Schur indices with class S line operators from networks and further skein relations
IIPMU 17-0017
Schur indices with class S line operators fromnetworks and further skein relations
Noriaki Watanabe ♪♪ Kavli Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Abstract
We compute the Schur indices in the presence of some line operators based on our con-jectural formula introduced in [1]. In particular, we focus on the rank 1 superconformalfield theories with the enhanced global symmetry and the free hypermultiplets with theelementary pants networks defined on the three punctured sphere in the class S context.From the observations on the concrete computations, we propose new kinds of the class Sskein relations in the sense that they include the generic puncture non-trivially. We alsogive a general formula to unify all the relations we have exhibited. a r X i v : . [ h e p - t h ] F e b ontents q -deformed Yang-Mills 3 q -deformed Yang-Mills . . . 52.2 Good theories for q -series expansion . . . . . . . . . . . . . . . . . . . . . . 72.3 Breakdown of unitarity bound for defects . . . . . . . . . . . . . . . . . . . 82.4 Free hypermultiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 Bi-fundamental type . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Second rank anti-symmetric type . . . . . . . . . . . . . . . . . . . 122.4.3 Exceptional type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Rank 1 SCFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.1 Rank 1 E SCFT case . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.2 Rank 1 E SCFT case . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.3 Rank 1 E SCFT case . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Superconformal QCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 A case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 A case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 General punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A Lie algebra convention 41
A.1 A -type Lie algebra convention . . . . . . . . . . . . . . . . . . . . . . . . . 41 B The formula for Coulomb and Higgs branch dimension associated toregular punctures 42 Introduction
Recently, defects in field theories can be analyzed in various set-ups and in various ways.In supersymmetric cases, we can compute them by using the localization methods [2–6]. Inconformal cases, we also apply the unbroken symmetry to restrict the correlator form [7–9].In particular, there are topological defects in conformal field theories (CFTs) [10, 11] ortopological field theories (TQFTs) [12] and they are determined by the isotropy class and,in many examples, some data such as representations of some algebra or group.Marvellously, interesting phenomena were discovered that there are relations betweenthe topological defects in special 2D CFTs (Liouville/Toda) or some TQFTs and BPSdefects in 4D N =2 superconformal field theories (SCFTs) called “class S theories”. Theclass S theories are 4D N =2 systems obtained by the twisted compactifications via someRiemann surfaces of 6D N =(2 ,
0) SCFTs with several codimension two defects localizedon the surfaces [13, 14]. These 4D/2D duality relations were first observed in [15–17] inthe absence of defects and, in particular, they identify the class S Schur indices with thepartition functions of the 2D q -deformed Yang-Mills theory in [18]. This conjecture allowsus to compute the superconformal indices (SCIs) of the Lagrangian unknown theories.This relation was extended to the correspondence between the 4D line operators and the2D Wilson loop/network defects [19–28]. Assuming this 4D/2D duality relation, we definethe class S line operators by specifying the networks on the punctured Riemann surfaceeven in the Lagrangian unknown theories. See Fig. 1. In the our previous paper [1], weproposed the conjectural formula to compute the Schur indices with such line operatorswithout any derivation. To justify the formula, we checked two things there. One is theproof that they satisfy some fundamental skein relations and the other is to see that theSchur index of rank 1 SCFT with the E global enhanced symmetry in the presence of theloop operators is written with E characters only at the leading order in the q -expansion.In Sec. 2 of this paper, we give more evidences that the formula would be correct bycomputing the Schur indices with class S line operators in many examples.Figure 1: The codimension four defects are put on the 6D bulk as the networks on the2D punctured Riemann surface (Left), a localized point in the space direction ( S in ourcase) (Middle) and the straight line or the loop in the time direction ( S in our case)(Right).In particular, we focus on the rank 0 SCFTs which are considered to be free hyper- In our paper [26], by using the skein relation (digon type without any puncture), we derived theformula to compute the Schur index with the elementary pants network in the T theory (See Sec. 2.5.1).However, the extension to higher rank cases ( N >
3) is highly non-trivial. In the first case, there are no non-trivial line operators but just flavor Wilson lines whichare just the classical functions, namely, not operators. This suggests that, in this case,all the Schur indices with the non-trivial networks are always factorized into the Schurindices without the line operators and the flavor Wilson line factor. From the conjecturalformula, it is almost impossible to see this factorization and this check is highly non-trivial. In the latter case, we expect that the only one Schur index with an elementarypants network respects the global enhanced symmetry and the expression is written withthe characters of the enhanced symmetry. We check these two expectations in three cases( E , E and E global symmetry) by expanding the Schur indices as q -series. Finally, weexhibit the superconformal QCD applications briefly.In Sec. 3, we propose the new kinds of class S skein relations in the sense that theyinclude the (non-degenerate) punctures. In particular, we focus on the digon-type skeinrelations with one puncture. These relations account for the previous factorization inthe rank 0 or 1 SCFTs in terms of the geometrical relations. We exhibit the concreteexamples in Sec. 3.2, 3.3 and 3.4 and then propose the unified form of for any puncturein Sec. 3.5. In Sec. 3.6, we see one simple application of these skein relations. q -deformedYang-Mills As explained in the introduction briefly, the 4D/2D duality relation says that there isthe map from the Riemann surfaces to the 4D SCFTs, which just means the twistedcompactification of the 6D su ( N )-type N =(2 ,
0) SCFT for fixed N = 2 , , . . . . Noticethat the Riemann surface may have (regular) punctures each of which has a type andthe holonomy around it determined from the type. We denote this map by T S g and T S g = su ( N ) [ C ( Y , Y , . . . , Y (cid:96) )] denotes the resulting 4D N =2 theory in the IR after the twistedcompactification on the Riemann surface C with punctures whose types are specified by Y , Y , . . . , Y (cid:96) . We can extend this map from the pair of a punctured Riemann surface andnetworks on it to the 4D SCFT with line operators. Elementary pants networks
In this section, we analyse three cases : rank 0 SCFTs (free hypermultiplets), rank 1SCFTs and superconformal QCDs in class S theories.Among the possible networks on the trinion, there are the minimal ones depictedin Fig. 2. They were discussed explicitly at first in [24] and shown to be elementarygenerators of the line operator algebra in [27] which were called pants networks. In thispaper, we refer to the network shown in Fig. 2 as ( a AB , a BC , a CA )-type elementary pantsnetwork and denote it by ℘ ( a AB , a BC , a CA ). The computation of the SCI in the simplest The rank means the complex dimension of the 4D Coulomb branch. AB a BC a CA R A R B R C λ A λ B λ C Figure 2: The ( a AB , a BC , a CA )-type elementary pants network ℘ ( a AB , a BC , a CA ) and theassociated dual decorated and oriented lattice on the two sphere.case, that is to say, ℘ (1 , ,
1) in the T theory, was done in our paper [26]. However, theextension to other types of pants networks had been highly non-trivial before our work [1].We briefly review the necessary results in Sec 2.1. Notice that these pants networks areexpected to generate all the possible networks not touching on the punctures. See Fig 3.Figure 3: An example of complicated networks on the trinion (three punctured sphere)which appears after the crossing resolutions in the product of two elementary pants net-works. See [26, 27].Through this and next sections, there is an important assumption : there is the samenumber of independent elementary pants networks as the rank of IR charge lattice whichequals to the Coulomb branch dimension. The first example we can check readily is the T N -theory [13]. The Coulomb branch dimension of the T N -theory equals to ( N − N − .On the other hand, the number of possible junctions in the type A N − case is given bythe number of the possible partitions of N into three parts, that is, just ( N − N − . Next, let us see the rank 0 SCFTs, namely, free hypermultiplets. They have noCoulomb branch moduli and there is no dynamical gauge field. Therefore, if all lineoperators given by pants networks are neutral under the background magnetic charges,they are expected to be flavor Wilson lines which are just the classical holonomies becausethe gauge fields are just the background fields. At the computation level, this fact is In the other types, this is not true. For example, in the g = so (2 N ) case, the Coulomb branch real dimension is given as 2 N ( N − N = 4 or so (8), sixteen independent pants networksare expected if we follow the above assumption. However, the actual number of possible junctions is 10.Here the reason why we consider the real dimension is just these networks are invariant under chargeconjugation. q -deformedYang-Mills In this section, we review the necessary facts to compute the q -deformed Yang-Millscorrelators or the Schur indices. Puncture
The punctures are the codimension two defects in the 6D N =(2 ,
0) SCFTs and the im-portant classes (regular,untwisted) are labelled by the principal embedding ρ : su (2) → su ( N ) [13,14,29]. This classification is equivalent to the partition of N when the Lie alge-bra type of the 6D SCFT N =(2 ,
0) is su ( N ), namely, the decomposition of N dimensionalirreducible representations of su ( N ) into the representation of su (2). We represent thisby Y = [ n n . . . n k ] where (cid:80) i n i = N . In the language of the 4D N =2 SCFT, this corre-sponds to the choice of the nilpotent VEV to the SU (2) R highest part of the flavor currentmoment map µ + for a SU ( N ) flavor symmetry. This VEV breaks SU (2) UVR × SU ( N ) sym-metry at the original SCFT into a subgroup G Y × U (1) and, after the flowing to anotherIR SCFT, we have new global symmetry G Y × SU (2) IRR . In other words, the SU (2) UVR singlet matter in N for the SU ( N ) symmetry are charged under SU (2) IRR following thedecomposition N −→ (cid:76) i n i . Indeed, under the SU (2) IRR × G Y symmetry, the matter in N for SU ( N ) symmetry at UV belongs to N −→ (cid:77) i n i = (cid:77) a m a ⊗ d a (2.1)where Y = [ n n . . . n k ] = [ m d . . . m d t t ] and G Y = S ( (cid:81) a U ( d a )). This is also true for thegauge multiplet. Under G → SU (2) IRR × G Y (neither embedding nor subgroup),Adj −→ (cid:77) m m ⊗ R m (2.2)where R m is the representation under G Y and m runs over the irreducible representationsof SU (2) IRR , namely, positive integers.
Expressions without defects
Now, let us review the relation between the 2D q -deformed Yang-Mills theory correlatorand the 4D Schur indices. The difference consists of the factor K Y ( a ; q ) associated to each5uncture Y and the overall factor N ( q ). For the [1 N ]-type puncture called full puncture,the factor from the q -YM to the Schur is given by K ( a ) = K [1 N ] ( a ) = 1 I Schurvector ( a ; q ) / = (cid:89) α ∈ Adj qa α ) ∞ (2.3)where I Schurvector ( a ; q ) is the Schur index for the free vector multiplet of SU ( N ). This is justa basis change of the class functions, at least, in the Schur limit.Next, let us see the other types of punctures. Recalling that the expression for K [1 N ] ( a ; q ) is the product over all the weights of the vector multiplet, the reduced ex-pression should be given by the product of the factor ( q I a w ; q ) ∞ ( j − j = 1) over allthe weights w of R m and over possible m . Therefore, the reduced expression is given by K Y ( a IR ; q ) = (cid:89) m (cid:89) w ∈ Π( R m ) q m +12 a wIR ) ∞ (2.4)where a IR is the fugacity of G Y . The relation between a IR and a UV = a Y follows from(2.1). See Table 10 in Sec. 3.5. Notice that the highest of m for SU (2) IRR has theadditional m − charge compared to the UV where the gaugino (not all) contribute toSCIs as qa w . This reproduces the rule given in [18]. See the derivations in [30, 31] fordetails.By introducing the new function for each Y defined as ψ ( Y ) R ( a ; q ) := K Y ( a ; q ) χ R ( a Y ) , (2.5)we can write down the complete expression of general class S Schur indices as I Schur T S [ C ( Y ,Y ,...,Y (cid:96) )] ( { a } ; q ) = N ( q ) (cid:96) − χ C (cid:88) λ ∈P su ( N )+ (dim q R ( λ )) χ C − (cid:96) (cid:96) (cid:89) i =1 ψ ( Y ) R ( a G Yi i ; q ) . (2.6)where χ C is the ordinary Euler number of C on ignoring the punctures and dim q R ( λ ) isthe quantum dimension defined as (A.2). On the other hand, the overall factor is givenby N ( q ) := K [ N ] ( q ) − = N (cid:89) i =2 ( q i ; q ) . (2.7) Expressions in the presence of network operators
For the loop operators in the 2D system, see [32–34] for example. Since we are mainlyinterested in the elementary pants network operators in this paper, we recall the formulafor the elementary pants network. The claim in [1] is that the 2D q -deformed Yang-Mills6orrelators on C ( Y A , Y B , Y C ) with the elementary pants network ℘ ( a AB , a BC , a CA ) is givenby Z q YM T S [ C ( Y A ,Y B ,Y C )] w/. ℘ ( a AB ,a BC ,a CA ) ( a, b, c ; q ) = (cid:88) λ A ,λ B ,λ C ∈P su ( N )+ χ R ( λ A ) ( a Y A ) χ R ( λ B ) ( b Y B ) χ R ( λ C ) ( c Y C ) N − (cid:89) h =1 N − h (cid:89) α =1 [(ˆ λ ABC ) h ; α + h ] q [ h ] q . (2.8)where P su ( N )+ is the dominant weight set or all the finite dimensional irreducible represen-tations set and [ n ] q is the q -number defined by [ n ] q := q n/ − q − n/ q / − q − / . (ˆ λ ABC ) h ; α is definedas (ˆ λ ABC ) h ; α := mj (cid:32) α + h − (cid:88) s = α λ A,s , α + h − (cid:88) s = α λ B,s , α + h − (cid:88) s = α λ C,s (cid:33) (2.9) mj ( a, b, c ) := a b = a or c = ab a = b or c = bc a = c or b = c ∞ otherwise (2.10)where “ ∞ ” in this formula simply means that the contribution for the triple ( λ A , λ B , λ C )in (2.8) vanishes and see Appendix. A for λ X,s . λ X ∈ P su ( N )+ ( X = A, B, C ) just meansthat each λ X,s runs over non-negative integers for s = 1 , , . . . , N −
1. In the absenceof the elementary pants networks ( a AB = N and a BC = a CA = 0 for example), thisreduces to the well-known 2D topological q -deformed Yang-Mills partition function. Thecorresponding Schur index is obtained by replacing χ R ( x Y ) by ψ ( Y ) R ( x G Y ; q ) in the aboveexpression and multiplying N ( q ). q -series expansion In the actual computations, we expand the above expression (2.8) in order of ascendingpowers of q . The important point is that the expansions are well-defined only when wechoose the types of punctures appropriately. For simplicity, let us consider the no networkdefect cases.We can expand the inverse of q -dimensions as1dim q R ( λ ) = q ρ α λ α (1 − { α | λ α (cid:54) = 0 } + O ( q )) (2.11)where ρ α = α ( N − α ). Therefore, the leading exponent for fixed λ is given by g R := 12 N − (cid:88) α =1 α ( N − α ) λ α . (2.12)7n the other hand, the characters is expanded as χ R ( λ ) ( a Y ) =: q ρ αL ; Y λ α ( f R,Y ( a G Y ) + O ( q / )) (2.13)where ρ αL,Y ≤ α = 1 , , . . . , N −
1. For Y = [1 N ], ρ αL,Y = 0. For Y = [ N ], ρ αL,Y = − α ( N − α ). See Table. 6, 7 and 8 for example.In the total expressions, (cid:88) λ ∈P su ( N )+ (cid:81) ni =1 χ R ( λ ) ( a i,Y i )(dim q R ( λ )) (cid:96) − χ = (cid:88) λ ∈P su ( N )+ q ( − χρ α + (cid:80) ρ αL ; Yi ) λ a (cid:0) f R ( { a } ) + O ( q / ) (cid:1) (2.14)and we can see that the leading exponent is linear in the dominant weight. To have awell-defined q -series, we must require that L α := ( − χρ α + (cid:80) ρ αL ; Y i ) > α . Weconjecture that this is always true for any good or ugly class S theories whose formalnumber of the Coulomb branch operators at each scaling dimension is non-negative. Itis expected that this is also equivalent to the condition that the explicit symmetry at UVis unbroken at the IR SCFT [35]. In particular, for bad theories with vanishing ρ αL,Y , wecan sometimes compute the Schur indices but some δ functions may appear. It causes thesymmetry breaking. We do not consider this ugly case in this paper. When we add somenetwork defects, it just shifts the leading order by some constant and the q -expansion isstill valid. Finally, we make a comment on the breakdown of unitarity bound. As we see in the laterexamples, some Schur indices have the negative exponents of q . In the absence of loopoperators, it is guaranteed that the exponents of q are always non-negative because thereare three (Poincar´e) supercharges anti-commuting the supercharges to define the Schurindices. However, the insertion of line defects breaks the full superconformal algebra intothe subalgebra and there left no supercharges anti-commuting the defining supercharge.See [36] for the subalgebras. This no longer ensures the non-negativity of the exponentsof q .Let us consider a simple example of the breakdown recalling the discussion in [37, 38].If we have a purely electric line operator and another purely magnetic line operator, theyclassically generate the Poynting vector around the axis through two on ignoring theEuclidean time direction. This contributes to the Schur indices as some negative powerof q when we put the line operators in the appropriate order. Notice also that the N =(2 ,
2) type surface operators also break this bound becausethey create angular momentum around them when we realize them as vortex strings [30].At mathematical level, they act on the superconformal indices as difference operators andmany times applications lead to the negative power of q . The author thanks Y.Tachikawa for a few discussions on these subjects. If we exchange the positions of two line operators, the sign of the contribution in the exponent of q is reversed. .4 Free hypermultiplets In our analysis, we assume that all the SCFTs without any Coulomb branch are freehypermultiplets specified by the representation of the flavor symmetry. Under the con-dition that only untwisted regular punctures are allowed, we can classify them. Usingthe formula (B.5) and the condition d C,k = 0 leads to the constraint p Ak + p Bk + p Ck =2 k − k = 2 , , . . . , N . We use the property that the regular punctures must satisfy p k +1 − p k = 0 or 1 in addition. These constraints determine all the possibility of p A,B,Ck and all the three punctures combinations are listed in Table. 1. We categorize them intothree classes. From the analysis later, we call each bi-fundamental type, second rankanti-symmetric type and exceptional type.To determine what representations of the explicit global symmetry given by the threepunctured sphere, we use the information about the dimension of the Higgs branch andflavor central charges for non-Abelian simple group.theory explicit flavor symmetry d H flavor central charges T S [ C ([1 N ] , [1 N ] , [ N − , SU ( N ) × SU ( N ) × U (1) N k SU ( N ) = k SU ( N ) = 2 N T S [ C ([1 N ] , [ n ] , [ n, n − , ( N =2 n ≥ SU ( N ) × SU (2) × U (1) N ( N +3) k SU ( N ) = N , k SU (2) = 2 N T S [ C ([1 N ] , [ n , [ n +1 , n ])] ( N =2 n +1 ≥ T S [ C ([1 ] , [2 ] , [42])] SU (6) × SU (3) × U (1) k SU (6) = 12, k SU (3) = 12 T S [ C ([21 ] , [2 ] , [3 ])] SU (4) × SU (3) × SU (2) × U (1) k SU (4) = 10, k SU (3) = k SU (2) =12 Table 1: Classification of the class S free hypermultiplets and their data
Bi-fundamental type : F (bf) A N − The first case F (bf) A N − = T S [ C ([1 N ] , [1 N ] , [ N − , SU ( N ) × SU ( N ) × U (1)-symmetry where U (1) is the baryon symmetry. In term of N =1 chiralmultiplets, the representation is given as ( N , N , ⊕ ( N , N , −
1) and this means k SU ( N ) i =2 N ( i = 1 , H also equals to d H = N . We can also see thisfact from the 4D/2D duality relation conjecture stated above by computing it concretely. Second rank anti-symmetric type : F (as) A N − In the case that N = 2 n ( N ≥ n ≥
2) is even, the theory F (as) A N − = T S [ C ([1 N ] , [ n ] , [ n, n − , N = 2 n + 1 ( N ≥ n ≥
2) is odd, the theory F (as) A N − = T S [ C ([1 N ] , [ n , , [ n + 1 , n ])] is also rank 0. Although the class S realizations ofthe even and odd cases look different, we can discuss both cases at the same time becausethe 4D physical properties are uniformly treated except U (1) charges. We can read offexplicit flavor symmetry as SU ( N ) × SU (2) × U (1) for N ≥ SU (4) × SU (2) × U (1)when N = 4. In both cases, k SU ( N ) = 2 N and k SU (2) = 2 N ( k SU (2) = 8 and k SU (2) = 6when N = 4) holds true. The dimension of Higgs branch is given by d H = N ( N + 3).9he matter content, namely, the representation in terms of the chiral multiplets, sat-isfying these conditions is following :( N , , ∗ ) ⊕ ( N , , ∗ ) ⊕ ( ∧ N , , ∗ ) ⊕ ( ∧ N , , ∗ ) when N ≥ , , , ∗ ) ⊕ ( , , , ∗ ) ⊕ ( ∧ , , , ∗ ) when N = 4 (2.16)where ∗ represents an undetermined U (1) charge.By computing the Schur indices from the q -deformed Yang-Mills partition function,we can fix U (1) charges as follows. N = 4( , , , ⊕ ( , , , − ⊕ ( , , ,
0) (2.17) N : even( N , , (1 , ⊕ ( N , , ( − , ⊕ ( ∧ N , , (1 , ⊕ ( ∧ N , , ( − , − N : odd( N , ( , , ⊕ ( N , ( , − , − ⊕ ( ∧ N , ( , , − N − ) ⊕ ( ∧ N , ( − , N − ) . (2.19) Exceptional case 1 : F (ex1) A = T S [ C ([ ] , [ ] , [ ])]In this case, the flavor symmetry is SU (6) × SU (3) × U (1), the flavor central chargesare k SU (6) = 12 and k SU (3) = 12 and the Higgs branch dimension is d H = 28. Thecandidates satisfying the above conditions are ( ∧ = , , ∗ ) ⊕ ( , , ∗ ) ⊕ ( , , ∗ ) or( ∧ = , , ∗ ) ⊕ ( ∧ = , , ∗ ) ⊕ ( , Adj = , ∗ ) ⊕ ( , Adj = , ∗ ). To the best of theauthor’s knowledge, we cannot determine which candidate is actually true. However,by the 4D/2D computation, we find that( , , ⊕ ( , , ⊕ ( , , −
1) (2.20)is the correct answer.
Exceptional case 2 : F (ex2) A = T S [ C ([ ] , [ ] , [ ])]In this case, the flavor symmetry is SU (4) × SU (3) × SU (2) × U (1), the flavor centralcharges are k SU (4) = 10, k SU (3) = 12 and k SU (2) = 12 and the Higgs branch dimension is d H = 24. Indeed, the matter content is given by( ∧ = , , , ⊕ ( , , , ) ⊕ ( , , , − ) ⊕ ( , , , ⊕ ( , , , − . (2.21) Naively speaking, if the latter case is true, there may exists a SU (2) symmetry but such symmetrydoes not appear. .4.1 Bi-fundamental type A bi-fundamental Let us see the simple case at first. This is the A bi-fundamentalhypermultiplet whose flavor symmetry is given by SU (3) × SU (3) × U (1). There, weconsider the ( a AB , a BC , a CA ) = (1 , , T S [ C ([1 ] , [1 ] , [2 , F (bf) A = Hyper( , , a, b and c be the holonomies of SU (3) × SU (3) × U (1). Thereare six sectors. Up to q / -order, we can see that the q -deformed Yang-Mills correlatorsreceive the contributions from all the triple of dominant weights at punctures listed inTable. 2. (( λ A ) , ( λ A ) ) (( λ B ) , ( λ B ) ) (( λ C ) , ( λ C ) ) q , q / (cid:26) (0 ,
0) (0 ,
1) (1 , ,
0) (0 ,
0) (0 , q / (2 ,
0) (1 ,
0) (1 , ,
1) (0 ,
1) (0 , ,
0) (1 ,
1) (0 , ,
1) (0 ,
2) (1 , ,
1) (1 ,
0) (0 , ,
1) (1 ,
0) (1 , q and q / terms.The result is given by I Schur F (bf) A ℘ (1 , , ( a, b, c ) = (cid:2) χ ( a ) c − + χ ( b ) c (cid:3) + q / (cid:2) χ ( a ) χ ( b ) + χ ( b ) c − + χ ( a ) c + χ ( a ) χ ( b ) c − + χ ( a ) χ ( b ) c + χ ( a ) χ ( b ) + χ ( a ) χ ( b ) (cid:3) + q (cid:2) c χ ( a ) χ ( b )+ c − χ (cid:48) ( a ) χ ( b ) + 2 c − χ ( a ) χ ( b ) + c − χ (cid:48) ( a ) χ ( b ) + c − χ ( a ) χ ( b ) + cχ ( a ) χ ( b )+ c − χ ( a ) χ ( b ) + c χ ( a ) χ (cid:48) ( b ) + 3 cχ ( b ) + 3 c − χ ( a ) + cχ (cid:48) ( b ) + 3 c − χ ( a ) χ ( b )+ c − χ ( a ) χ ( b ) + c χ ( a ) χ ( b ) + 2 cχ ( a ) χ ( b ) + cχ ( a ) χ (cid:48) ( b ) + 3 cχ ( a ) χ ( b )+ c − χ (cid:48) ( a ) + c − χ ( a ) + c − χ ( a ) χ ( b ) + c χ ( a ) χ ( b ) + cχ ( b ) (cid:3) + O ( q / )= (cid:2) χ ( a ) c − + χ ( b ) c (cid:3) I Schur F (bf) A ( a, b, c ) (2.22)where I Schur F (bf) A ( a, b, c ) = 1 + q / (cid:2) cχ ( a ) χ ( b ) + c − χ ( a ) χ ( b ) (cid:3) + q [1 + χ ( a ) + χ ( b ) + χ ( a ) χ ( b )+ c χ ( a ) χ ( b ) + c − χ ( a ) χ ( b ) + c χ ( a ) χ ( b ) + c − χ ( a ) χ ( b ) (cid:3) + O ( q / )(2.23)and (cid:48) = R (2 ω + ω ). Then, we find the flavor Wilson line factor. In terms of therepresentations of the global symmetry, it is given by W A bi − fund ℘ (1 , , = ( , , − ⊕ ( , , . (2.24) The sum in (2.8) splits into the possible pairs of λ B − λ A and λ C − λ A (finite sum) and λ A for thefixed pair (infinite sum). We refer to the former pair as “sector”. bi-fundamental We can check that all the elementary pants networks in the A = su (6) bi-fundamental free hypermultiplet are factorized as the SCI in the absence of linesand the following factors : W A bi − fund ℘ (4 , , = c − χ ( a ) + cχ ( b ) (2.25) W A bi − fund ℘ (3 , , = [2] q cχ ( b ) + c − χ ( a ) (2.26) W A bi − fund ℘ (3 , , = [2] q c − χ ( a ) + c χ ( b ) (2.27) W A bi − fund ℘ (2 , , = [3] q cχ ( b ) + c − χ ( a ) (2.28) W A bi − fund ℘ (2 , , = [3] q (cid:0) c − χ ( a ) + c χ ( b ) (cid:1) (2.29) W A bi − fund ℘ (2 , , = [3] q c − χ ( a ) + c χ ( b ) (2.30) W A bi − fund ℘ (1 , , = [4] q cχ ( b ) + c − χ ( a ) (2.31) W A bi − fund ℘ (1 , , = ([5] q + 1) c χ ( b ) + [4] q c − χ ( a ) (2.32) W A bi − fund ℘ (1 , , = ([5] q + 1) c − χ ( a ) + [4] q c χ ( b ) (2.33) W A bi − fund ℘ (1 , , = [4] q c − χ ( a ) + c χ ( b ) (2.34) A N − bi-fundamental In the general A N − = su ( N ) case ( T S su ( N ) [ C ([1 N ] , [1 N ] , [ N − , F (bf) A N − = Hyper( N , N ,
1) = Hyper( N , N , ⊕ Hyper( N , N , −
1) ), we conjec-ture that the ( p, r, s )-type ( N = p + r + s ) pants network gives the flavor Wilson lines W A N − bi − fund( p,r,s ) = (cid:20) r + s − r (cid:21) q W flavor ( ∧ p N , , − r ) ⊕ (cid:20) r + s − s (cid:21) q W flavor ( , ∧ p N , s )(2.35)where W flavor ( R A , R B , q C ) is the flavor line in the representation R A ⊗ R B ⊗ C q C under SU ( N ) A × SU ( N ) B × U (1) C flavor symmetry of the bi-fundamental hypermultiplet. Wehave also introduced the q-binomial coefficient (cid:20) x + yy (cid:21) q := [ x + y ] q ![ x ] q ![ y ] q ! = (cid:81) x + yi =1 [ i ] q (cid:81) xi =1 [ i ] q (cid:81) yi =1 [ i ] q . (2.36) Here we focus on two cases g = su (4) and su (5). The higher rank computations are totallysimilar. A second rank anti-symmetric We focus on F (as) A = T S [ C ([1 ] , [2 ] , [21 ])] whichconsists of ( , , , ⊕ ( , , , − ⊕ ( , , , SU (4) × SU (2) B × U (1) × SU (2) C and the associated fugacities are chosen as a [1 ] = ( a , a , a , a )( a a a a = 1), b [2 ] = ( q / b, q − / b, q / b − , q − / b − ) and c [21 ] = ( q / c, q − / c, c − c (cid:48) , c − c (cid:48) − ).12 A as ℘ (2 , , = [2] q c + c − χ SU (2) ( c (cid:48) ) + χ ( a ) χ SU (2) ( b ) (2.37) W A as ℘ (1 , , = [2] q cχ SU (2) ( b ) + c − χ SU (2) ( b ) χ SU (2) ( c (cid:48) ) + χ ( a ) (2.38) W A as ℘ (1 , , = c − χ ( a ) + χ SU (2) ( b ) χ SU (2) ( c (cid:48) ) + cχ ( a ) (2.39) A second rank anti-symmetric Next, we focus on F (as) A = T S [ C ([1 ] , [2 , [32])]which consists of ( , , ⊕ ( , − , − ⊕ (c . c . ). The global symmetry is SU (5) × SU (2) × U (1) B × U (1) C and the associated fugacities are chosen as a [1 ] = ( a , a , a , a , a )( a a a a a = 1), b [2 = ( q / bb (cid:48) , q − / bb (cid:48) , q / bb (cid:48)− , q − / bb (cid:48)− , b − ) and c [32] = ( qc, c, q − c, q / c − / , q − / c − / ). The flavor Wilson lines are given by W A as ℘ (3 , , = [2] q b c + b − cχ SU (2) ( b (cid:48) ) + b c − / + bχ ( a ) χ SU (2) ( b (cid:48) ) + b − c − / χ ( a ) (2.40) W A as ℘ (2 , , = [2] q b − c + [2] q (cid:16) b − c − / + b cχ SU (2) ( b (cid:48) ) (cid:17) + b − cχ SU (2) ( b (cid:48) ) + b χ ( a )+ b c − / χ SU (2) ( b (cid:48) ) + b − c − / χ ( a ) χ SU (2) ( b (cid:48) ) (2.41) W A as ℘ (2 , , = [2] q b − c − / + c − χ ( a ) + b cχ ( a ) + b c − / χ SU (2) ( b (cid:48) ) + b − c + b − c / χ ( a ) χ SU (2) ( b (cid:48) ) (2.42) W A as ℘ (1 , , = [2] q b − cχ SU (2) ( b (cid:48) ) + [2] q (cid:16) b − c − / χ SU (2) ( b (cid:48) ) + b c (cid:17) + b c − / + c − / χ ( a )(2.43) W A as ℘ (1 , , = [2] q b − c − / χ SU (2) ( b (cid:48) ) + [2] q (cid:16) c / χ ( a ) + b − c χ SU (2) ( b (cid:48) ) + b c − / (cid:17) + c − χ ( a ) + b c (2.44) W A as ℘ (1 , , = [2] q (cid:16) b − c / χ SU (2) ( b (cid:48) ) + c − χ ( a ) (cid:17) + c / χ ( a ) + b c / . (2.45) For the two exceptional type free hypermultiplets, we can also check the factorizations atorder-by-order in q . The factorized factors are the flavor Wilson lines and the results arefollowing. A exceptional type 1 The first case is given by F (ex1) A = T S [ C ([1 ] , [2 ] , [42])]. The global symmetry is given by SU (6) × SU (3) × U (1) and the associated fugacities are chosen as a [1 ] = ( a , a , a , a , a , a )( a a a a a a = 1), b [2 ] = ( q / b , q − / b , q / b , q − / b , q / b , q − / b ) ( b b b = 1) and c [42] = ( q / c, q / c, q − / c, q − / c, q / c − , q − / c − ). The flavor Wilson lines are given by W A ex1 ℘ (4 , , = [2] q cχ SU (3) ( b ) + c − χ ( a ) + χ ( a ) χ SU (3) ( b ) (2.46)13 A ex1 ℘ (3 , , = [3] q [2] q c + [2] q (cid:16) cχ SU (3) ( b ) + c − (cid:17) + χ ( a ) χ SU (3) ( b ) + c − χ ( a ) χ SU (3) ( b )(2.47) W A ex1 ℘ (3 , , = [2] q (cid:0) c − + c (cid:1) + c − χ ( a ) + χ ( a ) χ SU (3) ( b ) + cχ ( a ) χ SU (3) ( b ) (2.48) W A ex1 ℘ (2 , , = [3] q [2] q cχ SU (3) ( b ) + [2] q (cid:16) cχ SU (3) ( b ) + c − χ SU (3) ( b ) (cid:17) + c − χ ( a ) χ SU (3) ( b ) + χ ( a ) (2.49) W A ex1 ℘ (2 , , = [3] q (cid:0) c + c − (cid:1) χ SU (3) ( b ) + [2] q (cid:16) χ ( a ) χ SU (3) ( b ) + cχ ( a ) (cid:17) + c χ SU (3) ( b ) + c − χ ( a ) + c − χ SU (3) ( b ) (2.50) W A ex1 ℘ (2 , , = [2] q (cid:16) χ SU (3) ( b ) + c − χ ( a ) (cid:17) + cχ ( a ) χ SU (3) ( b ) + c χ ( a ) (2.51) W A ex1 ℘ (1 , , = [3] q [2] q cχ SU (3) ( b ) + [2] q c − χ SU (3) ( b ) + c − χ ( a ) (2.52) W A ex1 ℘ (1 , , = [3] q [2] q (cid:0) c + c − (cid:1) χ SU (3) ( b ) + [3] q χ ( a ) + c − χ ( a ) (2.53) W A ex1 ℘ (1 , , = [3] q [2] q χ SU (3) ( b ) + [3] q (cid:0) c + c − (cid:1) χ ( a ) + [2] q c χ SU (3) ( b ) (2.54) W A ex1 ℘ (1 , , = [3] q c − χ ( a ) + [2] q cχ SU (3) ( b ) + c χ ( a ) . (2.55) A exceptional type 2 The second case is given by F (ex2) A = T S [ C ([21 ] , [2 ] , [3 ])]. The global symmetry isgiven by U (1) × SU (4) × SU (3) × SU (2) and the associated fugacities are chosen as a [21 ] = ( q / a, q − / a, a − / a (cid:48) , a − / a (cid:48) , a − / a (cid:48) , a − / a (cid:48) ) ( a (cid:48) a (cid:48) a (cid:48) a (cid:48) = 1), b [2 ] = ( q / b , q − / b , q / b , q − / b , q / b , q − / b ) ( b b b = 1) and c [3 ] = ( qc, c, q − c, qc − , c − , q − c − ). The flavor Wilson lines are given by W A ex2 ℘ (4 , , = [2] q (cid:16) a − + a / χ SU (4) ( a (cid:48) ) (cid:17) χ SU (3) ( b ) + [2] q a + χ SU (3) ( b ) χ SU (2) ( c ) + a − / χ SU (4) ( a (cid:48) )(2.56) W A ex2 ℘ (3 , , = [2] q χ SU (2) ( c ) + [2] q (cid:16) cχ SU (3) ( b ) + a − χ SU (3) ( b ) (cid:17) + a − / χ SU (4) ( a (cid:48) ) χ SU (3) ( b )+ a / χ SU (4) ( a (cid:48) ) χ SU (3) ( b ) + χ SU (3) ( b ) χ SU (2) ( c ) (2.57) W A ex2 ℘ (3 , , = 2[2] q + a / χ SU (4) ( a (cid:48) ) χ SU (3) ( b ) + χ SU (4) ( a (cid:48) ) χ SU (2) ( c ) + a − / χ SU (4) ( a (cid:48) )+ a − χ SU (3) ( b ) χ SU (2) ( c ) + (cid:16) a / + a − / χ SU (3) ( b ) (cid:17) χ SU (4) ( a (cid:48) ) + aχ SU (3) ( b ) χ SU (2) ( c )(2.58) W A ex2 ℘ (2 , , = [2] q χ SU (3) ( b ) χ SU (2) ( c ) + [2] q (cid:16) aχ SU (3) ( b ) + a − (cid:17) + a − / χ SU (4) ( a (cid:48) ) χ SU (3) ( b ) + χ SU (3) ( b ) χ SU (2) ( c ) + a / χ SU (4) ( a (cid:48) ) (2.59) W A ex2 ℘ (2 , , = [2] q χ SU (3) ( b ) + [2] q (cid:16) a / χ SU (4) ( a (cid:48) ) + a − χ SU (2) ( c ) + aχ SU (3) ( b ) χ SU (2) ( c ) (cid:17) + χ SU (3) ( b ) + χ SU (3) ( b ) χ SU (2) ( c ) + a − / χ SU (4) ( a (cid:48) ) χ SU (3) ( b ) χ SU (2) ( c )14 a / χ SU (4) ( a (cid:48) ) χ SU (2) ( c ) + a + a − χ SU (4) ( a (cid:48) ) (2.60) W A ex2 ℘ (2 , , = [2] q (cid:16) aχ SU (3) ( b ) + a / χ SU (4) ( a (cid:48) ) χ SU (2) ( c ) + a − (cid:17) + a − / χ SU (4) ( a (cid:48) ) χ SU (3) ( b )+ χ SU (3) ( b ) χ SU (2) ( c ) + a − χ SU (4) ( a (cid:48) ) χ SU (2) ( c ) + a χ SU (2) ( c ) + a / χ SU (4) ( a (cid:48) )(2.61) W A ex2 ℘ (1 , , = [2] q χ SU (3) ( b ) χ SU (2) ( c ) + [2] q a + a − / χ SU (4) ( a (cid:48) ) (2.62) W A ex2 ℘ (1 , , = [2] q χ SU (3) ( b ) (cid:16) [3] q + χ SU (2) ( c ) (cid:17) + [2] q (cid:16) [2] q a + a − / χ SU (4) ( a (cid:48) ) (cid:17) χ SU (2) ( c ) (2.63) W A ex2 ℘ (1 , , = [3] q [2] q a + [3] q (cid:16) a − / χ SU (4) ( a (cid:48) ) + χ SU (3) ( b ) χ SU (2) ( c ) (cid:17) + [2] q aχ SU (2) ( c )+ a − / χ SU (4) ( a (cid:48) ) χ SU (2) ( c ) + χ SU (3) ( b ) χ SU (2) ( c ) (2.64) W A ex2 ℘ (1 , , = [3] q aχ SU (2) ( c ) + [2] q (cid:16) a − / χ SU (4) ( a (cid:48) ) χ SU (2) ( c ) + χ SU (3) ( b ) (cid:17) + aχ SU (2) ( c ) . (2.65) In this section, we focus on the special rank 1 SCFTs with enhanced global symmetries,namely, E , , symmetries. E SCFT case
The first result is already discussed in [26] in a direct way. There is only one typeelementary pants network specified by (1 , , ℘ (1 , ,
1) is given by I Schur rank 1 E SCFT w/. ℘ (1 , , ( a, b, c ) = q / χ E ( a, b, c ) + q / χ E ( a, b, c )+ q / (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) (cid:1) + q / (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c )+2 χ E ( a, b, c ) + χ E ( a, b, c ) + 2 χ E ( a, b, c ) (cid:1) + O ( q / ) . (2.66)Here all the irreducible representations appearing in characters are non-trivially chargedunder its center group Z .Notice that the Schur indices in the absence of defects up to q order is given as I Schur rank 1 E SCFT ( a, b, c ) = 1 + qχ E ( a, b, c ) + q (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + 1 (cid:1) + q (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + 2 χ E ( a, b, c ) + 1 (cid:1) + O ( q ) . (2.67)15 ω ω ω ω ω Figure 4: E Dynkin diagram. ω i gives i mod 3 charge under the center group Z .See [39] and its Mathematica package.dimension
27 351 1728 17550 46332 51975
Dynkin labels (100000) (000100) (100001) (000101) (100002) (101000)dimension
Dynkin labels (100003) (000001) (000002) (001000) (000003)Table 3: The dimensions of irreducible representations and their Dynkin labels in E . E SCFT case
In this theory, we find that the (2 , , E -symmetry. We computed this case in [1] at the leading orderof q . Indeed, the computation up to q / gives I Schur rank 1 E SCFT w/. ℘ (2 , , ( a, b, c ) = q / χ E ( a, b, c ) + q / χ E ( a, b, c )+ q / (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) (cid:1) + q / (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c )+2 χ E ( a, b, c ) + χ E ( a, b, c ) + 2 χ E ( a, b, c ) (cid:1) + O ( q / ) . (2.68)Notice that all the irreducible representations appearing in characters are non-triviallycharged under its center group Z .The no defect Schur index of this SCFT is given by I Schur rank 1 E SCFT ( a, b, c ) = 1 + qχ E ( a, b, c ) + q (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + 1 (cid:1) + q (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + 2 χ E ( a, b, c ) + 1 (cid:1) + O ( q ) . (2.69)The other two pants networks are just flavor Wilson lines, that is to say, factorizedinto the Schur index without any networks and the following factors. ℘ (1 , ,
1) : χ ( b ) χ SU (2) ( c ) + χ ( a ) We do not exactly check the terms at q / but only see the match of the values at the trivial fugacities a = b = c = 1. Notice also that there is the common structure between the previous E SCFT and this E SCFT on replacing the fundamental weights ω E , ω E , ω E , ω E by ω E , ω E , ω E , ω E respectively.The similar structure also appears in the SU (2) N f = 4 SQCD, namely, the SO (8) SCFT with or withoutthe fundamental Wilson line, for example. However, it seems to be not true in the E SCFT as we seelater. (1 , ,
2) : χ ( a ) χ SU (2) ( c ) + χ ( b ) ω ω ω ω ω ω ω Figure 5: E Dynkin diagram. ω , , gives non-trivial charge under the center group Z .dimension
56 912 6480 86184 320112 362880
Dynkin labels (0000010) (0000001) (1000010) (1000001) (2000010) (0100010)dimension
Dynkin labels (3000010) (1000000) (2000000) (0100000) (3000000)Table 4: The dimensions of irreducible representations and their Dynkin labels in E . E SCFT case
In this SCFT, only the (3 , , E -symmetry. The result is following. I Schur rank 1 E SCFT w/. ℘ (3 , , ( a, b, c ) − [2] q I Schur rank 1 E SCFT ( a, b, c ) = q / (cid:0) χ E ( a, b, c ) + 1 (cid:1) + q / (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + 1 (cid:1) + q / (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + 2 χ E ( a, b, c )+ χ E ( a, b, c ) + 3 χ E ( a, b, c ) + 1 (cid:1) + O ( q / ) (2.70)where, to express this as possible as simply, we have subtracted the Schur index withoutthe defects expressed as I Schur rank 1 E SCFT ( a, b, c ) = 1 + qχ E ( a, b, c ) + q (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + 1 (cid:1) + q (cid:0) χ E ( a, b, c ) + χ E ( a, b, c ) + χ E ( a, b, c ) + 2 χ E ( a, b, c ) + 1 (cid:1) + O ( q ) . (2.71)Notice that in the presence of (3 , , q is − .The other pants networks are written as the flavor Wilson lines as follows : ℘ (1 , ,
4) : [2] q χ ( a ) χ SU (2) ( c ) + [2] q χ SU (3) ( b ) ℘ (1 , ,
3) : [3] q χ ( a ) + χ ( a ) χ SU (2) ( c ) + ([2] q ) χ SU (3) ( b ) χ SU (2) ( c ) ℘ (1 , ,
2) : [3] q [2] q χ SU (3) ( b ) + [2] q χ SU (3) ( b ) χ SU (2) ( c ) + [2] q χ ( a ) χ SU (2) ( c )17 (1 , ,
1) : ([2] q ) χ SU (3) ( b ) χ SU (2) ( c ) + χ ( a ) ℘ (2 , ,
3) : χ ( a ) χ SU (2) ( c ) + χ ( a ) χ SU (3) ( b ) + χ ( a ) + χ SU (3) ( b ) χ SU (2) ( c ) ℘ (2 , ,
2) : ([2] q ) χ SU (3) ( b ) + χ ( a ) + χ ( a ) χ SU (3) ( b ) χ SU (2) ( c ) + χ ( a ) χ SU (2) ( c )+ χ SU (3) ( b ) χ SU (2) ( c ) + χ SU (3) ( b ) ℘ (2 , ,
1) : ([2] q ) χ SU (3) ( b ) χ SU (2) ( c ) + χ ( a ) χ SU (3) ( b ) + χ ( a ) + χ SU (3) ( b ) χ SU (2) ( c ) ℘ (3 , ,
1) : ([2] q ) χ SU (2) ( c ) + χ ( a ) χ SU (3) ( b ) + χ ( a ) χ SU (3) ( b ) + χ SU (3) ( b ) χ SU (2) ( c ) ℘ (4 , ,
1) : χ ( a ) + χ ( a ) χ SU (3) ( b ) + χ SU (3) ( b ) χ SU (2) ( c ) ω ω ω ω ω ω ω ω Figure 6: E Dynkin diagram. The center group is trivial.dimension
248 3875 27000 30380 1763125 4096000
Dynkin labels (00000010) (10000000) (00000020) (00000100) (00000030) (000000110)Table 5: The dimensions of irreducible representations and their Dynkin labels in E .In each case of the rank 1 SCFTs, it is interesting problem to rewrite them in terms ofthe affine characters in the admissible representations of ( e ) k = − , ( e ) k = − and ( e ) k = − asconjectured in [36, 40]. The other non-trivial checks are the “IR computations” discussedin [36, 41] by using the techniques and results on the spectral networks in the Minahan-Nemeschansky theory in [42, 43] and on the quantum traces in [27, 28]. SU (2) N f = 4 SQCD
Here we introduce the following symbol. χ ( R A , R B , R C , R D ) = χ SU (2) R A ( a ) χ SU (2) R B ( b ) χ SU (2) R C ( c ) χ SU (2) R D ( d ) . (2.72)We also written down the Schur index expression of SU (2) N f = 4 SQCD for com-parison. I Schur SU (2) N f =4 SQCD ( a, b, c, d ) = 1 + qχ SO (8) + q (cid:104) χ SO (8) + χ SO (8) + 1 (cid:105) B R A R C R D R B R A R C R D R B R A R C R D Figure 7: For the A or SU (2) SCQCD cases obtained by the four punctured sphere,we show a 4D Wilson loop in (cid:3) = R ( ω ) = (Left), a 4D ’t Hooft loop labelled by (cid:3) = R ( ω ) = (Middle) and both Wilson and ’t Hooft loops (Right).+ q (cid:104) χ SO (8) + χ SO (8) + χ SO (8) + 2 χ SO (8) + 1 (cid:105) + O ( q ) . (2.73)In this system described by the Lagrangian, χ SO (8) corresponds to the meson operator M AB = (cid:15) ab Q aA Q bB ( a, b : SU (2) gauge indices, A, B : SO (8) indices) with ∆ = 2 and I = 1 for example.When we add the Wilson line W (cid:3) , the Schur index computed from the q -deformedYang-Mills formula is given by I Schur SU (2) N f =4 SQCD w/. W (cid:3) ( a, b, c, d ) = q / [ χ ( , , , ) + χ ( , , , )] + q / [ χ ( , , , )+ χ ( , , , ) + χ ( , , , ) + χ ( , , , ) + χ ( , , , ) + χ ( , , , ) + χ ( , , , )+ χ ( , , , ) + χ ( , , , ) + χ ( , , , ) + χ ( , , , ) + χ ( , , , )] + O ( q / )= q / χ v ( a, b, c, d ) + q / χ v ( a, b, c, d ) + O ( q / ) (2.74)which matches with the known result in [36] for example. Next, let us see the dual ’t Hooft loop operators. Although its field theoretical defini-tion is not known yet, from the geometrical point of view, this acts on the above Wilsonline expression simply as the permutation of two [1 ]-type punctures. In particular, thispermutation is equivalent to the triality action of SO (8), that is to say, the exchange ofthe simple roots α and α . Therefore, its Schur index expression is I Schur SU (2) N f =4 SQCD w/. T dual (cid:3) ( a, b, c, d ) = q / χ s + q / χ s + q / [ χ s + χ s + χ s + χ s ] + O ( q / ) . (2.75)According to [36] for example, we can interpret the Schur index as the count of localoperators playing the role of line changing operators between the fundamental Wilson lineand this ’t Hooft line.Finally, let us evaluate the two loops coexisting case on the right side in Fig. 7. Byrepresenting the four punctured sphere as the disk by removing one point in the sphere, Of course, this is obviously true once the expression of the free bi-fundamental hypermultiplets isgiven by the correlator of the 2D q -deformed Yang-Mills theory. The insertion of the correspondingcharacter and the integration over the gauge group are same operations.
19e can geometrically compute the OPE of the Wilson loop and the dual ’t Hooft loopwith the crossing resolutions (see [20, 26] for example) as follows:
AB CD = + − q / − q − / (2.76)= + − q / − q − / (2.77)The first two terms give χ ( b ) χ ( d ) + χ ( a ) χ ( c ) = χ SO (8) c ( a, b, c, d ) (2.78)and the latter two have the same Schur index ( different at the 4D line operators level )up to the prefactors q ± / . The final expression is I Schur SU (2) N f =4 SQCD w/. T dual (cid:3) ◦ W (cid:3) ( a, b, c, d ) = I Schur SU (2) N f =4 SQCD w/. W (cid:3) ◦ T dual (cid:3) ( a, b, c, d )= χ c I Schur φ − [2] q I Schur
W T (cid:3) = qχ c + q [ χ c + χ c ] + O ( q ) . (2.79)Since we have no duality frame where both loops are simultaneously magneticallyneutral, it is difficult to interpret this result based on the Lagrangian description. SU (3) N f = 6 SCQCD case
The SU (3) N f = 6 SQCD is given by T S [ C (2 · [1 ] , · [2 , ]-type) and two simplepunctures ([21]-type).In the absence of line defects, I Schur SU (3) N f =6 SQCD ( a, b, c, d ) = 1 + q (2 + c − d − χ ( a ) χ ( b ) + χ ( a )+ cdχ ( a ) χ ( b ) + χ ( b )) + q / ( c − dχ ( a ) χ ( b ) + cd − χ ( a ) χ ( b )+ c − d χ ( a ) χ ( b ) + c d − χ ( a ) χ ( b ) + c − + c + d + d − ) + O ( q ) (2.80)20 B R A R C R D a R B R A R C R D b R B R A R C R D a b Figure 8: For the higher rank cases, Wilson loop in ∧ a (cid:3) (Left), A ’t Hooft loop labelledby ∧ b (cid:3) (Middle) and both Wilson and ’t Hooft loops (Right). Notice that the orderingof the insertions of two loops is irrelevant in this case.= 1 + q ( χ U (6)Adj + 1) + q / ( χ U (6) ∧ + χ U (6) ∧ ) + O ( q ) (2.81)where χ U (6) = χ U (6)[10000] = cχ ( b ) + d − χ ( a ) and χ U (6) = χ U (6)[00001] − = dχ ( a ) + c − χ ( b ).The Schur indices with the fundamental Wilson loop W (cid:3) (Left in Fig. 8) and the dual’t Hooft loop T dual (cid:3) (Right in Fig. 8) are given as I Schur SU (3) N f =6 SQCD w/. W (cid:3) ( a, b, c, d ) = q / (cid:0) cχ ( b ) + d − χ ( a ) (cid:1) + q (cid:0) c − dχ ( a ) χ ( b ) + c − χ ( b ) + d χ ( a ) (cid:1) + q / (cid:0) d − χ ( a ) χ ( b ) + cχ (cid:48) ( b )+ 3 cχ ( b ) + d − χ (cid:48) ( a ) + c − d − χ ( a ) χ ( b ) + 3 d − χ ( a ) + c dχ ( a ) χ ( b )+2 cχ ( a ) χ ( b ) + cχ ( b ) + c dχ ( a ) χ ( b ) + d − χ ( a ) + c − d − χ ( a ) χ ( b ) (cid:1) + O ( q )(2.82)= q / χ U (6) + q χ U (6) ∧ + q / (cid:104) χ U (6)[20001] − + χ U (6)[01001] − + χ U (6) (cid:105) + O ( q ) (2.83)and I Schur SU (3) N f =6 SQCD w/. T dual (cid:3) ( a, b, c, d ) = cd + q / (cid:0) cd − + c − d (cid:1) ++ q (cid:0) c − d − χ ( a ) χ ( b ) + cdχ ( b ) + cdχ ( a ) + 3 cd + c d χ ( a ) χ ( b ) + c − d − +2 χ ( a ) χ ( b )) + q / (cid:0) cd + 2 c d − χ ( a ) χ ( b ) + c d + cd − χ ( b )+ d χ ( a ) χ ( b ) + c − dχ ( b ) + c − χ ( a ) χ ( b ) + 2 c − d χ ( a ) χ ( b ) + 3 c − d + c − dχ ( a ) + c χ ( a ) χ ( b ) + 3 cd − + cd − χ ( a ) + d − χ ( a ) χ ( b ) (cid:1) + O ( q ) (2.84)= cd + q / cd ( c − + d − ) + q cd (cid:104) χ U (6)Adj + c − χ U (6) ∧ (cid:105) + q / cd (cid:104) ( c − + d − )( χ U (6)Adj + 1) + χ U (6) ∧ + χ U (6) ∧ (cid:105) + O ( q ) (2.85)respectively. Notice that χ U (6)[0 ... k ... ] p := ( χ U (6) ∧ ) p χ U (6) ∧ k and χ U (6) ∧ = c d − .When we add both fundamental Wilson loop and the dual ’t Hooft loop as shown inFig. 8 (Right) with a = b = 1, the answer is I Schur SU (3) N f =6 SQCD w/. T dual (cid:3) ◦ W (cid:3) ( a, b, c, d ) = q / ( cχ ( a ) + c dχ ( b ))21 q ( c − χ ( a ) + cd − χ ( a ) + c d − χ ( b ) + 2 c − dχ ( b ) + cd χ ( a ) + d χ ( a ) χ ( b ))+ q / (3 d − χ ( a ) χ ( b ) + 2 c − d − χ ( b ) + c d χ ( a ) χ ( b ) + c − d χ ( a ) χ ( b ) + c − dχ ( b )+ 5 cχ ( a ) + 3 cχ ( a ) χ ( b ) + c dχ ( b ) + d − χ ( a ) χ ( b ) + c − d χ ( a ) + c d χ ( a ) χ ( b )+ 2 c dχ ( a ) χ ( b ) + c − d − χ ( a ) χ ( b ) + c dχ (cid:48) ( b ) + 2 d − χ ( a ) χ ( b ) + 4 c dχ ( b )+ cχ (cid:48) ( a ) + cχ ( a ) + c − d − χ ( a )) + O ( q ) . (2.86)= q / cdχ U (6) + q (cid:104) cdχ U (6) ∧ + cd ( c − + d − ) χ U (6) (cid:105) + q / ( c − d ) (cid:104) d χ U (6)[20001] + d χ U (6)[01001] + χ U (6)[10100] + (1 + c d − ) χ U (6) ∧ + c χ U (6) (cid:105) + O ( q )(2.87)where we use the Boltzmann weight result for each crossing given in [1]. However, we canderive this result in a different way as shown in Sec. 3.6. In this section, we show several new kinds of skein relations based on the computationswhich the formula in (2.8) gives. They are new in the sense that both codimension twodefects (punctures) and codimension four defects (networks) are included in the relationsand, in addition, non-trivially related. In particular, we focus on the skein relation whatwe call digon type skein relations. They are associated to one puncture as shown in Fig. 9.
Ycca b
Figure 9: The local digon network with a puncture Y inside it.The strategy to find the skein relations is to compute the q -deformed Yang-Millscorrelators with every elementary pants network of the theory T S [ C ([1 N ] , [1 N ] , Y )] foreach puncture Y . We analyze the punctures other than the full, simple (instead given inSec. 3.1) and null/trivial punctures in A = su (4) (Sec. 3.2), A = su (5) (Sec. 3.3) and A = su (6) (Sec. 3.4) cases. From many examples, In Sec.3.5, we give a unified formulato generate all the skein relations. From the argument in the first part of Sec. 2, we havea natural conjecture that the number of the linearly independent skein relations (See theexplanation later) is equal to d [1 N ] C − d YC . See (B.1).Notice that we assume that all the skein relations admit the mirror operation defined22s M : Y, xcca b ←→ Y, x ∗ ccb a (3.1) M : x ←→ x ∗ = x − (3.2) M : cY ←→ c Y (3.3)and some skein relations are invariant and others gives the new relations. This operationis clearly involution and no more relations appear. We can check that the whole skeinrelations are closed under this operation in all the example below just by computing them.Hereafter, when we exhibit a skein relation, we abbreviate its mirror operated relation. Class S skein relations
We recall the concept of class S skein relations. This is the same as those used in thetheory of knots in mathematics. However, in our set-up, we have no three geometry aslong as we add surface operators or consider the multiple lines. For concrete examples,see the following sections or [26].Let W Γ ( { a } ; q ) be the expectation value of the Wilson operator associated with a net-work Γ on the punctured Riemann surface. { a } are all fugacities for the flavor symmetriesor holonomies around punctures of C .And let us consider two sub graphs γ A and γ B which allow the common puncturesinside them. For any pair of two graphs Γ A and Γ B which include γ A and γ B respectivelybut are same on removing these sub graphs, when the equality shown just below alwaysholds true, we identify γ A and γ B and write this as γ A ∼ γ B . The equality is W Γ A ( { a, c } ; q ) = F γ A → γ B ( c ; q ) W Γ B ( { a, c } ; q ) (3.4)where F γ A → γ B ( c ; q ) is a function of q and { c } which are holonomies around the puncturesincluded in γ A or γ B , independent of holonomies around other punctures and also deter-mined by γ A and γ B only. Now we have the equivalence relations ∼ and refer to them asthe “(class S) skein relations” hereafter.Finally, we make a few comments. γ B or Γ B may consist of sum of several networksand W is the homomorphism from the networks to the functions of q and all holonomies.23elated to this, when we present the skein relations, we must take some basis. Forexample, multiplying both hand sides by the some common factors gives the same relation.When there are two independent relations, we can take some linear combination of twobut the new relation is not linearly independent. This point will be important later inSec. 3.5.In all examples we know, F ( c ; q ) is a polynomial of q ± and c . In addition, we can takethe basis such that F ( c ; q ) = F ( c ; q − ) and the invariance under the permutations of c inany fixed (simple) flavor symmetry always hold true. The former property comes from thesymmetry (an assumption, however) of the 2D q -deformed Yang-Mills theory. Actually,we can compute the skein relations in the 2D topological q -deformed Yang-Mills theorynot in the Schur indices because they differ by just the overall factors. Furthermore, inmany cases, this relation is enough local and independent of the choice of C .The final comment is related to the class S set-up. Under the parameter identification q = e πib [26, 44] where b is a physical parameter in the Liouville/Toda CFT, the skeinrelations are common both in the CFTs and in the 2D q -deformed Yang-Mills theory. Thisis because the skein relations are expected to be the local relations about codimensionfour defects in the 6D N =(2 ,
0) SCFTs and to be independent of the four dimensionalglobal background geometries, namely, the choice of S b or S × q S . This puncture corresponds to Y = [ N − , U (1) N /U (1) ⊂ SU ( N )fugacity is expressed as c [ N − , = ( q N − c, q N − c, . . . , q − N +42 c, q − N +22 c, c − ( N − ) where c is the U (1) fugacity. In this case, we have already seen the examples in Sec. 2.4 and theconjecture (2.35) for general rank. We can read off the following relations from them. Ya + ba + ba b = c − a (cid:20) a + b − a (cid:21) q a + bY + c b (cid:20) a + b − b (cid:21) q a + b Y (3.5)where 1 ≤ a, b and a + b ≤ N − A case Let us see the non-trivial skein relations in the A = su (4) case. There are five types ofregular punctures, two of which are subjects. See the Table 6.First of all, we list up all the good theories in the sense discussed in Sec. 2.2.241 ] [21 ] [2 ] [31] [4][0, 0, 0] [-1, -1, -1] [-1, -2, -1] [-2, -2, -2] [-3, -4, -3][1, 2, 3] [1, 2, 2] [1, 1, 2] [1, 1, 1] [0, 0, 0]Table 6: The list for g = su (4) ( A ) : the puncture type Y , 2 ρ αL,Y and the numbers ofCoulomb branch operators with scaling dimensions 2 , , rank 3 T (3) A = T S [ C ([1 ] , [1 ] , [1 ])] (3.6) rank 2 T (2) A = T S [ C ([1 ] , [1 ] , [21 ])] (3.7) rank 1 T (1 − A = T S [ C ([1 ] , [1 ] , [2 ])] T (1 − A = T S [ C ([1 ] , [21 ] , [21 ])] (3.8) rank 0 F (bf) A = T S [ C ([1 ] , [1 ] , [31])] F (as) A = T S [ C ([1 ] , [21 ] , [2 ])] (3.9) Y = [ ]The flavor symmetry associated with this puncture Y = [2 ] is SU (2). The associatedfugacity is given by c [2 ] = ( q / c, q − / c, q / c − , q − / c − ). It is expected that there are 2independent skein relations. i Y
332 1 = 3 Y + χ SU (2) ( c ) 3 Y (3.10)and its mirror operated relation. Y = [ ]The flavor symmetry associated with this puncture Y = [21 ] is S ( U (1) × U (2)) (cid:39) U (1) × SU (2). The associated fugacity is given by c [21 ] = ( q / c , q − / c , c , c ). In particular,we introduce the natural recombined fugacities c := c and c (cid:48) := c / ( c c ) / = c c . It isexpected that there is 1 independent skein relation.25 c / Y
332 1 − c − / Y
331 2 = c / Y − c − / Y (3.11) Check for T (1 − A The above skein relations give some constraints on the elementary pants networks of T (1 − A = T S [ C ([1 ] , [21 ] , [21 ])]. The explicit flavor symmetry is given by SU (4) × U (1) B × SU (2) B × U (1) C × SU (2) C and we write the associated fugacities as a [1 ] = ( a , a , a , a )( a a a a = 1), b [21 ] = ( q / b, q / b − , b (cid:48) /b, /bb (cid:48) ) and c [21 ] = ( q / c, q / c − , c (cid:48) /c, /cc (cid:48) ).The relation (3.11) leads to the following equalities : I Schur T (1 − A w/. ℘ (1 , , ( a, b (cid:48) , c (cid:48) )= c − I Schur T (1 − A w/. ℘ (1 , , ( a, b (cid:48) , c (cid:48) ) + (cid:16) [2] q b − c + bcχ SU (2) B ( b (cid:48) ) − c − χ SU (4) ( a ) (cid:17) I Schur T (1 − A ( a, b (cid:48) , c (cid:48) )= b I Schur T (1 − A w/. ℘ (2 , , ( a, b (cid:48) , c (cid:48) ) + (cid:16) [2] q b − c + b − c − χ SU (2) C ( c (cid:48) ) − b χ SU (4) ( a ) (cid:17) I Schur T (1 − A ( a, b (cid:48) , c (cid:48) ) . (3.12)In this case, three elementary pants networks are equivalent up to flavor loops andthere is only one independent elementary pants network which coincides with the rankone. Check for F (as) A Let us focus on the rank 0 free SCFT F (as) A = T S [ C ([1 ] , [21 ] , [2 ])]. We can show thatall the elementary pants networks reduce to the flavor Wilson lines by applying the aboveskein relations and they reproduce the result (2.39). A case The next case we consider is the A = su (5) case. There are seven types of regularpunctures, four of which are subjects. See the Table 7.261 ] [21 ] [2
1] [31 ] [32] [41] [5][0,0,0,0] [-1,-1,-1,-1] [-1,-2,-2,-1] [-2,-2,-2,-2] [-2,-3,-3,-2] [-3,-4,-4,-3] [-4,-6,-6,-4][1, 2, 3, 4] [1, 2, 3, 3] [1, 2, 2, 3] [1, 2, 2, 2] [1, 1, 2, 2] [1, 1, 1, 1] [0, 0, 0, 0]Table 7: The list for g = su (5) ( A ) : the puncture type Y , 2 ρ αL,Y and the numbers ofCoulomb branch operators with scaling dimensions 2 , , , Y = [ ]The flavor symmetry associated with this puncture Y = [32] is U (1). The associatedfugacity is given by c [32] = ( qc , c , q − c , q / c , q − / c ). In particular, we introduce thenatural recombined fugacities c := c . It is expected that there are 4 independent skeinrelations. i-1& i-2 Y
442 2 = (cid:0) c + [2] q c − / (cid:1) Y + (cid:0) c − + [2] q c / (cid:1) Y . (3.13) Y
443 1 = (cid:0) c − / + [2] q c (cid:1) Y + c − / Y (3.14)and its mirror operated relation. ii c − / Y
331 2 − c / Y
332 1 = c − / Y − c / Y . (3.15)27 = [ ]The flavor symmetry associated with this puncture Y = [31 ] is S ( U (1) × U (2)) (cid:39) U (1) × SU (2). The associated fugacity is given by c [31 ] = ( qc , c , q − c , c , c ). In particular,we introduce the natural recombined fugacities c := c . It is expected that there are 3independent skein relations.Notice that the mirror operated relation for the second one in the category i is linearlydependent on the first and the second. i c Y
443 1 − c − Y
441 3 = [2] q c Y − c − Y (3.16)and Y
442 2 − [2] q c − Y
441 3 = c Y − [3] q c − Y . (3.17) ii We have the same skein relation (3.15). Y = [ ]The flavor symmetry associated with this puncture Y = [2
1] is S ( U (2) × U (1)) (cid:39) U (1) × SU (2). The associated fugacity is given by c [2 = ( q / c , q − / c , q / c , q − / c , c ). Inparticular, we introduce the natural recombined fugacities c := ( c c ) / and c (cid:48) i := c i /c for i = 1 ,
2. It is expected that there are 2 independent skein relations.28 Y
442 2 − cχ SU (2) ( c (cid:48) ) Y
443 1 = c − Y − c χ SU (2) ( c (cid:48) ) 4 Y (3.18)and c Y
443 1 − c − Y
441 3 = c χ SU (2) ( c (cid:48) ) 4 Y − c − χ SU (2) ( c (cid:48) ) 4 Y . (3.19) Y = [ ]The flavor symmetry associated with this puncture Y = [21 ] is S ( U (1) × U (3)) (cid:39) U (1) × SU (3). The associated fugacity is given by c [21 ] = ( q / c , q − / c , c , c , c ). In particular,we introduce the natural recombined fugacities c := c .It is expected that there is 1independent skein relation. i c Y
443 1 + c − Y
441 3 − Y
442 2 = c Y + c − Y . (3.20) A case The final example is A = su (6). There are eleven types of regular punctures, eight ofwhich are subjects. See the Table 8. 291 ] [21 ] [2 ] [2 ][0, 0, 0, 0, 0] [-1, -1, -1, -1, -1] [-1, -2, -2, -2, -1] [-1, -2, -3, -2, -1][1, 2, 3, 4, 5] [1, 2, 3, 4, 4] [1, 2, 3, 3, 4] [1, 2, 2, 3, 4][31 ] [321] [3 ] [41 ][-2, -2, -2, -2, -2] [-2, -3, -3, -3, -2] [-2, -4, -4, -4, -2] [-3, -4, -4, -4, -3][1, 2, 3, 3, 3] [1, 2, 2, 3, 3] [1, 1, 2, 2, 3] [1, 2, 2, 2, 2][42] [51] [6][-3, -4, -5, -4, -3] [-4, -6, -6, -6, -4] [-5, -8, -9, -8, -5][1, 1, 2, 2, 2] [1, 1, 1, 1, 1] [0, 0, 0, 0, 0]Table 8: The list for g = su (6) ( A ) : the puncture type Y , 2 ρ αL,Y and the numbers ofCoulomb branch operators with scaling dimensions 2 , , , , Y = [ ]The flavor symmetry associated with this puncture Y = [42] is S ( U (1) × U (1)) (cid:39) U (1).The associated fugacity is given by c [42] = ( q / c , q / c , q − / c , q − / c , q / c , q − / c ).In particular, we introduce c := c . It is expected that there are 7 independent skeinrelations. i-1 & i-2 Y
551 4 = ([3] q c − + c ) 5 Y + c Y (3.21) Y
552 3 = [3] q ( c + c − ) 5 Y + ([3] q + c ) 5 Y (3.22)30 i Y
442 2 − [2] q c Y
443 1 = c − Y − [3] q c Y . (3.23)and its mirror operated relation. iii c Y
332 1 − Y
331 2 = c Y − c − Y . (3.24) Y = [ ]The flavor symmetry associated with this puncture Y = [41 ] is S ( U (1) × U (2)) (cid:39) U (1) × SU (2). The associated fugacity is given by c [41 ] = ( q / c , q / c , q − / c , q − / c , c , c ).In particular, we introduce c := c . It is expected that there are 6 independent skeinrelations. i c − / Y
551 4 − c / Y
554 1 = [3] q c − / Y − c / Y (3.25)31nd c Y
553 2 − [3] q c Y
554 1 = 5 Y − c Y . (3.26) ii We have the same skein relation (3.23). iii
We have the same skein relation (3.24). Y = [ ]The flavor symmetry associated with this puncture Y = [3 ] is S ( U (2)) (cid:39) SU (2). Theassociated fugacity is given by c [3 ] = ( qc, c, q − c, qc − , c − , qc − ). It is expected that thereare 6 independent skein relations. i Y
554 1 = 5 Y + [2] q χ SU (2) ( c ) 5 Y (3.27)and Y
552 3 = ([3] q + χ SU (2) ( c )) 5 Y + [2] q χ SU (2) ( c ) 5 Y (3.28)and their mirror operated relations. 32 i χ SU (2) ( c ) Y
443 1 − Y
442 2 = χ SU (2) ( c ) 4 Y − Y (3.29)and its mirror operated relation. Y = [ ]The flavor symmetry associated with this puncture Y = [321] is S ( U (1) × U (1) × U (1)) (cid:39) U (1) × U (1) . The associated fugacity is given by c [321] = ( qc , c , q − c , q / c , q − / c , c − c − ).It is expected that there are 4 independent skein relations. i-1 [2] q c / Y
554 1 − c / Y
553 2 + c − / Y
551 4= [3] q c / Y + [2] q c − / Y (3.30)and its mirror operated relation. i-2 c / Y
554 1 − c / Y
553 2 + c − / Y
552 3 − c − / Y
551 433 c / Y − c − / Y . (3.31) ii c − Y
441 3 + c Y
443 1 − Y
442 2 = c − Y + c Y . (3.32) Y = [ ]The flavor symmetry associated with this puncture Y = [31 ] is S ( U (1) × U (3)) (cid:39) U (1) × SU (3). The associated fugacity is given by c [31 ] = ( qc , c , q − c , c , c , c ). In particular,we introduce the natural recombined fugacities c := c . It is expected that there are 3independent skein relations. i c − / Y
551 4 + [2] q c / Y
554 1 − c / Y
553 2= [3] q c / Y + [2] q c − / Y (3.33)and its mirror operated relation. 34 i We have the same skein relation (3.32) with the replacement of c by c . Y = [ ]The flavor symmetry associated with this puncture Y = [2 ] is S ( U (3)) (cid:39) SU (3).The associated fugacity is given by c [2 ] = ( q / c , q − / c , q / c , q − / c , q / c , q − / c )( c c c = 1). It is expected that there are 3 independent skein relations. i Y
553 2 − χ SU (3) ( c ) Y
554 1 = 5 Y − χ SU (3) ( c ) 5 Y (3.34)and its mirror operated relation. Y
551 4 − Y
554 1 = χ SU (3) ( c ) 5 Y − χ SU (3) ( c ) 5 Y . (3.35) Y = [ ]The flavor symmetry associated with this puncture Y = [2 ] is S ( U (2) × U (2)) (cid:39) U (1) × SU (2) × SU (2) . The associated fugacity is given by c [2 ] = ( q / c , q − / c , q / c , q − / c , c , c ).In particular, we introduce the natural recombined fugacities c := ( c c ) / and c (cid:48) i := c i /c (or c (cid:48) = ( c /c ) / and c (cid:48) = ( c /c ) / ). It is expected that there are 2 independent skeinrelations. i c − Y
551 4 + cχ SU (2) ( c (cid:48) ) Y
554 1 − Y
553 235 c − χ SU (2) ( c (cid:48) ) 5 Y + c χ SU (2) ( c (cid:48) ) 5 Y (3.36)and its mirror operated relation. Y = [ ]The flavor symmetry associated with this puncture Y = [21 ] is S ( U (1) × U (4)) (cid:39) U (1) × SU (4). The associated fugacity is given by c [21 ] = ( q / c , q − / c , c , c , c , c ).In particular, we introduce the natural recombined fugacities c := c .It is expected thatthere is 1 independent skein relation. i c / Y
554 1 − c − / Y
551 4 + c − / Y
552 3 − c / Y
553 2= c / Y − c − / Y . (3.37)
Reproduction of previous results on flavor Wilson loops
Using the above skein relations, we can reproduce (2.65), (2.55) and (2.72).
At this stage, we write down a formula to unify the above skein relations. We expect thatthis formula is valid for every puncture in each rank in the A = su type.36 otations and symbols Let Y be [ n , n , . . . , n k ] = [ m d m d · · · m d t t ]-type. Then, wehave a Young diagram whose i -th column is given by n i . To each box y = (cid:3) ( a y ,h y ) ∈ Y ( a y and h y specify the position of a box y from the left and the below, respectively ), weassign infinitely many digon type skein relations as we explain later. They are not linearlyindependent each other and, indeed, there are only finite independent skein relations. Y = [32 d = { , , } abc de fg h y ( a,h ) a (1 , b (1 , c (1 , d (2 , e (2 , f (3 , g (3 , h (4 , f y g y (cid:96) y c y A = su (8). The last line in the right table represents the number of independent skeinrelations. When c y = 8, the relations are trivial in the sense of the simplicity shown in(3.42). Otherwise, they are non-trivial ones. (cid:96) y is the number of linearly independent skeinrelations. In this case, there are five non-trivial skein relations. Notice that d [1 ] C − d YC = 5.See also the later explanations.To state the rule generating skein relations, let us introduce symbols f y = f ( (cid:3) ( a,h ) ) := { n ∈ { , , . . . , t } s.t. (cid:80) n − s =1 d s < a ≤ (cid:80) ns =1 d s } , g y = g ( (cid:3) ( a,h ) ) := a − (cid:80) f y − s =1 d s and (cid:96) y := m f y − h + 1 = n i − h + 1. This assignment corresponds to the g y -th weight ofthe f y -th flavor symmetry U ( d f y ) and the (cid:96) y -th weight of the m f y -dimensional irreduciblerepresentation of SU (2) IRR . In addition, we use c y := N + 1 − h y . See Table. 9 for anexample.Using the same notations introduced above, the fugacity x Y associated with the punc-ture Y is given by { q mfy +12 − (cid:96) y x U ( d fy ) fy g y = q mfy +12 − (cid:96) y x SU ( d fy ) fy g y x U (1) fy } y ∈ Y . x SU ( d s ) ds i =1 , ,...,N is thefugacity of SU ( d s ) i.e. belongs to the Cartan of it and satisfies (cid:81) Ni =1 x SU ( d s ) ds i = 1. x U (1) s is that of the s -th U (1) for s = 1 , , . . . , t but there is a constraint (cid:81) ts =1 ( x U (1) s ) m s d s = 1coming from the fact that x Y is the fugacity of SU ( N ).Next, we assign the U ( N ) character for any element of the su ( N ) weight lattice. Usingthe Weyl’s character formula, we can define it as (cid:101) χ U ( d ) λ ( x ) := (cid:88) w ∈W σ w x w ( ρ + λ ) (cid:88) w ∈W σ w x w ( ρ ) = ( x ) | λ | (cid:88) w ∈W σ w X w ( ρ + λ ) (cid:88) w ∈W σ w X w ( ρ ) (3.38)where x = x U ( N ) = ( x X , x X , . . . , x X N ) = x X SU ( N ) with the constraint (cid:81) Ni =1 X i = 1, W is the Weyl group of su ( N ) which is equivalent to the order N permutation group and σ w = ( − (cid:96) ( w ) . (cid:96) ( w ) is the length of w i.e. the minimum number of generators of W togenerate w . 37 − z z qz q − z q z q − z q z z −→ ⊗ C U (1) ⊕ ⊗ U (2) ⊕ ⊗ C U (1) under SU (2) R × S ( U (1) × U (2) × U (1) )Table 10: For Y = [32 SU (8) fugacity x [32 consists of the above ones. Here weintroduce the simplified notations as z = x U (1) = c , z = x U (2) = c c , z = x U (2) = c c − and z = x U (1) = c where z z z z = c c c = 1. On the right hand side, thedecomposition of the define representation ( (cid:3) = = R ( ω )) is shown.Let us see some examples. In the case u (2), (cid:101) χ ( n − ω = x n − χ SU (2) n (cid:101) χ − ω = 0 (cid:101) χ ( − n − ω = − x − n − χ SU (2) n (3.39)for n = 1 , , . . . . In the case u (3), (cid:101) χ nω + mω = x n + m × χ R ( nω + mω ) I : n, m ≥ − χ R (( − n − ω +( n + m +1) ω ) II : n ≤ − , n + m ≥ − χ R ( mω +( − n − m − ω ) III : m ≥ , n + m ≤ − − χ R (( − m − ω +( − n − ω ) IV : n, m ≤ − χ R (( − n − m − ω + nω ) V : n ≥ , n + m ≤ − − χ R (( n + m +1) ω +( − m − ω ) VI : m ≤ − , n + m ≥ −
10 otherwise . (3.40) Conjectural unified form of skein relations
Now, the skein relations are given by
DSR Y { b } ,β : c y (cid:88) p =0 ( − p (cid:96) y − (cid:89) i =1 [ p + b i ] q (cid:101) χ U ( d fy ) pω + ω gy + β ( x ) Yc y c y p c y − p = 0 (3.41)where b , b , . . . , b (cid:96) y − are arbitrary integers and β is an arbitrary element of the su ( N )root lattice. We may absorb ω g y to β and instead β is any element of weight lattice, wheretwo distinct y with same f y and h y give the same skein relations. (cid:81) i =0 [ p + b ] q =: 1 when38 m IIIIII IV V VI33 8 n ( C )( B )( A )Figure 10: The su (3) weight lattice (Left). • gives 0. • , (cid:63) and (cid:78) respectively correspondto the charges 0 , Z . On the walls, namely, theboundaries of the Weyl chambers, the characters vanish. The prefactors of the skeinrelations (3.34), its mirror and (3.35) correspond to the dotted boxes ( C ),( A ) and ( B )respectively (Right). (cid:96) y = 1 and ω := 0. Notice also that Ypp p = pY Yppp p Y . (3.42) Linearly independent choice
Although we generate infinitely many skein relationsfor digons with generic puncture inside it, there are linearly dependent. Let us remark onthe choices of linearly independent skein relations.Since [ k ] q [2] q = [ K +1] q +[ k − q for any integer k , the three relations DSR Y { b } ,β , DSR Y { b (cid:48) } ,β and DSR Y { b (cid:48)(cid:48) } ,β where b j = b (cid:48) j + 1 = b (cid:48)(cid:48) j for some j and b i = b (cid:48) i = b (cid:48)(cid:48) j for all i ( (cid:54) = j ) are notlinearly independent. Then we can restrict b form Z (cid:96) − onto { , } (cid:96) − for example. Fur-thermore, there is the permutation symmetry acting on b i . Combining two properties,we expect that the representative skein relations are labelled by the set { , } (cid:96) − dividedby the permutation. The complete invariant on this set is the summation of { b } and wesee that there are (cid:96) relations at least to generate all b . In addition, we can find thatthey are linearly independent in general cases. In the same way, we see that (cid:101) χ U ( d ) pω + ω g with g = 0 , , , . . . , d − β . 39otice also that, when h = 1, we also have the trivial skein relations : YNNN − a a = Ya = χ SU ( N ) ∧ a (cid:3) = R ( ω a ) ( x Y ) Y . (3.43)In summary, each y with h y > (cid:96) y = n a y − h y + 1 non-trivial ones which arelinearly independent each other. Then, the total number of non-trivial skein relations isgiven by (cid:88) y ( n a y − h y + 1) = 12 (cid:88) a =1 n a ( n a −
1) = 12 (cid:32)(cid:88) a =1 n a − N (cid:33) (3.44)and this is exactly equals to d [1 N ] C − d YC which is the difference between the complexdimension of the 4D local Coulomb branch contributed by the puncture Y and that bythe full puncture. See (B.1).Let us make two comments. First, only the [ k, N − k ]-type punctures ( N > k ≥ N/ , p, N − − p )-type elementary pantsnetworks in the 4D good theory T S [ C ( Y , Y , [ k, N − k ])] for any Y and Y . Second, it isexpected that these skein relations are valid for the Liouville-Toda CFT. This is becausethe skein relations are determined locally around the line operators and then independentof the choice of the 4-manifolds on which the SCFTs are defined. If we consider the S partition functions instead of the superconformal indices, the corresponding 2D theoriesare considered to be the Liouville-Toda CFT [15, 16]. We see one simple application of the above skein relations. They allow us to computethe q -deformed Yang-Mills correlator with a complicated network operator easier. Let usrewrite the network on the right hand in Fig. 8. By using the crossing resolution typeskein relation (See [26] for example)11 = q N
11 21 1 − q N −
11 (3.45)40nd then applying (3.5) several times, we have= cd [ N − q + d − − [2] q c − ( N − + (cid:2) c − ( N − dχ (cid:3) ( b ) + c − ( N − χ (cid:3) ( a ) − c [ N − q χ (cid:3) ( a ) (cid:3) (3.46)where we have used χ (cid:3) ( c [ N − , ) = [ N − q c + c − ( N − and the topological property ofcorrelators allowing the punctures to move around continuously. In particular, in N = 3 case, they reproduce the result (2.87) again. Acknowledgements
The author is supported by the Advanced Leading GraduateCourse for Photon Science, one of the Program for Leading Graduate Schools lead byJapan Society for the Promotion of Science, MEXT and also the World Premier Interna-tional Research Center Initiative (WPI), Kavli IPMU, the University of Tokyo.
A Lie algebra convention
First of all, Π( R ), P g + and ρ denote weights of the representation R , the set of dominantweights and the Weyl vector = rk g (cid:88) a =1 ω a = 12 (cid:88) α ∈ ∆ + g α respectively. And R ( λ ) denotes theirreducible representation associated with a dominant weight λ , λ R does the dominantweight to R conversely.In this paper, we use several conventions on representations. To express the irre-ducible representation, for example, we use the Dynkin weights (labels) [0 , . . . , k , . . . ,
0] or(0 · · · k · · · R ( ω k ) and the dimension ∧ k N as necessary. For g = su ( N )case, we also use the Young diagram ∧ k (cid:3) . A.1 A -type Lie algebra convention Let { α a } a =1 , ,..., rk g = N − be a set of chosen positive simple roots. Notice all choices areequivalent under the Weyl reflection actions. ω α =1 , ,...,N − are fundamental weights, h i =1 , ,...,N are weights in Π( R ( ω ) = (cid:3) ). Using the orthonormal basis e i =1 , ,...,N in R N satisfying ( e i , e j ) = δ i,j , there are relations among them as h a = ω a − ω a − where ω N = ω = 0, α a = e a − e a +1 and h i = e i − N N (cid:88) i =1 e i . Notice that there is another crossing at the point we have removed for the disk representation.
The perfect order of the indices of h i is determined by the partial order in the weight lattice. λ a := ( λ, α a ). In other words, λ = (cid:80) N − a =1 λ a ω a . Then,the dominant weight set is given by P su ( N )+ = { λ | λ a ∈ Z ≥ for ∀ a } . (A.1)Notice that ρ a = 1 for all a . Notice also that λ ˆ i := ( λ R , h i − h N ) means the number of theboxes in the i -th row of the corresponding Young tableau of the irreducible representation R ans satisfies λ i = λ ˆ i − λ ˆ i +1 . We also define a symbol | λ | := N (cid:88) i =1 λ ˆ i .The quantum dimension of R ( λ ) is given bydim R = (cid:89) α ∈ ∆ + g [( λ R + ρ, α )] q [( ρ, α )] q . (A.2) B The formula for Coulomb and Higgs branch di-mension associated to regular punctures
Let B Coulomb and M Higgs be the 4D Coulomb branch and the 4D Higgs branch respectively.According to [13, 45, 46], the contribution from the Y -type puncture is given as d YC := 12 (cid:32) N − (cid:88) i n i (cid:33) (B.1) d YH := 12 (cid:32)(cid:88) i s i − N (cid:33) (B.2)where [ n ] is the partition of N and { s } is its dual representation. See Fig. 11.Figure 11: N = 13 example : the corresponding standard partition is given by [ n ] =[4 , , , ,
1] = [4 ] and its dual representation is { s } = { , , , } = { } . d YC = p Y = { , , , , , , , , , , , , } .The total dimensions are given bydim C B Coulomb = (cid:88) A dim C B Coulomb ( Y A ) − χ C dim R g (B.3)dim H M Higgs = (cid:88) A dim H M Higgs ( Y A ) + rk g (B.4)where χ C is the Euler number of the Riemann surface on ignoring the punctures. Keepin mind that the analysis up to now are locally and do not give any global information.42owever, the 4D Coulomb branch has more information about the scaling dimensionof the Coulomb branch operators, namely graded by the scaling dimension. By analyzingthe Seiberg-Witten curve and its relation to Hitchin systems, we see d YC,k = p Yk = k − min { a | a (cid:88) b =1 s b ≥ k } for k = 2 , , . . . , N (B.5)and dim C B ( k )Coulomb = (cid:88) A d YC,k − (2 k −