Wilson loops in circular quiver SCFTs at strong coupling

We study circular BPS Wilson loops in the $\mathcal{N}=2$ superconformal $n$-node quiver theories at large $N$ and strong 't Hooft coupling by using localization. We compute the expectation values of Wilson loops in the limit when the 't Hooft couplings are hierarchically different and when they are nearly equal. Based on these results, we make a conjecture for arbitrary strong couplings.


Introduction
Holographic duality is remarkable while testing the duality is usually involved with the nonperturbative aspects of interacting field theory. The N = 4 super-Yang-Mill (SYM) theory offers an ideal playground where many observables can be obtained analytically beyond the perturbation theory. One such observable of interest is the BPS Wilson loop. The expectation value of the BPS Wilson loop in N = 4 SYM can be reduced to computation to a Gaussian matrix model [1,2], which can be derived by using the supersymmetric localization technique [3]. This exceptional result provides a nontrivial consistency check for the duality relation between the Wilson loop expectation value and the string worldsheet path integral [4,5].
In this work we study half-BPS circular Wilson loops in N = 2 superconformal A n−1 quiver gauge theories at strong coupling. There are n independent 't Hooft couplings for each gauge group factor, which gives rise to interesting dynamics but still allows analytical control. The field contents are graphically represented in fig. 1. When the couplings are equal, they are equivalent to the Z n orbifolds of N = 4 SYM [6]. Therefore they are probably the next simplest theories after the N = 4 SYM. Localization technique is also are applicable in such theories. Unlike the N = 4 SYM, the resulting matrix model is interacting. Many efforts have been devoted to study these theories by exploiting the localization matrix model [7][8][9][10][11][12][13][14][15].
These quiver theories admit a dual holographic description as type IIB string theory on the AdS 5 × (S 5 Z n ) geometry [16], where Z n acts on R 4 ⊂ R 6 the embedding space of S 5 . There are n − 1 independent collapsed two-cycles hidden in the fixed point. It is convenient to parameter the 't Hooft couplings as: λ l = 2πλ nθ l , l = 1, 2, ..., n (1.1) where the θ-parameters are constrained by n l=1 θ l = 2π, (1.2) and λ is the effective coupling: From the holographic perspective λ is related to the string tension T via λ = 4π 2 T 2 [6,17,18] and the θ-parameters are proportional to the fluxes of the NSNS B-field through appropriate collapsed two-cycles of the orbifold [6,17]. We are interested in the supergravity limit where λ is large with fixed θ-parameters. Figure 1: n-node quiver diagram. Each node represents an SU (N ) gauge group factor and the node n + k is identified with the node k. Each arrow represents a bifundamental hypermultiplet.
At strong coupling the leading order Wilson loop expectation value was obtained in [7] which is essentially the same as that in N = 4 SYM and independent of the θ-parameters. The "one-loop" correction beyond the leading exponential in the two-node theory was computed in [13]. The goal of this paper is to extend the result to the n-node theory. Hopefully, our result will provide important insight into the dynamics of string theory on the orbifold with nontrivial B-field fluxes.
This paper is organized as follows. In section 2 we briefly review Wilson loops at strong coupling in the localization approach. We then compute the expectations of the Wilson loops in the cases when the 't Hooft couplings are hierarchically different in section 3 and nearly equal in section 4. We make a conjecture for arbitrary values of strong 't Hooft couplings in section 5. Conclusions are given in section 6.

Setup
In this section we will investigate Wilson loop at strong coupling in the n-node quiver theory following similar analysis as in [13]. So we will be brief in the description and refer the reader to that reference for all details.

Localization
We consider an A n−1 quiver theory with gauge group SU (N ) n . There are n vector multiplets and n bifundamental hypermultiplets. The partition function of the theory of interest on a four-sphere can be localized to be the matrix model [3]: where H(x) is given by We are interested in the expectation value of Wilson loops: where C is the equatorial circle of the four-sphere. In the matrix model, they can be compute as In the following we will be interested in the large N limit in which the distribution of the eigenvalues a lj is characterized by the densities: They satisfy the saddle-point equations [19] The expectation values of Wilson loops are then given by When λ l = λ for all l, the solution is given by the Wigner distribution . (2.10) In this case the Wilson loop expectation values are the same as that in the N = 4 SYM When the 't Hooft couplings are unequal but strong, the bulk of the distributions can still be approximated by the Wigner distribution [7,13]. The leading order Wilson loop expectation values are still given by (2.11). The differences of the 't Hooft couplings lead to O(λ 0 ) corrections to the position of the endpoints. The boundary behavior of the density distributions is complicated and determines the Wilson loop expectation values to the next order in 1 √ λ. To obtain boundary behavior we adopt the method used in the study of the N = 2 * theory [20]. We need to solve the distributions in the bulk and then at the boundary and finally match the two solutions.

Bulk
When the effective coupling λ is large, µ l is of order √ λ. Therefore one can use the large-distance asymptotics approximation of the kernel K in the integral equations: The observation in [13] µ −µ suggests that strong-coupling expansions of the densities take the following form: The unit normalization of the densities gives (2.14) Plugging the asymptotic kernel into the saddle-point equations (2.6), we find The sum of the n equations determines A: Then the normalization (2.14) leads to because B l should stay finite in the limit of large λ . Therefore the shifts of the endpoints from the Gaussian-model prediction µ = √ λ 2π at strong coupling are Substituting (2.16) and (2.17) into (2.15), the difference between the lth and (l + 1)th equation in the large λ limit imposes the constraint Therefore we have n − 1 constraints on the 2n coefficients B l and C l . The remaining unknowns are fixed by the boundary behavior.

Boundary
To describe the boundary behavior of the densities, we define the scaling functions At large distance ξ → ∞ they should match the bulk solution (2.13): (2.21) To derive the integral equations for f from the saddle-point equations (2.6), one can use the trick in [20] to extract the contribution from the boundary region. Equations (2.6) can be written as where * denotes convolution. The perturbative bulk solution satisfies an exact equation: where R ∞ is R with K replaced by K ∞ and ρ ∞ is given by the first three terms of the right hand side of (2.13). Subtracting (2.23) from (2.22), we get To obtain the integral equations for the scaling functions, we take x = µ j − ξ and y = µ l − η. For ξ ∼ O(1), we find only η ∼ O(1) contributes because the integrand on the left hand side decay as η −3 2 log η away from the boundary and that on the right hand side as η −3 2 . Therefore convolution integrals can be extended to infinity and we get an integral equation of the Wiener-Hopf type: where the explicit form of the kernel is 26) and the same for R ∞ with K replaced by K ∞ . The shift of the argument in K is a result of different positions of the end points. It is advantageous to work in the Fourier space. The Fourier imagef is analytic in the upper half plane of ω. The equation (2.25) only holds for ξ > 0, so we have to introduce an unknown function X which is nonzero for ξ < 0. After taking the Fourier transform, convolutions in configuration space become products: whereX is a negative-half-plane analytic function of ω. To simplify expression, we denote Thenf andX are analytic in the left and right half-plane of u respectively. The explicit expressions for Fourier images appearing in (2.28) arê where a small > 0 is used to express the analytic form of sign ω: The densities and scaling functions should vanish like a square root at the boundary, so we requiref The Taylor expansion of the scaling functions at u = 0 are determined by because they should match the bulk solution at large ξ.
In [13] the Wiener-Hopf problem (2.28) for n = 2 was reduced to the analytic factorization of the kernel R, and f l can be express through the Wiener-Hopf factors of R via contour integration. We will use a more direct approach to express f l here. Let us first consider an auxiliary problem:Rf =X, (2.37) whereX(u) is analytic in the right half-plane andf (u) is analytic in the left half-plane except at u = 0. The asymptotic behavior off l at small and large u is required to bẽ Thenf can be obtained asf One can check that this is a solution to the Wiener-Hopf problem (2.28) with appropriatê X and satisfies the required asymptotic conditions at small and large u.
We now turn to Wilson loops. One can replace the density in equation (2.8) by the scaling function and extend the integration to ξ → ∞ because the contribution from the large ξ region is suppressed exponentially. Therefore the strong-coupling expectation values of Wilson loops can be computed as We are interested in the Wilson loop expectation value W l normalized by that in the N = 4 SYM at strong coupling: This quantity is useful for matching gauge theory and string theory results. In the string calculation there are subtleties related to the string path integral measure. To avoid this problem, it is convenient to consider the ratio of Wilson loops [21,22]. When n = 2, it was observed in [13] that the problem (2.37) is related to the scattering theory of the Pöschl-Teller potential and therefore can be solved exactly. However, it is difficult to find an analogous relation for general n. In the following two sections, we will solve the problem in the limit when the couplings are hierarchically different or nearly equal. Based on these results, we can make a conjecture for the ratios w l with arbitrary θ-parameters.

Leading order
Motivated by the n = 2 result in [13], we propose an ansatz when ∆ l − ∆ j ≫ 1 for any l ≠ j where a ± j and b ± lj depend on u and can be written as fractional power series in the ratios of the η-parameters defined as Plugging this ansatz into the problem (2.37), we find for any l and m satisfying η l > η m . Solving b ± lj and a − l in terms of a + l .
where S (l) is a submatrix of S containing elements with indices belonging to {m ∈ Z n η l > η m }. For concreteness we concentrate on the case when η 1 ≪ η 2 ≪ ... ≪ η n . We denote γ i ≡ η i η i+1 ≪ 1 for i = 1, ...n − 1. Writing explicitly these equations for the first few integers n, we find All other b ± jk vanish. Equations (3.7) and (3.8) can be solved by where we have chosen the coefficients in front of c such that When the γ-parameters are small, singularities off j (X j ) at large real negative (positive) values of u are suppressed by the (η l η j ) u factors. The coefficients c j (u) are independent of u in the limit γ j → 0 , j = 1, 2, ..., n − 1. In principle, they can be solved perturbatively in small γ l using the analytic properties off andX. We will show how to do that in the n = 3 case, but at this moment let us focus on the leading order contribution. Denoting c (0) j the leading order of c j (u), The condition that singularity off j at u = −(j +1) −1 , j = 1, 2, ..., n−1 is removable leads to We get (3.17) Expandingf at small u using (3.7)-(3.14), we find When l = n, the last term in (3.20) is an empty sum and hence equal to zero by convention. We will follow the same convention below. The endpoint positions are then given by which are consistent with µ l − µ j = ∆ l − ∆ j . Solving the constraints (2.19) we find

22)
To compute the ratios between the expectation values of Wilson loops and that in the N = 4 SYM using (2.42), we need to evaluatef at u = −2: Then the ratios are given by When γ i ≪ 1 for all l, using (3.22) we find θ 1 ≪ θ 2 ≪ ... ≪ θ n or equivalently λ 1 ≫ λ 2 ≫ ... ≫ λ n ≃ nλ and  For the first few cases n ⩽ 5, we have ). (3.30) The first node has the largest 't Hooft couplings and w 1 diverges as λ 3 1 . This limit can be considered as the supergravity decouple limit discussed in [13]. This is different from the true decoupling limit where λ l → 0 for l ≠ 1 and the theory becomes equivalent to the super-QCD. In the true decoupling limit, there should be cubic as well as log behavior [23].
In the rest of this section, we will consider some simple cases where it is easy to compute higher order corrections.

Higher order corrections in γ n−1
It is a relatively easy task to consider higher order corrections in γ n−1 . When γ n−1 is finite but the rest of the γ-parameters are small, the couplings satisfy λ 1 ≫ λ 2 ≫ ... ≫ λ n−2 ≫ λ and λ n−1,n ∼ O(λ). In this limit W j → ∞ and c j → 0 for j = 1, 2, ..., n − 2. We are going to compute the Wilson loop expectation values associated with the last two nodes. Keeping only the γ n−1 correctionsf andX becomẽ f = (a + 1 , ..., a + n−2 , a + n−1 + (sin πu tan Focusing on the last two nodes, the factorization problem reduces to The undesired singularities of at u = −(1 + 2m) n, m = 1, 2, ... and where g n−1,n (m, k) are constants and the overall factor K is determined by (3.18). When the undesired singularities are removed, we find the recurrence relations ng n−1 (p, k − p) 2(m + p + 1) .
When n = 2, there is only one γ-parameter and c j can be solved exactly: is the standard Pochhammer symbol. One can check that all the undesired singularities off andX are removed. Using (3.22) and (3.24) we get which are the same as the results obtain in [13]. Based on the result in the case of n = 2, we assume that w n−1 and w n take the form w n−1 = T + n (θ n−1 ) + (2π − θ n−1 )T − n (θ n−1 ), w n = T + n (θ n−1 ) − θ n−1 T − n (θ n−1 ), (3.47) where T + n (θ) = T + n (2π − θ) and T − n (θ) = −T − n (2π − θ). From the expansions (3.40), (3.41) and (3.42), one can try to conjecture the exact expressions of w n−1 and w n as functions of θ n−1 . The difference w n−1 − w n can be conjectured to be w n−1 − w n = 2πT − n (θ n−1 ) = π4 1−n (n − 1)n csc 2n θ n−1 2 sin θ n−1 . At this stage it is difficult to conjecture the expression of T + n for general n.

(3.59)
Requiring the undesired singularities to be removable, we find the recurrence relations

n = 4 with two couplings
It is a formidable task to deal with three independent γ-parameters in the 4-node case.
Here we consider a simple case where λ 1 = λ 2 and λ 3 = λ 4 . We parameterize the 't Hooft couplings as Because of the symmetry of the four-node quiver, it is reasonable to expect η 1 = η 2 and η 3 = η 4 so the factorization problem reduces to (3.69) where γ 3 = η 1 η 3 and M 3 is given by (3.34). Therefore one can use the result in subsection 3.2. Taking account into the extra scalar factor, we have where c 2,3 are given by (3.37) with n = 3. Expanding f l at small u and solving the constraint 2.19, we find (3.71) The normalized Wilson loop expectation values are (3.72) One can compute the first few terms in small γ by using the recurrence relation (3.39). The exact expression can be conjectured to be w 1 = 1 4 (cos 2θ 1 + 5) csc 4 θ 1 + 3 4 (π − θ 1 ) sin 2θ 1 csc 6 θ 1 , (3.73) These results are plotted in fig. 3.

Nearly equal couplings
In this section we consider the limit ∆ j → 0 so the couplings are expected to be nearly equal. We also consider the limit n → ∞ with ∆ j fixed. We find that the differences between the θ-parameters are of order n −2 and hence the couplings are also nearly equal.

Small ∆ limit
To solve the problem (2.37) in the small ∆ limit, it is convenient to decomposef into a superposition of the eigenvectors ofR given by where we denote x i = −π∆ i . The eigenvalues are with j = 0, ..., n−1. We have analytically factorized the eigenvalues. When the x-parameters are small and of the same order we propose an ansatz where P j (u) depends on the variables x l . The expansion of P j for small x l is where P (0) and P (1) j (u) and P (2) j (u) are linear and quadratic in the x-parameters respectively. This ansatz already assures the desired analytic properties off andX except at 0 and ∞. Then the coefficients P (n) j (u) are determined by the conditions (2.38) and (2.39). Expanding r j at small and large u, we find Expanding (4.5) at small u, we find When the x l 's are small, up to the first nontrivial order we have We denote α l = θ l − 2π n as the fluctuation of θ l around 2π n. Using (4.17) and (4.19) we find When all the α-parameters go to zero, we get w l → 1 as expected.
To check the conjecture (3.66), we compute w l up to the second nontrivial order in the n = 3 case. Without loss of generality we set x 1 + x 2 + x 3 = 0 because only differences between the x-parameters appear in the kernel R. Using (4.11) and (4.12) we get Substituting into (4.11) and solve x 1 and x 2 in small α l leads to (4.23) Using (4.19) we find which are consistent with the conjecture (3.66) expanded around θ l = 2π n.

Large n limit
The limit of large n is interesting because it is related to the dimensional deconstruction [24,25]. In this case, the theory becomes effectively five-dimensional and the quiver can be interpreted as the fifth dimension. We require that the x-parameters stay finite and vary slowly along the quiver, i.e. x j+1 − x j ∼ O(n −1 ) . To compute higher order corrections to P j , we make the ansatz where Q is a function to be determined. Then we havẽ (4.26) Using the asymptotics of r + j at infinity (4.28) In the limit of large n with s ≡ l n fixed, we get 1 n 2 πr (4.29) For finite s, the n −2 corrections are neglectable, but when s → 0 or s → 1 these corrections can be large. It is reasonable to assumer + l ≃ 0 to all order of the large u expansion when s is not close to 0 or 1. To test this assumption, we computer + l (−2) with n = 1000 and the result is shown in fig. 4.  has been taken into account. It is worth mentioning that (4.30) is not valid for general u. For instance, when u → 0 we havẽ In the case of u < 0, we havẽ where x(l n) ≡ x l . When Q is independent of u we get the desired asymptotics at infinity. Taking into account the small u expansion, we find Q(u, s) = 1. Sof at negative real u does not depend on the x-parameters in the large n limit, so we have n jm e uxm . The right hand side are of order n −2 , so α l ∼ O(n −2 ). Therefore the 't Hooft couplings are nearly equal in this limit.
Solving the x-parameters in terms of the α-parameters, we get

A conjecture
Based on the results above, we can actually make the following conjecture: obtained from the large n approximation (4.42) with the α-parameters given by (5.6). The dots represent the same quantity obtained from (5.1) with n = 100.
It would also be interesting to consider the limit of large n with finite λ l λ ∼ O(n 0 ). By applying the conjecture (5.1) to some explicit examples, we find log w 1 ∼ O(n) and equation (4.42) is no longer valid.

Conclusions
In this paper, we investigated the expectation values of the circular Wilson loops in the n-node superconformal quiver theories at strong coupling using localization. We computed the Wilson loops expectation values when the couplings are hierarchically different or nearly equal, and finally, made a conjecture for arbitrary strong couplings. It would be interesting to derive the conjecture (5.1) rigorously. One possible way is to relate the matrix factorization problem to a solvable quantum system as the n = 2 case in [13].
To compare our result with the string prediction, one needs to compute the one-loop correction to the string worldsheet integral in a similar way as was done in AdS 5 × S 5 [21,22,[28][29][30]. In the orbifold case, the θ-dependence is expected to arise from worldsheet instanton wrapping the collapsed two-cycles. We hope the comparison can be achieved in the nearby future.