Wilson loop and its correlators in the limit of large coupling constant

In this paper we study Wilson loops in various representations for finite and large values of the color gauge group for supersymmetric ${\cal N}=4$ gauge theories. We also compute correlators of Wilson loops in different representations and perform a check with the dual gravitational theory.


Introduction
Supersymmetric Wilson loops as well as their correlators both with chiral primary operators and with other Wilson loops are remarkable observables of the supersymmetric Yang-Mills gauge theories (SuSy YM) and provide stringent tests of AdS/CFT correspondence. Due to the fact that the propagator is constant on a circle the computation on the CFT side can be performed with a matrix model as it was demonstrated with the help of localization [1].
As discussed in a number of papers [2,3] in the case of large coupling constant the computation of a Wilson loop's vacuum expectation value in the framework of a matrix model can be significantly simplified. In the present paper we consider the very same limit of large coupling constant and perform some matrix model calculations for the vacuum expectation value of the Wilson loop in different representations, for its correlators with another Wilson loop and with chiral operators both for finite number of colours N of the gauge theory and for large N . Furthermore, we compare a correlator between two 1 2 -BPS Wilson loops, one of which is in the fundamental representation of the gauge group and the other in a representation associated with a Young tableau with several long lines, with the corresponding quantity on the AdS side and find perfect agreement.
The paper is organized as follows: in Section 2 we calculate a vacuum expectation value of a Wilson loop in a general representation and perform calculation for finite number of colours N . We proceed with considering the large N limit of the previous case. In Section 3 we turn to the correlator of a symmetric Wilson loop with primary chiral operators, again both for finite and large N . Finally in Section 4 we study the correlator of the two Wilson loops discussed in the previous sections both from the point of view of the matrix model and on the AdS side.

Wilson loops in arbitrary representations
We consider a 1 2 -BPS circular Wilson loops in N = 4 super Young-Mills theory with gauge group U (N ). The vacuum expectation value of the WL defined as due to localization can be found as a U (N ) matrix model integral [1] (2. 2) The averages in the matrix model are defined as (a u − a v ) . (2.5) A representation R of the U (N ) group is specified by the Dynking labels λ = (λ 0 , λ 1 , . . . λ N −2 ) and central charge Q, or equivalently by a Young tableau with rows of length K u given by Let us introduce the orthonormal basis {e i } with e i ∈ R N and write the U (N ) simple roots as α i = e i − e i+1 for i = 0, . . . N − 2. The character of a representation is given by the Weyl formula with the sum running over the set of weights {α} defining the representation R. The determinant in the numerator can be written as while that in the denominator can be explicitly computed and written in the form det u,v e g au(N −v) = u<v (e g au − e g av ) . (2.9) It can be noted that (2.9) is invariant under permutation up to a sign of the permutation, so we can replace a u , a v by a σu , a σv and get rid of (−) σ in the nominator.
We also notice that the integrals over the eigenvalues are equal for each one of the N! permutations, so we can choose one permutation (for example the trivial one) and rewrite (2.10) as follows (2.11) In the strong coupling limit g ≫ 1 the product in the denominator of (2.11) is equal to 1 in the region Ω ⊂ R n such that a v < a u for all v > u, u = 0, ... N − 1 and diverges otherwise, so only in Ω the integrand is not suppressed exponentially by the denominator. Hence we actually have Thinking of a representation associated with a Young tableau with g+1 groups consisting of n i rows of the same length (including rows of zero length) one can notice that the integrand of (2.12) stays the same under the permutation u → σ u such that K u = K σu . It implies that the integral (2.12) over Ω can be replaced with the integral over the union of the images of Ω under all such permutations (let us denote it asΩ) divided by the number of the permutations n 1 ! n 2 ! . . . n g+1 !. ClearlyΩ includes all regions where a v < a u whenever K v < K u . We then get (2.14) Finally, we can extend the integral (2.13) back to R n fromΩ since the integrand is sup-pressed exponentially everywhere exceptΩ. We finally get 1 with H n (x) being the "physicists" Hermite polynomials 2 The Hermite polynomials satisfy the integral relations standing for the Laguerre polynomials and In the case of the antisymmetric representation associated with a Young tableau with one column of the length l one finds We notice that in this case the result is exact.
In particular for the fundamental representation one gets standing for the generalized ones. Using these integral relations one can compute the integrals as follows In the following subsections we specify to some simple cases.

The completely symmetric representation
Let us consider the K-symmetric representation, characterised by a Young tableau with a single row of length K, i.e. K u = K δ u0 . Formula (2.24) reduces to There are various limits one can consider.
• Large N keeping K finite: In this limit one can write where we used the definition of the Bessel function (2.28) • N and K tend to infinity with K/N finite. In this limit one can use the large N asymptotic formula of the Laguerre polynomials [4] So the WL in this limit is The exact expression for the symmetric WL (2.26) can be obtained with the help of diagrams in the frame of the matrix model. One would expect that the large N limit can be restored by only planar diagrams, however, it can be checked that for any K the planar diagrams sum to (2.27) and never give (2.29), i.e. the planar diagram approach cannot be applied for the case of finite K/N .

Two-row Young tableau
Let us consider now a representation defined by a Young tableau made of two rows of lengths K = {K 0 , K 1 }. We write In this case

so one finds for the Wilson loop
Although it is difficult to extract the large N limit of a WL in the representation associated with a Young tableau with two rows directly from (2.34), we will need this result later in Section 4. We then use an alternative method to perform such limit for the case of a Young tableau with long rows (the lengths of the rows K u are of order N ). The WL in the K-symmetric representation can be written as an integral over N × N hermitian matrices As it was shown in [6] the WL in the limit of large coupling constant λ can be rewritten as an integral over N − 1 × N − 1 hermitian matrices Assuming now, that the length of the row K is of order of N one can compute the integral with measure dµ[v 0 ] in (2.36) using standard perturbation theory for large N and the other integral with the saddle point method. Taking only the leading order in the large N expansion one will find then that at the saddle point v 0 = 2 √ N 1 + y 2 (2.36) gives the correct exponential behavior Taking the leading and the first subleading orders in large N expansion one will restore also the correct pre-exponential factor [6] W K ≈ 1 √ 2πN Let us now use the same method to find the exponential behavior of a WL in the representation defined by a Young tableau with two rows with lengths K 0 , K 1 ∼ N . Acting in the same way as in [6] one will find that where dµ[ṽ] is the Gaussian measure in the space of N − 2 × N − 2 hermitian matrices. If K 0 = K 1 one will see immediately from the previous case that the saddle points are v 0 = 2 √ N 1 + y 2 0 , v 1 = 2 √ N 1 + y 2 1 and in the leading order of the large N expansion If now K 0 = K 1 one has to be more careful with the factor (v 0 − v 1 ) 2 and to find the saddle points with the corrections v 0 = 2 √ N 1 + y 2 + a, v 1 = 2 √ N 1 + y 2 − a, a 2 = y 1 + y 2 and hence only the pre-exponential factor is changed, but the exponential behavior (2.40) stays the same.
Similarly for a WL in the representation associated with a Young tableau with several lines (number of lines n ≪ N ) with lengths K u ∼ N one gets (2.41)

Examples of the representations of U(3) gauge group
Let us show on some simple examples that the leading order of (2.10) for U(3) group given by the approximation (2.17) in the large λ limit. The character in this case is χ = e 2g a 1 + e 2g a 2 + e 2g a 3 + (2.43) +e g (a 1 +a 2 ) + e g (a 2 +a 3 ) + e g (a 1 +a 3 ) .
The first three terms are leading in some regions of the space of eigenvalues (in agreement with c R = 3! 1!2! = 3), while the last three terms are never the leading ones. The WL with all terms (2.10) in this case is As it can be seen in the λ ≫ 1 limit the difference between (2.44) and (2.45) is exponentially suppressed.
The analytical answer (2.34) is

Correlators of a symmetric Wilson loop and chiral primary operators
Let us find now a connected correlator between a WL in a symmetric representation and a chiral primary operator. This correlator can be written via matrix model integrals [5].
tr K e a tr a n c = tr K e a tr a n − tr K e a tr a n (3.1) where tr K e a tr a n = 1 Both integrals above can be evaluated in the same way as Z( K) (2.24). Having it done, one gets tr K e a tr a n c = t=0 .
If n ≪ N the terms with i < n−δ 2 can be dropped and (3.3) can be rewritten with hypergeometric functions tr K e a tr a n c = where ǫ = n−s 2 , ρ = n+s 2 .

Large N, finite K/N limit
To calculate (3.4) in the first non-vanishing order, for N → ∞ and finite K/N , an integral representation of Laguerre polynomials can be used.
Let us consider the second term in (3.4) in the limit of large N (3.6) Here Using (3.5) one gets a sum over i inside B Substituting the result back into (3.6) and using (3.5) again one finally gets the second term of (3.4) in the form and so tr K e a tr a n c = For the large order Laguerre polynomial with negative argument there is an asymptotic form [4] L l+s+1 N −s−1 (−νỹ 2 )e νỹ 2 2 = α 2 l+s+1 F (ỹ) l+s+1 I l+s+1 (νF (ỹ)) (3.12) And since the argument of the Bessel function is large, the asymptotic of Laguerre polynomial can be written as Further simplification can be done if K/N tends to a non-zero value. In this case of finite K/N the differential operator in the first term of (3.11) acts only on the exponent in the asymptotic representation (3.13). Otherwise one needs also to take derivatives of other terms since they give terms proportional to N/K. Using (2.29) one gets 1 N tr K e a tr a n c tr K e a ≈ λ In order to get rid of the operator mixing appearing on S 4 one should use the following relation [6] tr K e a : tr a n : c = tr K e a : tr a n : c tr K e a ≈ 2 1−n sinh(n arcsinh(y)) .
which is in the agreement with [7] up to a normalization and coincides with [6] exactly.

Large N, K/N → 0 limit
Although it's difficult to find the N → ∞, K/N → 0 limit of (3.3) for general n, one can always check for particular values of n, that this limit of (3.3) is 1 N tr K e a : tr a n : c tr K e a = n 2 n λ which is in agreement with [6]. It's worth noting that unlike the case of finite K/N , the highest order in the large-N expansion of (3.3) vanishes if K/N → 0.
It also should be noted that (3.18) can not be found as the K/N → 0 limit of (3.17) since in the derivation of (3.17) it was essential that K/N was strictly positive, but can be found as the K/N → 0 limit of (3.11) with the help of the relation (3.15).

Correlators of two Wilson loops
In this section we consider the correlator of two 1 2 -BPS Wilson loops preserving the same subgroup of supersymmetries, i.e. taken over the same circle and sharing the orientation in the internal space. For such Wilson loops the correlator also can be written as a matrix model integral due to localization [1]. (4.1) Using representation theory [8] one can write that tr R e g a tr R ′ e g a = tr R⊗R ′ e g a = where R i stands for an irreducible components in R ⊗ R ′ and C R⊗R ′ ,R i are the corresponding Clebsch-Gordan coefficients.
For example, for a product of two symmetric Wilson loops (assuming K ≥ K ′ ) one gets a sum of traces in the two-rows representations (4.3) So the correlator of two symmetric Wilson loops can be written as (4.4) and hence in the limit of large coupling constant can be found exactly in terms of N with (2.34).

Large N limit
Let us now consider a correlator of a WL in the fundamental representation with a WL in the representation associated with a Young tableau with several lines (number of lines n ≪ N ) of the same length K = {K 1 , ..., K n }, K i = K.
Letting K to be of order N one can apply (2.41) and write As for the second term, it is clear that in the leading order of large N (4.7) Comparing (4.6) and (4.7) we see that (4.6) is the leading term in the sum (4.5) for any y > 0, so (4.7) should be omitted.
A correlator between a WL in the fundamental representation and a WL in the representation R associated with a Young tableau with many rows (number of lines n ∼ N ) of the same length was found in [9] W In this case an of two terms can be the leading one for some values of the parameters. In order to compare the correlator (4.8) with the corresponding quantity on the AdS side, one has to find it also in the theory with SU (N ) gauge group. In the theory with SU (N ) group of symmetries there is an additional factor of (det e g a ) − |R| N with |R| = g i=0 K i n i . In the case of several lines in a Young tableau (n ≪ N ) |R| N ≈ 0, so the correlator does not change.

String in degenerated genus one background
As recently discussed in [9], according to the AdS/CFT correspondence, the correlator of Wilson loops of the form W R, f /W R can be computed in the large 't Hooft coupling constant limit as the on-shell action of a fundamental string in the bubbling geometries arising due to the backreaction of a WL in the representation R.
The metric of the bubbling geometries is the one associated with AdS 2 , S 2 and S 4 fibration over a 2-dimensional complex Riemann surface Σ. All the geometric functions and fluxes can be expressed in terms of two holomorphic functions A, B defined on the Riemann surface Σ.
Following the notations and approach of [9] we take Σ as a torus described by coordinates (z,z) with periods 2ω 1 and 2ω 3 and write that in the case of genus one background corresponding to a rectangular Young tableau on the CFT side the functions A, B are the following where ζ stands for the Weierstrass ζ-function, a primitive of the Weierstrass ℘-function and κ 1 , κ 2 are the constants defined by a requirement that the geometry reduces asymptotically to AdS 5 × S 5 as The functions ζ(z), ℘ depend on the periods of the torus, which in their turn are specified by the two branch pointsẽ 1 ,ẽ 2 . Introducing alsoẽ 3 = −(ẽ 1 +ẽ 2 ) we write the half-periods ω 1 , ω 3 as where K is the complete elliptic integral of the first kind. Let us also introduce 14) The parameters of the Young tableau defining the representation of the WL causing the geometry are related to the half-periods by the equations where n stands for the number of rows and K for the length of rows of the rectangular Young tableau. The string configuration extremizing the string action is the one with the world sheet extending all along the AdS 2 , sitting at an arbitrary point both on S 2 and S 4 and at the points z = ω α , α = 0, 1, 2, 3 on the complex plane Σ.
The on-shell action is To compare the minimalized string action with the correlator on the CFT side one has to express ω α through the parameters of the Young tableau n, K with the help of equations (4.15) and to substitute them in (4.16). The relations (4.15) were inverted in [9] for the Young tableau with the number of lines n being of order of N and the length of the lines K being of order of N or larger. Let us do it for a small number of lines (n ≪ N ).
In this case the interval [ẽ 1ẽ2 ] collapses and hence the AdS 5 × S 5 geometry is recovered [10,11]. The half-period ω 1 tends to infinity ω 1 = O (ln(ẽ 1 −ẽ 2 )) = O (ln(N )) and therefore the Weierstrass elliptic functions can be written as The half-period ω 3 can be found from the second equation of (4.15). .
(4.18) Substituting (4.18) and infinite ω 1 in (4.16) one will find Again, as in the previous subsection, the first contribution to the action is suppressed comparing to the second one for any y > 0. Hence we get in the leading order of the large N expansion which coincide exactly with (4.8).