The planar limit of N=2 superconformal quiver theories
aa r X i v : . [ h e p - t h ] J un Prepared for submission to JHEP
The planar limit of N = 2 superconformalquiver theories Bartomeu Fiol, Jairo Mart´ınez-Montoya and Alan Rios Fukelman
Departament de F´ısica Qu`antica i Astrof´ısica iInstitut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028Barcelona, Catalonia, Spain
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We compute the planar limit of both the free energy and the expectationvalue of the 1 / N = 2 superconformal quivertheories, with a product of SU( N )s as gauge group and bi-fundamental matter. Super-symmetric localization reduces the problem to a multi-matrix model, that we rewrite inthe zero-instanton sector as an effective action involving an infinite number of double-trace terms, determined by the relevant extended Cartan matrix. We find that theresults, as in the case of N = 2 SCFTs with a simple gauge group, can be writtenas sums over tree graphs. For the c A case, we find that the contribution of each treecan be interpreted as the partition function of a generalized Ising model defined on thetree; we conjecture that the partition functions of these models defined on trees satisfythe Lee-Yang property, i.e. all their zeros lie on the unit circle. ontents N = 2 quiver CFT 4 N limit 15A Planar free energy up to th order 19B Wilson loop up to λ The emergence of quantum gravity from a gauge theory is one of the most fascinatingissues that can be addressed with the AdS/CFT correspondence. Since the work of [1]it has been clear that not every conformal field theory (CFT) in the large N limit canbe dual to a gravitational theory described by a two derivative Einstein-Hilbert action.For instance, for four dimensional CFTs a necessary condition is that the two centralcharges coincide in the large N limit, a = c [1]. For instance, this property is satisfiedby N = 4 super Yang-Mills, but it is not satisfied by N = 2 SU( N ) with n F = 2 N hypermultiplets in the fundamental representation, thus ruling out that the large N limit of this CFT has a holographic dual well described by gravity.Since the early days of the holographic correspondence, it has been important tofind further examples of CFTs with holographic duals, beyond the original exampleof N = 4 SYM. Four dimensional quiver gauge theories with N = 2 superconformalsymmetry satisfy an ADE classification [2], and for certain values of the marginalcouplings, they are orbifolds of N = 4 SYM and have a gravity dual [3, 4]. These quivergauge CFTs constitute thus an interesting laboratory, as variation of their marginal– 1 –ouplings allows to connect CFTs with and without gravity duals in the large N limit[5–12].In this work we will consider N = 2 SCFTs with gauge group a product ofSU( N )s, paying special attention to the simplest case, the c A theory, with gauge groupSU( N ) × SU( N ). This theory has two marginal couplings ( g , g ) and varying them onecan reach an orbifold of N = 4 SYM and N = 2 SU( N ) SQCD. Our main technical toolwill be supersymmetric localization [13]. Thanks to this tool, the planar free energyand expectation value of the 1/2 BPS circular Wilson loop are known to all orders inthe ’t Hooft coupling for the limiting theories mentioned above N = 4 SYM and N = 2SU( N ) SQCD [14–19].Four dimensional N = 2 quiver CFTs have already been studied using localization[7, 10–12, 20]. The novelty of this work is that we evaluate various quantities of thesetheories in the planar limit, to all orders in the ’t Hooft couplings λ i . We do so byapplying the same strategy developed for CFTs with simple gauge groups in [19]. Forthese quiver CFTS, supersymmetric localization [13] reduces the evaluation of variousquantities to matrix integrals. Compared to the case of N = 2 SCFTs with a simplegauge group, the main novelty is that the resulting matrix models are multi-matrixmodels. In the simplest case, the model to solve is a two-matrix model. As in ourrecent work [19], we rewrite the 1-loop factor as an effective action involving an infinitenumber of double-trace terms, in the fundamental representation of the respective gaugegroups. We then show that this double-trace form of the potential implies that theperturbative series considered admit a combinatorial formulation, as sums over treegraphs.While we will present results valid for all N = 2 quiver CFTs, we will pay specialattention to the simplest theory, c A . This theory has a Z symmetry exchangingthe two nodes of the quiver. Since the ranks of the gauge groups are equal, this Z symmetry amounts to exchanging g ↔ g . We will be particularly interested inobservables that transform nicely under this symmetry: the free energy and particularlinear combinations of the usual 1/2 BPS circular Wilson loop defined for each node[7]. In Section 2, after introducing the theories we will consider, we derive the per-turbative series of the planar free energy, to all orders in the ’t Hooft couplings λ i .– 2 –et’s present here the answer for the c A theory. It is convenient to define F ( λ , λ ) = F ( λ , λ ) − F ( λ ) N =4 − F ( λ ) N =4 . The perturbative series is given by a sum overtree graphs, F ( λ , λ ) = ∞ X m =1 ( − m ∞ X n ,...,n m =2 ζ (2 n − . . . ζ (2 n m − n . . . n m ( − n + ··· + n m n − X k =1 (cid:18) n k (cid:19) · · · n m − X k m =1 (cid:18) n m k m (cid:19) X unlabeled treeswith m edges | Aut(T) | m +1 Y i =1 ¯ V i , (1.1)where the product at the end of the last line runs over the vertices of a tree, and ¯ V i are factors to be defined below. This expression is formally identical to the one foundfor N = 2 SQCD in [19], except for the fact that now the factors ¯ V i depend on two ’tHooft couplings, λ and λ . The terms in (1.1) with a single value of the ζ functionhave already appeared in [20]. In the perturbative expansion of F ( λ , λ ) above, eachproduct of values of the ζ function is accompanied by a polynomial in λ and λ , thatcan be rewritten as a palindromic polynomial in λ /λ . Intriguingly, up to the orderwe have checked explicitly, all such polynomials have all roots on the unit circle of thecomplex λ /λ plane. This is of course reminiscent of the seminal work by Lee andYang [21] for the zeros of the partition function of the ferromagnetic Ising model on agraph. We are able to prove this property for all the terms in (1.1) with a single valueof ζ , and formulate two conjectures for general trees.In Section 3, we compute the planar limit of the expectation value of the 1 / c A theory, and in the fundamental representation. The answer is now given as a sum overrooted trees. This Wilson loop is defined for one of the two nodes of the quiver, so itdoes not transform nicely under the Z symmetry of the theory. For this reason weconsider h W i ± = h W i ± h W i (with the N = 4 results subtracted). For h W i ± we findagain that, up to the orders we have checked explicitly, all the polynomials in λ /λ that appear have all roots on the unit circle.In the appendices, we write the first terms in the explicit expansion of the planarfree energy and expectation value of various Wilson loop operators.This work leaves open a number of interesting problems. First, there are generalarguments that the perturbative series of the planar limit of quantum field theories– 3 –ave finite radius of convergence [22]. We have been able to determine the domain ofconvergence of just a small subset of the perturbative series found in this paper - see also[20] - but rigorously determining the full domain of convergence of the full perturbativeseries seems like a much harder problem. Second, in the main text we formulate twoconjectures on the zeros of the polynomials that appear in the perturbative series ofthe planar free energy and expectation values of Wilson loops. It would be interestingto prove these conjectures, and further investigate if this property is related to theintegrability of the planar limit of these theories [6, 9, 23, 24]. N = 2 quiver CFT In this section we introduce the theories we are going to study, and recall how super-symmetric localization reduces the evaluation of selected quantities to matrix integrals.In particular, we will study first the planar free energy of the theory. Following [25–27],the integrals are performed over the full Lie algebra instead of restricting to a Cartansubalgebra, and the 1-loop factor is rewritten as an effective action. We will focuson the planar limit and in this limit, as in [19], we will unravel the underlying graphstructure of the perturbative expansion.Let us start by briefly reviewing the classification and field content of N = 2superconformal quiver gauge theories with SU( N ) gauge groups. They are in one-to-one correspondence with simply-laced affine Lie algebras \ ADE , and thus follow an ADEclassification [2]. The gauge sector and matter content are encoded in the extendedCartan matrix of the affine Lie algebra. The gauge group is Y i SU( n i N ) , (2.1)where n i is the Dynkin index of the i -th node of the affine Dynkin diagram. Thehypermultiplets transform in the representations ⊕ a ij (cid:0) n i N, n j N (cid:1) , (2.2)where a ij is the adjacency matrix of the Dynkin diagram.These theories have a marginal coupling for each gauge group and, in the particularcase where the complexified couplings satisfy τ i = n i τ , (2.3)– 4 –he quiver theory can be obtained as an orbifold of N = 4 SU( N ) super Yang-Mills bythe discrete subgroup Γ of SU(2) [2], which also follow an ADE classification. Thesetheories can be engineered in string theory via a suitable brane configuration and evenmore, in a suitable limit, they admit a weakly curved gravity dual in terms of the AdS × S / Γ geometry [3, 4]. On the other hand, when all the couplings are set to zeroexcept one, say g , the quiver theory reduces to N = 2 SQCD.After having reviewed N = 2 superconformal quiver theories, let’s discuss super-symmetric localization for them. Following [13] it is possible to localize the \ ADE theories on S . It is also possible to localize the theory on a squashed sphere of pa-rameter b for which in the limit b → Z = Z d a I Z ( a I , b ) | Z inst ( a I , b ) | e − P nI =1 8 π g I Tr a I , (2.4)where a I denotes the eigenvalues of the vector-multiplet scalars Φ I restricted to theconstant mode on S . In what follows we will be mostly interested in quantities thatare relevant in the b ≃ |Z inst | , and expanding (2.4) in b we obtain Z = Z d a I Z ( a I ) e − P nI =1 8 π g I Tr a I + O (( b − ) , (2.5)higher order terms in b were studied before in [10] and we refer the reader there formore details. The factor Z is the 1-loop contribution determined by the mattercontent. For instance for the [ A n − theory it is given by Z = n Y I =1 Q i 2, and bi-fundamentalmatter: ( X, Y, X † , Y † ) : D µ X = ∂ µ X + A µ X − XA µ . The l-loop factor reduces to Z = Q i 1. The overall sign of a given product of correlators is then − − α j to all vertices of the tree corresponding to– 8 –orrelators of, say, the second gauge group (this choice is arbitrary and the final resultis independent of it). To convince oneself that these two rules are the same, write everysign on top of the edges of the tree: if it is a − − − 1) and again assign one − V ( x , . . . , x α ) = V ( x , . . . , x α ) (cid:16) ˜ λ P i x i + ( − α ˜ λ P i x i (cid:17) , (2.17)and the generic expression (2.16) simplifies to F ( λ , λ ) = ∞ X m =1 ( − m m ! ∞ X n ,...,n m =2 ζ (2 n − . . . ζ (2 n m − n . . . n m ( − n + ··· + n m n − X k =1 (cid:18) n k (cid:19) · · · n m − X k m =1 (cid:18) n m k m (cid:19) X directed treeswith m labeled edges m +1 Y i =1 ¯ V i . (2.18)Finally, by exactly the same arguments as in our previous paper [19], the last sum canbe reduced to a sum over unlabeled trees F ( λ , λ ) = ∞ X m =1 ( − m ∞ X n ,...,n m =2 ζ (2 n − . . . ζ (2 n m − n . . . n m ( − n + ··· + n m n − X k =1 (cid:18) n k (cid:19) · · · n m − X k m =1 (cid:18) n m k m (cid:19) X unlabeled treeswith m edges | Aut(T) | m +1 Y i =1 ¯ V i . (2.19)Let’s mention a further property of F ( λ , λ ). Since F ( λ , λ ) = F ( λ , λ ) and F ( λ , λ ) = 0, it follows that F ( λ , λ ) has a double zero, F ( λ , λ ) = ( λ − λ ) f ( λ , λ ) , (2.20)this implies that at the orbifold point λ = λ not just the free energy, but also its firstderivative with respect to λ coincides with the N = 4 result. To see that this propertyis implied by our result (2.19), we are going to prove that the contribution of every treeto (2.19) is of the form (˜ λ − ˜ λ ) v odd p (˜ λ , ˜ λ ) , (2.21)where v odd is the number of vertices of the tree with odd degree, and p (˜ λ , ˜ λ ) isa symmetric polynomial in ˜ λ and ˜ λ with positive coefficients. This follows from– 9 –nspection of the factor attached to each vertex, (2.17). When the degree α of a vertexis odd, ˜ λ = ˜ λ is a simple root of that factor. After pulling out these factors, whatis left is a polynomial with positive coefficients. As a check, notice that v odd is alwayseven: for a tree with m + 1 vertices, P m +1 i =1 α i = 2 m , and since P i α even i is even, P i α odd i must be even also, which implies that v odd is even. This concludes the argument for(2.21). Now, since every tree has at least two vertices of degree one, v odd ≥ 2, and(2.20) follows.To illustrate (2.19), let’s work out the first terms. The m = 1 terms in (2.19) areterms with a single value of ζ [20]. To write them, it is convenient to first recall thedefinition of the Narayana numbers N ( n, k ) = 1 n (cid:18) nk (cid:19)(cid:18) nk − (cid:19) , (2.22)and the Narayana polynomials C n ( t ) = n − X k =0 N ( n, k + 1) t k , (2.23)that satisfy C n (1) = C n with C n the Catalan numbers. At this order, we have to considertrees with two vertices. There is just one such tree, and both vertices have degree one.Then, F ( λ , λ ) | ζ = − ∞ X n =2 ζ (2 n − n ( − n n − X k =1 (cid:18) n k (cid:19) C n − k C k (cid:16) ˜ λ n − k − ˜ λ n − k (cid:17) (cid:16) ˜ λ k − ˜ λ k (cid:17) = − ∞ X n =2 ζ (2 n − n ( − n C n ˜ λ n " λ n ˜ λ n ! C n +1 − C n +1 ˜ λ ˜ λ ! , (2.24)where to avoid confusion, the first term in the parenthesis involves the Catalan number C n +1 , and the second one the Narayana polynomial C n +1 (˜ λ / ˜ λ ). A first question wecan ask about this series is what is its domain of convergence in C . As pointed outin [19, 20], when λ = 0 it is straightforward to prove that the radius of convergenceis λ = π , and the same holds, mutatis mutandi , when λ = 0. When both couplingsare different from zero, since F ( λ , λ ) = 0 the series trivially converges when bothcouplings are equal. When the two couplings are different, one of them is larger, say λ , applying the quotient criterion it follows that for any | λ | < | λ | ≤ π , the series is– 10 –onvergent. All in all, this series is convergent in | λ | ≤ π , | λ | ≤ π plus the λ = λ line.For N = 2 superconformal field theories with a simple gauge group, terms with afixed number of values of the ζ function form an infinite series. In [19] we sketched anargument that all these series have the same radius of convergence. It seems possiblethat this property extends to quiver theories.Let’s work out a couple more of terms in (2.19). Terms with two values of the ζ function are given by a sum over trees with two edges. There is just one tree withtwo edges, and its vertices have degrees (1 , , ζ function are given by a sum over trees with three edges. There are twosuch unlabeled trees. The degrees are (1 , , , 1) for the first tree, and (3 , , , 1) for thesecond, all these trees are despicted in fig. (1) and (2). Up to this order, F ( λ , λ ) = − ∞ X n =2 ζ (2 n − n ( − n n − X k =1 (cid:18) n k (cid:19) V ( n − k ) V ( k )(˜ λ n − k − ˜ λ n − k )(˜ λ k − ˜ λ k )+ 12 ∞ X n i =2 ζ (2 n i − n n ( − n + n n i − X k i =1 (cid:18) n i k i (cid:19) V ( k ) V ( n − k , n − k ) V ( k )(˜ λ k − ˜ λ k )(˜ λ n − k + n − k + ˜ λ n − k + n − k )(˜ λ k − ˜ λ k ) − ∞ X n i =2 ζ (2 n i − n n n ( − n + n + n n i − X k i =1 (cid:18) n i k i (cid:19) h V ( n − k ) V ( k , n − k ) × V ( k , n − k ) V ( k )(˜ λ n − k − ˜ λ n − k )(˜ λ k + n − k + ˜ λ k + n − k ) × (˜ λ k + n − k + ˜ λ k + n − k )(˜ λ k − ˜ λ k ) + V ( n − k , n − k , n − k ) V ( k ) V ( k ) V ( k ) × (˜ λ n − k + n − k + n − k − ˜ λ n − k + n − k + n − k )(˜ λ k − ˜ λ k )(˜ λ k − ˜ λ k )(˜ λ k − ˜ λ k ) i + O ( ζ ) . (2.25)As a first check, when either of the two couplings vanishes, we recover the result of N = 2 SCQD presented in [19]. Also, in this expression we can see rather explicitlythat at every order the contribution has at least a double zero ( λ − λ ) . In AppendixA we have written the outcome of these sums, up to order λ .– 11 – .2 The Lee-Yang property of the planar free energy expansion. We would like to discuss one further property of the perturbative expansion (2.19).Notice that the contribution of a given tree is obtained by summing over all the possibleways to assign one gauge group, 1 or 2, to each vertex in the tree, see figures (1) and (2).This is reminiscent of the Ising model defined on that tree, where on each vertex we canhave a spin up or down. It is indeed possible to construct a generalized Ising-type model,with inhomogeneous external magnetic field, whose partition function yields each treecontribution in (2.19). This generalized Ising model is admittedly a bit contrived, butfollowing the classical work by Lee and Yang [21], it motivates the study of the zerosof its partition function.In more detail, every tree graph contributes to the planar energy in (2.19) a ho-mogeneous polynomial in λ and λ . Being homogeneous, these polynomials can bethought of as polynomials of a single variable λ /λ . Inspired by the classical work byLee and Yang [21] on the ferromagnetic Ising model, we are going to put forward twoconjectures regarding the zeros of these polynomials: first, that for a given tree, allthe zeros of the corresponding polynomial are on the unit circle in the complex λ /λ plane. Second, that when we sum the contributions from different trees with the samenumber of nodes, the same property holds.To provide context, let’s start by briefly recalling the definition of the Ising modelon a graph and the Lee-Yang theorem. Let G be a finite graph, E its set of edges and V its set of vertices. The Ising model on G is defined by assigning to each vertex i ∈ V ,a σ i = ± H = − J X i − j ∈ E σ i σ j − H X i ∈ V σ i , (2.26)with J the coupling among spins and H the external magnetic field. The partitionfunction can be written as Z ( βJ, βH ) = X all states e − β H = e βJ | E |− βH | V | X all states e − βJe ± e βHv ↑ , (2.27)where e ± is the number of edges connecting different spins, and v ↑ the number of spinsup in a given configuration. Define τ = e − βJ , x = e βH . The last sum defines a– 12 –olynomial palindromic in x , P ( τ, x ) = X all states τ e ± x v ↑ . (2.28)In [21], Lee and Yang proved that for τ ∈ [ − , P ( τ, x ) haveall their x roots on the unit circle. In fact, they proved it for arbitrary ferromagneticcouplings J ij ≥ 0, and different magnetic fields per site H i .To construct an Ising-type model whose partition function yields the polynomialsthat appear in (2.19), proceed as follows. Take the graph G to be a tree T,1. Assign a positive integer n i to each of the e edges of the tree graph.2. For every edge, split n i into two positive integers, n i = k i + ( n i − k i ) and assigneach of these two integers to one of the vertices at the ends of that edge.3. Then, if a vertex has degree d j this procedure assigns to that vertex d j integers.Let m j be the sum of these integers at a given vertex; the magnetic field at thatvertex is then m j H .So far, for a fixed partition of all n i , this is a peculiar way to assign external magneticfields that are different at each vertex. This defines P ( τ, x, k i , n i ) = X all states τ e ± Y verticeswith spin up x m j , (2.29)Lee and Yang already proved (lemma in Appendix II of [21]) that all the zeros of thesepolynomials are on the unit circle. Finally, consider the sum over all the partitions ofeach of the n i into two P ( τ, x, n , . . . , n e ) = n − X k =1 · · · n e − X k e =1 ρ ( k i , n i ) X all states τ e ± Y verticeswith spin up x m j , (2.30)where ρ ( k i , n i ) is a distribution that weights different configurations. The contributionof every tree to the planar free energy in (2.19) is obtained from the free energy of thisIsing-type model, by setting τ = − x = λ /λ , and the distribution ρ ( k i , n i ) = (cid:18) n k (cid:19) . . . (cid:18) n m k m (cid:19) m Y i =1 V i . (2.31)– 13 –he main reason we have defined this family of Ising-type models is that there is nu-merical evidence that suggests that they share the Lee-Yang property with the originalIsing model. This leads us to formulate the following two conjectures:Conjecture 1: For any tree with e edges, any fixed positive integers n , . . . , n e andarbitrary ρ ( k i , n i ) > P ( τ, x, n , . . . , n e ) have all their x roots on theunit circle.Conjecture 2: If we sum the polynomials of all the trees with the same number ofedges, the resulting polynomial still has the Lee-Yang property.We can prove the first conjecture in the particular case of the simplest tree. In thiscase, (2.30) is simply P ( τ, x, k, n ) = x n + τ x n − k + τ x k + 1 , (2.32)that for | τ | ≤ P ( τ, x, k, n ) = n − X k =1 ρ ( n, k ) (cid:0) x n + τ x n − k + τ x k + 1 (cid:1) , (2.33)with arbitrary ρ ( n, k ) > 0. To prove that these polynomials have their roots on theunit circle, we make use of the following theorem [30]: if P ( x ) = A n x n + A n − x n − + · · · + A x + A is a palindromic polynomial and 2 | A n | ≥ P n − j =1 | A j | , then all its zerosare in the unit circle. In our case, the inequality in the theorem is satistifed as longas | τ | ≤ 1, so the result follows. Back to the free energy of the quiver theory, one cancheck indeed that the polynomials in the expansion (2.24) have the Lee-Yang property.We haven’t been able to prove these two conjectures for arbitrary tree graphs.After the seminal work [21], the proof of the Lee-Yang unit circle theorem has beenextended to many other systems, see e.g. [31, 32]. It would be interesting to see if anyof these arguments can be adapted to prove our conjectures. Figure 1 : Trees contributing to the first and second order expansion of the free energy.– 14 –a) (b) Figure 2 : The two trees with three edges: ( a ) Tree with vertices of degrees (1,2,2,1).( b ) Tree with vertices of degrees (3,1,1,1). There are 16 ways to color each of them. N limit For each of the gauge groups of the quiver theory, we can define a 1/2 BPS Wilsonloop, with circular contour in Euclidean signature. The evaluation of its expectationvalue reduces to a matrix integral thanks to supersymmetric localization. We will nowevaluate the planar limit of this expectation value and show that the perturbative seriesinvolves a sum over rooted trees. While the Wilson loop can be defined for arbitraryrepresentations of the gauge group, in order to take advantage of the results of [19, 29],we will restrict its study to the fundamental representation h W I i = h N Tr F P exp I C ds (cid:0) iA Iµ ( x ) ˙ x µ + Φ I ( x ) | ˙ x | (cid:1) i , (3.1)where I = 1 , · · · , n . The theory can be localized [13] on the sphere with squashingparameter b , where b = 1 corresponds to S , in such case the vev of the 1 / h W ± I i = 1 Z Z da I Tr (cid:16) e − πb ± a I (cid:17) e − P nI =1 8 π g I Tr a I Z ( a I , b ) |Z inst ( a I , b ) | , (3.2)now ± represents the two different trajectories in which we can compute the Wilsonloop on the squashed sphere [11]; from now on we will avoid the ± to make the notationless cumbersome, bearing in mind that in order to switch between trajectories we needto make the replacement b → b − in the following results. Once again we will considerthe 1-loop contribution as an effective action, given by (2.10), and as discussed on theprevious section we will compute the large N limit of this interacting theory whilerestricting ourselves to the zero-instanton sector. We are interested in observables thatare only sensitive to the linear dependence of h W b i in ( b − Z ( a I , b ) is quadratic in b − 1, for our purposes we can compute h W b i directly on S [33], h W ± I i = 1 Z Z da I Tr (cid:16) e − πb ± a I (cid:17) e − P nI =1 8 π g I Tr a I Z ( a I ) + O (( b − ) . (3.3)Let us expand the Wilson loop insertion h W I i = ∞ X l =0 (4 π b ) l (2 l )! h N − Tr a lI e − S ih e − S i . (3.4)As argued in our recent work [19], the large N expansion of this expectation value scaleslike N , so given the overall normalization factor 1 /N , the relevant terms to keep from (cid:10) Tr a lI S m (cid:11) are products of m + 1 connected correlators. Now there are 2 m + 1 traces tobe distributed in m + 1 correlators, but since (cid:10) Tr a lI (cid:11) can’t be by itself, we effectivelyhave to distribute 2 m traces into the m + 1 connected correlators, which is the bynow familiar sign that the possibilities are given by tree graphs. As in [19], one ofthe vertices is singled out by the presence of (cid:10) Tr a lI (cid:11) , so these are rooted trees. Thecorrelator that contains (cid:10) Tr a lI (cid:11) is a correlator of a I operators, so it involves the λ I coupling; by convention, the root vertex corresponding to this correlator will be referredas the vertex 1. The remaining m correlators can be either products of a I traces or a J traces. As we found in the evaluation of the planar free energy in the previous section,this is accounted for by modifying the numerical factor of the correlator by a weightedsum over the coupling. eq. (2.17). All in all, for the case of c A h W i − h W i = X l =1 (2 πb ) l (2 l )! ∞ X m =1 ( − m ∞ X n ,...,n m =2 ζ (2 n − . . . ζ (2 n m − n . . . n m ( − n + ··· + n m n − X k =1 (cid:18) n k (cid:19) · · · n m − X k m =1 (cid:18) n m k m (cid:19) X unlabeled rooted treeswith m edges | Aut(T) | ˜ λ d V m +1 Y i =2 ¯ V i , (3.5)In the language of Ising-type models on trees introduced in the previous section, wecan think of the Wilson loop insertion as a spin that is pinned to be up, at the rooted– 16 –ertex. To illustrate this result, let’s expand it up to second order, h W i − h W i = ∞ X l =1 (4 π b ) l (2 l )! ( − ∞ X n =2 ζ (2 n − n ( − n n − X k =1 (cid:18) n k (cid:19) V ( l, n − k ) V ( k )˜ λ l + n − k (cid:16) ˜ λ k − ˜ λ k (cid:17) + 12 ∞ X n ,n =2 ζ (2 n − ζ (2 n − n n ( − n + n n i − X k i =1 (cid:18) n k (cid:19)(cid:18) n k (cid:19) × (cid:20) V ( l, n − k ) V ( k , n − k ) V ( k )˜ λ l + n − k (cid:16) ˜ λ k + n − k + ˜ λ k + n − k (cid:17) (cid:16) ˜ λ k − ˜ λ k (cid:17) + 4 V ( l, n − k , n − k ) V ( k ) V ( k )˜ λ l + n − k + n − k (cid:16) ˜ λ k − ˜ λ k (cid:17) (cid:16) ˜ λ k − ˜ λ k (cid:17)(cid:21)) , (3.6)for which the corresponding rooted trees can be seen in figure (3).In Appendix B, we present the result of these sums up to order λ . We have checkedthat they reproduce the results of [10, 11]. Contrary to what happened for the freeenergy, the expectation value of this Wilson loop does not have nice properties underthe exchange λ ↔ λ . The reason is obvious, the Wilson loop is defined for one of thetwo gauge groups in the quiver, thus breaking the Z symmetry. For this reason, let’sconsider the linear combinations h W i ± h W i , which were referred in [7] as twisted anduntwisted. These are symmetric and antisymmetric under the λ ↔ λ exchange, sowe can introduce h W i + h W i − h W i − h W i = ( λ − λ ) w + ( λ , λ ) , (3.7) h W i − h W i − h W i + h W i = ( λ − λ ) w − ( λ , λ ) , (3.8)with w ± symmetric under λ ↔ λ . What is more, to the orders we have checkedexplicitly, again all the polynomials that appear in the expansion of w ± have all theirroots in the unit circle of the complex λ /λ plane. We again conjecture that this istrue for the polynomials generated by every tree.For the polynomials that appear in w + ( λ , λ ), this would follow from our firstconjecture if it is true. In particular, since in the previous section we proved the firstconjecture for the simplest tree, it follows that it holds also for w + , for the simplest tree.– 17 –a)(b) Figure 3 : Rooted trees corresponding to the Wilson loop in the large N , we see thatinserting the operator selects from figure (1) trees with the same color as the operatorthat we are inserting, trees containing two different colors arise from interaction termsin (2.10). ( a ) Terms corresponding to V ( l, n − k ) V ( k ). ( b ) Trees corresponding to V ( l, n − k ) V ( k , n − k ) V ( k ) and V ( l, n − k , n − k ) V ( k ) V ( k ).For w − the argument does not apply immediately, since h W i − h W i − h W i + h W i produces antipalindromic polynomials.To conclude, we can use these results to compute the one-point function of theenergy-momentum tensor with these 1/2 BPS Wilson loops. This one-point functionis fixed up to a coefficient h W [34], which can be obtained from the expectation valueof the deformed Wilson loop h W b i by the formula [33, 35] h W = 112 π ∂ b ln h W b i| b =1 . (3.9)finally, we can also compute the Bremsstrahlung function B [36] using the relation B = 3 h W [33, 37, 38], valid for any N = 2 superconformal field theory [39]. The resultswe obtain agree with those of [11]. Acknowledgments Research supported by Spanish MINECO under projects MDM-2014-0369 of ICCUB(Unidad de Excelencia “Mar´ıa de Maeztu”) and FPA2017-76005-C2-P, and by AGAUR,grant 2017-SGR 754. J. M. M. is further supported by ”la Caixa” Foundation (ID100010434) with fellowship code LCF/BQ/IN17/11620067, and from the EuropeanUnion’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 713673. A. R. F. is further supported by an FPI-MINECOfellowship. – 18 – Planar free energy up to th order Here we present the explicit form of the planar free energy in terms of λ i = λ i π F ( λ , λ ) = ( λ − λ ) h − ζ + 20 ζ ( λ + λ ) − ζ (cid:0) λ + 3 λ λ + 2 λ (cid:1) + 84 ζ ( λ + λ ) (cid:0) λ + 10 λ λ + 13 λ (cid:1) − ζ (cid:0) λ + 116 λ λ + 141 λ λ + 116 λ λ + 61 λ (cid:1) + 36 ζ (cid:0) λ + λ (cid:1) − ζ ζ ( λ + λ ) (cid:0) λ − λ λ + 3 λ (cid:1) + 840 ζ ζ (cid:0) λ + 5 λ λ + 2 λ λ + 5 λ λ + 8 λ (cid:1) + 200 ζ (cid:0) λ + 12 λ λ + 4 λ λ + 12 λ λ + 19 λ (cid:1) − ζ (cid:0) λ − λ λ + 6 λ λ − λ λ + 5 λ (cid:1)i + O ( λ ) . (A.1)Up to the order we have explicitely checked, the polynomials have all unimodularroots. B Wilson loop up to λ Here we present the explicit expansion of the circular Wilson loop corresponding toan insertion in the first node of the quiver; it is possible to obtain the insertion in thesecond node by making the change λ ↔ λ . For simplicity, in the expansion we haveset b = 1 and λ i = λ i π . If one wishes to restore the powers of b that appear in theperturbative expansion of h W b i evaluated on S , one only needs to add in each term asmany powers of b as powers of π there are.– 19 – W i − h W i = ( λ − λ ) h − π ζ λ − π ζ λ − π ζ λ − π ζ λ − π ζ λ + 80 π ζ λ (3 λ + λ ) + 803 π ζ λ (13 λ + 4 λ )+ 323 π ζ λ (17 λ + 5 λ ) + 649 π ζ λ (7 λ + 2 λ ) − π ζ λ (cid:0) λ + 5 λ λ + λ (cid:1) − π ζ λ (cid:0) λ + 55 λ λ + 10 λ (cid:1) − π ζ λ (cid:0) λ + 29 λ λ + 5 λ (cid:1) + 336 π ζ λ (5 λ + λ ) (cid:0) λ + 8 λ λ + 3 λ (cid:1) + 672 π ζ λ (cid:0) λ + 41 λ λ + 17 λ λ + 2 λ (cid:1) − π ζ λ (cid:0) λ + 56 λ λ + 36 λ λ + 11 λ λ + λ (cid:1) + 288 π ζ λ (cid:0) λ − λ λ + λ (cid:1) + 192 π ζ λ (cid:0) λ − λ λ + 2 λ (cid:1) + 192 π ζ λ (cid:0) λ − λ λ + λ (cid:1) − π ζ ζ λ (cid:0) λ − λ λ + λ λ + 5 λ (cid:1) − π ζ ζ λ (cid:0) λ − λ λ + λ λ + 20 λ (cid:1) + 3360 π ζ ζ λ (cid:0) λ − λ λ − λ λ + 11 λ λ + 11 λ (cid:1) + 1600 π ζ λ (cid:0) λ − λ λ − λ λ + 14 λ λ + 13 λ (cid:1) − π ζ λ (cid:0) λ − λ λ + 5 λ λ − λ λ + 2 λ (cid:1)i . (B.1)Note that we are inserting the operator in only one of the two nodes of the quiver thusbreaking the Z invariance of the theory. This is the reason why the vev (B.1) does notexhibit the same properties as the free energy. It is possible to retain the Z invariance– 20 –f we consider the sum and the difference, for the case of the sum we have w + ( λ , λ ) = h − π ζ ( λ + λ ) − π ζ (cid:0) λ + λ λ + λ (cid:1) − π ζ ( λ + λ ) (cid:0) λ + λ (cid:1) − π ζ (cid:0) λ + λ λ + λ λ + λ λ + λ (cid:1) − π ζ ( λ + λ ) (cid:0) λ + λ λ + λ (cid:1) + 80 π ζ (cid:0) λ + 4 λ λ + 3 λ (cid:1) + 803 π ζ ( λ + λ ) (cid:0) λ + 4 λ λ + 13 λ (cid:1) + 323 π ζ (cid:0) λ + 22 λ λ + 22 λ λ + 22 λ λ + 17 λ (cid:1) + 649 π ζ ( λ + λ ) (cid:0) λ + 2 λ λ + 7 λ λ + 2 λ λ + 7 λ (cid:1) − π ζ ( λ + λ ) (cid:0) λ + 5 λ λ + 8 λ (cid:1) − π ζ (cid:0) λ + 146 λ λ + 156 λ λ + 146 λ λ + 91 λ (cid:1) − π ζ ( λ + λ ) (cid:0) λ + 29 λ λ + 54 λ λ + 29 λ λ + 49 λ (cid:1) + 336 π ζ (cid:0) λ + 118 λ λ + 138 λ λ + 118 λ λ + 65 λ (cid:1) + 672 π ζ ( λ + λ ) (cid:0) λ + 41 λ λ + 68 λ λ + 41 λ λ + 51 λ (cid:1) − π ζ ( λ + λ ) (cid:0) λ + 56 λ λ + 96 λ λ + 56 λ λ + 61 λ (cid:1) + 288 π ζ ( λ + λ ) (cid:0) λ − λ λ + 2 λ (cid:1) + 192 π ζ (cid:0) λ + 2 λ λ + 4 λ λ + 2 λ λ + 5 λ (cid:1) + 192 π ζ ( λ + λ ) (cid:0) λ − λ λ + 4 λ λ − λ λ + 3 λ (cid:1) − π ζ ζ (cid:0) λ + 10 λ λ + 6 λ λ + 10 λ λ + 15 λ (cid:1) − π ζ ζ ( λ + λ ) (cid:0) λ − λ λ + 78 λ λ − λ λ + 77 λ (cid:1) + 3360 π ζ ζ ( λ + λ ) (cid:0) λ − λ λ + 30 λ λ − λ λ + 48 λ (cid:1) + 1600 π ζ ( λ + λ ) (cid:0) λ − λ λ + 34 λ λ − λ λ + 57 λ (cid:1) − π ζ ( λ + λ ) (cid:0) λ − λ λ + 8 λ λ − λ λ + 5 λ (cid:1)i . (B.2)– 21 –or the case of the difference we have w − ( λ , λ ) = h − π ζ (cid:0) λ + λ (cid:1) − π ζ (cid:0) λ + λ (cid:1) − π ζ (cid:0) λ + λ (cid:1) − π ζ (cid:0) λ + λ (cid:1) − π ζ (cid:0) λ + λ (cid:1) + 80 π ζ ( λ + λ ) (cid:0) λ − λ λ + 3 λ (cid:1) + 803 π ζ (cid:0) λ + 4 λ λ + 4 λ λ + 13 λ (cid:1) + 323 π ζ ( λ + λ ) (cid:0) λ − λ λ + 12 λ λ − λ λ + 17 λ (cid:1) + 649 π ζ (cid:0) λ + 2 λ λ + 2 λ λ + 7 λ (cid:1) − π ζ (cid:0) λ + 5 λ λ + 2 λ λ + 5 λ λ + 8 λ (cid:1) − π ( λ + λ ) (cid:0) λ − λ λ + 46 λ λ − λ λ + 91 λ (cid:1) − π ζ (cid:0) λ + 29 λ λ + 5 λ λ + 5 λ λ + 29 λ λ + 49 λ (cid:1) + 336 π ζ ( λ + λ ) (cid:0) λ − λ λ + 38 λ λ − λ λ + 65 λ (cid:1) + 672 π ζ (cid:0) λ + 41 λ λ + 17 λ λ + 4 λ λ + 17 λ λ + 41 λ λ + 51 λ (cid:1) − π ζ (cid:0) λ + 56 λ λ + 37 λ λ + 22 λ λ + 37 λ λ + 56 λ λ + 61 λ (cid:1) + 288 π ζ (cid:0) λ − λ λ + 2 λ λ − λ λ + 2 λ (cid:1) + 192 π ζ ( λ + λ ) (cid:0) λ + λ (cid:1) (cid:0) λ − λ λ + 5 λ (cid:1) + 192 π ζ (cid:0) λ − λ λ + λ λ + λ λ − λ λ + 3 λ (cid:1) − π ζ ζ ( λ + λ ) (cid:0) λ − λ λ + 26 λ λ − λ λ + 15 λ (cid:1) − π ζ ζ (cid:0) λ − λ λ + λ λ + 40 λ λ + λ λ − λ λ + 77 λ (cid:1) + 3360 π ζ ζ (cid:0) λ − λ λ + 4 λ λ + 22 λ λ + 4 λ λ − λ λ + 48 λ (cid:1) + 1600 π ζ (cid:0) λ − λ λ + 3 λ λ + 28 λ λ + 3 λ λ − λ λ + 57 λ (cid:1) − π ζ (cid:0) λ − λ λ + 7 λ λ − λ λ + 7 λ λ − λ λ + 5 λ (cid:1)i . (B.3)The series w ± ( λ , λ ) are symmetric. At the considered orders, the polynomialsthat appear also have all unimodular roots.– 22 – eferences [1] M. Henningson and K. Skenderis, The holographic Weyl anomaly , JHEP , 023(1998), [ arXiv:hep-th/9806087 ].[2] S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4-D N=2 gaugetheories: 1 , Adv. Theor. Math. Phys. , 53-114 (1998), [ arXiv:hep-th/9706110 ].[3] S. Kachru and E. Silverstein, , Phys.Rev. Lett. , 4855 (1998), [ arXiv:hep-th/9802183 ].[4] A. E. Lawrence, N. Nekrasov and C. Vafa, On conformal field theories infour-dimensions , Nucl. Phys. B , 199 (1998), [ arXiv:hep-th/9803015 ].[5] A. Gadde, E. Pomoni and L. Rastelli, The Veneziano Limit of N = 2 SuperconformalQCD: Towards the String Dual of N = 2 SU(N(c)) SYM with N(f ) = 2 N(c) ,[ arXiv:0912.4918 [hep-th] ].[6] A. Gadde, E. Pomoni and L. Rastelli, “Spin Chains in N=2 Superconformal Theories:From the Z Quiver to Superconformal QCD , JHEP , 107 (2012)[ arXiv:1006.0015 [hep-th] ].[7] S. J. Rey and T. Suyama, Exact Results and Holography of Wilson Loops in N=2Superconformal (Quiver) Gauge Theories , JHEP , 136 (2011),[ arXiv:1001.0016 [hep-th] ].[8] E. Pomoni and C. Sieg, From N=4 gauge theory to N=2 conformal QCD: three-loopmixing of scalar composite operators , [ arXiv:1105.3487 [hep-th] ].[9] A. Gadde, P. Liendo, L. Rastelli and Y. Wenbin, On the Integrability of Planar N = 2 Superconformal Gauge Theories , JHEP , 015 (2013), [ arXiv:1211.0271 [hep-th] ].[10] V. Mitev and E. Pomoni, Exact effective couplings of four dimensional gauge theorieswith N = , Phys. Rev. D , 125034 (2015),[ arXiv:1406.3629 [hep-th] ].[11] V. Mitev and E. Pomoni, Exact Bremsstrahlung and Effective Couplings , JHEP ,078 (2016), [ arXiv:1511.02217 [hep-th] ].[12] K. Zarembo, Quiver CFT at strong coupling , JHEP , 055 (2020),[ arXiv:2003.00993 [hep-th] ]. – 23 – 13] V. Pestun, Localization of gauge theory on a four-sphere an supersymmetric Wilsonloops , Commun. Math. Phys. , 71–129 (2012), [ arXiv:0712.2824 [hep-th] ].[14] J. Erickson, G. Semenoff and K. Zarembo, Wilson loops in N=4 supersymmetricYang-Mills theory , Nucl. Phys. B , 155-175 (2000) [ arXiv:hep-th/0003055 ].[15] N. Drukker and D. J. Gross, An Exact prediction of N=4 SUSYM theory for stringtheory , J. Math. Phys. , 2896 (2001), [ arXiv:hep-th/0010274 [hep-th] ].[16] F. Passerini and K. Zarembo, Wilson Loops in N=2 Super-Yang-Mills from MatrixModel , JHEP , 102 (2011), [ arXiv:1106.5763 [hep-th] ].[17] J. G. Russo and K. Zarembo, Large N Limit of N=2 SU(N) Gauge Theories fromLocalization , JHEP , 082 (2012), [ arXiv:1207.3806 [hep-th] ].[18] B. Fiol, J. Mart´ınez-Montoya and A. Rios Fukelman, Wilson loops in terms of colorinvariants , JHEP , 202 (2019), [ arXiv:1812.06890 [hep-th] ].[19] B. Fiol, J. Mart´ınez-Montoya and A. Rios Fukelman, The planar limit of N = 2 superconformal field theories , JHEP , 136 (2020), [ arXiv:2003.02879 [hep-th] ].[20] A. Pini, D. Rodriguez-Gomez and J. G. Russo, Large N correlation functions N = , JHEP , 066 (2017), [ arXiv:1701.02315 [hep-th] ].[21] T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions.2. Lattice gas and Ising model , Phys. Rev. , 410 (1952).[22] J. Koplik, A. Neveu and S. Nussinov, Some Aspects of the Planar Perturbation Series ,Nucl. Phys. B , 109 (1977).[23] E. Pomoni, Integrability in N = 2 superconformal gauge theories , Nucl. Phys. B ,21-53 (2015), [ arXiv:1310.5709 [hep-th] ].[24] E. Pomoni, N = 2 SCFTs and spin chains , [ arXiv:1912.00870 [hep-th] ].[25] M. Bill`o, F. Fucito, A. Lerda, J. F. Morales, Ya. S. Stanev and C. Wen, Two-pointCorrelators in N=2 Gauge Theories , Nucl. Phys. B926 , 427–466 (2018),[ arXiv:1705.02909 [hep-th] ].[26] M. Bill`o, F. Galvagno, P. Gregori and A. Lerda, Correlators between Wilson loop andchiral operators in N = 2 conformal gauge theories , JHEP , 193 (2018),[ arXiv:1802.09813 [hep-th] ]. – 24 – 27] M. Bill`o, F. Galvagno and A. Lerda, BPS wilson loops in generic conformal N = 2SU(N) SYM theories , JHEP , 108 (2019), [ arXiv:1906.07085 [hep-th] ].[28] W. T. Tutte, A census of slicings , Can. J. Math. , 708–722 (1962).[29] R. Gopakumar and R. Pius, Correlators in the Simplest Gauge-String Duality , JHEP , 175 (2013), [ arXiv:1212.1236 [hep-th] ].[30] P. Lakatos and L. Losonczi, Self-inversive polynomials whose zeros are on the unitcircle , Publ. Math. Debrecen , 409–420 (2004).[31] T. Asano, Theorems on the Partition Functions of the Heisenberg Ferromagnets ,Journal of the Physical Society of Japan. (2): 350-359 (1970).[32] D. Ruelle, Zeros of Graph-Counting Polynomials , Comm. in Math. Phys. Exact Bremsstrahlung Function in N = 2 Superconformal Field Theories , Phys. Rev. Lett. , 081601 (2016),[ arXiv:1510.01332 [hep-th] ].[34] A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories andS-duality , Phys. Rev. D74 , 025005 (2006), [ arXiv:hep-th/0501015 ].[35] L. Bianchi, M. Bill`o, F. Galvagno and A. Lerda, Emitted Radiation and Geometry ,JHEP , 075 (2020), [ arXiv:1910.06332 [hep-th] ].[36] D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation ofa moving quark in N=4 super Yang Mills , JHEP , 048(2012),[ arXiv:1202.4455 [hep-th] ].[37] B. Fiol, B. Garolera and A. Lewkowycz, Exact results for static and radiative fields of aquark in N=4 super Yang-Mills , JHEP , 093 (2012), [ arXiv:1202.5292 [hep-th] ].[38] A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and theenergy radiated by a quark , JHEP 025 (2014), [ arXiv:1312.5682 [hep-th] ].[39] L. Bianchi, M. Lemos and M. Meineri, Line defects and radiation in N = 2 theories ,Phys. Rev. Lett. , 141601 (2018), [ arXiv:1805.04111 [hep-th] ].].