Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks
Jiakang Bao, Omar Foda, Yang-Hui He, Edward Hirst, James Read, Yan Xiao, Futoshi Yagi
DDessins d’Enfants, Seiberg-Witten Curves and
Conformal Blocks
Jiakang Bao, a Omar Foda, b Yang-Hui He, a,c,d,e
Edward Hirst, a James Read, f YanXiao, g Futoshi Yagi ha Department of Mathematics, City, University of London, EC1V 0HB, UK b School of Mathematics and Statistics, University of Melbourne, Royal Parade, Parkville,VIC 3010, Australia c Merton College, University of Oxford, OX1 4JD, UK d London Institute of Mathematical Sciences, 35a South St, Mayfair, London, W1K 2XF, UK e School of Physics, NanKai University, Tianjin, 300071, P.R. China f Pembroke College, University of Oxford, OX1 1DW, UK g Department of Physics, Tsinghua University, Beijing, 100084, China h School of Mathematics, Southwest Jiaotong University, West Zone, High-Tech District,Chengdu, Sichuan, 611756, China
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , futoshi [email protected] Abstract:
We show how to map Grothendieck’s dessins d’enfants to algebraic curvesas Seiberg-Witten curves, then use the mirror map and the AGT map to obtain thecorresponding 4d N = 2 supersymmetric instanton partition functions and 2d Virasoroconformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures onthe sphere. We find that the parametrizations obtained from a dessin should be relatedby certain duality for gauge theories. Then we will show that some dessins couldcorrespond to conformal blocks satisfying certain rules in different minimal models. a r X i v : . [ h e p - t h ] F e b ontents (4) ∩ Γ(2) 383.6 Minimal Models and General Dessins 40
A The B-model and Omega Deformations 43B Brane Configurations 44
B.1 The Type IIA Brane Configuration 44B.2 The M-theory Brane Configuration 46
C Congruence Subgroups of the Modular Group 46D Elliptic Curves and j -Invariants 47E Elliptic Functions and Coulomb Moduli 49 E.1 The Elliptic Integral of First Kind 49E.2 The Elliptic Logarithm 50
References 51 – 1 – edicated to the memory ofour dear friend,Professor Omar Foda, a gentleman and a scholar . . .
Introduction and Summary
Consider a 4-point conformal block (CB) in a 2d conformal field theory (CFT) based on W × H , where W is the Virasoro algebra and H is the Heisenberg algebra. Using theAlday-Gaiotto-Tachikawa (AGT) correspondence [1], this is identified with an instantonpartition function in an N = 2 supersymmetric Yang-Mills (SYM) theory, with anSU(2) gauge group and four fundamental hypers.The low energy physics of this gauge theory is described in terms of a Seiberg-Witten (SW) curve and the SW differential on it [2, 3]. Then in [4], a method ofinstanton counting was introduced to find these low energy solutions to SW theories.Later, the S-duality for N = 2 supersymmetric systems was studied in [5]. In recentworks [6–12], connections between Grothendieck’s dessins d’enfants on the one handand 4d N = 2 SYM on the other were studied. In this note, we further explore theseconnections and extend them to 2d conformal field theory. We focus on six specifictrivalent dessins with 4 punctures on the sphere, which, as we will see, are related toa simple and important class of 4d N = 2 SYM theories and conformal blocks in 2dconformal field theory.From these dessins, we obtain algebraic curves that we interpret as SW curves of 4dSU(2) N = 2 N f = 4, SYM theories. These curves are given in terms of six parameters,four mass parameters ( µ , µ , µ , µ ), a parameter ζ and a modulus U . We write thesecurves in the form that appears in [13, 14], and use their mirror map to translate theabove parameters to those characterizing the 4d instanton partition function of a 4d N = 2 gauge theory. In particular, we map the modulus U to the Coulomb parameter a . Following that, we use the AGT dictionary to interpret the result in 2d CFT terms.Let us take a closer look at the six parameters for the SU(2) gauge theory. With N f = 4, the theory has an SO(8) ⊃ SU(2) flavour symmetry. Then the mass parame-ters of the four hypers could be indentified as the charges of the primaries in Liouvilletheory under AGT correspondence [1]. Following [15, 16], the AGT map could also leadto diagonal minimal models by further restrictions on the partition pairs. As usual,we would arrange the poles of the SW curves at z = 0 , , ∞ and ζ . This ζ is nothingbut the UV gauge coupling τ via ζ = e πiτ . For each dessin, we find that ζ could haveseveral different values but these values enjoy certain triality.Recall that the Coulomb parameter a denotes the vev of the adjoint scalar φ ,or equivalently, a could be obtained by integrating the SW differential along the so-called A -cycle on SW curve. Such supersymmetric vacua can be gauge invariantlyparametrized by u = (cid:104) tr φ (cid:105) / a up to quantum corrections. As we will discuss in § U , which will appear in the parametrization of the curve, is linearin the Coulomb moduli u . In fact, as we will see, each dessin gives a family of solutions– 3 –or the gauge theory parameters, and indeed, we would have the same correspondingdessin under the change m i → km i , a → ka , U → k U for k ∈ R . This is consistentwith their mass dimensions.The above discussions can go the other way as well. Starting from the CBs inCFTs, we can write down the Nekrasov partition functions under the AGT dictionary.This 4d partition function can also be lifted to 5d, which leads to topological stringpartition functions and SW curves. As the SW differential and the Strebel differentialfrom the dessin side are both quadratic, the gauge theories are naturally related todessins.Since the instanton partition functions with extra conditions on the Young tableauxpairs could be mapped to conformal blocks in minimal models [15, 16], we can thencheck whether (the parametrizations from) the dessins could correspond to such CBsin minimal models. As we will see in § ∼ § Proposition 1.1.
There is a subset of trivalent dessins with four punctures on thesphere such that each dessin therein corresponds to a family of 4-point conformal blocksfor (A-series) minimal models.
In principle, there are countably infinite such dessins although only a small partof them have been studied in details. As we will show, not all the dessins would giveminimal models. For those in the above subset, we will also determine the familiesof CBs they correspond to. In particular, we will illustrate this proposition with sixwell-known dessins as examples. Notice that we can only say that this correspondenceis not for all dessins. On the minimal model side, it is still not determined whetherthis subset of dessins could recover all the CBs or just part of them.Now that the dessins are rigid, it would be interesting to understand whether theparametrizations from these dessins are special for gauge theories and (R)CFTs infuture. It would also be natural to extend this for dessins with more faces and othergauge theories. So far, we only get the values for ζ from the dessins which are relatedby triality as briefly mentioned above, and the calculations should be non-perturbative.More details are worth studying for future works. It would be nice to even know moreabout the bulk of AdS through holographic duality. On the other hand, we wouldlike to know whether the physical theories could in turn help us learn more about thedessins. For example, we know that not all dessins can give CBs in minimal models,but what kind of dessins have this correspondence is still unknown. Moreover, the– 4 –orenthedieck-Teichmuller group, which is related to the Galois group Γ( ¯ Q : Q ) anddessins, may be related to the monodromy of CBs in RCFT [17]. This might lead todeeper connections between the mathematics and the physics.The paper is outlined as follows. In §
2, we start from the CFT side and reviewthe AGT correspondence to get the corresponding partition functions. Then fromA-model topological strings, we obtain the SW curve for SU(2) with 4 flavours andthence the dessins. In §
3, we reverse the discussion and contemplate six of the dessinswhich would yield specific parametrizations for SW curves. Then we will study if theseparametrizations would give conformal blocks in minimal models following the AGTdictionary. In the appendices, we give some background on brane systems as well aselliptic curves and elliptic functions.This work is dedicated to the memory of Professor Omar Foda, who was instru-mental in initiating this project.
Before we derive the results in 2d CFT from the 6 dessins with 4 punctures on thesphere, we give a brief review of different subjects including CBs, topological strings,SW curves and dessins, following a route map from CBs to dessins.
The connection between 2d Liouville CFTs and SU(2) supersymmetric gauge theoriesin 4d with N = 2 was first raised in [1]. The free parameters of the two areas arenaturally mapped to each other under AGT correspondence. Later in [15, 16], theAGT correspondence was also extended to minimal models by further restrictions onthe partitions/Young diagrams. We first start with the CFT side under the Coulomb-gas formalism. Conformal Blocks
Conformal blocks form a basis of the vertex operator (VO) al-gebra, used when performing a particular operator product expansion (OPE) of a cor-relation function. They are a key ingredient in the conformal bootstrap approach tocalculating these correlators in 2d CFTs. Global conformal Ward identities of theCFT allow 2-point functions to be completely determined, whilst 3-point functions tohave fixed results up to their respective structure constants. Thus when calculatingan N ≥ { , ζ, , ∞} , leaving a single‘cross-ratio’ coordinate ζ for the conformal blocks to be a function of. Explicitly (cid:104) (cid:89) i =1 V i (cid:105) = (cid:88) R,R C R C R B R ( ζ ) B R ( ζ ) (2.1)for Virasoro operators V i . The sum is over the fields’ representations in the left andright Virasoro algebras, denoted R, R ; with the sum including structure constants, C ijk ,and conformal blocks, B . These highest-weight representations of the Virasoro algebraare described in terms of Verma modules. They are generated by primary states andare irreducible in the absence of degenerate fields.Since fields in a correlator can be permuted without change to the result, this trans-lates into allowing different OPEs of the same correlator as different conformal blockbases are used for expansion. The equivalence of these OPEs leads to ‘crossing symme-try’ and introduces additional consistency constraints which allow structure constantsand block dimensions to be calculated. This is the conformal bootstrap methodol-ogy, and leaves calculation of conformal blocks as the final ingredient for computingcorrelators [19].Conformal blocks are traditionally computed via Zamolodchikov recursion meth-ods, however in the cases of degenerate fields in the correlators the BPZ equationsprovide a shortcut to finding them. In special cases these blocks can be expressedsimply - for example a 4-point function on the sphere with one degenerate field (witha 2nd order null vector) can be expressed in terms of hypergeometric functions.The AGT correspondence then makes a connection between the conformal dimen-sions of the fields in the correlator, and the coordinate ζ , with parameters arising in– 6 –ekrasov instanton partition functions, as subsequently described. With the help ofthis correspondence we then compute the conformal blocks associated to the 6 dessinsconsidered in this study. The Nekrasov Partition Function
For generic vev a , the general N = 2 lowenergy effective action reads S eff = (cid:90) d x d θ F (Ψ)(+c.c.) , (2.2)where Ψ is the N = 2 V-plet, and the holomorphic function F is known as the prepo-tential . First conjectured in [4] and then proven in [20], the prepotential can be solvedby F = lim (cid:15) , → (cid:15) (cid:15) log Z Nek , (2.3)where (cid:15) i ’s are known as the deformation parameters, and Z Nek is the
Nekrasov partitionfunction , which reads Z Nek = Z tree Z Z inst , where Z tree/1-loop is the tree/1-loop levelpartition function and Z inst denotes the contribution from instantons.We will now focus on the instanton partition function Z inst . For SU(2) quivertheories, the Coulomb branches are parametrized by the Coulomb moduli (cid:126)a = ( a , a ) =( a, − a ). Each Coulomb modulus is associated with a Young tableau Y , in which everybox is labelled by a pair s = ( i, j ) to denote its position. Hence, the instanton partitionfunction depends on (cid:126)Y = ( Y , Y ), the vev a , and possibly the mass m of matter in thetheory. Let us define [14] E ( a, Y , Y , s ) ≡ a − (cid:15) L Y ( s ) + (cid:15) ( A Y ( s ) + 1) (2.4)with L Y ( s ) = k i − j, A Y ( s ) = k (cid:48) j − i, (2.5)where k i is the length of i th row of Y , and k (cid:48) j is the height of j th column of Y . Let I, J label the gauge nodes. Then z bifund ( a I , (cid:126)Y I ; a J , (cid:126)Y J ; m ) ≡ (cid:89) i,j =1 (cid:89) s ∈ Y Ii ( E ( a Ii − a Jj , Y Ii , Y Jj , s ) − m ) (cid:89) s ∈ Y Jj ( (cid:15) − E ( a Jj − a Ii , Y Jj , Y Ii , s ) − m ) , (2.6)where (cid:15) = (cid:15) + (cid:15) . For (anti-)fundamentals, z fund ( (cid:126)a, (cid:126)Y , m ) ≡ (cid:89) i =1 (cid:89) s ∈ Y i ( φ ( a i , s ) − m + (cid:15) ) , z antifund ( (cid:126)a, (cid:126)Y , m ) ≡ z fund ( (cid:126)a, (cid:126)Y , (cid:15) − m ) , (2.7)– 7 –here φ ( a i , s ) = a i + (cid:15) ( i −
1) + (cid:15) ( j − z adj ( (cid:126)a, (cid:126)Y , m ) ≡ z bifund ( (cid:126)a, (cid:126)Y ; (cid:126)a, (cid:126)Y ; m ) , z vec ( (cid:126)a, (cid:126)Y ) ≡ z adj ( (cid:126)a, (cid:126)Y , . (2.8) The AGT correspondence
The (chiral) VO can be written as V α =: e αφ : forsome free scalar φ . If we introduce some background charge Q , by considering theOPE between stress tensor and the VO, we get the conformal dimension of V α , whichreads ∆ α = α ( Q − α ). Likewise, the OPE between stress tensors yields the centralcharge c = 1 + 6 Q .Now we are ready to bridge the CBs and instanton partition functions. Originally,this was done for Liouville theory in [1]. We can fix the scale by setting b = (cid:15) √ (cid:15) (cid:15) ,where Q = b + b and b is the parameter coming from the Liouville potential. Therefore, Q = (cid:15) + (cid:15) √ (cid:15) (cid:15) ≡ (cid:15) √ (cid:15) (cid:15) . Consider a quiver consisting of an SU(2) gauge group with 2SU(2) antifundamentals and 2 SU(2) fundamentals with mass parameters µ , and µ , respectively. Then the instanton partition function reads Z inst = (cid:88) Y , e π i τ ( | Y | + | Y | ) (1 − e π i τ ) ( µ + µ )(2 (cid:15) − ( µ + µ )) z vec ( (cid:126)a, (cid:126)Y ) z matter , (2.9)where the denominator correpsonds to the decoupling of a U(1) factor, and z matter = z antifund ( (cid:126)a, (cid:126)Y , µ ) z antifund ( (cid:126)a, (cid:126)Y , µ ) z fund ( (cid:126)a, (cid:126)Y , µ ) z fund ( (cid:126)a, (cid:126)Y , µ ) . (2.10)The instanton number | Y i | is the number of boxes in Y i . Then under the following AGTdictionary, µ √ (cid:15) (cid:15) = α + α − Q , µ √ (cid:15) (cid:15) = α − α + Q , µ √ (cid:15) (cid:15) = α + α − Q ,µ √ (cid:15) (cid:15) = α − α + Q , a √ (cid:15) (cid:15) = α int − Q , e π i τ = ζ, (2.11)the instanton partition function is equal to B α int ( α i | ζ ), where the conformal block from (cid:104) V α V α V α V α (cid:105) as in (2.1) can be written as B = ζ ∆ α int − ∆ α − ∆ α B α int ( α i | ζ ) and “int”stands for (the VO in) the intermediate channel. One may check this perturbatively,and at level | Y | max , B α int and Z inst should agree up to O (cid:0) ζ | Y | max +1 (cid:1) [1, 21]. Notice thatwhen we have c = 1 CBs, viz, Q = 0, the AGT relation is simplified to µ √ (cid:15) (cid:15) = α + α , µ √ (cid:15) (cid:15) = α − α , µ √ (cid:15) (cid:15) = α + α , µ √ (cid:15) (cid:15) = α − α , a √ (cid:15) (cid:15) = α int . (2.12)– 8 –t is also possible to build a similar correspondence between gauge theories andminimal models. In [15], it was shown that Z inst should recover the CBs for minimalmodels if we put further restrictions to the Young tableaux pairs known as the Burgecondition . For a minimal model, we may write the central charge as c = 1 − p (cid:48) − p ) p (cid:48) p , (2.13)where p (cid:48) , p are coprime integers and p (cid:48) > p >
1. The spectrum is finite and all the VOshave conformal dimensions ∆ r,s = ( p (cid:48) r − ps ) − ( p (cid:48) − p ) p (cid:48) p (2.14)for integers r, s and 1 ≤ r < p , 1 ≤ s < p (cid:48) . In other words, they should all live in theKac table. Then the instanton partition function leads to well-defined A-series minimalmodels under (2.11) if the partitions Y , are restricted to be Burge pairs , that is, theysatisfy [15] Y ,R − Y ,R + s − ≥ − r, Y ,R − Y ,R + p − s − ≥ − p (cid:48) + r, (2.15)where Y i,R denotes (the number of boxes in) the R th row of Y i . In particular, thedeformation parameters can now be written in terms of the screening charges as (cid:15) √ (cid:15) (cid:15) = − i (cid:114) pp (cid:48) , (cid:15) √ (cid:15) (cid:15) = i (cid:115) p (cid:48) p . (2.16) For type II string/M-theory (whose brane configurations are discussed in Appendix B)compactified on a Calabi-Yau 3-fold, the amplitudes at genus g correspond to the A-model string amplitudes of the CY which enumerates the holomorphic functions fromgenus g Riemann surfaces to the CY [22, 23]. The topological amplitudes for toric CYthreefolds can be computed by topological vertices introduced in [24–26]. A topologicalvertex is a trivalent vertex as the (black) dual graph of the (grey) toric diagram: C Y Y Y Y Y Y , (2.17)– 9 –here Y i ’s are the Young tableaux associated to the legs, and C Y Y Y ( q ) is the factorassociated to the vertex, which can be expressed in terms of Schur and skew-Schurfunctions [24]. Albeit not labelled explicitly, each leg also has a direction such thatthe three legs attached to the same vertex all have outcoming or incoming directions.Then each leg is assigned a vector v i = ( v i , v i ) in that direction, such that the sum ofthe three vectors vanishes due to charge conservation and det( v i , v i +1 ) = ± i ∈ Z ).Now two topological vertices Y Y Y (cid:48) Y (cid:48) Y Q (2.18)can be glued as (cid:88) Y C Y T1 Y T2 Y T0 ( q )( − ( n +1) | Y | q − nκ ( Y ) / Q | Y | C Y (cid:48) Y (cid:48) Y , (2.19)where κ is related to quadratic Casimir of the representation corresponding to | Y | ,namely, κ ( Y ) = (cid:80) i y i ( y i − i + 1) with y i being the number of boxes in the i th row.The framing number n equals det( v in , v out ), where the two vectors are chosen suchthat v in · v out >
0. The parameter Q is the (exponentiated) K¨ahler parameter for the2-cycle corresponding to the line in the dual toric diagram.In [25], the above is extended to a refined topological vertex as C Y Y Y ( q, t ) = (cid:16) qt (cid:17) ( || Y || + || Y || ) / t κ ( Y ) / P Y T0 (cid:0) t − ρ ; q, t (cid:1) × (cid:88) η (cid:16) qt (cid:17) ( | η | + | Y |−| Y | ) / s Y T1 /η (cid:0) t − ρ q − Y (cid:1) s Y /η (cid:16) t − Y T0 q − ρ (cid:17) , (2.20)where P Y T0 ( t − ρ ; q, t ) is the Macdonald function and s α/β ’s are the skew-Schur functions.The squared double slash denotes the quadratic sum of the number of boxes in eachrow of the Young tableau. Notice that the three Young tableaux are not cyclicallysymmetric and Y corresponds to the preferred leg for gluing. One may check thatwhen the Ω-background parameters satisfy q = t , we would recover the unrefinedtopological vertex.Define the framing factors, f Y ( q, t ) = ( − | Y | q || Y T || / t −|| Y || / , ˜ f Y ( q, t ) = ( − | Y | q ( || Y T || + | Y | ) / t − ( || Y || + | Y | ) / , (2.21)– 10 –nd the edge factor, ( − Q ) | Y | × [framing factor]. Then the topological string partitionfunction takes the sum over all the Young tableaux of internal legs as Z topo = (cid:88) Y i (cid:89) edges [edge factor] (cid:89) vertices [vertex factor] . (2.22)Again, let us contemplate the SU(2) gauge theory with 4 flavours. The dual webdiagram is Q B Q F Q Q Q B Q F Q Q . (2.23)Following the gluing process, the partition function reads Z topo = (cid:88) λρν ( − Q F ) | λ | ˜ f λ ( q, t )( − Q F ) | λ | ˜ f λ ( t, q )( − Q B ) | ρ | f ρ T1 ( q, t )( − Q B ) | ρ | f ρ T2 ( t, q ) × ( − Q ) | ν | ( − Q ) | ν | ( − Q ) | ν | ( − Q ) | ν | C λ T1 ν T1 ρ T1 ( q, t ) C ν λ ρ ( q, t ) × C λ T2 ν T2 ρ T2 ( t, q ) C ν λ ρ ( t, q ) C ν T3 ∅∅ ( q, t ) C ∅ ν ∅ ( q, t ) C ν T4 ∅∅ ( t, q ) C ∅ ν ∅ ( t, q ) . (2.24)Recall the 4d instanton partition function (2.9), which can be lifted to 5d as [28] Z inst,5d = 1 Z U(1),5d (cid:88) Y , e π i τ ( | Y | + | Y | ) z vec,5d ( (cid:126)a, (cid:126)Y ) z matter,5d . (2.25)It is discussed in [28–32] that under the parameter identification q = e − R(cid:15) , t = e R(cid:15) , Q i = e − R ( µ i − a ) , Q B = e π i τ e R (cid:18) − a + (cid:80) i µ i (cid:19) , Q F = e − Ra , (2.26)where R is the radius of the compactified dimension S , Z topo reproduces Z inst,5d up toperturbative part and U(1)/extra factor [30, 33–38]. Notice that when (cid:15) = − (cid:15) , i.e., Q = 0, we have the unrefinement q = t . In particular, under the 4d limit R →
0, the 4dtopological A-model partition function would give the 4d instanton partition function. The Young tableaux of external legs would be ∅ . More generally, for SU( N c ) gauge group with N f = 2 N c , the partition function was given in [27]. Often Q i would be written as e − R ( − µ i − a ) for i = 2 ,
4. However, due to invariance under Weylgroup symmetry, the Nekrasov partition function should not change under µ , → − µ , . – 11 – .3 From Topological String Partition Functions to Seiberg-Witten Curves As aforementioned, the low energy effective theory for 4d N = 2 can be encoded bythe prepotential F , where F = lim (cid:15) , → (cid:15) (cid:15) log Z Nek in terms of the Nekrasov partitionfunction which is in turn naturally related to topological partition functions in theA-model.On the other hand, in the SW solution, the prepotential can be determined bythe SW curve. Given such auxiliary curve Σ, it is possible to translate into the form λ = q ( z ), where λ is the Seiberg-Witten differential , and q ( z ) is the meromorphicquadratic differential on the Gaiotto curve C [5, 39]. In this subsection, we demonstratehow this translation runs for the theory with a single SU(2) factor and N f = 4. Thistheory will constitute our running example throughout the paper.To begin, following § N f = 4 theory in hyperellipticform is f ˜ z (˜ x − ˜ µ )(˜ x − ˜ µ ) + ( f (cid:48) ˜ z )(˜ x − ˜ µ )(˜ x − ˜ µ ) = ˜ x − u, (2.27)where f and f (cid:48) are complex numbers and u parametrizes the space of supersymmetricvacua, viz, the u -plane. We first choose the coordinate of ˜ z so that f (cid:48) = 1. Completinga square in ˜ x by defining x = ˜ x + f ˜ z (˜ µ + ˜ µ ) + ˜ z (˜ µ + ˜ µ )2 (cid:0) − ˜ z − f ˜ z (cid:1) , (2.28)we obtain x = g (˜ z ) where g has double poles at c , ( f ). Now rescale z = ˜ z/c ( f ) sothat the poles are at z = 1 , ζ , and we get x = P ( z )( z − ( z − ζ ) (2.29)for some quartic polynomial P ( z ) determined by ˜ µ i , ζ and u . Then a and its magneticdual a D can be obtained by integrating the SW differential λ ≡ x d z/z along A - and B -cycles: a = (cid:73) A λ, a D = ∂ F ∂a = (cid:73) B λ. (2.30) Construction of SW Curves from Toric Diagrams
For 5d gauge theories, the 5-brane web diagrams can be used to construct SW curves. In fact, such a web diagram isexactly the same as the dual toric diagram in the geometric engineering in § The finite non-zero deformation parameters could also physically make sense for SW curves, forexample in the context of topological B-models as in Appendix A. – 12 – c c c c c c c c ˜ m ˜ m t t ˜ m ˜ m t t Figure 2.1 : The toric diagram and its dual diagram for SU(2) with 4 flavours. in [42] and elaborated in [43]. Here, we will still focus on SU(2) with 4 flavours, wherethe web/dual diagram is reproduced in Figure 2.1, along with its toric diagram.For each vertex ( i, j ) in the toric diagram, we assign a non-zero number c ij . Withthese coefficients, the SW curve is given as (cid:88) i,j c ij t i w j = 0 . (2.31)By multiplying an overall constant to this equation, we can impose c = 1 (2.32)without the loss of generality. There are four boundaries, so the boundary conditionsaccording to the toric diagram now are | w | (cid:29) , c w + c tw + c t w = c w ( t − t )( t − t ) , | w − | (cid:29) , c t + c t + c = c ( t − t )( t − t ) , | t | (cid:29) , c t + c t w + c t w = c t ( w − ˜ m )( w − ˜ m ) , | t − | (cid:29) , c w + c w + c = c ( w − ˜ m )( w − ˜ m ) , (2.33)where t and w can be thought of as the horizontal and vertical coordinates of thediagram respectively. In order for the curve to satisfy all the conditions consistently,we need the compatibility condition which reads˜ m t − ˜ m t = ˜ m t ˜ m t − . (2.34)Since the SW differential is invariant under the rescaling of t , we can impose t = 1 (2.35)– 13 –or simplicity. Also, by rescaling w , we can further impose˜ m t − ˜ m t = 1 . (2.36)This condition turns out to correspond to the traceless condition of the vacuum expec-tation value of the SU(2) vector multiplet [29]. The instanton factor is the geometricaverage of t i , which is ζ ≡ (cid:18) t t t t (cid:19) . (2.37)The only undetermined coefficient c = − U (cid:48) is interpreted as Coulomb moduli param-eter. Defining a parameter S ≡ ˜ m ˜ m ˜ m ˜ m , we have t = ˜ m ˜ m , t = 1 , t = ˜ m ˜ m ( S ) − ζ, t = ( S ) ζ, (2.38)and thus, the SW curve for 5d N = 1 SU(2) gauge theory with N f = 4 ist ( w − ˜ m ) ( w − ˜ m ) − t (cid:16) w ˜ m ˜ m (cid:16) ζ ( S ) − (cid:17) + wU (cid:48) + ˜ m ˜ m (cid:16) ζ ( S ) (cid:17)(cid:17) + ζ ( S ) − ( ˜ m ˜ m ) ( w − ˜ m ) ( w − ˜ m ) = 0 . (2.39) The 4d limit curve
Till now, the SW curve is a 5d curve, its 4d SW curve can beobtained by taking the vanishing limit of size of compactification circle β →
0, wherewe have w = e − βv , ˜ m i = e − βµ i . (2.40)We then expand the 5d Coulomb parameter U (cid:48) as U (cid:48) = ∞ (cid:88) k =0 u k β k . (2.41)In particular, u = − ζ ) , u = 2( ζ + 1)( µ + µ ) . (2.42)For the N f = 4 curve, we then have the 4d limit t ( v − µ ) ( v − µ ) + ζ ( v − µ ) ( v − µ ) + t (1 + ζ ) (cid:32) − v + ζv ζ (cid:88) i =1 µ i + U (cid:33) = 0 , (2.43)where U = − u ζ − ( µ + µ ) − ζ ζ ) (cid:32) (cid:88) i =1 µ i (cid:33) . (2.44)– 14 – eparametrization Let us rewrite the curve as t ζ ( v − µ ) ( v − µ ) − v + ζv (cid:80) i =1 µ i ζ + U + ζt
11 + ζ ( v − µ ) ( v − µ ) = 0 . (2.45)Redefining f / ˜ z = t ζ , f (cid:48) ˜ z = ζt (1 + ζ ) , ˜ x = v − ζ ζ ) (cid:88) i =1 µ i , ˜ µ i = µ i − ζ ζ ) (cid:88) i =1 µ i ,u = (cid:32) ζ ζ ) (cid:88) i =1 µ i (cid:33) + U (2.46)recovers the curve of form (2.27), which is reproduced here: f ˜ z (˜ x − ˜ µ )(˜ x − ˜ µ ) + ( f (cid:48) ˜ z )(˜ x − ˜ µ )(˜ x − ˜ µ ) = ˜ x − u. (2.47) We begin with a refresher on some preliminary definitions and key results [44, 45].
Definition 2.1. A dessin d’enfant , or child’s drawing , is an ordered pair ( X, D ) , where X is an oriented compact topological surface and D ⊂ X is a finite graph, such that1. D is a connected bipartite graph, and2. X \D is the union of finitely many topological discs that are the faces of D . There is a bijection between the dessins and Belyi maps known as the
Grothendieckcorrespondence [46], where
Definition 2.2. A Belyi map β is a holomorphic map from the Riemann surface X to P ramified at only 3 points, which can be taken to be { , , ∞} ∈ P . Recall that ramification means that the only points ˜ x ∈ X where ddx β ( x ) (cid:12)(cid:12) ˜ x = 0 aresuch that β (˜ x ) = 0 , ∞ . In other words, the local Taylor expansion of β ( x ) aboutthe pre-images ˜ x of { , , ∞} have (at least) vanishing linear term.– 15 – rom Belyi maps to dessins We can associate β ( x ) to a dessin via its ramificationindices : the order of vanishing of the Taylor series for β ( x ) at ˜ x is the ramificationindex r β (˜ x ) ∈{ , , ∞} ( i ) at that i th point. By convention, we mark one white node forthe i th pre-image of 0 with r ( i ) edges emanating therefrom. Similarly, we mark oneblack node for the j th pre-image of 1 with r ( j ) edges. We then connect the nodes withthe edges, joining only black with white, such that each face is a polygon with 2 r ∞ ( k )sides. In other words, there is one pre-image of ∞ corresponding to each polygon of D . Moreover, there is a cyclic ordering arising from local monodromy winding aroundvertices, i.e., around local covering sheets that contain a common point.The power of dessins comes from Belyi’s remarkable theorem. Theorem 2.1.
There exists an algebraic model of X (as a Riemann surface) definedover ¯ Q iff there exists a Belyi map on X . Thus, the existence of a dessin on X is equivalent to X admitting an algebraicequation over the algebraic numbers. Moreover, the Galois group Gal( ¯ Q : Q ) actsfaithfully on the space of dessins. Quadratic Differentials A (holomorphic) quadratic differential q on a Riemannsurface X is a holomorphic section of the symmetric square of the contangent bundle.In terms of local coordinates z , q = f ( z )d z ⊗ d z , for some holomorphic function f ( z ).A curve γ ( t ) ⊂ X can be classified by q as • Horizontal trajectory: f ( γ ( t )) ˙ γ ( t ) > • Vertical trajectory: f ( γ ( t )) ˙ γ ( t ) < Strebel differential : Definition 2.3.
For a Riemann surface X of genus g ≥ with n ≥ marked points { p , . . . , p n } such that − g < n , and a given n -tuple a i =1 ,...,n ∈ R + , a Strebel differ-ential q = f ( z )d z is a quadratic differential such that • f is holomorphic on X \{ p , . . . , p n } ; • f has a second-order pole at each p i ; • the union of all non-compact horizontal trajectories of q is a closed subset of Xof measure 0; – 16 – every compact horizontal of q is a simple loop A i centered at p i such that a i = (cid:72) A i √ q . (Here the branch of the square root is chosen so that the integral has apositive value with respect to the positive orientation of A i that is determined bythe complex structure of X .) The upshot is that [47]
Theorem 2.2.
The Strebel differential is the pull-back, by a Belyi map β : X → P , ofa quadratic differential on P with 3 punctures, q = β ∗ (cid:18) d ζ π ζ (1 − ζ ) (cid:19) = (d β ) π β (1 − β ) = ( β (cid:48) ) π β (1 − β ) d z , (2.48) where z and ζ are coordinates on X and P respectively. Recall the definition of the SW differential λ = v d zz . (2.49)Then q = λ = v d z z =: φ ( z )d z (2.50)is the quadratic differential on C . For our purposes, the important point to note isthat the SW curve (2.29) can be written in the form (2.50) [39]. This construction willprove essential in what follows. SW curves and Dessins
As mentioned above, the SW curve Σ is related to thequadratic differential q . Moving in the moduli space of the theory in question will alterthe parameters in the SW curve, thereby altering the parameters in q [9]. Following[47], it was found in [9] that at certain isolated points in the Coulomb branch U g,n ,where g is the genus of the Gaiotto curve C with n marked points, q is completely fixedand becomes a Strebel differential q = φ ( t )d t = d β π β ( t )(1 − β ( t )) .As examples for SU(2) with N f = 4, we will discuss 6 Strebel points in U g,n × R n , forwhich the Belyi maps are presented in Table 2.1. These six Belyi maps are those foundin [7, 48] to be associated to the six genus zero , torsion-free , congruence subgroups ofthe modular group Γ = PSL(2 , Z ) ∼ = Z ∗ Z , where ∗ denotes the free product .From the Belyi maps in Table 2.1, we can compute the associated dessins as dis-played in Figure 2.2. The dessins d’enfants associated to each Strebel point of thegeneralised quiver theory in question turn out to have an interpretation as so-called ribbon graphs on the Gaiotto curve C . For details, the readers are referred to [9, 47]. For the background on the congruence subgroups of Γ, see Appendix C. It remains an openquestion whether dessins associated to other subgroups of the modular group, perhaps of higherindex, arise for other N = 2 generalised quiver theories in a parallel manner. – 17 –raph β ( t ) Ramification Strebel q Γ(3) t ( t +6) ( t − t +36) t − ( t +3 t +9) [3 | | ] − t ( t +216)4 π ( t − Γ (4) ∩ Γ(2) ( t +224 t +256) t ( t − ( t +4) [3 | | , ] − t +896 t +10244 π t ( t − Γ (5) ( t +248 t +4064 t +22312 t +40336) t +5)( t − t − [3 | | , ] − t +248 t +4064 t +22312 t +403364 π ( t +5) ( t − t − Γ (6) ( t +7) ( t +237 t +1443 t +2287) t +3) ( t +4) ( t − [3 | | , , , − ( t +7)( t +237 t +1443 t +2287)4 π ( t +5) ( t +3) ( t +4) Γ (8) ( t +240 t +2144 t +3840 t +256) t ( t +4) ( t − [3 | | , , ] − t +240 t +2144 t +3840 t +2564 π t ( t − Γ (9) ( t +6) ( t +234 t +756 t +2160) t +3 t +9)( t − [3 | | , ] − ( t +6)( t +234 t +756 t +2160)4 π ( t − Table 2.1 : The list of the six genus-zero, torsion-free, congruence subgroups of the modular group Γ,of index 12. The corresponding Belyi maps β ( t ) and their ramification indices, as well as the Strebeldifferentials are also shown. Note that the ramification indices for all 6 are such that there are 4pre-images of 0 of order 3 and 6 pre-images of 1 of order 2. The pre-images of ∞ (aka the cusp widths )all add to 12, as do the ramification indices for 0 and 1. This is required by the fact that all thesubgroups are of index 12 within Γ. Γ(3) Γ (4) ∩ Γ(2) Γ (5)Γ (6) Γ (8) Γ (9) Figure 2.2 : The dessins d’enfants associated to the six Strebel points of the SU(2), N f = 4 theory. – 18 – From Dessins to Conformal Blocks
Let us now complete the cycle of the route map above by considering what gauge theoryand CFT data we can obtain starting from these 6 dessins.
Given that all our graphs in Figure 2.2 are drawn on the Riemann surface (genus zero)with 4 marked points (one for each face), we can naturally interpret these as Gaiottocurves [6, 9], and thence N = 2 gauge theories.To begin, the Seiberg-Witten curve Σ for the SU(2) N f = 4 theory in algebraicform is standard [39]. For future convenience, we start with the SW curve of form(2.43) and write the SW differential as [13, 14] λ SW = (cid:112) P ( z ) z ( z − z − ζ ) d z, P ( z ) = m
20 4 (cid:89) i =1 ( z − λ i ) = m
20 4 (cid:88) i =0 z − i S i , (3.1)under the substitution λ SW = v d z/z, t = z,µ = m + m , µ = m − m , µ = m + m , µ = m − m , (3.2)The parameters S i are given in terms of the flavour mass and coupling parameters m , , , , ζ, U ∈ C so that S = 1 for the top coefficient and m S = − (cid:0) m + m ( ζ −
1) + m ζ + 2 m m ζ + (1 + ζ ) U (cid:1) ,m S = ( m + m − m + 2 m m ) ζ + m ( ζ − ζ + 2 m m ζ + m ζ + (1 + ζ ) U,m S = − (cid:0) ( m − m ) ζ + ( m + 2 m m + m ) ζ + ζ (1 + ζ ) U (cid:1) ,m S = m ζ . (3.3)Now the SW curve is of the form z ( v − ( m + m ))( v − ( m − m )) + z (1 + ζ ) (cid:18) − v + 2 ζ (1 + ζ ) ( m + m ) v + U (cid:19) + ζ ( v − ( m + m ))( v − ( m − m )) = 0 . (3.4)On the other hand, the S -parameters can be written in terms of the λ i as standardsymmetric polynomials, S k = (cid:88) ≤ j ≤ ... ≤ j k ≤ λ j . . . λ j k . (3.5)– 19 –ollowing Appendix E.1, we can then writed a ( U )d U = − π i 1 + ζm (cid:112) ( λ − λ )( λ − λ ) K ( r ) , (3.6)where r = ( λ − λ )( λ − λ )( λ − λ )( λ − λ ) , (3.7)and K ( r ) is the elliptic integral of the first kind. The right hand side of (3.6) implicitlydepends on U , through λ i and thence S i , thus we only need to integrate it to obtain a ( U ) as a function of U , which could be a daunting task analytically.Let us nevertheless attempt at some simplifications. First, we see that the righthand side depends only on the cross-terms in the four λ i , which we will denote as λ ( ij )( kl ) = ( λ i − λ j )( λ k − λ l ). Combining with (3.5), let us see whether these can bedirectly expressed in terms of S i , and thence, in terms of U . This is a standard algebraicelimination problem and we readily find the following: Lemma 3.1.
Consider the monic cubic polynomial, x + (cid:0) − S + 6 S S − S (cid:1) x + (cid:0) S − S S S + 24 S S + 9 S S + 144 S − S S S (cid:1) x +27 S S + 4 S S − S S S S − S S S + 4 S S S + 6 S S S − S S S + 192 S S S +80 S S S S + 27 S − S + 4 S S − S S S + 128 S S − S S − S S S . (3.8) Then the squares of the 3 cross-products x = λ , x = λ , x = λ (3.9) are the three roots of it. Of course, we can substitute the S i parameters in terms of the m i , ζ, U parametersfrom (3.1), though the expression become too long to present here. Now, we have a ( U ) − a ( U ) = − ζm π i (cid:90) U U d U (cid:48) (cid:112) x ( U (cid:48) ) K (cid:32) (cid:112) x ( U (cid:48) ) (cid:112) x ( U (cid:48) ) (cid:33) . (3.10)To determine the integral constant, we choose U such that a ( U ) = 0. We can findsuch U by solving the discriminant of P ( z ) where two branch points coincide and the A -cycle shrinks . Hence, a ( U ) = 1 + ζm π i (cid:90) U U d U (cid:48) (cid:112) x ( U (cid:48) ) K (cid:32) (cid:112) x ( U (cid:48) ) (cid:112) x ( U (cid:48) ) (cid:33) . (3.11) Alternatively, we may also integrate from U to ∞ as the large U behaviour can be determined asin Appendix E.2. – 20 –n general, when we integrate from some U to U , the positions of branch points andcuts might change. Therefore, this is really a sum of integrals: (cid:90) U U = (cid:90) U U + (cid:90) U U + · · · + (cid:90) U n = U U n − (3.12)such that x i does not change its expression for each integral on the right hand side.Recall the definition of the Seiberg-Witten differential from (2.50), we have that λ = φ SW ( z )d z (3.13)is a quadratic differential. This is the above mentioned meromorphic quadratic dif-ferential on C . Moving in the moduli space of the theory in question will alter theparameters in the Seiberg-Witten curve, thereby altering the parameters in q (cf. [9]).Following [47], it was found in [9] that at certain isolated points in the Coulomb branchof the moduli space U g,n of the gauge theory in question, where g is the genus of theGaiotto curve C with n marked points, q is completely fixed, which becomes a Strebeldifferential.We therefore have two forms of the Strebel differentials, φ β ( t ) coming from thedessin and φ SW ( z ) coming from the physics. Now, because dessins are rigid , they haveno parameters. The insight of Belyi and Grothendieck is precisely that the maps β have parameters fixed at very special algebraic points in moduli space. Thus, φ β ( t ) isof a particular form, as a rational function in t with fixed algebraic coefficients.On the other hand φ SW ( z ) from the gauge theory has parameters which we sawearlier, corresponding to masses, couplings etc. Therefore, up to redefinition of thevariables ( t, z ) and identifying φ SW ( z ) and φ β ( t ) it is natural to ask how the specialvalues of the parameters from the dessin perspective fix the physical parameters in thegauge theory and if a dessin implicates any interesting physical theory.We have now introduced all the necessary dramatis personae of our tale and ourstrategy is thus clear. There are also some further details that we should be carefulabout in the calculations. We will work through an example in detail to illustrate themin the following subsection. Γ(3)Let us take the dessin for Γ(3), whose Belyi map is β ( t ) = t ( t + 6) ( t − t + 36) t − ( t + 3 t + 9) . (3.14)– 21 –e can readily get the pre-images of 0, 1 and ∞ :Pre-image Ramification β − (0) − − i √ i √ β − (1) 3(1 − √
3) 23(1 + √
3) 2 (cid:0) + i (cid:1) (cid:0) √ − − i ) (cid:1) (cid:0) − − i (cid:1) (cid:0) √ i ) (cid:1) (cid:0) ( − i ) − (3 − i ) √ (cid:1) (cid:0) ( − i ) + (3 − i ) √ (cid:1) β − ( ∞ ) ∞
33 3 − i (cid:0) √ − i (cid:1) i (cid:0) √ i (cid:1) . (3.15)We can construct the corresponding dessin as in Figure 2.2. Subsequently, using (2.48),we see that the Strebel differential is q = φ β ( t )d t , where φ β ( t ) = − t ( t + 216)4 π ( t − . (3.16)We have marked φ with a subscript β to emphasize its dessin origin. On the other side,we have the Seiberg-Witten curve and quadratic differential for SU(2) with N f = 4from (3.1) and (3.13), to be φ SW ( z ) = P ( z )( z ( z − z − ζ )) , where P ( z ) = z m − z (cid:0) m + m ( ζ −
1) + m ζ + 2 m m ζ + (1 + ζ ) U (cid:1) + z (cid:0) ( m + m − m + 2 m m ) ζ + m ( ζ − ζ + 2 m m ζ + m ζ + (1 + ζ ) U (cid:1) − z (cid:0) ( m − m ) ζ + ( m + 2 m m + m ) ζ + ζ (1 + ζ ) U (cid:1) + m ζ . (3.17)Here, likewise we have marked φ with a subscript “SW” to emphasize its Seiberg-Witten origin. We have also explicitly written the differential coming from the Seiberg-Witten side in terms of the parameters m , , , , ζ, U .We need to match (3.16) with (3.17), up to an PGL(2 , C ) transformation on thecomplex variable z . The reason for this is that we are dealing in this example with– 22 – quadratic differential on the sphere . For curves of higher genus, such PGL(2 , C )transformations are generically not permitted, as they will not preserve the structureof the poles and zeros of the quadratic differential.We can therefore write z = at + bct + d , a, b, c, d ∈ C (3.18)and solve for complex coefficients a, b, c, d as well as the parameters m , , , , ζ, U so thatwe have identically for all t that φ β ( t ) = φ SW (cid:18) at + bct + d (cid:19) . (3.19)There are actually continuous families of 2 × φ SW → k φ SW with k ∈ R , where the square comes from the λ in the differential. Obviously, equatingthe numerators of φ β and φ SW as well as equating their denominators would give asolution. For future convenience, such solution will be referred to as the “basic” valuesof the parametrization. Then other parametrizations would simply follow φ SW = k φ SW,basic . (3.20)There are two points we should pay attention to: • As we will try to relate this to minimal models, due to modular invariance, wecan only allow primary states with pure imaginary charges [49]. Recall the AGTrelation (2.11), which in terms of m i is m √ (cid:15) (cid:15) + Q α , m √ (cid:15) (cid:15) + Q α , m √ (cid:15) (cid:15) = α , m √ (cid:15) (cid:15) = α , a √ (cid:15) (cid:15) + Q α int . (3.21)In fact, (cid:15) , are not completely free once Q = ( (cid:15) + (cid:15) ) / √ (cid:15) (cid:15) is chosen. More-over, to have real conformal dimensions, m i ’s and a should only be real or pureimaginary (depending on (cid:15) , ). This is also the reason why k should be real. • One may easily check that an SW differential/elliptic curve would have the same j -invariant under φ → k φ . As a result, the parameters, based on their massdimensions or by looking at P ( z ) and a ( U ), would follow m i → km i , a → ka, (cid:15) i → k(cid:15) i , U → k U ; ζ → ζ, α i, int → α i, int , Q → Q. (3.22)– 23 –herefore, rather than discrete parameters, we would have families of differen-tials. Importantly, we can see that the coupling ζ is invariant . Following theAGT map, the dimensionless CFT parameters, α i, int and Q , are also invariantunder the scaling though we still have the freedom to choose √ (cid:15) (cid:15) .Now expanding the above and setting all the coefficients of t to vanish identicallygives a complicated polynomial system in ( a, b, c, d, m , , , , ζ, U ) for which one can findmany solutions. For example, the following constitutes a solution (with k = 1), m = − m = m = − m = 12 √ π , ζ = 12 + i √ , U = 19 π (3.23)with ( a, b, c, d ) = (cid:16) √ / , i (cid:16) √ √ − / √ (cid:17) + √ √ + / √ , , (1 − i)3 / √ (cid:17) . The numerator ofthe SW differential takes the form P ( z ) = − z − (cid:0) √ − (cid:1) z + (cid:0) √ (cid:1) z − √ z + 3i √ − π . (3.24)We now need the roots λ i of P ( z ) as given in (3.1): z + (cid:18) − − √ (cid:19) z + (cid:16) √ (cid:17) z − √ z + 12 i (cid:16) √ (cid:17) = (cid:89) i =1 ( z − λ i ) . (3.25)The SW curve itself is genus 1 and is in fact an elliptic curve. We can recast (3.17) as y = z m − z (cid:0) m + m ( ζ −
1) + m ζ + 2 m m ζ + (1 + ζ ) U (cid:1) + z (cid:0) ( m + m − m + 2 m m ) ζ + m ( ζ − ζ + 2 m m ζ + m ζ + (1 + ζ ) U (cid:1) − z (cid:0) ( m − m ) ζ + ( m + 2 m m + m ) ζ + ζ (1 + ζ ) U (cid:1) + m ζ , (3.26)where the redefinition y = ( z ( z − z − ζ )) φ SW ( z ) = P ( z ) is used. FollowingAppendix D, as one may check, the j -invariant we get from the parameterization (3.23)agrees with the one directly from the Strebel differential (3.16): j = 0 . (3.27)Indeed, j = 0 corresponds to a special elliptic curve with Z / Z -symmetry, much likethe dessin for Γ(3) itself.In this case, we can integrate (3.11) numerically to obtain a ( U ) = √ π . Now wecan use the AGT relation (3.21) to get the parametrizations for CBs. If we take Q = 0,we have α = α = − α = − α = i2 √ π , α int = i3 √ π , (3.28) Numerical integration would often give decimals rather than precise values. For instance, herewe get a ≈ . a with the help of minimal – 24 –here we have chosen − (cid:15) = (cid:15) = 1 as an example.We can also have pure imaginary m i ’s and a for the above example such as m = − m = m = − m = i2 √ π , ζ = 12 + i √ , U = − π , a = i3 √ π . (3.29)Then we can still get the same CFT parameters for Q = 0 as in (3.28) with the choice − (cid:15) = (cid:15) = i. Here, we report all parameters from the six dessins in Table 3.1 ∼ ± ) φ SW,basic with pure imaginary m i ’s and a .There is actually a family for each parametrization following (3.22). ζ = e π i τ m m m m U (cid:80) i m i a = α int √ (cid:15) (cid:15) + Q − i √ π − i √ π i √ π − i √ π − π i √ π − i √ π (1 − i √ − i √ π i √ π − i √ π i √ π − π − i √ πi √ π − i √ π i √ π − i √ π − π i √ πi √ π i √ π − i √ π i √ π − π − i √ π i √ π − i √ π − i √ π i √ π − i √ π − π i √ π i √ π (1 + i √ − i √ π i √ π − i √ π i √ π − π i √ πi √ π − i √ π i √ π − i √ π − π − i √ πi √ π i √ π − i √ π i √ π − π − i √ π − i √ π Table 3.1 : The parameters obtained from Γ(3). Using (3.21), we can get the values for α i ’s.models. This can be done by checking multiple minimal models and finding certain (cid:15) , so that ∆ i would fit into their Kac tables. Then the correct closed form of a can be obtained if ∆ int also lives inthese Kac tables for all these minimal models. (To get the correct CBs under AGT map, we furtherneed the fusion rule, but even if the corresponding ∆ int does not satisfy the fusion rule for some CB,this could still be regarded as a verfication of fine-tuning a as long as ∆ int , along with ∆ i , belongs tothe Kac table.) – 25 – = e π i τ m m m m U (cid:80) i m i a − i π − i π − i π − i π − π i π i πi π − i π − i π − i π − π i π − i π − i π − i π − i π i π − π i π i π − i π i π − i π − i π − π i π i πi π − i π − i π i π − π i π − i π − i π − i π i π − i π − π i π i πi π i π − i π − i π − π − i π − i π i π − i π i π − π i πi π − i π i π − i π − π − i π − i π − i π i π i π − π i πi π i π − i π i π − π − i π − i π − i π i π i π − i π − π − i π i πi π − i π i π i π − π − i π − i πi π i π i π − i π − π − i π − i π − i π i π i π i π − π − i π i πi π i π i π i π − π − i π i π − i π − i π − i π − i π π i π i πi π − i π − i π − i π π iπ − i π − i π − i π i π − i π − π iπ i πi π − i π i π − i π − π i π − i π − i π − i π − i π i π − π i π i π − i π i π − i π − i π π i π i πi π i π − i π − i π π − i π − i π i π i π − i π − π i πi π − i π − i π i π − π − i π − i π − i π i π i π π i πi π − i π i π i π π − i π − i πi π i π i π − i π − π − i π − i π − i π i π − i π i π − π − i π i π – 26 – π i π − i π i π − π − iπ − i π − i π i π i π i π π − iπ i πi π i π i π i π π − i π − i π − i π − i π − i π i π i π − i π − i π i π − i π i πi π − i π − i π i π − i π − i π i π − i π Any 0 0 − i π i π − i π i π value − i π i π i π − i π i π i π − i π i π − i πi π i π i π − i π − i π Table 3.2 : The parameters obtained from Γ (4) ∩ Γ(2). Using (3.21), we can get the values for α i ’s. As the size of the table increases, we will give a more compact version for the remainingcases below. For each ζ , there are usually 2 = 16 possibilities. For a , as the sign of a only depends on the sign of m (in the following sense), “ ± ” in a means that a has thesame sign as m while “ ∓ ” in a indicates that m and a have opposite signs. ζ = e π i τ m m m m U a ± i √ π ± i √ π i √ π i √ π ( − √ π ± i √ π − √ − i √ π − i √ π ± i √ π ± i √ π i √ π − i √ π − √ π ∓ i √ π − i √ π i √ π ± i √ π ± i √ π i √ π i √ π ( − − √ π ∓ i √ π + √ − i √ π − i √ π ± i √ π ± i √ π i √ π − i √ π − − √ π ∓ i √ π − i √ π i √ π ± i (5 √ π ± i √ π i (5 √ π i √ π . ± i √ π Here, any complex number can be a basic value for U since all the terms of U in P ( z ) contain(1 + ζ ) as well. Moreover, the integral for a always vanishes. – 27 – + √ − i (5 √ π − i √ π ± i (5 √ π ± i √ π i (5 √ π − i √ π . ± i √ π − i (5 √ π i √ π ± i √ π ± i √ π i √ π i √ π − . ± i √ π − + √ − i √ π − i √ π ± i √ π ± i √ π i √ π − i √ π − . ∓ i √ π − i √ π i √ π ± i √ π ± i √ π i √ π i √ π − . ∓ i √ π − √ − i √ π − i √ π ± i √ π ± i √ π i √ π − i √ π − . ± i √ π − i √ π i √ π ± i (5 √ π ± i (5 √ π i √ π i √ π . ± i √ π − − √ − i √ π − i √ π ± i (5 √ π ± i (5 √ π i √ π − i √ π − . ± i √ π − i √ π i √ π Table 3.3 : The parameters obtained from Γ (5). Using (3.21), we can get the values for α i ’s and Q . ζ = e π i τ m m m m U a ± i π ± i √ π iπ i π π ± . i − iπ − i π ± i π ± i √ π iπ − i π − π ± . i − iπ i π ± i π ± iπ i √ π i π − √ π ± . i − i √ π − iπ ± i π ± iπ i √ π − i π − − √ π ± . i − i √ π i π ± i π ± i √ π i π iπ π ± . i − i π − iπ ± i π ± i √ π i π − iπ π ± . i – 28 – i π iπ ± i π ± iπ i π i √ π √ π ± . i − i π − i √ π ± i π ± iπ i π − i √ π − √ π ± . i − i π i √ π Table 3.4 : The parameters obtained from Γ (6). Using (3.21), we can get the values for α i ’s and Q . ζ = e π i τ m m m m U a ± i π ± i π i π i π π ± . − i π − i π ± i π ± i π i π − i π − π ± . − i π i π ± i π ± i π i π − iπ π ± . − i π iπ ± i π ± i π i π iπ π ± . − i π − iπ Table 3.5 : The parameters obtained from Γ (8). Using (3.21), we can get the values for α i ’s and Q . ζ = e π i τ m m m m U a ± i √ π ± i √ π i √ π i √ π − i (33 i +25 √ π ± ( − . . i ) − i √ − i √ π − i √ π ± i √ π ± i √ π i √ π − i √ π − i (3 i +8 √ π ± ( − . . i ) − i √ π i √ π ± i √ π ± i √ π i √ π i √ π i ( − i +25 √ π ± ( − . − . i ) i √ − i √ π − i √ π ± i √ π ± i √ π i √ π − i √ π i ( − i +8 √ π ± ( − . − . i ) − i √ π i √ π – 29 – able 3.6 : The parameters obtained from Γ (9). Using (3.21), we can get the values for α i ’s and Q . Based on the above calculations, there are some remarks we can make: • One may check that the elliptic curves parametrized by these m i , ζ and U have thesame j -invariants as in Table 3.7 for the six Belyi maps. Moreover, there are twoΓ(3) 0Γ (4) ∩ Γ(2) Γ (5) Γ (6) − (8) Γ (9) 0 Table 3.7 : The j -invariants that correspond to the six index-12 Belyi maps. cases with ζ = (1 ± i √ /
2, which are the cusp points for the fundamental diagramof SL(2, Z ). They are exactly the dessins whose Belyi maps have j -invariant 0. • It is obvious that for each dessin, the parametrizations for different ζ ’s are relatedby triality ζ ↔ ζ (cid:48) = 1 ζ ↔ ζ (cid:48)(cid:48) = 1 − ζ. (3.30)This is explicitly listed in Table 3.8. Modular invariance of the curve also leadsto the following transformations of mass parameters: ζ ↔ ζ : ( m , m , m , m ) ↔ | ζ | ( m , m , m , m ); ζ ↔ − ζ : ( m , m , m , m ) ↔ ( m , m , m , m ) . (3.31)In particular, the two rows for Γ (5) are also related by triality: 1 − (cid:0) + √ (cid:1) = − − √
5. – 30 –essin ζ ζ (cid:48) ζ (cid:48)(cid:48) Γ(3) (1 ± i √ (1 ∓ i √ (1 ∓ i √ (4) ∩ Γ(2) 2 − (5) − √ + √ + √ − + √ − − √ − √ (6) 2 -Γ (8) 2 -Γ (9) (1 ± i √
3) - (1 ∓ i √ Table 3.8 : The parametrizations for each case are related by triality. The hyphens indicate that such ζ either gives no solution to mass parameters (Γ (6) and Γ (8)) or does not satisfy the transformationsof masses (Γ (9)). Γ(3)As an example, let us match the parametrizations for Γ(3) obtained above to 4-pointCBs in minimal models. In fact, as we will see, such CB first appears for the tetracriticalIsing model when p (cid:48) = 6 and p = 5, that is, c = 4 /
5. As usual, we can write the 4-pointCB as α α α α α int . (3.32)Then the intermediate field φ k,l should satisfy the fusion rule φ r,s × φ m,n = min( m + r − , p − − m − r ) (cid:88) k = | m − r | +1 k − m + r − ∈ Z min( n + s − , p (cid:48) − − n − s ) (cid:88) l = | n − s | +1 l − n + s − ∈ Z φ k,l , (3.33)where the entire conformal family of a primary is implicit in the above abuse of notation.Let φ r i ,s i correspond to α , and φ m i ,n i correspond to α , ( i = 1 , φ k,l ∈ φ r ,s × φ m ,n , φ k,l ∈ φ r ,s × φ m ,n (3.34)with constraints on k, l indicated in (3.33).Before we insert the specific values of the parametrizations, we can make somesimplifications: • Recall that the mass parameters are real or pure imaginary. If we have someparametrization with m i ∈ R , without loss of generality we can choose (cid:15) < < – 31 – . Then since (cid:15) + (cid:15) √ (cid:15) (cid:15) = Q = i (cid:16)(cid:113) p (cid:48) p − (cid:113) pp (cid:48) (cid:17) , we have √ (cid:15) (cid:15) = − i (cid:15) (cid:113) pp (cid:48) . Likewise,for some parametrization with m i ∈ i R , without loss of generality we can choose (cid:15) / i < < (cid:15) / i. Such two cases related by m i → i m i should give the same (cid:15) , upto a factor of i. • If we make the choice in the above point for some specific m i , then m i → − m i should give the same CFT parameters with (cid:15) , → i (cid:15) , . If we only have m →− m or m → − m , then we should always get the same parametrization evenwithout changing (cid:15) , since the corresponding conformal dimension is Q − m , (cid:15) (cid:15) . • Swapping m ↔ m and swapping m ↔ m simultaneously should give the sameCFT parameters (for same (cid:15) , ) due to the AGT map. This simply correspondsto read the CB (3.32) from the left or from the right.In light of these points, it suffices to only contemplate one parametrization , say m = − m = m = − m = − √ , for Γ(3). When p (cid:48) = 6 , p = 5, we find that there isonly one possibility for ∆ and ∆ , that is,∆ = ∆ = 115 . (3.35)There are two possible solutions for the remaining mass parameters (and deformationparameters): (cid:15) = 2 √ π , ∆ = 140 , ∆ = 18 ; (3.36) (cid:15) = − √ π , ∆ = 18 , ∆ = 140 . (3.37)Moreover, for the intermediate channel, a = − √ π , ∆ int = 140 . (3.38)Hence, the intermediate channel ( k, l ) obtained from Γ(3) corresponds to (2 ,
2) or (3 , k, l ) satifying the fusion rule but not from the dessin is (2 ,
4) or (3 , Since ∆ = ∆ , when considering ζ ↔ /ζ , it is equivalent to swapping both m ↔ m and m ↔ m . Therefore, ζ = (1 ± i √ / | ζ | (cid:54) = 1, as longas ∆ = ∆ , swapping 2 ↔ / | ζ | can beabsorbed into √ (cid:15) (cid:15) . – 32 – (2,3) (2,3) (2,3) (2,3) (3,3) (3,3) (3,3) (3,3)∆ (2,2) (2,2) (3,4) (3,4) (2,2) (2,2) (3,4) (3,4)∆ (1,2) (4,4) (1,2) (4,4) (1,2) (4,4) (1,2) (4,4)∆ (3,3) (2,3) (2,3) (3,3) (2,3) (3,3) (3,3) (2,3)∆ int (3,4) (3,4) (2,2) (2,2) (2,2) (2,2) (3,4) (3,4) Table 3.9 : There are 8 possible combinations. Each column gives a CB. In the leftmost column, ∆ i ’sfollow the nomenclature correpsonding to (3.36). For (3.37), it just swaps 2 ↔ = ∆ ).Therefore, it essentially gives the same CBs. In other words, the two solutions just correspond toreading the 4-point CB (3.32) from left or from right.
138 23 18
75 2140 115 140 25
25 140 115 2140 75
18 23 138
31 2 3 4 5 (a)
154 167 3328 37 128
95 117140 835 − −
128 37 3328 167 154 (b)
227 127 57 17
238 8556 3356 556 156 38
43 1021 121 121 1021 43
38 156 556 3356 8556 238
17 57 127 227
51 2 3 4 5 6 (c)
92 9532 74 2732 14 −
115 187160 920 7160 −
120 27160 710
710 27160 −
120 7160 920 187160 115 −
132 14 2732 74 9532 92 (d)
152 16532 134 5732 34 532
347 675224 4528 143224 328 3224 514
97 99224 128 15224 1528 323224 3914
514 3224 328 143224 4528 675224 347
532 34 5732 134 16532 152 (e)
Figure 3.1 : Here we list the first five possible examples of CBs that Γ(3) corresponds to: (a) p (cid:48) =6 , p = 5, (b) p (cid:48) = 7 , p = 5, (c) p (cid:48) = 7 , p = 6, (d) p (cid:48) = 8 , p = 5, (e) p (cid:48) = 8 , p = 7. Those appeared in theCBs are in cyan in the Kac tables. For (e), we also have another combination of CBs in green. By looking at these examples, one might see some patterns of the minimal modelsand the positions of conformal dimensions in cyan appeared in the Kac tables. Now,– 33 –ases ConditionsAll ( r , s ) = ( p − r , p (cid:48) − s ) ∈ Z , Z )( r , s ) , ( r + 1 , s + 1) , ( r − , s − , ( r , s )( r , s ) , ( r + 1 , s + 1) , ( r + 1 , s + 1) , ( r , s ) r ≤ p − , s ≤ p (cid:48) − , ( r , s ) , ( r − , s − , ( r − , s − , ( r , s ) k = r , l = s ( r , s ) , ( r + 1 , s + 1) , ( r − , s − , ( r , s )( r , s ) , ( r + 1 , s + 1) , ( r − , s − , ( r , s ) (cid:0) p +12 ≤ r ≤ p − r , s ) , ( r + 1 , s + 1) , ( r + 1 , s + 1) , ( r , s ) or p +12 ≤ r ≤ p − or p = 2 r (cid:17) ( r , s ) , ( r − , s − , ( r − , s − , ( r , s ) and (similar relations with p → p (cid:48) , r → s )( r , s ) , ( r + 1 , s + 1) , ( r − , s − , ( r , s ) and k = p − r , l = p (cid:48) − s Table 3.10 : The possible CBs of minimal models that Γ(3) corresponds to. we are going to show
Proposition 3.2.
The dessin
Γ(3) corresponds to 4-point conformal blocks, where thecorresponding weights of the primaries satisfy the conditions in Table 3.10, in minimalmodels.
Following the specific values for m i and a , we can define M := m √ (cid:15) (cid:15) so that α = − M + Q , α = − M , α = M , α = M + Q , α int = 2 M Q . (3.39)There are two possible choices for ∆ in the Kac table. For future convenience, letus denote them as ∆ r ,s and ∆ r ,s . Then( p (cid:48) r i − ps i ) − ( p (cid:48) − p ) p (cid:48) p = Q − M = − ( p (cid:48) − p ) p (cid:48) p − M . (3.40)Therefore, M = − ( p (cid:48) r i − ps i ) p (cid:48) p . (3.41)It is also immediate from (3.39) that ∆ = ∆ . Hence, we can denote ∆ as ∆ m i ,n i without specifying whether ( m , , n , ) corresponds to ∆ or ∆ . We can plug this into∆ m i ,n i = ∆ = − M + QM and get( p (cid:48) m i − p (cid:48) n i + xn ) − x = ( p (cid:48) r j − p (cid:48) s j + xs j ) − x ( p (cid:48) r j − p (cid:48) s j + xs j ) , (3.42)where x := p (cid:48) − p is some positive integer. Its expansion gives p (cid:48) ( m i − n i ) +2 p (cid:48) ( m i − n i ) xn i + x n i − x = p (cid:48) ( r j − s j ) +2 p (cid:48) ( r j − s j )( s j − x + x s j − x s j . (3.43)– 34 –ince this is for general p (cid:48) , by comparing coefficients at different orders of p (cid:48) , we have m i − n i = ± ( r j − s j ) , n i = ± ( s j − , n i − s j − s j , (3.44)where ± can be seen from the symmetry of p (cid:48) and p (cid:48) terms in (3.43). Due to asimilar symmetry for ( m i , n i ) ↔ ( p − m i , p (cid:48) − n i ), it is possible to replace ( m i , n i ) with( p − m i , p (cid:48) − n i ) or ( r j , s j ) with ( p − r j , p (cid:48) − s j ) in (3.42). It turns out that they alsogive the same set of equations. The third equation is actually redundant, and hencewe have m i − n i = ± ( r j − s j ) , n i = ± ( s j − . (3.45)Strictly speaking, in (3.42), we should really have | p (cid:48) r j − p (cid:48) s j + xs j | on the right handside. Taking this into account, we would obtain another set of solutions with − m i = r j − , n i = s j − , (3.46)or m i = r j + 1 , n i = s j + 1 . (3.47)As we also have similar relations for ∆ and we have seen that ∆ m ,n (cid:54) = ∆ m ,n for Q (cid:54) = 0, we learn that( m i , n i ) = ( r j , s j ) ± (1 , , ( m , n ) (cid:54) = ( m , n ) , ( m , n ) (cid:54) = ( p − m , p (cid:48) − n ) . (3.48)For the intermediate channel, using a √ (cid:15) (cid:15) = M , we have( p (cid:48) k − p (cid:48) l + l ) = 49 ( p (cid:48) r j − p (cid:48) s j + s j ) , (3.49)so likewise, k = 23 r , l = 23 s , (3.50)or k = p − r , l = p (cid:48) − s , (3.51)where without loss of generality we have chosen j = 1 for convenience. As k, l areintegers, we must have r , s ∈ Z (or in other words, ( p − r ) , ( p (cid:48) − s ) ∈ Z ). As p = p (cid:48) −
1, it is straightforward to see that k, l ∈ Z for (3.50) while ( k, l ) ∈ (2 Z , Z + 1)or ( k, l ) ∈ (2 Z + 1 , Z ) for (3.51).We also need to take the fusion rule into account. In general, there are 2 × (cid:0) (cid:1) = 24possible choices of external legs, where 2 is the number of choices of ∆ and ∆ and (cid:0) (cid:1) corresponds to the choices of ∆ (cid:54) = ∆ . Therefore, we can discuss these possibilities caseby case. Here, we will provide the details for three representative cases as examples . Below we will use the correpsonding r ’s for external legs to denote each case. – 35 – xample 1: r , r + 1 , r − , r In such case, the fusion rule gives2 ≤ k ≤ min(2 r , p − r − ≤ k ≤ min(2 r − , p − r ) . (3.52)Putting them together, we have2 ≤ k ≤ min(2 r − , p − r − . (3.53)Therefore, r ≥ , p − r ≥ . (3.54)In fact, we can omit r ≥ r ∈ Z . Furthermore, we also require k − ( r + 1) + r − ∈ Z , that is, k ∈ Z . We can write similar conditions for l . Inparticular, l should also be even, so ( k, l ) should obey (3.50). Therefore, we also needto plug (3.50) into the above inequality. This gives r ≤ p − , s ≤ p (cid:48) − . (3.55)Comparing p − p − /
4, we find that p − ≤ p − / p ≤ p = 5). However, for p ≤
4, we cannot have p − r ≥ r ∈ Z . Hence, r ≤ min( p − , p − /
4) = 3( p − / s . In all, the conditions forthis case are r ≤ p − , s ≤ p (cid:48) − , k = 23 r , l = 23 s . (3.56) Example 2: r , r + 1 , r − , r In such case, it is not hard to see that k and l shouldsatisfy (3.51). Besides, the fusion rule gives | p − r − | + 1 ≤ k ≤ p − | p − r + 1 | + 1 ≤ k ≤ p − . (3.57)Putting them together, we havemax( | p − r − | + 1 , | p − r + 1 | + 1) ≤ k ≤ p − . (3.58)Since p − r − < p − r + 1, there are three possibilities:1. p − r − ≥
0: If p ≥ r + 1 , (3.59)then p − r + 2 ≤ k ≤ p − . (3.60)Plugging k = p − r into (3.60), one may check that (3.59) and (3.60) are indeedconsistent (they give the conditions r ≥ / r ≥ r ∈ Z ). – 36 –. p − r + 1 ≤
0: If p ≤ r − , (3.61)then 2 r − p + 2 ≤ k ≤ p − . (3.62)For this inequality to hold, we need p ≥ r + 2. Plugging k = p − r into theinequalities, we need r ≤ p − . Following the above same reasoning, it sufficesto keep r ≤ p − .3. p − r = 0: If p = 2 r , (3.63)then 2 ≤ k ≤ p − . (3.64)Plugging k = p − r into the inequalities, one may check that these inequalitiesare indeed consistent (they give the conditions r ≥ / r ≥ r ∈ Z ).The disussion for p (cid:48) , l, s is the same. Example 3: r , r + 1 , r − , r In such case, the fusion rule gives2 ≤ k ≤ min(2 r , p − r − ≤ k ≤ min( r + r − , p − r − r ) = min( p − , p ) = p − . (3.65)Putting them together, we have2 ≤ k ≤ min(2 r − , p − r − , p − . (3.66)Therefore, r ≥ , p ≥ , p − r ≥ , (3.67)where we can omit the first two conditions as we already have r ∈ Z . Furthermore,we also require k − ( r + 1) + r − ∈ Z , that is, k ∈ Z . We can write thesimilar conditions for l . In particular, l should also be even. However, we also have k − ( r −
1) + r − ∈ Z , that is, k − r − r = k − p = k − p (cid:48) + 1 ∈ Z . Likewise, l − p (cid:48) ∈ Z . This means that k, l cannot be even at the same time (i.e., they shouldsatisfy (3.51)). Hence, we reach an contradiction and this case is not possible.In fact, we can still reduce the number of cases to be checked. Since r = p − r ,we have r ± p − ( r ∓ r ± , r ∓ (cid:0) (cid:1) possibilities as ∆ (cid:54) = ∆ . Hence, there are 16 cases (including– 37 –he above three examples) overall. Moreover, just like in Example 3, we see that it failsto satisfy the fusion rule due to the parity of k, l . This can also be used to reduce thenumber of possible cases. One may check that r = r i , m = r j ± , i = j ⇒ ( k, l ) ∈ Z , Z ); r = r i , m = r j ± , i (cid:54) = j ⇒ ( k, l ) ∈ (2 Z , Z + 1) or (2 Z + 1 , Z ) . (3.68)This further reduces the number of possible cases (including the first two examples) to8. Although there are 8 distinct cases, there are only two conditions as in Example 1and 2. This is because for the combination r i , r i ± , r j ± , r j , we always have2 ≤ k ≤ min(2 r − , p − r − , (3.69)and for the combination r i , r j (cid:54) = i ± , r ι , r κ (cid:54) = ι ±
1, we always havemax( | p − r − | + 1 , | p − r + 1 | + 1) ≤ k ≤ p − . (3.70)This completes the proof, and the above conditions are summarized in Table 3.10.We can also see why the tetracritical Ising model is the one with smallest p (cid:48) for Γ(3).One way is to compute p (cid:48) = 3 , , p ) case by case, and none of themwould give parametrizations from Γ(3). Alternatively, it is straightforward to use theabove conditions as well. Likewise, we can deduce that the smallest possible p is 5.Moreover, this also tells us why we cannot have r = 6 or s = 6 for p (cid:48) = 6 , s = 6 is not allowed for ( p (cid:48) , p ) = (8 ,
5) as in Figure 3.1 etc.If a minimal model has CBs corresponding to Γ(3), then ( r , s ) = (3 ,
3) (and hence( r , s ) = ( p − , p (cid:48) − k, l ) is (2 , p − , p (cid:48) − or ∆ corresponds to (2 ,
2) or ( p − , p (cid:48) −
2) for all theeight cases. Therefore, we may use this to solve M and (cid:15) , . Suppose ∆ int = ∆ , then Q − M = − M + QM . (3.71)Hence, M = Q or Q with Q = i √ p (cid:48) ( p (cid:48) − . If we consider ∆ int = ∆ (which we haveseen that this would give no new CBs), then we have the opposite values, that is, M = − Q or − Q . Using M = m √ (cid:15) (cid:15) and √ (cid:15) (cid:15) Q = (cid:15) + (cid:15) , we may also solve (cid:15) , . Γ (4) ∩ Γ(2)Let us now discuss one more example, Γ (4) ∩ Γ(2). We first focus on the cases when ζ = 1 /
2. In terms of the simplifications we can make as above, there are only two caseswe need to consider. Again, we set M = m √ (cid:15) (cid:15) . In particular, one can find that the– 38 –wo cases only differ by ∆ . However, after some calculations, the fusion rule wouldalways lead to p (cid:48) , p ∈ Z , which is impossible for coprime p (cid:48) and p .Next, for ζ = 2, it is very similar to ζ = 1 / m , m and anoverall rescaling. We also have two distinct cases. For (+ , + , − , − ) , using the samemethod yields the CBs in minimal models with conditions in Table 3.11.Cases Conditions2 r , r ± , r ± , r r , r ± , p − ( r ± , p − r r ≤ p − ± ,p − r , p − (2 r ± , r ± , r k = 2 r p − r , p − (2 r ± , p − ( r ± , p − r p − r , r ± , p − ( r ± , r (cid:0) r < p ∓ or p − r , r ± , r ± , p − r p ∓ ≤ r ≤ p − / ∓ / or r = p − (cid:1) r , p − (2 r ± , p − ( r ± , r and k = p − r r , p − (2 r ± , r ± , p − r Table 3.11 : One set of possible CBs in minimal models that Γ (4) ∩ Γ(2) corresponds to. There aresimilar relations for s , l, p (cid:48) by a simple substitution of the corresponding letters, where we have set α = α r ,s . Likewise, the other case with ( − , + , − , +) gives the conditions in Table 3.12.Cases Conditions2 r , r ± , r ∓ , r r , r ± , p − ( r ∓ , p − r r ≤ p − ± ,p − r , p − (2 r ± , r ∓ , r k = 2 r p − r , p − (2 r ± , p − ( r ∓ , p − r p − r , r ± , p − ( r ∓ , r (cid:0) r < p ∓ or p − r , r ± , r ∓ , p − r p ∓ ≤ r ≤ p − / ∓ / or r = p − (cid:1) r , p − (2 r ± , p − ( r ∓ , r and k = p − r Here, it is still sufficient to choose two representatives for the two distinct cases. As differentparametrizations of the masses would only differ by signs of m i ’s, we will only use their signs todenote ( m , m , m , m ). This should be clear from the tables in § – 39 – r , p − (2 r ± , r ∓ , p − r Table 3.12 : The other set of possible CBs in minimal models that Γ (4) ∩ Γ(2) corresponds to. Thereare similar relations for s , l, p (cid:48) by a simple substitution of the corresponding letters, where we haveset α = α r ,s . It is not hard to see that for (+ , + , − , − ), the first CB appears in the minimalmodel with p (cid:48) = 5 , p = 4, viz, the tricritical Ising model. For ( − , + , − , +), the first CBappears in the minimal model with p (cid:48) = 4 , p = 3, viz, the (critical) Ising model. TheKac tables and corresponding CBs are shown in Figure 3.2.3
32 35 110
716 380 380 716
110 35 32
110 380380 380 (a)
12 116
01 0
116 12 (b) Figure 3.2 : Here we list the first possible examples of CBs that Γ (4) ∩ Γ(2) corresponds to: (a) Thefirst CB for (+ , + , − , − ). For reference, the one in grey is the CB from ( − , + , − , +) for this minimalmodel. (b) The first CB for ( − , + , − , +). Finally, let us consider ζ = −
1. Since a always vanishes, ∆ int = Q = − ( p (cid:48) − p ) p (cid:48) p .Hence, p (cid:48) k − pl = 0, that is, p (cid:48) /p = l/k . However, as gcd( p (cid:48) , p ) = 1 and k < p, l < p (cid:48) ,this is impossible.Now that we have found two dessins that corresponds to CBs in minimal models,we can consider their CBs in the same minimal model. Such example would first appearwhen p (cid:48) = 6 , p = 5 as in Figure 3.3. Following the above steps, we can derive the results for any dessin in general.
Proposition 3.3.
Suppose for a dessin, we have the gauge theory parameters withrelation m = ± / ∓ k m , m = ± / ∓ k m , m = ± / ∓ k m , a = ± / ∓ k int m , (3.72)– 40 – 3
138 23 18
75 2140 115 140 25
25 140 115 2140 75
18 23 138
31 2 3 4 5
115 115140 18140 140
115 140140 140 Figure 3.3 : The CBs from Γ(3) (cyan) and Γ (4) ∩ Γ(2) (green) in the tetracritical Ising model. Theones in orange appear for both of the dessins. The three CBs, from left to right, come from Γ(3),(+ , + , − , − ) and ( − , + , − , +) in Γ (4) ∩ Γ(2) respectively. where k i, int > . Then the dessin corresponds to 4-point CBs satisfying conditions inTable 3.13 in minimal models. Cases Conditions k r , k r ± , k r ± / ∓ , r max( | (cid:78) ± | , | (cid:77) ± | ) + 1 k r , k r ± , p − ( k r ± / ∓ , p − r ≤ k int r ≤ min( (cid:70) − , p − − (cid:70) ) ,p − k r , p − ( k r ± , k r ± / ∓ , r and k = k int r p − k r , p − ( k r ± , p − ( k r ± / ∓ , p − r p − k r , k r ± , p − ( k r ± / ∓ , r max( | ( k + k ) r ± | , | ( k + 1) r ± | ) + 1 p − k r , k r ± , k r ± / ∓ , p − r ≤ k int r ≤ min( p − − | (cid:78) | , p − − | (cid:77) | ) ,k r , p − ( k r ± , p − ( k r ± / ∓ , r and k = p − k int r k r , p − ( k r ± , k r ∓ , p − r Table 3.13 : The set of possible CBs in minimal models that a general dessin corresponds to. Thereare similar relations for s , l, p (cid:48) by a simple substitution of the corresponding letters, where we have set α = α r ,s and (cid:78) = ( k − k ) r , (cid:77) = (1 − k ) r , (cid:70) = (cid:18) −| k + k − k − | + (cid:80) i k i (cid:19) r . In particular, k i, int r ∈ N ∗ is a necessary condition. With these conditions, we can check what CBs in minimal models we can obtainfrom a dessin. For instance, when ζ = + √ (5), we have k = 1 , k = k =5 , k int = 2. It is not hard to find that the first CB it corresponds to appears when p (cid:48) = 7 , p = 6 as in Figure 3.4. – 41 – 5
227 127 57 17
238 8556 3356 556 156 38
43 1021 121 121 1021 43
38 156 556 3356 8556 238
17 57 127 227
51 2 3 4 5 6
556 156156
Figure 3.4 : The CB on the right has conformal dimensions coloured cyan in the Kac table.
Examples not giving minimal models
From Proposition 3.6, it is straightforwardto see that there could be dessins that do not correspond to CBs in minimal models.Besides the inequalities in Table 3.13, a necessary condition is that k i, int r and k i, int s should be positive integers. Let us verify this with some examples.For Γ (6), there are two big classes of parametrizations. If m or m has the factor √ m has the factor √ = Q − M . However, if we now express M in terms of the labels ( r , s ) for ∆ and insert this into ∆ , we find that( p (cid:48) r − ps ) = 4 × p (cid:48) r − ps ) , (3.73)where 109 is not a square number, and hence no integer solutions (except when 0 = 0which is excluded for minimal models). Therefore, it is not possible to get CBs inminimal models for Γ (6).For Γ (8), m i and a are non-zero and cannot simultaneously be real/pure imaginaryas in Table 3.5. Without loss of generality, suppose m i √ (cid:15) (cid:15) is pure imaginary and then a √ (cid:15) (cid:15) is real. This yields ∆ int = Q − a (cid:15) (cid:15) < Q . (3.74)Therefore, ( p (cid:48) k − pl ) − ( p (cid:48) − p ) p (cid:48) p < Q − ( p (cid:48) − p ) p (cid:48) p . (3.75)In other words, ( p (cid:48) k − pl ) < . (3.76)Hence, it is not possible to get CBs in minimal models for Γ (8).For Γ (9), since the a ’s are not real or pure imaginary, it should not give CBs inminimal models. – 42 – cknowledgement OF wishes to thank R. Santachiara for useful discussions on the topic of this note, aswell as related topics. The research of JB is supported by the CSC scholarship. OFis supported by the Australian Reasearch Council. YHH would like to thank STFCfor grant ST/J00037X/1. EH would like to thank STFC for the PhD studentship.YX is supported by NSFC grant No. 20191301017. FY is supported by the NSFCgrant No. 11950410490, by Fundamental Research Funds for the Central Universi-ties A0920502051904-48, by Start-up research grant A1920502051907-2-046, in part byNSFC grant No. 11501470 and No. 11671328, and by Recruiting Foreign ExpertsProgram No. T2018050 granted by SAFEA.
A The B-model and Omega Deformations
When mapping gauge theory/SW geometry parameters to CFT parameters, we needto include a factor of √ (cid:15) (cid:15) , which would lead to divergence under the flat space limit (cid:15) , →
0. Here, we discuss a way in terms of topological B-model so that the SWgeometry is still physically meaningful when (cid:15) , are non-zero.Recall that we have related N = 2 gauge theories to A-model topological strings.The mirror in B-model is defined by the equation vw + f ( x, y ) = 0 , (A.1)which is a CY that can be considered as fibration of uv = c for some constant c overthe Riemann surface f ( x, y ). In particular, f ( x, y ) = 0 can be identified as the SWcurve Σ. Denote the multiplicity of a BPS state in this 5d theory as N β , where β isessentially the charge of the BPS state . Mathematically, the BPS configuration canbe defined by a (complex) one-dimensional sheaf F (plus certain section in H ( F ))such that β = ch ( F ) , n = χ ( F ) , (A.2)where β ∈ H ( M , Z ) and n ∈ Z .The topological string amplitude then has the expansion F ( (cid:15) , (cid:15) , t ) = log( Z ) = ∞ (cid:88) n,g =0 ( (cid:15) + (cid:15) ) n ( (cid:15) (cid:15) ) g − F ( n,g ) ( t ) , (A.3) More precisely, we should also include the indices denoting the SU(2) L × SU(2) R spin represen-tations, but for our purpose here, it suffices to label it with the topological data β only. For moredetails, see for example [50]. – 43 –here Z is known as the (refined) Pandharipande-Thomas (PT) partition function, and g stands for the genus while t denotes the K¨ahler parameter measuring the volume of acurve in β , which can be identified as the Coulomb parameter a as we are focusing onSU(2) gauge group in this paper [50–52]. In particular, when n = g = 0, F (0 , is theprepotential F . In the limit (cid:15) , →
0, the PT partition function is naturally identifiedas the Nekrasov partition function at leading order:log( Z ) = ( (cid:15) (cid:15) ) − F (0 , . (A.4)Moreover, F (0 , and F (1 , can also be determined using the metric on M and thediscriminant of Σ as in Equation (3.22) and (3.23) in [52]. Then F ( n,g ) with higher( g + n ) can be deduced from the (generalized) holomorphic anomaly equation [50–52]¯ ∂ ¯ i F ( n,g ) = 12 ¯ C jk ¯ i (cid:32) D j D k F ( n,g − + (cid:88) m,h (cid:48) D j F ( m,h ) D k F ( n − m,g − h ) (cid:33) , g + n > , (A.5)where the three-point coupling ¯ C jk ¯ i is given in [51, 52], and D i is the covariant derivative.The prime in the sum indicates the omission of ( m, h ) = (0 , , ( n, g ). We also requirethe first term on the right hand side to vanish if g = 0.Therefore, the non-zero (cid:15) , would also make sense for the SW theory physicallyas the prepotential generates the topological string amplitudes. Hence, we could avoidthe divergence when mapping the gauge theory parameters to CFT parameters as in § B Brane Configurations
B.1 The Type IIA Brane Configuration
A type IIA configuration of parallel NS/D5-branes joined by D4-branes can be repre-sented in M theory as a single M5-brane with a more complicated world history.Before we write the rule for finding the Seiberg-Witten curve, we need to find outwhether we have a U( N ) or an SU( N ) gauge theory. This is discussed in [53], and goesas follows.First, consider D5-branes and D4-branes in type IIA superstring theory. The world-volume of a D5-brane is described as follows. D5-branes are located at x = x = x = 0and, in a semi-classical approximation, at fixed values of x . The world-volume ofD5-branes are parameterised by values of x , x , · · · , x . In addition, D4-branes areparameterised by x , x , x , x and x . D4-branes have their x -coordinate finite so thatthey terminate on D5-branes. We need to introduce a complex variable v = x + ix .– 44 –lassically, every D4-brane is located at a definite value of v . Since a D4-brane endingon a D5-brane creates a dimple in the D5-brane, the value x is the value measured at v = ∞ , far from the disturbance created by the D4-brane. By minimizing the volumeof the D5-brane, at large v , we obtain x = k ln | v | + const . (B.1)This is not well-defined for large v . Nevertheless, with D4-branes attached to the leftand to the right of the D5-brane, we have x = k q L (cid:88) i =1 ln | v − a i | − k q R (cid:88) j =1 ln | v − b j | + const , (B.2)where a i and b j are the v -values, or x -coordinates of D4-branes ending on the left andright respectively. Now x is well-defined for large v if and only if q L = q R , that is, ifthe forces on both sides are balanced. For infrared divergence, we need to consider themotion of the D4-branes, whose movement causes the D5-brane to move. The motionof a D5-brane contributes to the kinetic energy of the D4-brane. The D5-brane kineticenergy is given by (cid:82) d x d v (cid:80) µ =0 ∂ µ x ∂ µ x . Therefore, with x in (B.2), we have k (cid:90) d x d v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:32)(cid:88) i ∂ µ a i v − a i − (cid:88) j ∂ µ b j v − b j (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (B.3)This integral converges if and only if ∂ µ (cid:32)(cid:88) i a i − (cid:88) j b j (cid:33) = 0 , (B.4)so that (cid:88) i a i − (cid:88) j b j = q α , (B.5)where q α is characteristic of α -th plane. From the D4-brane point of view, (B.5) meansthe U(1) part of U( k ) for k D4-branes between two D5-branes are frozen. This isbecause (cid:80) i a i is the scalar part of U(1) vector multiplet in one factor U( k α ) and (cid:80) j b j is the scalar part of the U(1) vector multiplet in the factor U( k α +1 ). Since, following(B.5), the difference is fixed by supersymmetry, the entire U(1) vector multiplet ismissing, and we have SU( N ). – 45 – .2 The M-theory Brane Configuration The world-volume of the M5-brane is such that,1. It has arbitrary values in the first M coordinates x , · · · , x , and is located at x = x = x = 0;2. In the remaining four coordinates, which parametrize a 4-manifold Q ∼ = R × S ,D5-brane worldvolume spans a 2d surface Σ;3. The N = 2 supersymmetry means we give Q the complex structure in which v = x + ix and s = x + ix are holomorphic, then Σ is a complex Riemannsurface in Q . This makes M × Σ a supersymmetric cycle in the sense of [54] andso it ensures spacetime supersymmetry.When projected to type IIA brane diagrams, Σ has different components describedlocally by saying that s is constant (the D5-branes) or that v is constant (the D4-branes). In type IIA, different components can meet and singularity appears in there.However, in going to M theory, singularities disappear. Hence, for generic values ofparameters, Σ will be a smooth Riemann surface in Q . C Congruence Subgroups of the Modular Group
In this appendix, we very briefly recall some essential details regarding the modulargroup Γ ≡ Γ (1) = PSL (2 , Z ) = SL (2 , Z ) / {± I } , the group of linear fractional trans-formations Z (cid:51) z → az + bcz + d , with a, b, c, d ∈ Z and ad − bc = 1. It is generated by thetransformations T and S defined by T ( z ) = z + 1 , S ( z ) = − /z . (C.1)The presentation of Γ is (cid:10) S, T | S = ( ST ) = I (cid:11) .The most important subgroups of Γ are the congruence subgroups, defined byhaving the entries in the generating matrices S and T obeying some modular arithmetic.Of particular note are the following: • Principal congruence subgroups:Γ ( m ) := { A ∈ SL(2; Z ) ; A ij ≡ ± I ij mod m } / {± I } ; • Congruence subgroups of level m : subgroups of Γ containing Γ ( m ) but not anyΓ ( n ) for n < m ; – 46 – Unipotent matrices:Γ ( m ) := (cid:40) A ∈ SL(2; Z ) ; A ij ≡ ± (cid:18) b (cid:19) ij mod m (cid:41) / {± I } ; • Upper triangular matrices:Γ ( m ) := (cid:26)(cid:18) a bc d (cid:19) ∈ Γ ; c ≡ m (cid:27) / {± I } . In [6, 48], attention is drawn to the conjugacy classes of a particular family of subgroupsof Γ: the so-called genus zero, torsion-free congruence subgroups: • Torsion-free means that the subgroup contains no element of finite order otherthan the identity. • To explain genus zero , first recall that the modular group acts on the upper half-plane H := { τ ∈ C , Im ( τ ) > } by linear fractional transformations z → az + bcz + d .Then H gives rise to a compactification H ∗ when adjoining cusps , which arepoints on R (cid:116) ∞ fixed under some parabolic element (i.e. an element A ∈ Γnot equal to the identity and for which Tr ( A ) = 2). The quotient H ∗ / Γ is acompact Riemann surface of genus 0, i.e. a sphere. It turns out that with theaddition of appropriate cusp points, the extended upper half plane H ∗ factoredby various congruence subgroups will also be compact Riemann surfaces, possiblyof higher genus. Such a Riemann surface, as a complex algebraic variety, is calleda modular curve . The genus of a subgroup of the modular group is the genus ofthe modular curve produced in this way.The genus zero torsion-free congruence subgroups of the modular group are very rare:there are only 33 of them, with index I ∈ { , , , , , } , as detailed in [48]. D Elliptic Curves and j -Invariants Given the Weierstrass function ℘℘ ( z | ω , ω ) = 1 z + (cid:88) n + m (cid:54) =0 (cid:18) z + mω + nω ) − mω + nω ) (cid:19) , (D.1)where ω and ω are complex-valued vectors that span the lattice Λ = { mω + nω : m, n ∈ Z } , and we can write ℘ ( z | ω , ω ) = ℘ ( z | Λ). The embedding of a torus, as anelliptic curve over C in the complex projective plane, follows from( ℘ (cid:48) ( z )) = 4 ( ℘ ( z )) − g ℘ ( z ) − g , (D.2)– 47 –here ℘ (cid:48) ( z ) is the derivative of ℘ ( z ) with respect to z . Naturally defined on a torus C / Λ, ℘ is doubly-periodic with respect to lattice Λ. This torus can be embedded inthe complex projective plane by z (cid:55)→ [1 : ℘ ( z ) : ℘ (cid:48) ( z )]. Close to the origin, ℘ ( z ) can beexpanded as ℘ ( z | Λ) = 1 z + g z
20 + g z
28 + O (cid:0) z (cid:1) , (D.3)where g = 60 (cid:88) ( m,n ) (cid:54) =(0 , (cid:18) mω + nω (cid:19) ,g = 140 (cid:88) ( m,n ) (cid:54) =(0 , (cid:18) mω + nω (cid:19) . (D.4)The summed terms in g and g are the first two Eisenstein series respectively. TheEisenstein series G k with weight 2 k are modular forms of weight 2 k , that is, theytransform as G k ( τ ) (cid:55)→ ( cτ + d ) k G k ( τ ) under SL(2 , Z ) with τ = ω /ω in upper half-plane H . If two lattices are related by a multiplication by a non-zero complex number c , then the corresponding curves are isomorphic. The j -invariants are defined as j ( τ ) = 1728 g g − g . (D.5)This definition shows that the j -invariant is a weight-zero modular form. From theabove discussion, we can see that each isomorphism class of elliptic curves over C hasthe same j -invariant.As the SW curves and Strebel differentials we have are of quartic form, y = az + bz + cz + dz + q , we can make the substitution (for q (cid:54) = 0) z = 2 q ( X + c ) − d / (2 q ) Y , y = − q + 12 q q ( X + c ) − d / (2 q ) Y (cid:18) q ( X + c ) − d / qY − d (cid:19) (D.6)so that the elliptic curve can be expressed in the standard Weierstrass form Y + a XY + a Y = X + a X + a X + a , (D.7)where a = dq , a = c − d q , a = 2 bq, a = − aq , a = ad − acq . (D.8)Using SAGE [55], we can compute its j -invariant j = − (( a + 4 a ) − a a − a ) ( a a − a a a + a a − a + 4 a a )( a + 4 a ) + 8( a a + 2 a ) − a + 4 a )( a a + 2 a )( a + 4 a ) + 27( a + 4 a ) . (D.9) If q = 0 such as the Strebel differential for Γ(3), we can replace z and y with 1 /z and y/z respectively to obtain a quartic form with a non-vanishing constant term [56].– 48 – Elliptic Functions and Coulomb Moduli
E.1 The Elliptic Integral of First Kind
We first give a quick review on deriving (3.6). From (3.1), we haved λ SW d U = − ζ (cid:112) P ( z ) . (E.1)Then d a d U = 12 πi (cid:73) A d λ SW d U = − πi (cid:90) λ λ ζ (cid:112) P ( z ) d z. (E.2)Therefore, the integral boils down to solving (cid:90) λ λ d z (cid:112) ( z − λ )( z − λ )( z − λ )( z − λ ) . (E.3)First, we make a PSL(2, Z ) transformation, z = At + BCt + D , such that λ , , are mapped to0 , , ∞ respectively. Then A = ( λ − λ ) λ , B = ( λ − λ ) λ , C = λ − λ , D = λ − λ gives a solution, and λ = ( λ − λ )( λ − λ )( λ − λ )( λ − λ ) =: t . After substitution of variables and somealgebra, the integral becomes 1 (cid:112) ( λ − λ )( λ − λ ) (cid:90) d t (cid:112) t (1 − t )(1 − t − t ) . (E.4)In particular, (cid:90) d t (cid:112) t (1 − t )(1 − t − t ) = π F (cid:18) ,
12 ; 1 , t − (cid:19) = 2 K ( t − / ) , (E.5)where K is the elliptic integral of the first kind K ( x ) = (cid:90) d t (cid:112) (1 − t )(1 − x t ) . (E.6)Hence, (cid:90) λ λ d z (cid:112) ( z − λ )( z − λ )( z − λ )( z − λ ) = 2 K (cid:16)(cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:17)(cid:112) ( λ − λ )( λ − λ ) . (E.7)– 49 – .2 The Elliptic Logarithm Here we present an alternative way to obtain a from d a/ d U by integrating from U to ∞ . Therefore, we need to determine a when U → ∞ . At large U , we have P ( z ) U (cid:12)(cid:12)(cid:12)(cid:12) U →∞ = ( − − ζ ) z + (1 + ζ ) z + ( − ζ (1 + ζ )) z, (E.8)which yields P ( z ) | U →∞ = − (1 + ζ ) U z + (1 + ζ ) U z − ζ (1 + ζ ) U z. (E.9)If | ζ | ≤
1, thend a d U (cid:12)(cid:12)(cid:12)(cid:12) U →∞ = − ζ πi (cid:90) λ λ (cid:0) ( − (1 + ζ ) U )( z − (1 + ζ ) z + ζz ) (cid:1) − / d z = √ ζ π √ U (cid:90) ∞ ( z − (1 + ζ ) z + ζz ) − / d z = − √ ζπ √ U EL(1 , − − ζ, ζ ) , (E.10)where EL is the elliptic logarithm defined asEL( x, y ; a, b ) = 12 (cid:90) x ∞ d t √ t + at + bt , y = √ x + ax + bx. (E.11)Therefore, a ( U ) | U →∞ = − π (cid:112) ζ EL(1 , − − ζ, ζ ) √ U . (E.12)However, notice that the above steps are not rigorous. We need to be careful aboutthe branches of square roots. Taking this into account, when 1 + ζ <
0, there shouldbe an minus extra sign , that is, a ( U ) | U →∞ ,ζ< − = 2 π (cid:112) ζ EL(1 , − − ζ, ζ ) √ U . (E.13)Henceforth, we will not repeat this point below. As a sanity check, we can see whatwould happen at weak coupling. When ζ = 0, EL(1 , − ,
0) = π/
2. We learn that a ( U ) | U →∞ ,ζ → = −√ U , (E.14) In practice, we usually choose a large cutoff (which can be either positive or negative) for U instead of ∞ when performing numerical integrals. Therefore, the branches of square roots with U inside are also important if we take U to a large negative number. We may also have an extra minus sign for all the expressions of a ( U ) | U →∞ here. Mathematically,this should correspond to choosing a different branch in the redefinition of square root. Physically,this is due to the action of Weyl group of the gauge symmetry. – 50 –hich is the familiar behaviour in the (semi)classical limit.If | ζ | >
1, then we can just replace (cid:82) ∞ with (cid:82) ∞ ζ , and hence a ( U ) | U →∞ = − π (cid:112) ζ EL( ζ, − − ζ, ζ ) √ U . (E.15)Likewise, we can also write down a similar expression for a D at large U , a D = − π (cid:112) ζ EL(0 , − − ζ, ζ ) √ U . (E.16)If we take ζ = 0, then EL(0 , − ,
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