirotaka Hayashi 1609.08034 1 The Authors. a, c D-branes, Extended Supersymmetry, Field Theories in Lower Dimensions

We consider mixed branches of 3 , [email protected] La Caixa-Severo Ochoa Scholar. 1 Universidad Aut´onomade Madrid, Cantoblanco, 28049Tokai Madrid, University, Spain 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan E-mail: Departamento and de Te´orica Instituto F´ısica UAM-CSIC, de Te´orica F´ısica b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: understood directly from thecharges type of the IIB monopoles brane andthe picture brane Higgs by position branch a moduli. part relation Wecases of between also show a apply the complete the given magnetic agreement restriction mixed with rule the branch to results by calculated exploiting by 3d different mirror methods. symmetry. Both Abstract: the Hilbert series ofacting the Coulomb on branch the part Hilbert ofcharge the series summation mixed of only branch the from tothe full a the Coulomb restriction Coulomb subset rule branch branch of sublocus that BPS will where dressed truncate the the monopole mixed magnetic operators branch that stems. arise in This restriction can be Federico Carta Hilbert Series and Mixedtheories Branches of JHEP02(2017)037 ]). Generically, 2 , 1 3 ]. Although the Higgs branch 6 – 3 14 17 9 = 4 theories 15 8 N 19 25 28 29 – 1 – 2] 1] 2] , , , )] theory 5 21 N = [2 = [3 = [3 32 34 36 7 = 4 mirror symmetry [ ρ ρ ρ 2] 1] 2] 21 , , , 37 N [SU( [SU(3)] T 32 25 = [2 = [3 = [3 T 17 ρ ρ ρ 1] of , 1 5.2.3 The mixed branch 5.2.1 The5.2.2 mixed branch The mixed branch 6.3 The mixed branch 6.1 The mixed6.2 branch The mixed branch 5.1 The case5.2 of [2 Other explicit checks 3.3 Hilbert series for the Higgs branch factor 4.1 The restriction4.2 rule The restriction rule with an example 2.2 Higgs branch moduli space 3.1 The brane3.2 realization of the Hilbert mixed series branch for the Coulomb branch factor 2.1 Coulomb branch moduli space = 4 supersymmetric field theories have a Coulomb branch and a Higgs branch, which are moduli space can beto quite determine different from the theN quantum classical moduli one. space Hence,exchanged exactly. with it each other is For by typically 3d example,is difficult classically three-dimensional exact, (3d) the Coulomb branch moduli space is not protected against quantum One of the importantand aspects structure of of supersymmetric theprovides quantum insight moduli field on theory space strongly-coupled are ofthe regimes the (for vacua classical geometry general and moduli reviews, the space see associated may [ chiral receive ring, quantum corrections which and often therefore the quantum 7 Conclusion and discussion 1 Introduction 6 Higgs branch examples 5 Coulomb branch examples 4 The restriction rule for the Hilbert series 3 Mixed branches of the Contents 1 Introduction 2 Hilbert series for moduli spaces of 3d JHEP02(2017)037 , ρ ]. )] As 33 14 N 1 (1.1) ]. [SU( )] theory 28 T – N The mixed 21 2 )] theory and [SU( N T [SU( = 4 T N = 2 gauge theories, non- = 4 N N ]. The Hilbert series (HS) is a 7 , ρ boxes, or equivalently a partition )] theory is also related to regular . ]. Mixed branch of the 3d ρ ]. In 3d N N × H stands for an origin. Similarly, we call the full 36 31 ρ , is the Higgs branch factor. – , we call the Coulomb branch as the full Coulomb C } = 4 supersymmetric field theories usually ] of the monopole operators as well as the } 0 = 4 theories, for example, in [ } × H 35 [SU( ρ ρ 0 { 29 0 N 20 T ∪ { H N – ×{ ρ = ]. – 2 – C 18 )] ] to construct the Coulomb branch chiral ring and , 37 = 4 supersymmetric gauge theories by using the N 11 ]. The Hilbert series computation becomes possible – theories [ ]. The full moduli space is then given by 14 N 8 , ]. The 17 )] can also play an important role to determine the [SU( 7 S – 18 37 T , N 12 = 4 supersymmetric gauge theories, one needs to take into , which was initiated in [ M theories [ ] can be generated and lift the classical Coulomb branch [ 35 , [SU( N S 32 18 T = 2 gauge theories in [ N = 2 gauge theories theories can be written by a summation of magnetic )] theory arises as the S-dual to a half-BPS boundary condition of an as done in [ N N N [SU( ] it was further pointed out that the restriction may be also applied to the is the Coulomb branch factor and Hilbert series approach T 30 ρ C The In [ The Hilbert series approach has been successfully extended to the calculation of the Recently there has been much progress in the systematic determination of the exact A different approach hasFor the taken special case where in the mixed [ branch is = 4 super Yang-Mills theory [ 1 2 ]. The quantum corrections restrict the values of the Coulomb branch moduli and the mixed branch of the 4d class its quantization. branch since the branchHiggs has branch maximal for dimension. the Here Higgs branch part of the case of of the integer where branch structure of the 3d and indeed find agreement with the results calculated fromN a different method. punctures of four-dimensional class theory may be classified by a Young tableaux with charges, which is restricted by the effects of superpotentials. computation of the Hilbertthis is series indeed of the direction mixed we taketo branched in the this of computation paper. 3d of Namely, the we Hilbert apply the series restriction of technique mixed branch of the 3d moduli space of 3d perturbative superpotentials [ 34 restriction in turn is relatedseries to of the the 3d restriction of magnetic charges. Therefore, the Hilbert by adopting the dimension formulafact [ that there isThen the a Hilbert unique series can BPS beThis bare expressed method as monopole has a operator summation been over for applied all possible to every magnetic various magnetic charges. 3d charge [ generating function counting chiral operators,global graded symmetry by group the of chargestheir the they relations, theory. carry and under hence It the we containsfor can the the reconstruct Coulomb information the branch moduli of of space 3d chiralaccount from operators monopole the and Hilbert operators, series. which(IR) can superconformal be field theory defined [ as a disorder operator in the infrared the exact Coulomb branch moduligives space insight of into 3d strongly-coupled dynamics of the theories. Coulomb branch moduli spaceso-called of 3d corrections. Those theories are strongly-coupled at low energies, and the understanding of JHEP02(2017)037 ρ )] is N D ×H ρ ρ ]. Here C [SU( 43 T – = 4 gauge )] theory by with the one 41 ] where N N , )] theory. We 2 N 36 27 , , [SU( T 35 26 [SU( , , the restriction rule to T 18 24 from the Hilbert series ] to a 3d we focus on the mixed [ 4 ρ 3 D C 40 ρ , , we also compare the Hilbert 39 'C 6 can be also calculated with the ρ ρ H )] theory by utilizing the restriction = 4 theories H N N we briefly review the main underlying , which yields a check of the restriction ρ as well as the 3d mirror symmetry. We 2 [SU( H 4 T ] by utilizing the 3d mirror symmetry. We , we give some examples of how this procedure 38 )] theory, by restricting the summation of the ] for the full Coulomb branch. This method was – 3 – 7 5 . We then use in section N and also give a speculative argumet for computing 2 7 [SU( ] by making use of a gauge theory whose full Coulomb T 23 , and also in [ 22 = 4 supersymmetric gauge theories, which was first developed ρ C N . We also give a way to compute the Hilbert series of the Higgs ρ )] theory. In section )] theory. We present a method of computing the Hilbert series of the N N by applying the technique developed in [ = 4 supersymmetric gauge theory. In section α [SU( N [SU( H T T d ] for the full Higgs branch and in [ )] theory. Hence, our method does not use the information of the IR gauge theory has been computed in [ ) from the restriction technique. Originally, the Hilbert series of the Coulomb branch N ρ 40 , C This paper is organized as follows. In section The main aim of this paper is to compute the Hilbert series of the mixed branch 1.1 39 [SU( in [ successfully tested in different contexts andfor already example produced the some computation interesting applications, ofwe the moduli recall spaces the of minimal instantonsmore in notions details. [ needed in the following, and we refer to the literature for rule and then gluing the different mixed branches altogether. 2 Hilbert series forIn moduli this spaces section, of we will 3d space of briefly three-dimensional review the Hilbert Series technique for studying the moduli works, and we perform someseries consistency of checks. the In Higgs section branchobtained moduli by space using computed the by the restrictionfinally method summarize rule our in in results section in section section the Hilbert series of the full moduli space of the using the techniques reviewed incompute section the Hilbert series oftheory the Coulomb from branch the part of Hilbertdescribe a how series mixed we branch of can of the obtain the branch the full of restriction the Coulomb rule branch from the of brane the picture realizing the mixed idea to the “HilbertHilbert Series series for Program”, the as full Coulomb wellLagrangian and as the 3 full the Higgs branch procedure ofbranch to the of moduli the compute space explicitly of a the Coulomb generic branch and the Higgs branch parts of the mixed branch of the dual to the partition branch factor theory whose full Higgs branchrule is as isomorphic well to as the 3d mirror symmetry. magnetic charges in thebe latter calculation. obtained directly In from fact,T we the argue brane that configurationbut the realizing restriction only the rule uses can mixed thebranch. branch brane The of Hilbert configuration the series asrestriction 3d of rule well the by as Higgs using the branch the factor Hilbert mirror symmetry series relation of the full Coulomb in ( part branch moduli space gives here compute the Hilbertof the series full of Coulomb the branch Coulomb of branch the part JHEP02(2017)037 H M = 3 (2.1) (2.2) d will be , where M C is parameterized C = 4 supersymmetric N , d i k i under the symmetry which x n N =1 i Y n . In a 3 of holomorphic functions defined k via usual techniques in algebraic ··· M , O 2 M , ,k n 1 t k is defined as a set of gauge-inequivalent n a a M N and a Coulomb branch factor k n X X H , a very fruitful approach is to count all the ··· – 4 – is the number of chiral operators having respec- ) = 2 M symmetries. t k in the Taylor expansion ( X N k global symmetries. For each one of them, we choose N n 1 a ··· k , HS X 2 N ,k 1 = 4 supersymmetric theories the geometry of k ) = i a ) is the main tool used for this counting purpose. It is a N t . This can be refined to the case in which one whishes to t, x ( under the t ( is labeled by vacuum expectation values (vevs) of a set of scalar N HS k will be defined as a scheme locally isomorphic to the spectrum of HS of a supersymmetric field theory is defined as a ring of chiral opera- M . Now, if the ring of holomorphic functions on an unknown algebraic ··· M R , R 2 as a grading parameter. Then the Hilbert series will be given by , k 1 k ,N is known, one can reconstruct and define , with Zariski’s topology. While in principle this strategy will work, it is in ··· R , . In particular, we can associate a holomorphic function on the moduli space M = 1 M The algorithmic procedure to compute the Hilbert series from the data defining The Hilbert series The chiral ring In order to study the geometry of For a generic field theory, the moduli space = 4 gauge theories varies, depending on the fact that we want to compute the Hilbert , i i and the interpretation istively that charges N grade the chiral operatorschiral operators by are more charged than under x one symmetry. For example, suppose that the ring. In more details, the coefficient will be equal to theis number of weighted chiral by operators having a charge fugacity approach of simply counting chiralsymmetries operators, that grading the them theory by undergeneral their study much charges enjoys. simpler under than This all is computing the a the well chiral defined ring problem exaclty. generating which function is that in keeps track, in a systematic way, of all the operators of the chiral variety geometry. Namely, the ring general hard to explicitlythe determine all holomorphic the functions elements on of the the chiral moduli ring space), (or so equivalently all one settles down to a more modest strategy to employ. tors. This ring is believedover to be isomorphic toto the every ring element in is parameterized by theby vev of the the vev scalarsHyperK¨ahlervarieties. of in the the hypermultiplets, scalars and in the vectorgauge-invariant multiplets. chiral operators, Moreover,global grading both symmetries them moduli of by spaces the their are theory. charges under In the all following the we different briefly review why this is a good vacua. Each point of fields of the theory, which thereforetheory, give the coordinates scalar on fieldsIt belong is either also to knownlocally the that vector a for multiplets product 3d or of to a the Higgs hypermultiplets. branch factor JHEP02(2017)037 . x (2.4) (2.5) (2.3) ) at the G ) is defined 7→ x ( ]. In this article, m symmetry is euqal 18 V is a HyperK¨ahler R C ], i.e. disorder oper- . Therefore we will C 17 – , | 12 ) m ( . are the weights of the matter i ρ G i | , defined as the flux of the gauge ρ i of m ρ surrounding the insertion point X r dϕ, 2 . is the gauge connection on the north- ], and satisfies a Dirac quantization ) i S θ G X  -charge. Then the conformal dimension 44 is the rank of the gauge group, and all [ 1 2 A R r cos G + ) = 1 − L | ) 1 )] theory, which we will define later. m  N – 5 – πim ( where ( α = 4 supersymmetric gauge theory is characterized r 2 | m ]. [SU( + or the Coulomb branch factor T N 15 ∼ ∆ exp (2 ∈ d X H  ) are used and α A 3 − ] of a 3 20 r, θ, ϕ – ) = C ’t Hooft operators. A bare monopole operator m 18 d , ∆( 14 , 7 . Namely x ] 45 are the positive roots of the gauge algebra, and is the magnetic charge of the monopole operator, which takes values in the weight α m The relevant operators in the chiral ring of the full Coulomb branch are however not A bare monopole operator carries a magnetic charge Unlike the Higgs branch, the Coulomb branch receives quantum corrections. An ap- The dimension formula for a monopole operator will be valid when the UV U(1) is broken to the maximal torus U(1) 3 turn on a vevwithout for spoiling a the complex BPS condition scalar [ in the adjoint representationto of the IR the superconformal vectorwe R-symmetry. focus multiplet, Those on theories a are particular called good “good” theory or called “ugly” in [ where representations. bare monopole operators, but dressed monopole operators. Indeed, it is also possible to has a conformal dimension, determinedof by a its BPS IR bare monopoledimension operator formula is [ given in terms of the magnetic charge by the following lattice of the Langlandscondition [ (GNO) dual group field through a sphere surrounding the insertion point of the monopole operator. It also where spherical coordinates ( ern (respectively southern) hemisphereHere of a sphere proach to the study ofators, this analogous branch to employs the monopole 4 as operators a [ boundary condition inconnections the onto which Euclidean the path path integral,tions integral by having is requiring a performed that Dirac will the monopole’s beinsertion set restricted singularity point of to (specified gauge a by set an of embedding connec- U(1) vev of the dualG photons. On athe generic W-bosons and vacuum charged of mattervariety the fields of Coulomb get the massive. branch quaternion the dimension The geometry equal gauge of to group the rank split the discussion in two. 2.1 Coulomb branch moduliThe (full) space Coulomb branch by giving nonzero vev to the triplet of scalars in the vector multiplets, and also by the series for the Higgs branch factor JHEP02(2017)037 , . m m H H (2.7) of factors, and i symmetry. The which commutes G n J . i G 1 Q = 30 , Casimir operators of G r 24 ) counts the gauge invariant ) (2.6) , 18 , , l 20 2 t, m m, t , 14 ( ( , , − G 18 G ) 12 + 1 l l l , , 12 P P m 2 2 2 , ) 9 ( i , l , , , , 14 . Then one can consistently turn on m ) is a correction factor taking care of d , 8 10 12 1 t m , , , ∆( ··· ··· ··· ··· 6 8 12 8 t − , , , , , , , , m, t G ( 1 3 4 4 4 5 6 8 6 6 L , , , , , , , , , G Degrees 2 2 2 2 2 2 2 2 2 r W P =1 i Y / consists of a product ) – 6 – , defined as the subgroup of G g G X L G ) = Γ( ⊂ ∈ t, m m ) are all the degrees of the m . The explicit expression is given by ( ] . Now, ) is included due to the following reason. When the m G H m ( 46 i P [ ) = H t d m, t 1 4 3 ( 2 ( G L G ≥ ≥ ≥ ≥ ]. The Hilbert series with this latter fugacities included, called P HS and W 47 m , l , l , l , l l l 4 2 6 7 8 l l H Simple Lie Algebra a b c d e e e f g . Degrees of the Casimir invariants of the simple Lie algebras. runs over all the lattice points of a Weyl chamber, i.e. over the weight symmetry is a symmetry which induces in the semiclassical picture the m J which keep track of charges under the 3d topological U(1) Table 1 ) of the Langlands (GNO) dual group of the gauge group modded out by the i G z L is a fugacity keeping track of the conformal dimension of the monopoles. The is the rank of t r ]. However, there are still different ways in which it can be dressed. Given this, the In the case in which the gauge group In details, the factor In order to count those operators grading them by their conformal dimension, it is 14 [ some of them arefugacities not simply connected, onetopological can U(1) further refine thisshift counting of by the including dual photon [ where As a reference, the degrees of the Casimir operators are given in table generically broken to a subgroup with the magnetic fluxa with vev the for magnetic awithout charge complex spoiling scalar the in BPS conditions theoperators for adjoint the of representation monopole. the of residual this group residual gauge group lattice Γ( action of the Weylthe group different dressings. vev of a bare monopole operator is turned on in the background, the gauge group is where magnetic charge crucial that there ism exactly one bare BPSHilbert series monopole is operator defined for as every magnetic charge JHEP02(2017)037 , i X (2.9) (2.8) topo- is the (2.10) ]: ) such n J h to avoid 40 N , x ˜ , t )) fugacities 39 ··· f , G 1 -th U(1) x i ( f = 4 hypermultiplet . )) is the character of x N ! ( , ) 0 i ) -th chiral multiplet i k n R x ) (resp. rk( . We introduced m, t g , 1 2 ( ··· k G # G , ˜ t k 1 ) P ) x = 4 hypermultiplets. x ) of the ( ( 0 i m 0 i ( = 4 supersymmetric gauge theory f i R R N J i -term conditions need to be satisfied ) (resp. char z N =1 appears since a ∞ k X F w d h n ( =1 is the number of non-simply connected

i Y i )char N ) R (resp. w n ( m i i R R ∆( t G – 7 – , and L )] := exp char n n W h / N ) =1 , x i ··· X 2 G , " X L is again a fugacity counting the conformal dimension of ··· Γ( t , ∈ = 1 PE 1 m x i ( f ) = where , is then given by t ( 1 2 = 2 chiral multiplets. Here PE is the Plethysitic exponential, a PE[ t ) is a collective notation for all the rk( N HS = x ˜ t = 2 chiral multiplets belonging to 0) = 0 as N ··· (resp. , w (0 . ) represents the charge of the monopole operator under the f G m of relations arising from the fact that the ( i r is defined as J ˜ In this second step, one hasof to scalars take generated into account in the theN fact step that above the symmetric are products not independent, but subject to a number generating function for symmetrizations, definedthat for any function t some operators. Note thatthe a appearance scalar of in fractional 3d powersset has in dimension of the the expressions. Thenumber of sum hypermultiplets is and done 2 in overis front all made of the of two where of the gauge (resp. flavor)the group. gauge Also (resp. char flavor) representation To do this one computes Refined Hilbert Series The F-term prefactor. Generating all the possible symmetric products of the scalars in the chiral multiplets. In the Higgs branch, the relevant operators of the chiral ring are gauge invariant 2. 1. by the following three step procedure if a Lagrangian of the theory is known [ vacuum of the Higgsbranch, branch the the Higgs gauge branch is groupgeometry protected is can against completely be quantum corrections, studied broken. and in therefore Unlike the its the classical exact theory. Coulomb operators composed of hypermultiplets, subject to F-term conditions. One can count them factors of 2.2 Higgs branchThe moduli (full) space Higgs branchis of characterized the by moduli giving space nonzero of vev a to 3 the scalars in the hypermultiplets. On a generic where logical symmetry, where here now JHEP02(2017)037 = 4 )] = t ( (2.13) (2.11) (2.12) g ]) N 49 ) + t , ( -th relation, f i , ! k i α ˜ t k ) w z ( of the 0 i r =1 R Y k 00 i R − 4 , . 1 1 = 2. The F-term relations G )char −

i w # d + ( i i d ∆ ˜ t R ∈ Y ) α w r ( r char 00 i h w R dw ··· PE char (0) = 0. 1 g i 1 r ]). Y =1 N w i X dw ) ˜ 48 t " )] theory (0) = =1 – 8 – w, f r | , defined for any Lie group as (see e.g. [ N ) by a factor w | G . This implies that only gauge singlets give non- I Pfc( ij δ ) := PE 2.10 ˜ t G ··· [SU( dµ w, T =1 ) such that ) = 1 t G | ( w Z w g | ( Pfc( = 4 theories have many mixed branches where we can turn j I ¯ R r ) = ) N . = 2 notation) involves a trace on the flavor indices, and this trace ) and 1 t πi ˜ 6 t, z ( ( N f )char ) is the character of the gauge representation (2 w w ( ( = HS i 00 i R R is the set of positive roots of the Lie algebra of G is the Haar measure of + dµ G char )], for any is its degree in the conformal dimension: typically µ G t ( i Z g G d dµ )]PE[ where ∆ where zero contribution after theand 2 integration. over the Therefore, fullonly gauge integrating the group the will gauge discard result invariant all ones. of the stepIn gauge-variant operators, conclusion 1 and the keep Hilbert series of a Higgs branch is given by In order to countric products, only we the need gaugethe to invariant gauge project operators, singlets. all and thegauge This not representations is group.R that all done the of by PE integrating This the generates the symmet- indeed onto gauge works fugacities over since the from whole representation theory it holds that However, the F-term conditionCoulomb is branch moduli. automatically satisfiedextracted Characters easily since and from we degrees the dodo of superpotential, this not we the in turn will section classical on show relations detailed can examples be of how to and will not usually depend on(in flavour terms fugacities, of due the toalways the appears fact that also the inunder superpotential the a hypermultiplet F-term may equations. give rise One to might an F-term think equation that that the has variation flavor indices. one has to multiply equation ( where char by every vacuum of the Higgs branch. To enforce this fact in the counting procedure, t The Molien-Weyl projection, (see e.g. [ ( Here we have used the well known property of the Plethystic exponential that PE[ f 3. 4 So far we havetheories. focused on In a general, full 3d Coulomb branch andPE[ a full Higgs branch of 3d 3 Mixed branches of the JHEP02(2017)037 N (3.1) is the α )] theory, H -direction. 6 N x is parametrized [SU( by scalars in the N )-plane gives the α T 9 α C 9 H , x - x x x 8 , x 7 8 - x x x x 7 - x x x 1) gauge group. 6 - x x x − . Those two full branches intersect at N 5 - x x α x . In order to visualize the mixed branch ) flavor group. In other words, there are is the Coulomb branch factor and 1 } × H N 4 0 - × H α x x x ) is clearly not a disjoint union, as in general { = 4 supersymmetric gauge theories was first C )] theory. Here each circle node with a number α )-plane and in the ( = 4 theory. In this table “x” means that the brane – 9 – 5 C N 3.1 = 4 supersymmetric theory. 3 N - x x α , x x N [ 4 3 N are Hyperk¨ahlervarieties, where [SU( , x ) of the gauge group, and a line between two gauge nodes T 2 α - - - 3 k x x H denotes a SU( 1 - - - x and N = 4 theory in fact has the structure α -direction. Therefore, the worldvolume theory on the D3-branes C 0 2 - - - N 6 x )] theory, it is useful to engineer it with a brane construction in x N = 4 theory after the dimensional reduction along the D5 D3 NS5 N [SU( and a full Higgs branch is T } 1 denotes a factor U( 0 1 − ]. The configuration of the branes in the ten-dimensional spacetime is shown 5 N C × { . In this configuration, some D3-branes are suspended between NS5-branes, and . The quiver graph for the ··· SO(3) R-symmetry of the 3d labels the different mixed branches, , . The brane system realizing a 3d 2 × α = 1 In this paper, we are mainly interested in the mixed branches of the in table are of finite length in the is effectively a 3d The rotational symmetry inSO(3) the ( type IIB superstring theory and we will3.1 heavily make use of The it. braneThe realization of type the IIB mixed braneanalyzed branch in system [ yielding 3d different mixed branches intersectbranch with is one another.a With single this point, notation, where a typically full the theory Coulomb is awhich superconformal is field realized theory. bystructure the of linear the quiver of figure where Higgs branch factor. Both by the vev of scalarshypermultiplets. in the The vector union multiplets and in the equation dual ( photon and special locus of a fulland Coulomb there branch, open we may upgeneric turn some three-dimensional on vevs directions for in scalars a in hypermultiplets Higgs branch. Then, the full moduli space of a Table 2 is pointlike in that direction, while “-” means that iton is extended vevs in for that scalars direction. both in vector multiplets and hypermultiplets. For example, at some hypermultiplets in the fundamental representation of U( Figure 1 k, k stand for one hypermultipletrightmost in node the with bi-fundamental representation a of number the two gauge groups. The JHEP02(2017)037 . for are ⊗ 5 2 topo- , x N 1 1 − , x = 4 super N J 0 )] theory is x N )] theory, we N N , the vertical axis 6 [SU( x )] theory yield the T [SU( N T ) flavor symmetry, and + 1)-th NS5-brane k N [SU( T D3-branes between NS5-branes give -th and the ( ) by the effect of monopole operators. k k N NS5-branes in fact realize non-perturbative N )] theory. In this picture the directions – 10 – topological global symmetry. The U(1) N 1 − N J [SU( T on which the NS5-branes are stretched, and the “out of the page 9 ]. From the quiver description of the D3-branes between the , x 8 , on which the flavor D5-branes are stretched. Hence, horizontal lines k 18 5 D3-branes are attached only to the last NS5-brane. At the end ), and we call them “color D3-branes”. On the other hand, the , x k 7 , x 4 N x )] theory is self-mirror and the full Coulomb branch moduli space ]. The brane configuration which yields the 3d )] theory can be realized by using a brane system in type IIB string , x N 3 N 18 ) flavor symmetry, the x D5-branes in the brane configuration for the N [SU( 1, and . We have [SU( N T 2 − T ,N . The brane picture for the ··· ) global symmetry [ , One nice feature about the brane picture is that the Coulomb branch moduli space, the While the Also the N The order is counted from left to right. Namely, the leftmost NS5-branes is the first NS5-brane. D3-branes attached to the rightmost NS5-brane realize the SU( 5 = 1 Moreover, the is isomorphic to the full Higgs branch moduli space. Higgs branch moduli space and all the mixed branches can be pictorically understood from perturbative SU( SU( know that at least welogical have global the symmetry U(1) is in fact enhanced to SU( of each rightmost D3-brane, wewill may put be one useful D5-brane. for reading Therise introduction off to of the a the Higgs gauge D5-branes branch.N group The U( we call them “flavor D3-branes”. theory, which arises as theYang-Mills S-dual theory of [ a half-BPSgiven in boundary figure condition ofk a 4d Figure 2 suppressed, since they are sharedcorresponds to among directions all the branes.axis” corresponds The to horizontal axisand is vertical lines represent D3-branes and NS5-branes respectively. D5-branes are denoted by JHEP02(2017)037 3 )-directions, 9 , x 8 , x 7 [SU(3)]. x [SU(3)]. T )-directions are related to T 9 , x 8 , x 7 x [SU(3)] theory is shown in figure T – 11 – )] theory may arise when only a part of the positions N [SU( T ) global symmetry is associated to the Coulomb branch and )-directions. These latter degrees of freedom correspond to the 5 N , x ) flavor symmetry is associated to the Higgs branch. The full 4 . The brane picture for the full Higgs branch of N . The brane picture for the full Coulomb branch of , x 3 x Figure 4 Figure 3 When we tune the positions of the color D3-branes in the ( 6 respectively. A mixed branch of the 4 One the other hand, the positions of the flavor D3-branes in the ( 6 Coulomb branch and theand full Higgs branch of the of the color D3-branes are tuned.the mass At parameters some of the subloci fundamental hypermultiplets. of the full Coulomb branch moduli D5-branes in the ( moduli parametrizing the HiggsD3-branes branch. are In tuned particular, to whenthe all zero, non-perturbative the the SU( positions full ofthe Higgs the perturbative branch color SU( opens up. Due to this construction, brane motions. The D3-branesbranes. suspended These degrees between of NS5-branes freedom correspondtheory. can to the move Coulomb along branch moduli thethe of NS5- the flavor 3d gauge D3-branes may be fractionated between D5-branes and can move between the JHEP02(2017)037 . 7 )] , 1] N , , n N N (3.2) = ··· = [2 1] gives , i [SU( , ρ a . T ρ boxes for the = 1 Y ··· =1 i n i , i a 1 as P , ρ with the maximal 1]. = [1 and , ρ n a } × H = [2 0 ρ { ]. The correspondence goes 37 ≥ · · · ≥ , ] gives 2 a N 35 , . In particular, ≥ 18 = [ [ N 1 , ρ a ρ N × H ] means that the Young diagram has , and n ρ ], the full moduli space is given by C . ] with C – 12 – , a n 5 37 ρ , [ , a ··· , 2 35 , ··· , a , ), and can be classified by a Young diagram with 1 2 18 a N ( , a = [ 1 su a ρ boxes [ = [ . The brane picture for the mixed branch )] theory the mixed branch structure may be completely specified . Due to this correspondence, we will use a partition and the corresponding ρ N N , n ··· [SU( with , ]. Therefore, when some flavor D3-branes are put on one D5-brane, some T Figure 5 ρ 5 = 1 flavor D3-branes are put together on one D5-brane for each i i a is all the possible partitions of the integer [SU(3)] theory is shown in figure with the maximal Coulomb branch ρ T } 0 Since for the When some of the positions of the color D3-branes are fixed, some of the flavor D3- In terms of a Young diagram, 7 -th column for where C × { i Young diagram interchangeably and write the Young diagram associated to a partition by the partition branes may be fractionated betweentheory D5-branes and can hence a be mixedmixed branch realized. branch, of the one Note alsohypermultiplets, needs that An to example in tune of the order theof mass mixed to the branch parameters corresponding realize of to the the the remaining partition maximal fundamental Higgs branch of a brane can be suspendedsupersymmetry between [ an NS5-braneof and the a flavor D5-brane D3-branes in shoulddoes order connect not to to break some preserveCoulomb color the the branch D3-branes s-rule. moduli. so that In the configuration this way, the Young diagram classification can tune the boxes or equivalently aas partition follows. of the A integer means partition that Note that this restrictionthe does positions not of only color fix D3-branes. the This positions is of due flavor to D3-branes the s-rule but which also states fix that only one D3- space, a Higgs branchgiven opens by up. nilpotent In orbits fact, of the subloci where a Higgs branch opens up are JHEP02(2017)037 . ] 0 n N , b (3.6) (3.7) (3.8) (3.3) (3.4) (3.5) = i a ··· , should be 1 . =1 b n i 0 5 ρ of the mixed , yielding the P ] D ρ ρ 37 H can be computed , and ] which is dual to ρ 0 C n n 35 ]. For example, one , a n , b 18 , a D3-branes ending on the ) as ··· , , . Then it is possible to see in a different mixed branch k ··· 1 3.2 b 0 , b , n ≥ · · · ≥ ρ 1 , in a mixed branch specified by , = 2 C the dimension can be counted a 2 ρ
! ! a = [ ) ··· 2 i 2 i 1] H , 2 , b 0 = [ a ≥ [2 ρ 0 is given by the partition [ ρ 1 n =1 n H + 1 =1 i 0 = 1 X a i X . )] theory . The partition ρ 2 i , ρ − − , N D (2 2 ρ 2 D for ρ − = 4 theory implies [ N × H N ] with 2 k × C [SU(

n 3

D , then there are N 'C ρ ρ k T 1 2 ≥ 2 1 , a b C – 13 – ρ 1 2 H i ], a ρ H is ρ [ ··· ) = [ ) = , ρ k ) = ρ 37 , which is associated to the transpose of the Young 1 , of the C 1] H ρ a , ( ( 0 ≥ [2 H ρ H 35 D3-branes end on the rightmost NS5-brane. In general, , i C should be related to the number of D3-branes put on one ( a 0 n 18 H ρ dim satisfying dim i a dim is the entry of the partition is the entry of the partition is given by [ 0 may be given by ρ defined in this way is nothing but the dual partition satisfying , n , n ρ 0 i ρ a )] theory is self-mirror, a Higgs branch ) also implies that the dimension of the Higgs branch ··· )] theory is mapped to a Coulomb branch ··· N , . The dimension of the Coulomb branch moduli space , N 3.5 H = 1 = 1 . This property can be inferred from the brane configuration. In terms of the [SU( is the dual partition to ρ [SU( ). T is the number of ], which exchanges NS5-branes with D5-branes but keep D3-branes unchanged. Y , i , i T i D 5 i k + 1)-th NS5-brane. Therefore, we find that 3.5 b a ρ b k Due to this feature, one can write the full moduli space ( In fact, the mirror symmetry of the 3d − of the . For example regarding the Higgs branch factor N where ρ or The relation ( branch specified by that the partition claim ( NS5-brane. Suppose This means that forif example the number of ( where Since the ρ specified by a differentrelated partition to theHence, number in of the flavor original theory, D3-branes put on one D5-brane in the mirror picture. where diagram brane configuration, the mirrortheory symmetry [ is realized by the S-duality in type IIB string can check which agrees with the number of color D3-branes that are not frozen in figure from the associated partition [ where Higgs branch JHEP02(2017)037 is to to 1] 6 , 2 [2 )] the- H N in a mixed ρ [SU( , one obtains C ρ ρ T H . Since the mixed and the Higgs branch 2 ρ )] theory, which should C N )] theory should be the same . [SU( ρ N . From the brane configuration 6 T [SU(3)] theory. 7 1] , [SU( 1]. Then the dimension of )] theory. An example of the brane [2 ρ , T T N does not affect the Coulomb branch part ρ [SU( H ρ T – 14 – . ρ C [SU(3)] case after the Hanany-Witten transitions is given 1] , )] theory, we can use the method described in section [2 N T . The brane picture of the ]. Although it is non-trivial to read off the gauge theory con- [SU(3)] theory is shown in figure [SU( 1] ρ , by utilizing the method described in section 23 [2 T , . T ρ 22 5 Figure 6 = 4 theory which we call ), which agrees with the number of mobile D3-branes suspended between N 3.4 . To read off the gauge theory content we moved the two D5-branes in figure , the value of the vevs parameterizing )] theory [ ρ 7 N H , one can consider the Hilbert series of the Coulomb branch for the The brane picture of the Since the Coulomb branch moduli space of the ρ . In terms of the brane picture, we send the pieces of D3-branes between D5-branes to C [SU( . Hence, in particular we can consider infinitely large vevs for the scalars parameterizing ρ ρ ρ compute the Hilbert series ofcoincide with the the Coulomb branch Hilbert of series the for in figure the left and obtain antoher brane configuration in figure can move the D5-brane toannihilation the of left D3-branes until isgives no a due D3-branes hypermultiplet are to in attached the thecolor to Hanany-Witten fundamental D3-branes the transitions. representation in D5-brane. under the The the Then cellory gauge where description the group the of D5-brane given D5-brane the by is located. Once we obtain the gauge the- picture realizing the as ory. The Hilbert seriesT can be calculated bytent going from to the the original gauge brane theory picture description with of several the D3-branes on top of one D5-brane, one factor C H infinity. At low energiesa at different the 3d infinitely large vev of the Higgs branch 3.2 Hilbert series forIt the is Coulomb possible branch to factor branch compute specified the by Hilbert seriesbranch is for locally the given Coulomb by a branch product factor of the Coulomb branch factor by using the dualagain partition 2 from which ( isD5-branes the in same figure as [2 JHEP02(2017)037 ρ ) is and l (3.9) n (3.10) (3.11) = a i a of a mixed , ρ ) 1 , H ≥ · · · ≥ − 1 2 . The method in N − ρ a N C ≥ | = 1)] − ,N 1 i 1 a a (0) = 0 and #( ··· − , H [# ( N ( = 1 ] with U(1). Similarly, we will use a U n × , a . , k [1] )) ··· [SU(3)] after Hanany-Witten transitions , . k − · · · − − 1 1] )]. , a , n − )] [2 N k T ) N = [ ··· k ( U(1) − , ρ N − [SU( − | N ρ i − – 15 – = 1 a T = ( k i i ( U(1) a H U for − )) l k [# ( [1] − by utilizing the method for computing the full Higgs branch hypermultiplets in the fundamental representation of the . ρ N n 6 )] theory where ( N − · · · − − i a 1)] ( ) [SU( implies 1 ρ − n =1 i N T X | − N ] which is equal to − = n [1] is one hypermultiplet charged under the U(1) to which the line is = i k a i (1 − a N is given by the following linear quiver theory, U or ) is the Heaviside step function with the convention N the gauge theory description can be inferred as [# ( x . The brane picture for the IR theory ( = 7 − i H a In the next section, we will describe a different technique, namely the restriction pre- In general, the =1 n i and does not use the IR gauge3.3 theory of Hilbert seriesIt for is the also Higgs possible branchbranch to specified factor calculate by a the partition Hilbert series of the Higgs branch factor where the number of scription, to compute thefact Hilbert directly series of uses the the Coulomb brane branch picture factor realizing the mixed branch specified by a partition with gauge group to which thehypermultiplet in line the is bi-fundamental connected representation and of a theP line two between gauge two groups. gauge groups means a Here [1] connected. The otherfundamental representation line under between the thenotation gauge two where group U(1)’s [ U(1) denotes a hypermultiplet in the bi- Figure 7 compared with the one in figure in figure JHEP02(2017)037 N = (3.12) i a =1 n i P is isomorphic 1] , [2 and H n a )] theory. After knowing , ] N N ≥ · · · ≥ [ )] theory. The resulting brane 2 [SU( − N )] theory should be the same as a ρ ) is zero then we remove the gauge ˜ 1 T N ≥ k − [SU( 1 [SU(3)] obtained by decoupling all the N N ρ a 1] , ˜ [SU( N T [2 ρ ˜ ˜ T T U( is independent of the value of the Coulomb ] with to the gauge theory corresponding to the n ρ ). When 2 , a H – 16 – − · · · − 3.11 ) ··· 2 , 1 N a U( . Therefore, the Higgs branch factor )] theory is the one at the origin of the Coulomb branch = [ 8 − N ρ ) 1 of the mixed branch. Similarly, both the Hilbert series should N [SU( ρ 1 is given by ( ρ U( ˜ T H − . We can again make use of the locally product structure of the mixed . In particular, we can take infinitely large vevs for the Coulomb ,N ρ 2 C ··· )] theory. By moving to a generic point of the Coulomb branch moduli , N [SU(3)] theory, decoupling the Coulomb branch moduli yields the U(1) gauge = 1 T [SU( , k )] theory. The full Higgs branch of the . The brane picture for the IR theory of ρ k ˜ N T N In general, the IR theory at the infinitely large vev for the Coulomb branch part of For example, as for the Higgs branch of the mixed branch specified by the partition 1] of the [SU( , ρ ˜ where node as well as the line attached to it. to the full Higgs branch of the U(1) gauge theorythe with mixed 3 branch flavors. specifiedis by give by the following linear quiver theory, the Higgs branch factor be the same. [2 theory with 3 flavors as in figure configuration of the the of the space, one can readthe off gauge the theory gauge description, theory oneof content can of apply the the the Higgs technique forT branch computing introduced the Hilbert in series section branch. Namely, the Higgsbranch branch moduli factor of branch moduli. In termsD3-branes of the to brane infinity. picture, At wemoduli, send one low obtains the energies a non-fixed different at positions theory the which of we infinitely the call color large vev of the Coulomb branch Figure 8 unfrozen Coulomb branch moduli of the UV theory. described in section JHEP02(2017)037 (4.1) , one can ρ C . ρ M = 2 vector multiplet inside the N . For that, we made use of the gauge ρ in the r σ m 2 ∼ – 17 – σ ) at the insertion point of a monopole operator, is the radial coordinate. On the other hand, vevs ) means that the corresponding magnetic charges 2.3 r 4.1 ] )] obtained after certain Hanany-Witten transitions of N 15 [SU( ρ T , we described a way to compute the Hilbert series of the Coulomb branch 3.2 is the magnetic charge and m Hence, the restriction of the positions of the color D3-branes given by the partition At a point in the Coulomb branch factor of a mixed branch, we tune the positions of Due to the boundary condition ( = 4 vector multiplet satisfies [ can be translated into the condition that the corresponding magnetic charges are zero. simply insert the conditionfor the that full some Coulomb magneticfrom branch. charges which And color are the D3-branes zero restrictioncharges are of into to frozen. the the a Physically, magnetic subset Hilbert the chargesbranch corresponding restriction series can factor. to truncates be the BPS read magnetic monopole off operators that arise in the Coulomb are zero. Thenalso the have BPS to condition be ( zero. ρ Then, when one computes the Hilbert series of the Coulomb branch factor moduli and the positions ofhypermultiplets, the the flavor tuning D3-branes implies are thatparameters. the the Coulomb mass In branch parameters moduli order for are fundamental tothe equal flavor obtain to D3-branes the a are mass mixed alignedmasses branch, along to one we line. zero turn without Then, offpositions loss it the of is of mass the possible generality. parameters color to This D3-branes and set in or the all equivalently turn values the of means corresponding the that Coulomb branch the moduli value of the frozen therefore relate, in theof brane monopole picture, operators. color D3-brane positions with the magnetic charges some of the colorending D3-branes on so D5-brane. that they Since coincide the with positions the of positions the of color D3-branes flavor D3-branes are the Coulomb branch N where of the scalars in the vector multiplet are related to color D3-brane positions. We can the corresponding brane diagram.Hilbert series of However, the we Coulombbut argue branch directly factor that from without we the going brane are to configuration the able realizing gauge to the theory compute mixed description the branch the BPS condition implies that the real scalar 4.1 The restriction rule In section factor in the mixed branchtheory specified description by of a the partition In this section weseries develop for the the main result Coulombseries of branch of this part the full article. of Coulomb a branch,explain We by how mixed conjecture performing this that branch restriction a rule specific the can is restriction Hilbert be easily of understood obtained the in latter. from terms We the of also the Hilbert type IIB brane picture. 4 The restriction rule for the Hilbert series JHEP02(2017)037 , N , (4.5) (4.3) (4.4) (4.2) can be ··· defined , k ρ N associated associated R N

= ρ ρ ! i R R ) a =1 n i m, t ) are set to zero in ( , ) k with the P k   ) . U( k ij ) )) N P M j and k m 1 − x k n − =1 ( 1 − ∆( Y a k t N . U − =1 ρ Y ρ j P N

N R ( | t, ) ij 1 − magnetic charges which we set to m   − . i j N a G k D ≥ · · · ≥ 1 ( ρ P L i − N 2 , z N a H W D 1 ρ / -th cell. The rule is that we consider all HS m 1 are fugacities for the non-perturbative − =1 ) )) ≥ k i Y N k magnetic charges of U( X × − 1 G ···≥ × C ), ) 2 a L k × i − ρ N ), namely can be written by C is replaced with ,N 3.6 m N ρ t, z ) – 18 – ( ≥ ( = k 1 . Furthermore, by using the product structure of 3.11 ρ ··· M ρ − ρ N ] with , U( ), the total number of color D3-branes which are i n m P ( a a M M HS ( ) = 1 except for the case where there is no Coulomb branch 3.10 k ··· ··· n =1 is )] theory ( , i , k i ) = X U( ρ i 1 j R ρ z P N | a ) = , y i ). 22 k = [ m N [SU( t, z 4.2 ρ X ≥ ( T magnetic charges in the 21 ρ should be composed of non-frozen Coulomb branch moduli. There- m k M ( ) k -th cell is given by ( ρ k R -th cell, the factor | U( HS 1 k P X magnetic charges among the m of the mixed branch k ρ N C ). ) = i is always smaller than 4.3 t, z k ( ) topological symmetry associated to the Coulomb branch moduli. We will now define ρ in ( the possible choices which are compatibleto with remain the in condition for the the same magnetic Weyl charges chamberThe Γ( change of the factor The factor fore, in the Hence, the summations of ( N moduli. Then we havezero several among ways the to choose From the quiver theoryfrozen in in the ( The restriction on the magnetic charges. In this way, we propose that the restriction rule gives the Hilbert series for the Coulomb Let us label the cells between adjacent NS5-branes of the brane diagram as 1, 2, In more detail, our conjecture of the Hilbert series of a Coulomb branch moduli space N • • HS the Hilbert series for the mixed branch branch part the mixed branch of the starting from the leftmost cell.to a From the partition brane of picture, theread the type off restriction as map follows: where the summations are modifiedto in the a way frozen prescribed color bySU( D3-branes. a restriction map this map, and explain how it is determined by the partition of a mixed branch specified by JHEP02(2017)037 (4.6) (4.7) ) Lie 1 where N ) flavor 0 when 0 when ( − N > > = 4 gauge su ) ) ,N N k k is in fact the ··· − − , ij (5 (5 M = 1 − − , i 1 2 a a +1 i e which give charges for the − ) = 1 magnetic charge. ) = 2 magnetic charges. ) = 1 magnetic charge. , i z k k k i i e )] theory, the restriction rule , H i − − − +1 N j e +1 (5 (5 (5 j ) Lie algebra as e − − − − [SU( j N 1 associated to the Cartan generators 1 1 2 − ( T e | a a a j ) − e su | N ( ij δ ,N su ij C = ··· i = , – 19 – i +1 j = 1 e +1 . j is an element of the Cartan matrix of the N e − , i ) R i j x − N e ( | j = 2. Then, z i e su ij | n ) Lie algebra can be expressed as H x i C as ]. In that case, the restriction of the magnetic charges or the N H ( 8 31 1 are the fugacities associated to the perturbative SU( ), – su − N 29 in the later computation which we will perform. )] theory where a Higgs branch opens up. Furthermore, the restriction ) = 5 and 1 where 1. On the other hand, for the Hilbert series computation for the Higgs ,N N N 1 associated to the Cartan generators ( − N − is an element of a matrix yielding a linear combination of the Cartan su ij = 4, in the 4th cell we set to zero = 4, in the 4th cell we set to zero = 3, in the 3rd cell we set to zero − ··· , .N C [SU( ,N k k k ij are orthonormal bases in T = 2, the restriction appears from the 4th cell since = 3, the restriction appears from the 3rd cell since ,N M ··· = 1 ··· 2 1 , , ,N 4. Then, 3. Then, a a For For For ··· , j , j 1. 1. 2. ≥ ≥ ··· = 1 = 1 x , k k For For = 1 The similar restriction has been made use of for computing the Hilbert series of 3d Although we focus on mixed branches of the In the Hilbert series computation for the Coulomb branch, we will use the fugacities which give charges to the simple roots of the = 1 i, j The simple roots of the • • i, j = 2 gauge theories [ 2]. In this case 8 x i , , i , i i i e The algorithmic rule definedinvolved above is at quite first. straightforward toapplied Hence apply, to however let it determine us can[3 the seem give frozen now an magnetic explicit charges, example in of a how nontrivial the case rule of should the be partition superpotentials which lift arestriction part of of the the Coulomb Coulomb branchbranch branch of arises the moduli. since we In considerof the a the current magnetic sublocus case, charges of the can the be full understood Coulomb from4.2 the frozen D3-branes in The the restriction brane rule picture. with an example will be applicable totheories the which computation have of the mixed type branches IIB of brane more construction general withoutN 3d orientifolds. corresponding Coulomb branch moduli occurs due to the generation of non-perturbative for algebra. Due toCartan these matrix choices of the Cartan generators,, the matrix for branch, we will use theH fugacities symmetry. generators of the flavor symmetry group, depending on thez definition of the fugacities. simple roots of the SU( where JHEP02(2017)037 2] , (4.9) (4.8) ), we see that 4.8 for U(3) satisfying 33 , m 32 , m 31 m . 44 , m 33 ≥ Number of mobile branes. m Number of frozen branes Number of the cell 43 . ≥ k m N 32 . ≥ k − m N k – 20 – k. 42 ≥ = = = m 31 ≥ m , 41 44 m = = , where one color D3-brane is fixed in the 3rd cell and three = 0 = 0. 44 m 4 3 1 41 42 9 m 43 44 m m ≥ m m 43 = m 2] example, and the different numbers of frozen branes in every cell. = 0 43 31 32 , = 3 1 2 m m 33 m and they are subject to be in the the same Weyl chamber of the weight 42 m = m 44 42 = . The [3 , m 22 21 0 2 2 m 41 43 m m ≥ m , m 41 42 1 m 0 1 1 Figure 9 m 0= A similar reasoning works also for the magnetic charge that should be set to zero in the Now, in the 4th cell we have 4 magnetic charges in total. Let’s call the , m 1. 2. 41 3rd cell. In the 3rd cell there are three magnetic charges and we see that in this case we have three ways to put one of the magnetic charges to zero, account all the possible ways.there By are looking only at two the ways. Weyl chamber We condition can ( have m space of U(4), therefore they satisfy Among them we should choose three to vanish and the rule is that we must take into to zero in theThis 4th information cell, can and be only alsobranch, understood 1 as shown in magnetic in a charge the clear should figure color way D3-branes be from are the put frozen brane to picture in zero of the in the 4th the [3 cell. 3rd cell. Therefore, in this case, we see that a total of 2 + 1 = 3 magnetic charges must be put JHEP02(2017)037 (4.11) (4.12) (4.10) where 11 . k m and in fact the sum k [SU(3)] theory defined N of the magnetic charges T k − ≥ · · · ≥ magnetic charges are zero. N 2 different sets of magnetic , k +2 k . k × N N ··· + , = 0 i k ) to sum up all the possibilities of m . N k = 0 + ≥ i i m 4.11 m are the magnetic charges for U(2) which +1 k = N 22 + i ··· , m ≥ · · · ≥ m 2 21 = . The brane picture is given in figure – 21 – ≥ m m 0 +2 10 i ≥ = 0 is repeated both in the first and the second way > m 1 i 34 m = + 1 possibilities of which m m k +1 ≥ [SU(3)] = i N 1 T m − 33 − , , i k m 33 33 m and perform some explicit checks that our conjecture holds. = 0. m m = 1] of ρ , 33 , we can consider the following set ≥ ≥ 32 k m m 32 N . The Hilbert series for the full Coulomb branch of is given by the general ≥ · · · ≥ = 0 ≥ m = − 22 1 32 32 ≥ 31 m m , k m m m 31 ≥ , in order to explain the rather abstract rule that defines the restriction map in ≥ ≥ ··· m 4 , 21 31 31 m m m = 0 0= For the practical computation of the restriction of the magnetic charges, we can divide Therefore, in this example, we see there are in total 3 i ) with the magnetic charges satisfying the Weyl chamber condition is the magnetic charge for U(1) and 2. 3. 1. k 1 5.1 The caseTo of begin, [2 let us thinkby of the the easiest following possible linear case.m quiver We of consider the figure satisfy 5 Coulomb branch examples In this section we willin work section out explicitly some examplesterms of of the the general partition procedure outlined These sets are all disjointof between the each sets other exhausts for allpractical all the calculation elements one in can thethe use summation restriction the after of disjoint the the sets restriction. magnetic ( charges. Hence, in the with In a Coulomb branch part ofare a zero. mixed branch, Then, the there ruleFor says are that for the 4th cell. the possibilities ofcrucially setting avoid which the magnetic overcountingU( charges problem to outlined zero above. into Let disjoint us sets. consider a This gauge will node branch will split incharges six are different chosen. sub-sums, Inall depending of the on these restriction the conditions of andvalue way the for sum in Hilbert over the which all series, magnetic the ofwe one charge non-zero them. see has is that However, repeated, to to take it avoid oversumming, into should if account be some counted only once. For example, charges that need to be put to zero, and therefore the Hilbert series of the full Coulomb JHEP02(2017)037 , ) = 0, , t (5.1) (5.2) (5.3) (5.4) , 22 22 ) | ] we must , m 22 5 [3]. > m 21 m | − m 21 ( m + 3 | U(2) . , U(2) 21 P 22 − 22 ) m | m , t 1 > m = + 3 m ( | 21 21 1 m m m U(1) P − for for 22 3 22 m , [SU(3)] theory. , m + t | ) T 21 2 1 t − + m 2 and the other is to set | 1 z − 1 1 , 1]. In order to satisfy the s-rule [ 12 m m 1 2 1 , z )(1 ) ) = ) t t − 2 [SU(3)] theory yielding U(1) 21 22 , t 22 1 1 − − T 21 m m – 22 – = [2 ,m m m shows how the brane system looks for the mixed (1 (1 ρ ( 21 | ( ,m        1 2 12 1 U(1) m + P | ) reads ) = ∆( 1 1 t . The quiver graph for 22 , t 2.5 m 22 m 22 m , as shown in figure − ≥ , m X 22 21 21 21 m m m m ( Figure 10 ≥ −| −∞ ∞ = X 1 U(2) 1]. . The brane picture for the = 0 P ) = m , 22 21 ) = m ) = [2 , m 2 ρ 21 , z 2.8 1 Figure 11 , m 1 t, z ( Now we should point out that there are two ways to set one of the two gauge branes to Then we focus on the mixed branch m HS ∆( Computationally, this is implementedto by the setting position to ofbranch that zero of brane. the magnetic Figure charge associated zero: one is set the position ofof one the of mass the parameters, two and D3-branes therefore in equal the to second the cell position to of be one exactly of equal the to flavor D5-brane. one and the classical factors are and where the dimension formula ( formula ( JHEP02(2017)037 ) in 5.1 = 0 once ) in which 22 5.1 m = 21 m . U(2) 1]. In this first case the restriction P , also restricts the classical factors, = [2 1] from ρ , [2 R U(1) P 1]. In this second case the restriction amounts to , – 23 – = [2 , the map ρ 4.1 = 0 = 0 21 22 . m 22 m 21 22 m . The physical intuition for this fact is that since one of the two m m plus the Hilbert series of the full Coulomb branch ( ≥ . Both these cases are allowed and we should sum over both of U(1) P 22 = 0 13 1 m 1 21 m m ≥ with m 1] will be given by the Hilbert series of the full Coulomb branch ( = 0. , 22 = 0. In this addition, we only count the magnetic charge = 0 U(2) . Another brane picture for . The brane picture realizing the mixed branch = [2 P 22 > m 21 ρ m 21 > m m By the second rule in section 21 and there is no overcounting. replacing branes is frozen tothe a classical specific dressing position, factor the will residual be gauge reduced group to becomes U(1). Therefore, as shown inthem. figure With thisbranch we mean that thewhich Hilbert series for them Coulomb branch part of mixed Figure 13 take Figure 12 amounts to take JHEP02(2017)037 9 (5.6) (5.8) (5.9) (5.5) (5.7) (5.10) , ) , t ) , we get 21 . t , t ) m 22 (2). By the basis ( , t 2 su m ( n are Dynkin labels ( . U(1) 2 2 P − ) U(1) 0). , n x U(1) 1 , P , t ) P [1] linear quiver theory. n 1 . On the other hand, by the 21 ) 1 , t − m − , , t 1 ( + 1 + z 1 , m ) | 1] mixed branch is therefore 2 m 1 )] theory. To do so we first n 2 , ( x ( , , m ) ) N n U(1) U(1) | = ] where P ) is written by the characters of 10 10 + 1 + 2 U(1) x 2 t t U(1) − + z ( ( P m | 2 P , n 5.7 [SU( 2 1 2 z . = O O 1 1] 1 n k 2 n m , 2 ) = ∆( n z t z z + + m 1 [2 z U(1) 1 ] 21 1 − z k k n T 1 ) t t m 2 1 − z z z z 21 k, k ) , m n ] ] [ ) | 2 1 ,m 22 ,n =0 ∞ 1 m + 1 X k k, k k, k ( | n ,m m [ [ 2 ( 1 1 2 – 24 – we use a Dynkin label notation for the characters of a ∆( n 9 9 ) = m =0 =0 | t ∆ ( k k X X 6 ( t 1 z, t 0 ∆ 1 2 ( t −∞ > ) = ) = ∞ and 0 = 21 X HS ) and ∆ 2 ≤ 5 ) = n ,m z, t z, t 22 22 2 ( ( , yielding the [1] X ), the character is given by [2] =0 ,m , n −∞ 7 , m X 1 22 ∞ 4.7 0 = HS HS =0 X n 1 , m Since the Hilbert series of ( 21 1 ), the character is given by [2] n m ∆( m 4.6 −∞ 10 ∞ = ) = X 1 2 −∞ ∞ = m X , z 1 1 + ) = ∆( m 22 t, z ( is the character of the representation [ ) = are the magnetic charges of the two U(1)’s. The Hilbert series of the full z , m 2 ] 1 2 2 HS , z m 1 ( , n , n (3) Lie algebra. 1 1 1 n t, z n su (3) Lie algebra, it implies that the topological symmetry is enhanced to SU(3). ( We now want to check that the restriction rule indeed works. We compare the Hilbert In detail, by performing this truncation of the sum at the ninth order in The Hilbert series for the Coulomb branch part of the [2 su Here and everywhere else in section This computation, writing the HS in terms of characters, was performed explicitly only up to order 9 HS 10 . However, the obtained result strongly implies that the same structure will continue to higher orders. 9 . representation of a Lieof algebra. the For Cartan example, generators [2]basis in means of ( the the character Cartan of generators the in adjoint ( of t Therefore we conjecture that the Hilbert series is given by Performing this computation explicitly gives us For this theory the monopole dimension is where Coulomb branch is given by series that we obtained byHilbert restricting series the of summation the overgo the full to magnetic the Coulomb charges, IR branch withbranch. theory, of the effectively The the giving resulting infinite brane vevwas configuration to already after the obtained a scalars in sequence parameterizing figure of the Higgs Hanany-Witten transitions where [ of the the where ∆ given by JHEP02(2017)037 − are | 21 22 (5.11) . U(2) m ) | − , m − 33 21 m 32 | m m | + 4 + | | 32 22 | m | m 33 m − [SU(4)] theory given by + 4 | T − 31 4 31 m 32 | ). This matching was checked m | m = 4 + | 5.7 [SU(4)] theory. + 4 21 N | − | T | m 33 d 22 by more non-trivial examples. m − m 3 31 32 33 4 − 31 are the magnetic charges of the U(3) and − m m m m 31 | 33 33 m m + | , m | – 25 – . This theory can be also realized in terms of + 32 is the magnetic charge of the U(1), 11 | − | | 1 m 32 14 2 21 22 [SU(4)] theory yielding the linear quiver U(1) 21 , m 2] m , T m − m m 31 m − m 22 − m = [2 31 | 33 . The quiver graph for the ρ m where m + | 1 11 + 15 1 | − | | m m 22 22 − m m 21 Figure 14 − − are the magnetic charges of the U(4). The monopole dimension formula m | 21 32 44 ( . m 1 2 m t | , m + − | + 43 . The brane picture for the , m ) = 42 [4]. ~m − , m ∆ ( 41 the quiver diagram depictedthe brane in picture figure inthe figure magnetic charges of them U(2), for the full Coulomb branch reads: 5.2 Other explicitWe checks then exemplify the restriction rules in section 5.2.1 The mixedIn branch this example we start by considering now the 3 U(3) and we see that thisat exactly order matches 30 with in the equation ( Figure 15 JHEP02(2017)037 ≥ 2] 2]. , , and . By = 0 (5.13) (5.12) 22 = [2 = [2 32 1 m 4.1 ρ ρ . m m 1 ) z ≥ ) = 10 t , t 21 ( 31 4 33 t O m m ) + , m , or one can choose z 8 9 32 t t 0] , 16 ) ) 4 , m z z ) , 31 33 0] 1] , , m ,m 8 8 + [0 ) satisfying ( , , 32 z 33 6 7 ,m t t 2] U(3) , + [0 + [1 31 , m ) ) 2 P z z z z ) , ,m 32 2] 3] 0] 1] , t 22 , , , , . = 0, as in figure 22 , m ) 6 6 2 6 6 + [2 ,m t , , , , 33 31 z 21 32 5 ) , m m t 4] z m ,m + , 21 , m 1 + [2 + [3 + [0 + [1 ) 0] 0 32 = z m z z z z , 22 , m ( m 2 1] 4] 5] 2] 3] ∆( , 31 + , 2] mixed branch. In this subcase, , , , , t , m – 26 – , 4 4 4 4 4 31 m U(2) , 33 , , , , 21 + ([4 m P + [0 ( m 3 3 ) t = [2 z .m z ≥ ) 1 + [1 , t + [2 + [3 + [4 + [5 ρ ) 32 2] 1 22 z x z z z z , m m X m 0 m = 1] 3] 4] 5] 6] 7] ≥ , ( + = 0 , , , , , , 33 31 = ( 2 2 2 2 2 2 31 21 . m 32 , , , , , , m m m U(1) ( ~m 2 m + ([2 22 P z 17 says that we can set two magnetic charges to zero in the 3rd cell. t · · m + [1 + [3 + [4 + [5 + [6 + [7 ≥ z z z z z z z X 4.1 21 1] 3] 5] 6] 7] 8] 9] , , , , , , , m 21 22 0 0 0 0 0 0 0 , m m , , , , , , . −∞ ∞ = 33 X 1 + ([3 + ([5 + ([6 + ([7 + ([8 + ([9 m m = 0 as in figure ≥ 1 . The brane picture for the ) = 1 + [1 33 32 ) := m t , m m z, t z ( ( = ≥ One can now compute the Hilbert series for the Coulomb branch part of the Now we are interested in studying the mixed branch given by the partition The Hilbert series for the full Coulomb branch of this theory is H . 31 32 HS 33 mixed branch, by restrictingdoing the this full one summation finds in a the series way explained in section The first rule in section Furthermore, one sees again thatcharges there in are the third two different cell:m ways one to can set choose to zero two magnetic with the magnetic charges m Figure 16 m JHEP02(2017)037 = 32 (5.14) > m . ) | 31 22 m n | + 2 (4) Lie algebra, it | su 21 n | + 2 | 2 n − 1 22 n | + | 2 n − 2 2 2] mixed branch. In this subcase, , 21 – 27 – n are the magnetic charges of the two U(1)’s and | [SU(4)] theory, obtained by sending to infinity all the = [2 2 2] + , | [2 ρ , n 1 1 T n n − = 1 ) is written by the characters of the = 0 31 32 22 m 33 n m | 5.13 m ), and + 22 | 1 , n | n 21 22 21 22 − n m m , n 21 2 − n | , n ( 21 1 n 1 2 n −| + 1 . The brane picture for the . The quiver graph for the = ( m . For this theory, the dimension formula of the monopole operators is given by are the magnetic charges of the U(2). ) = ~n 22 ~n 18 Let us then compare this result with the one obtained from the IR theory after taking = 0. , n ∆( 33 21 where n the limit where theperforming Higgs some branch Hanany-Witten transitions, vevsfigure the of quiver all theory the at unfrozen the branes IR will become be infinite. given by After Since the Hilbert seriesimplies of that ( the topological symmetry is enhanced to SU(4). Figure 18 unfrozen Higgs branch vevs. m Figure 17 JHEP02(2017)037 ) 5.12 (5.16) (5.17) (5.15) 1] of the , , ) ) , t 10 t , = [3 22 ( [SU(4)] theory 4 ρ , n T t O 21 ) = 0) + z n 8 9 ( t t 33 0] , ) ) m 4 z z , 3 U(2) n = 1] is 3 0] 1] P , z , , ) ) 8 8 32 + [0 , , , t 22 z m 2 n 6 7 n + t t 2] ( 1] mixed branch. , + [0 + [1 21 , ) ) 2 n z z z z , ( 2 z 0] 1] 2] 3] U(1) 1 = 0 or = [3 , , , , P n 1 2 6 6 6 6 ρ + [2 ) z t , , , , 32 ) z , t 5 ) m 22 1 t 4] z , ,n n + [0 + [1 + [2 + [3 ) = 0] ( 0 21 z z z z z , , 2 ,n 31 1] 2] 3] 4] 5] , 2 , , , , , – 28 – U(1) m 4 ,n 4 4 4 4 P 1 , , , , , + ([4 n · + [0 3 1] t z ∆( , t + [1 + [2 + [3 + [4 + [5 ) 2] x z z z z z , 22 0 n 3] 4] 5] 6] 7] 1] , = [3 = 0 = 0 ≥ , , , , , , 33 ). X 2 2 2 2 2 2 = 0) and ( 21 ρ m 31 32 , , , , , , n m m 22 5.13 + ([2 t . The brane picture for the m −∞ + [3 + [4 + [5 + [6 + [7 + [1 ∞ z = z z z z z z X 2 ). 1] 5] 6] 7] 8] 9] 3] n , , , , , , , = 0 22 0 0 0 0 0 0 0 , , , , , , , m 21 −∞ = 0 or 5.12 ∞ = m X 1 21 Figure 19 n m + ([5 + ([6 + ([7 + ([8 + ([9 + ([3 ( ) = 1 ) = 1 + [1 m z, t ( z, t ( [SU(4)] theory. The Hilbert series for the full Coulomb branch of the The restriction of the magnetic charges corresponding to [3 By computing explicitly the Hilbert series in this case we find The Hilbert series for the IR theory is given by H HS T giving in total 4and possible sum choices. the four We have resulting to sub-sums apply obtained. each After one performing of such them a to restriction equation to the ( 5.2.2 The mixedAnother branch example is the Coulomb3d branch moduli part ofis the again mixed given branch by ( which exactly agrees with ( JHEP02(2017)037 , . ) 44 . t , t 4.1 3 , m (5.19) (5.20) (5.21) (5.18) n ( 43 , m U(1) ) is written 42 P ) 5.18 , m , t 2 41 n ( m , [SU(5)] theory. The ) T | 1 5 3 , the magnetic charges U(1) n | P 22 ) + , m , t | 1 2 , . 21 2] of the n ) ) n , ( m [SU(4)] theory. 10 10 − [SU(5)]. t t 1] 1 4 ( ( , . We denote the magnetic charge 3 T U(1) = [3 [3 n O O | P T ρ 21 3 + + + n 3 | k k z t t 1 2 z z [SU(4)] theory, as explained in section n n 2 ] ] z T 1 − , k , k n 1 0 0 1 3 2 z ) n | k, k, 3 – 29 – [ [ ,n + 2 9 9 =0 =0 | k k X X 2] ,n 1 1 , n n | ( . The quiver graph for ) = ) = ∆( 1 2 t ). This matching has been checked up to order 12 in 2 1 = [3 z, t z, t ( ( ρ −∞ and the magnetic charges of the U(5) by ) = 5.18 ∞ . The quiver graph for the = 3 X 3 [SU(5)] is given by figure (4) Lie algebra. HS HS 33 n T , n 2 su Figure 21 , m −∞ , n 32 ∞ = 1 X 1 1 2 n n , the magnetic charges of the U(2) by , m Figure 20 . The dimension formula of this latter IR theory will read 11 ∆( 31 are the magnetic charges of the three U(1)’s. −∞ m 20 m ∞ 3 = X 1 n , n 2 , n ) = 1 n z, t ( By computing this explicitly we get The Hilbert series for the IR theory is given by On the other hand, the IR theory at the infinitely large Higgs vev is given by the linear HS 5.2.3 The mixed branch As a final example now wequiver consider description the mixed of branch the of the U(1) by of the U(3) by which precisely agrees with ( where by the characters of the quiver of figure We also see the enhancement of the topological symmetry to SU(4) since ( Hilbert series of the full Coulombwe branch find of the the following Hilbert series: JHEP02(2017)037 , . . ) 22 ) | 22 10 ) m t , t 21 ( (5.24) (5.23) (5.22) 44 | | | | | 44 4 m O ,m ≥ t 42 44 32 31 33 ) − 43 , m + m m m m m z 8 9 21 43 ,m 32 t t 0] − − − − − , 42 ) ) = 0. In total, m m 2 z z , m | 41 43 41 43 44 , ,m 33 0] 1] 2 42 m m m m m + 41 , , , | | | m | 4 4 ,m , , , m + + + 22 4 4 33 | − | | − | | | | 41 , , + [0 m ,m 33 44 31 33 32 z m 6 7 − ( 32 t t 2] = 0 or m m m m m = 0. For any of these two + [0 + [1 , ) ) ,m ) satisfying 31 1 z z z z − − − − − 32 44 U(4) , 31 44 m 0] 1] 2] 3] 1 | m P 32 42 41 42 44 , , , , m , ,m ) 3 3 3 3 , m m m m m m + 22 , , , , , t | | | = . | (5) Lie algebra, and this implies [SU(5)] is given by 3 3 3 3 ) 43 ,m ) splits, and we must sum over all 33 , , , , + [2 = 0, + + + 21 2 T 43 44 su t 21 | − | | − | | | | z , m m m 5 ) 31 m , m t 33 43 22 32 31 4] ,m z + 5.23 + [0 + [1 + [2 + [3 42 1 , ) m − 32 = 43 m m m m m 0] 0 z z z z z m , , m 31 , m 1] 2] 3] 4] 5] , 1 − − − − − 42 0 , m ∆( . + , , , , , , , ) t 41 m 44 | 2 2 2 2 2 m 1 31 | 42 44 31 42 33 42 44 , , , , , , m 44 m 2 2 2 2 2 m m m m m m m + , m ≥ ( , , , , , | | | + | m + ([4 – 30 – | 43 33 ≥ 41 + [0 + + + 11 3 m t m | − | | − | | | | z U(3) = 0 or + [1 + [2 + [3 + [4 + [5 ( 43 4 ≥ , m m ) + 5 P X z 32 44 21 31 33 2] z z z z z z | 42 ) ) 32 m , 43 − m m m m m 4] 5] 6] 7] 1] 3] m 0 43 33 , t , , , , , , m , ≥ ≥ 22 , m m 1 1 1 1 1 1 − − − − − m 0 22 | 41 , , , , , , + , = m 31 42 1 1 1 1 1 1 | 33 42 43 31 41 m 32 , , , , , , , m m 42 + 5 m m m m m m + 33 , m 21 | | | | | + m ≥ m + ([2 22 m + [4 + [5 + [6 + [7 + [1 + [3 42 31 + + + ≥ t 11 ( = z | | | | − | | − | 41 z z z z z z m 32 m , m m ( 3 | 1] 22 22 43 33 32 6] 7] 8] 9] 3] 5] m m X z 41 , 21 , , , , , , ) U(2) − ≥ 0 m m m m m 0 0 0 0 0 0 m P 22 + 5 , 31 , , , , , , ) 21 , m | 0 − − − − − m 0 0 0 0 0 0 and m , , , , , , , , t + m 11 41 | 1 21 32 41 43 41 22 33 ( 21 m m m m | m m m m m m 1 2 m | | | ( ( 2 ≥ + ([6 + ([7 + ([8 + ([9 + ([3 + ([5 X z 21 +5 + + + + − | − | ≥ 1 = ( m U(1) m 1 32 P z ) = 1 + [1 ) = ) := ~m · · m t −∞ , ~m ∞ z, t = z X ≥ ( 1 By performing the summation over all these subcases we find the following By going to the mixed branch we wish to analyze, we have the brane picture in figure The Hilbert Series for the full Coulomb branch of this theory is ( ∆( m 31 HS HS The Hilbert series isthat written the by topological the symmetry characters is of enhanced to the SU(5). we find six different sub-casesof into them. which equation ( Hilbert series. We see that bycell, using and the 1 first magnetic charge rule4th of we cell the we have 3rd can to cell. have cases, set Again, we to there have are zero three choices different 3 in ways magnetic to the do 3rd charges so: cell, of namely in the the 4th where m The dimension formula for the full Coulomb branch of JHEP02(2017)037 . | = ) | 1 | 22 41 m 22 . For (5.26) (5.25) n m m | − ) − 23 + 22 , t | 22 m | 22 21 n | n | + , m + | 1 + | 21 | m 21 m 22 ( m − m | − 21 U(2) + m 22 P ) | | 1 n ( ) | 21 ,n 1 2 1 , t + m 22 | 22 + | ,n | [SU(5)] theory. 22 + 21 , n 21 2] | , ,n m n 21 [3 22 22 n − T n − ( ,m 21 − 22 21 is the magnetic charge for the leftmost 2 1 n 1 n 1 U(2) | n 1 4 ,m n P 1 | z + ) m ) |−| m | + 22 , t 21 | n ∆( 21 2] mixed branch, for the subcase in which 2 t , + m 21 n = 0. – 31 – m ( 22 21 n − n 33 − ) are the magnetic charges for the leftmost (resp. m ( 3 = [3 − 2 1 ≥ z 22 22 U(1) ) 21 1 X ρ 21 n P m n n 22 > m | | m ) m 32 22 − | + + , t + = = n 1 and m = 0 21 44 for the rightmost U(1). The Hilbert series for the IR theory ≥ n 41 42 m X ( m 43 1 ≥ 21 21 ( 2 m m ) = . The Quiver graph for the n m n z n 1 31 1 1 U(1) m , n m 1 −∞ z P ∞ 22 = · · X 1 = 0 31 32 and n , n (resp. m m 33 21 44 m Figure 23 22 −∞ m , n ∞ = m X 1 22 ≥ m 21 22 , m m m and = 0 . The brane picture for the 21 ) = 21 43 , m 1 m m z, t 1 ( m Now we would like to check this result by using the same procedure of the other = m 42 HS ∆( is given by where we assign magneticU(1), charges as follows: rightmost) U(2) group, and performing some Hanany-Witten transitions,this we linear find quiver theory, the the quiver dimension theory formula of is figure Figure 22 m examples. After going to the IR by giving infinitely large vev to the hypermultiplets and JHEP02(2017)037 . . ), ) ˜ q 2.2 10 q, t (5.27) ( 4 t O )] theory ) + z . N 8 9 t t t 0] , ) ) . For that we 2 z z , [SU( 0] 1] 2 , , T , 3.3 4 4 , , 4 4 = 2 quiver notation, , , + [0 2] branch. The brane z N , 6 7 t t 2] + [0 + [1 , ) ) 1 z z z z , = 2 chiral multiplets ( 0] 1] 2] 3] 1 , , , , . In our notation, if an arrow , 3 3 3 3 N . We first compute the Hilbert , , , , 3 . 3 3 3 3 , , , , + [2 G 2 16 t z 5 ) t 4] z + [0 + [1 + [2 + [3 , ) 0] 0 z z z z z , , 1] 2] 3] 4] 5] 1 0 , , , , , , , 2 2 2 2 2 1 , , , , , , 2 2 2 2 2 , , , , , + ([4 – 32 – + [0 3 , and by using the method described in section t z + [1 + [2 + [3 + [4 + [5 by using the method described in ) 2] z z z z z z , 2] 3.3 , 3] 4] 5] 6] 7] 1] 2] 0 [2 , , , , , , , , ). 1 1 1 1 1 1 0 H , , , , , , , = 4 vector multiplet. From the quiver, we can read the 1 1 1 1 1 1 = 2 chiral multiplet Φ in the adjoint representation of the , , , , , , 3.12 , then a chiral multiplet associated to the arrow is in the = [2 N N G ρ + ([2 + [3 + [4 + [5 + [6 + [7 + [1 t ). This matching has been checked up to order 9 in z z z z z z z by using the two methods. We will in fact find the complete 1] 5] 6] 7] 8] 9] 3] , , , , , , , 5.24 0 0 0 0 0 0 0 5.2 , = 4 hypermultiplet is split in two 3d , , , , , , 0 0 0 0 0 0 0 , , , , , , , N , we use yet another way to compute the Hilbert series of the Higgs branch + ([5 + ([6 + ([7 + ([8 + ([9 + ([3 we describe the matter in this IR theory, by using a 3d 4.1 ) = 1 + [1 24 z, t ( By decoupling all the unfrozen Coulomb branch moduli (i.e. sending to infinity the By explicitly computing the refined Hilbert series we get HS U(2), which lies insidecharge a assignment of 3d all thethe different gauge chiral and multiplets, flavor and symmetryis we then groups, pointing associate according toward fugacities to a to fundamental table representation group under the symmetry group mobile color D3-branes) we3rd see that cell, we and are 4In left flavor figure D3-branes. only with The 2 IRin frozen theory color which is D3-branes one therefore in 3d givenand the by we U(2) explicitly with write 4 a flavors. 3d In this case, wepicture are for interested this in mixedseries branch the of is Higgs the given branch Higgs already part branchmake in factor use of figure of the the [2 IR theory of ( considered in section agreement between the two resultsas which the give 3d a mirror nice symmetry. check for the restriction6.1 rule as well The mixed branch The Hilbert series forcan the Higgs be branch computed part by of goingare not any to frozen, mixed the as branch IR explained of in byIn section the decoupling section all the Coulombby branch using moduli the which restrictioncompute rule explicitly as the well Hilbert as series for the the 3d Higgs mirror branch symmetry. part of In all this the section mixed we branches will tion rule in equation ( 6 Higgs branch examples and we see this is in perfect agreement with the Hilbert series found directly by the restric- JHEP02(2017)037 = 2 (6.4) (6.1) (6.2) (6.3) N and an i = 2 chiral , N  ˜ t x 0] , 0 ), as explained in , . Φ are 3d [1 ) 2 2 2 ˜ q, 2.12 w , we notice that on the w j ¯ q, i [1] − 1 4 − 1 , (1 3 3 w 2 1 ,x ,x 2 2 2 − + f w ,x ,x dw ˜ t 
1 1 1 x 2 x x 1 1 ˜ t 1] 0] 1] [SU(4)] theory, as seen from the quiver w dw , , , , + 2] SU(4) ) , 0 0 0 2 j ¯ , , , [2 ˜ , and carrying both an index t =1 ˜ ˜ q q q | ˜ t ˜ 2 [0 T [1 [0 j ¯ 2 i 2 w w | w Φ i I [2] g q  [1] 2 2 2 1 w w w =1 | – 33 – w 1 PE  g [2] [1] [1] w | SU(2) = tr ( I 2 PE 2 W ) = · U(2) g ) ˜ t 1 ) 1 , 1 ˜ t − 1 πi 1 2 , w w 2 (2 w U(1) w = 4 hypermultiplet. = ˜ Pfc( q q Φ N also has the following superpotential in terms of the 3d Pfc( U(2) Φ 24 as dµ are zero. Out of this information we can derive the F-term prefactor U(2) j ¯ 2.2 Z i dµ form a 3d ˜ q Z ) is an index of the fundamental (resp. anti-fundamental) representation of j ¯ q, =
. . . The quiver graph for the U(2) gauge theory with 4 flavors. ˜ t, x (resp. . The charge assignment of the fields of the 24 . This splits into two independent equations: one in the adjoint and the other in the 2.2 i j ¯ Therefore the Hilbert series is given by The quiver in figure HS where we recall that the Haar measure for a U(2) gauge group is given by which we willsection need to multiply to the integrand of formula of ( trivial representation of the U(2).since the The vevs other for F-term Φ conditionsdescribed are in automatically section satisfied where the U(2), and the trace isfrom performed this on the superpotential flavor by indices.Higgs taking By a deriving branch the derivative there F-term with equations is respectindex one to relation Φ of order 2 in in figure notation, Table 3 Figure 24 multiplets and JHEP02(2017)037 . 1 2 w w 25 (6.6) (6.7) (6.8) (6.5) . Note t . We also 2 4 U(2) , ) , 3 ) → , t 3 ( t 2 ( O O + factor. In particular 1]. The brane picture for We then move on to the + , 2 2 t 2 2]. Therefore we can reuse 2 2 t ) ˜ , q q x ) 2]. z for SU(2) 0] , , 0] [SU(4)] theory. 3 , 2 , for the topological symmetry the 2 2 2] w , 2 , i x , [2 z 1 . ˜ T − 1 to the U(2) + [0 1]. Therefore we compute the Hilbert ! x and 2 , + [0 2 x 3 1 Φ 2 z 7→ 2] − . , w , where we use the restriction rule and the 2] 2] is in fact [2 2 2 2 , 1 2 0 , ˜ t 2 , 0 U(2) w − , 4.1 = , z

t → 1 , – 34 – . + ([2 − 2 = 2 + ([2 1 1 t x ˜ 2 q q 2 1 t x . We first compute the Hilbert series by the method 2.2 A z x 1] 1] 1] , M 19 , , 0 7→ (3) Lie algebra is given by 0 , 1 ). This matching has been checked up to order 7 in , z su = [3 6.5 . The quiver graph for the ρ 1 ) = 1 + [1 ) = 1 + [1 by performing a redefinition of the Cartan generators as explained x, t x t, z ( we can read what are the matter fields. We assign gauge fugacities ( Figure 25 ). By keeping track of the fugacities ). In this case, the relations are 25 HS HS 1 5.13 4.7 factor of the gauge group, and and hence we send to infinity all the unfrozen color D3-branes. Doing this, Φ 1 3.3 is related to ) and ( z From figure 4.6 as the Higgs branch partseries, of using the the techniques mixed in branch section of [3 to the U(1) will be the fugacity for the overall U(1) The next example is the Higgsthis branch mixed part branch of is the mixed givenin branch in [3 section figure we end up with a braneOur diagram claim yielding a is theory that given by the the Hilbert quiver depicted series in figure for the full Higgs branch of this theory is the same since the Cartan matrix of the 6.2 The mixed branch which exactly coincides with ( that in ( other method of the3d computation mirror in symmetry. section Thethe dual result of partition to ( result [2 was where we write the expression in termsMatching of with the dual Coulomb branch part of [2 By performing explicitly this residue computation we get the following result, JHEP02(2017)037 ,  ˜ t (6.9) x (6.14) (6.11) (6.12) (6.13) (6.10) 0] , 0 , [1 1]. Hence we 3 , , , w . ) 1 ) ) 3 , 3 2 3 t [1] t ( 1 ( w − 2 O O − w We then move on the + + 1] is [2 (1 + , 2 , 2 3 3 t 2 3 3 t ˜ t 3 ,x ,x ) x x ) 2 2 x w 1 z 1]. dw f 1] ,x ,x . − 2 , 1] , 1] 1 1 2 1 2 x 1 , , x x 0 1 1 1 1 0 , − , 0 w 1] 0] dw , , 7→ , , [0 
1 SU(4) 3 0 0 2 1 3 ˜ t , , w z w + [1 + [1 dw [SU(4)] theory, as seen from the quiver [0 [1 [1] x z 1] U(2) gauge group is given by + 2 , 2 1 =1 0] 0] [3 | 2 − 3 w 2 × , , ˜ t 3 ˜ T x 2 2 2 2 3 3 2 2 w 2 2 | + , w w w w w , w 1 x I ˜ t 1 3 2 [1] [1] [1] [1] [2] [2] − 1 w SU(2)  =1 + [0 + [0 | x 2 z x [1] – 35 – w 2 | U(2) 2 PE 2] 2] · 7→ 1 1 I , , w 1 1 2 2 ) − − 2 2 2 1 1 1 0 0 ˜ t w w . The dual to the partition [3 , , − 1 w w =1 , by the restriction rule. We do not repeat the process | U(1) ). This matching has been checked up to order 6 in 3 ) = 1 w ˜ t 1] w , z w , , | 4.1 1 1 + 3 1 , we can also read off the superpotential that in this case , I + ([2 + ([2 6.12 − 2 1 ˜ t w [2 1 1 t t 3 3 x − 1 1 1 1 1 Pfc( C z ) x 2 1 25 w w w x 1 1] 1] U(1) πi , Pfc( . , [1] to the flavor SU(4) group. We summarize the matter fields and 2 1 0 by performing a redefinition of the Cartan generators and it is 0 (2 7→ ˜ t , , 3 U(2) − 2 1 2 1 2 2 1 x 1 × ˜ ˜ q q q q . = w Φ Φ = z , x 1 2 t 4 w U(1)  , x U(2) 1 × dµ x PE ) = 1 + [1 ) = 1 + [1 · Z U(1) t, z x, t = ( dµ ( is related to .
˜ t Z z . The charge assignment of the fields in the HS 25 HS x, Therefore the Hilbert series is given by From the quiver in figure HS . Here which completely agrees witht ( given by Matching with the dualthe Coulomb mirror computation branch in part section compute of the [2 Hilbert seriesof of the computation and quote the result By performing explicitly this residue computation we get to the following result: where we again used where we recall that the Haar measure for a U(1) in figure assign fugacities their charges in table leads to F-term constraints generating a prefactor Table 4 JHEP02(2017)037 1 2]. w , (6.15) (6.16) [SU(5)] 2] , [3 ˜ T 5 4 4 ,x ,x 3 3 ,x ,x . 2 2 to the U(3) gauge group, f 1 ,x ,x 4 − 2 2 1 1 ˜  x x q q w 1 1 1 1 2 ˜ t 0] 1] , , SU(5) [SU(5)] theory. 0 0 ). and , , 2] + 2 . , 0 0 2 3 [3 , , ˜ t [SU(5)] theory, as seen from the quiver    ˜ 3.12 T 4 [1 [0 2] 1 , w , 2 . In particular we assign a fugacity ,w 2 3 [3 − 3 Φ 5 ˜ w 4 3 3 3 3 w T 1 0 1 2 2 ,w ,w ,w ,w ,w 1] − − 3 2 2 2 2 , w w w w w 1 1 2 [1 2 0  1] 0] 1] 1] 0] 2 − , , , , , SU(3) – 36 – , we use the quiver diagram of the [1 [1 [0 [0 [1    1 1 ˜ U(3) q q = 3.3 2 2 1 1 ) = PE 2] 1 1 2 2 ˜ t A − − 2 2 , 1 1 , w w (3), 4 w w M U(1) su , w = [3 3 1 . The quiver graph for the 1 w ρ 1 to the SU(5) flavor symmetry. 1 1 − 1 1 1 1 1 w 4 w by using the general result ( U(1) Pfc( , x 3 26 2 1 2 2 1 1 , x ˜ ˜ q q q q Φ Φ 2 Figure 26 , x 1 x 1 Φ . . The charge assignment of the fields of the 26 Furthermore, from the quiver we can write down the superpotential, and by writing the From the quiver, we can read the matter fields and their charges under the global and F-term equations we see thatwhich there are is singlets one under relation the inthe gauge the following adjoint group. of form In SU(3) particular and the two prefactor relations in this example takes theory given in figure gauge symmetry groups. Thisto is the reported in U(1) table factorand of fugacities the gauge group, fugacities 6.3 The mixed branch In this last example weFor are doing interested the in computation the in Higgs section branch part of the mixed branch [3 from the Cartan matrix of the Table 5 in figure JHEP02(2017)037 .  2 3 4 (6.20) (6.21) (6.22) (6.18) (6.19) (6.17) w w we find − . , ) 2 1 ) ˜ t 3 3 , t t  ( , ( = ˜ t 2 4   x O t O x 2 4 3 1 0] w + w + , − 3 )] theory from the 2 0 2 x t − t , 4 N 4 ) 0 ) 1 x , z 7→ w + dw We then again compare  ˜ t 0] 4 [1 0] ) 4 3 [SU( , 4 , 3 4 1 1 ,w T ,w w , w 3 , dw 3 3 1 , z 1]. 1 w w 2 , 1 , 2 , w 1] − 4 0] 2 = 4 w , , . dw x , − 2 3 [0 1 + [0 [1 + [0 1 2 N      x 1 (1 x z 1 w 1 w · − 2 dw − 2 1] 1 1] U(3) gauge group is given by − w , , x − ). This matching has been checked up 1 =1 0 0 | 1 0 1 2 × w + , , 4 7→ 0 0 ˜ t − − w | , + , 6.19 x 3 are I 1 0 0 1 2 ˜ t 1] 4 x − − , =1 · + [1 ,w + [1 0 | , z 1 2 3 3 , z ) x 2 0 0 0 1 – 37 – w 0 ˜ w t − | − 3 and , 2] 2] , 0] I , , 3 x 1]. The Hilbert series of the Coulomb branch part ,      [0 z 2 2 0 0 , 4 w , , x [1 2 =1 | = 0 0 1 1 ,w , 2 3 , , 4 − 1 − 2 w w | (5) Lie algebra is Pfc( A x w I 1] 1 su M , 7→ w + ([2 + ([2 =1  [0 | 2] is [2 t 2 t 1 U(2) 2 , z x w × w | 1] 1] PE I , , + · , z 0 0 4 1 U(1) , ) , − 2 0 0 1 dµ x , πi , 2 1 (2 x Z = = 7→ ) with the result by using the 3d mirror symmetry and the restriction rule. . The relations between
1 t z U(3) )] theory in two ways. One way is to use the technique by making use of the 6.19 ) = 1 + [1 ˜ ) = 1 + [1 t, x × N t, z t, x ( ( is given by U(1) HS [SU( 1] We also computed the Hilbert series of the Higgs branch part of a mixed branch of the With this information, we can write the Hilbert series, which in this case is dµ , HS T HS 2 , Z [2 branch precisely gives an explicit waythe to classical truncate dressing factor. the magnetic We charges confirmedthe as the restriction method well with as by the comparing to result the reduce obtained result by obtained the by technique after going3d to the IR gauge theory. 7 Conclusion and discussion In this paper wethe have determined Coulomb the branch restriction partHilbert rule of series for a of computing mixed the the branch full Hilbert of Coulomb series branch. the for In 3d particular, the brane realization of the mixed which again yields the perfectto agreement order with 4 ( in since the Cartan matrix of the the result ( The dual to theC partition [3 By performing this computation explicitly, and expanding to low order in Matching with the dual Coulomb branch part of [2 where we recall that the Haar measure for a U(1) JHEP02(2017)037 by , 1] (7.2) (7.1) ) , t 1 ( − ] N N [ [ H H . Along the 1] HS , 1 , we sum over a 0, which implies ~ − 1] N , ) + [ = , there are special 1 t R ( } − | ~m ) 1] ~m , N [ 1 , t { = 4 theories. C 1]. The restriction rule − } , N [ 1 . ~m N ) H { t − ( ( ] f N HS N [ 1] H × N, [ . R )    HS . The Hilbert series of the Coulomb }| , which also includes the origin. The 1] , t , ~m )] theory. The restriction rule indeed 1 1] 4 } { 0, and add the Hilbert series of the full ~ − ) + N, ~m N [ N [ P = { , t R ( R } | }| ~m f ) ~m [SU( )] theory is that it is given by a sum of a set ~m } { ) = T , t ~m { N ( ), t } t X { ( – 38 – f ( ~m 1] ] , } . In order to use the method, we consider an IR )] theory. In fact, the restriction procedure seems 1 N { 0 ~ [ [SU( ( − ) = (7.3) N H t ) T N f }\{ ( [ 2.2 X ] C , t ~m N } { HS [ [SU( R ), we remove ~m HS HS opens up and we denote the Hilbert series for T }| { ( ) = ~m 7.1 1] t ( f \{ N, 11 [ 1] 1] , , 1 1 X H as HS − − } N N [ [ ~m R R { }| }| X ~m ~m { }\{    ~m and we denote the subset by { + } ). Then the Hilbert series with the mixed branch might be t ~m ( { ) = 1] is the restriction map introduced in section , t )] theory from the restriction rule, it is interesting to consider the Hilbert series of 1 ( − 1] N , N 1 [ By taking the product of the Hilbert series of the two branches, we are able to compute Although our computation determines the Hilbert series of a mixed branch of the 3d − HS H We suppress the flavor fugacities for simplicity. N 11 [ [SU( R branch part can besublocus, written a by Higgs branch HS This guess will alsoexample, there lead is to a asays mixed that way branch for of specified computing incorporating by thesubset a Hilbert of another partition series mixed [ of branch the further. Coulomb branch For the origin of the Coulomb branchthe moduli origin. space, we have There theNamely is full from Higgs a the branch summation natural which ( shares guessHiggs branch to which implement we the denote by intersection to the Hilbert series. The restriction rule sayssub-summations where that a among Higgs branch the opens possible up. summation For example, of when T the full moduli spaceto of suggest the a 3d naturalCoulomb way branch to moduli obtain space it. ofof The the magnetic basic charges structure of the Hilbert series of the full of the restriction rule as well as the mirrorthe symmetry Hilbert of series the of 3d gives a a mixed systematic branch way oftwo to the full obtain branches. the series from the product of the Hilbert series of the gauge theory by decoupling the Coulombthe branch moduli. method and With the wereother IR able theory, way we is to can apply to compute utilizeCoulomb the branch the part Hilbert 3d of a series mirror mixed symmetry ofexactly branch. and gives the Intriguingly, the the the completely Higgs same restriction different branch computation result rule part. including for flavor computing The fugacities. the This provides a non-trivial check Molien-Weyl projection discussed in JHEP02(2017)037 - n ]. C Phys. , ] SPIRE IN ][ ]. , in the SPIRE ]. )] theory although the hep-th/9509066 ]. It would be certainly IN , June 4–29, Boulder, [ 18 N ][ 50 hep-th/9611063 -planes to the brane picture, [  [SU( = 4 supersymmetric theories. 3 T (1996) 1 O N d = 4 ]. (1997) 101 45BC N hep-th/9611230 [ SPIRE B 493 IN Mirror symmetry in three-dimensional gauge ][ ]. – 39 – )] theories constructed in [ (1997) 152 SPIRE N Lectures on supersymmetric gauge theories and Mirror symmetry in three-dimensional gauge theories IN [ Nucl. Phys. [Sp( , ), which permits any use, distribution and reproduction in B 492 T Type IIB superstrings, BPS monopoles and three-dimensional Nucl. Phys. Proc. Suppl. hep-th/9607207 , [ )] and N CC-BY 4.0 Nucl. Phys. , An unorthodox introduction to supersymmetric gauge theory hep-th/0309149 Strings, Branes and Extra Dimensions (TASI 2001) 0. Therefore, the restriction rule yields a natural guess of computing the ~ This article is distributed under the terms of the Creative Commons [SO( (1996) 513 T = ] N ]. [ R }| B 387 ~m SPIRE { IN Lett. theories, quivers and D-branes gauge dynamics [ proceedings of U.S.A. (2003), electric-magnetic duality We hope the result obtained in this paper could be useful for future studies on the J. de Boer, K. Hori, H. Ooguri and Y. Oz, A. Hanany and E. Witten, M.J. Strassler, K.A. Intriligator and N. Seiberg, K.A. Intriligator and N. Seiberg, [4] [5] [2] [3] [1] any medium, provided the original author(s) and source areReferences credited. by the ERC Advanced Grantthe SPLE grant under FPA2012-32828 contract ERC-2012-ADG-20120216-320421, from“Centro by de the Excelencia MINECO, Severo Ochoa” and Programme. byOpen the Access. grant SEV-2012-0249Attribution of License ( the F.C. and H.H. would likeon to the thank manuscript Amihay Hanany andhospitality and fuitful during Stefano the discussions. Cremonesi workshop forwhen comments F.C on this “Geometric would paper correspondence was likeinternational finalized. of to programme “La The gauge thank Caixa-Severo work Ochoa”. theories the of The 2016”, ICTP work F.C. has is for been partially supported kind supported through a fellowship of the and therefore to determinebranches the of restriction the rule for computing the HilbertAcknowledgments series of mixed the above procedure isplanes consistent glued with at the a Hilbertinteresting point. series to A of prove similar this a gluing guess variety was and made first we from discussed leave two it inmixed for [ branches future of work. theOne moduli interesting direction space of the of generalization more is to general include 3 Hilbert series of theto full a moduli sublocus space by andsublocus. removing adding some The the magnetic repetition charges HilbertHilbert of corresponding series series the of procedures of the would thecombinatorics give of mixed full a dividing branch moduli systematic the which space summation way stems of to will from be compute the the more the 3d complicated. 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