Large $N$ and large representations of Schur line defect correlators

We study the large $N$ and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory. By means of the factorization property, the large $N$ correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan's general theta functions and the $q$-analogues of multiple zeta values ($q$-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large $N$ correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large $N$ correlators.


Introduction and summary
The superconformal index [1,2] of 4d N = 2 supersymmetric field theory which can be viewed as a supersymmetric partition function on S 1 × S 3 captures the protected states of the theory. It generally depends on three parameters coupled to the Cartan generators of the superconformal algebra as well as extra parameters coupled to the global charges. It admits the Schur index [3,4], that is a specialization of the index which only depends on one of the three parameters for the superconformal generators. For a 4d superconformal field theory of class S [5,6], one can interpret it as a correlation function of a 2d topological field theory on a Riemann surface [3,7]. It is observed in [8] that there is a correspondence between the protected subsector of N = 2 SCFT and vertex operator algebra (VOA) so that the unflavored Schur index is identified with the vacuum character of the associated VOA. In addition, closed-form expressions of the Schur indices are recently obtained in terms of the special functions endowed with nice elliptic and modular properties [9][10][11]. The Schur index can be decorated by the BPS line operators wrapping the S 1 and localized at a point along a great circle in the S 3 [12][13][14][15][16][17][18][19]. It can be understood as a correlation function of the line operators which is independent of each distance of the adjacent line operators along the great circle. We call it the Schur line defect correlation function.
In this paper, we study the large N limit of the Schur line defect correlation functions of the Wilson line operators transforming in various representations labeled by the Young diagrams for N = 4 U (N ) super Yang-Mills (SYM) theory. Such Wilson line operators indexed by the Young diagrams are conjectured to be holographically dual to the configurations of Type IIB string theory. For example, fundamental, symmetric and antisymmetric representations correspond to a fundamental string [20,21], a D3-brane [22][23][24][25][26] and a D5-brane [23,25,27,28] respectively. More general representations are conjectured to be holographically dual to a configuration with multiple D3-and D5branes and bubbling geometries [23,[29][30][31][32][33][34][35]. In fact, it is shown [13,18] that the large N Schur line 2-point functions of the Wilson line operators in the fundamental representation and that of the Wilson line operators in the large (anti)symmetric representation precisely encodes the gravity spectra [36,37], which fall into the representations of the 1d N = 8 superconformal group OSp(4 * |4).
Exact expressions for the large N correlators of the Wilson line operators in arbitrary representations can be calculated by means of the factorization property [18] for the large N correlators of the charged Wilson line operators as well as the relations of the symmetric functions. By taking the further limit where the number of boxes of the diagram becomes infinitely large, we obtain the large representation limit of the Schur line defect correlation functions which is expected to encode the fluctuation modes for the dual bubbling geometries. When the fugacities are specialized (i.e. in the unflavored limit or the half-BPS limit), several large N and large representation correlators turn out to be expressible in terms of certain infinite series such as Ramanujan's general theta functions [38] and the q-analogues of multiple zeta values (q-MZVs) [39][40][41][42][43][44][45][46][47][48][49][50][51].
In addition, we find that the large N 2-point functions of the Wilson line operators in the large hook representation and those in the large rectangular representation are given by the generating function [52,53] for the partitions with non-negative cranks [54,55] and the generating functions [56,57] for the conjugacy classes of general linear groups over finite fields respectively. According to the closed-form expressions, we evaluate the asymptotic growth of the states by applying the convolution theorem [58].
It follows that the asymptotic degeneracy of the bubbling geometry with genus one surface dual to the large rectangular representation grows much faster than that for any p-branes [59][60][61][62][63].
Also we find several conjectured properties satisfied by the large N Schur line defect correlation functions. Under the transformation q → q −1 (resp. q → q −1 ) and the conjugation of the Young diagrams associated with the Wilson line operators, the large N normalized 2-point functions (resp. their half-BPS limit) are invariant up to the overall factor. In particular, the correlation function of the Wilson line operators in the representations labeled by a set of self-conjugate partitions enjoys the automorphy. Moreover, the large N normalized correlator can be factorized to a certain polynomial with positive integer coefficients and hook-length denominator.

Future works
Here we list several open problems for future research.
• While we focus on the Wilson line operators in the hook and rectangular representations in this work, in the upcoming work [64], we plan to report the study of the excitations of the dual bubbling geometries containing higher genus surfaces by computing the large N and large representation Schur line defect correlators of the Wilson line operators labeled by more general Young diagram with multiple parts.
• The determination of fully explicit formulas for general flavored large N and large representation Schur line defect correlators is not all obvious whereas we find some of them.
• It would be nice to give physical explanations or/and mathematical proofs of the conjectured properties satisfied by the large N Schur line defect correlators. Also it would be interesting to explore further properties, e.g. elliptic and modular properties.
• The large N and large representation correlators for other gauge theories, e.g. N = 4 SYM theory with different gauge groups and N = 2 SCFTs, will be also calculable by employing the similar factorization property.

Structure
The organization of the paper is as follows. In section 2 we review basic features of the large N limit of the Schur line defect correlators of the Wilson line operators for N = 4 U (N ) SYM theory. It contains the factorization of the large N correlators of the charged Wilson line operators and the holographic dual description of the Wilson line operators. In section 3 we study the large N correlators of the Wilson line operators in the (anti)symmetric representations. This generalizes the analysis in [13,14,18] by examining more general correlators and their properties. In section 4 we investigate the large N correlators of the Wilson line operators in the hook representations. The generating function [56,57] for the partitions with non-negative crank shows up in the large representation limit of the 2-point function. In section 5 the large N correlators of the Wilson line operators in the rectangular representations are examined. We find that in the large representation limit the 2-point functions agree with the generating functions [56,57] for the conjugacy classes of general linear groups over finite fields. In section 6 we discuss several conjectured properties of the large N Schur line defect correlators.
2 Large N Schur line defect correlators

Schur line defect correlators
The Schur line defect correlation function of the Wilson line operators transforming in the representations R j , j = 1, · · · , k for N = 4 U (N ) SYM theory is given by [13] where χ R j is the character of the representation R j of the U (N ) gauge group. In the absence of the Wilson line operator, it reduces to the Schur index I U (N ) (t; q) realized as a certain supersymmetric partition function on S 1 × S 3 . The Wilson line operator in the representation R j wraps S 1 and localizes at a point on a great circle in S 3 [15]. Under the conformal map S 1 × S 3 → R 4 , it turns into a half-line, a ray emanating from the origin in R 4 . We define the normalized Schur line defect correlator by The Schur line defect correlators which decorate the ordinary Schur index can be viewed as the "index" counting contributions of the BPS local operators due to the insertion of the Wilson line operators. The matrix integral (2.1) is invariant under an exchange of the characters with positive powers of gauge fugacities σ i and those with negative powers of σ i . For example, the 2-point function satisfies where λ and µ are the Young diagrams labeling the representations of the Wilson line operators and the notation µ stands for the representation labeled by the diagram µ for which the character is a symmetric function of the negative powers of gauge fugacities. According to the Gauss law, the correlator (2.1) does not vanish if the sum of the weights of the partitions associated with the characters with positive powers of gauge fugacities is equal to that of the weights of the partitions for which the characters have negative powers of gauge fugacities. For example, the 2-point function (2.3) is non-trivial when |λ| = |µ|.
When the two Young diagrams are equal, λ = µ, a pair of two half-lines conformally maps to a straight superconformal line along R in R 4 so that the q-series expansion of the 2-point function of the Wilson line operators starts with 1 + · · · and contains positive powers of q.

Half-BPS limit
The Schur indices of N = 4 SYM theories admit the special limit of fugacities keeping q := q 1 2 t 2 being finite and setting q to 0 [11]. For U (N ) SYM this results in which enumerates the half-BPS local operators [73], which correspond to partitions whose length is no greater than N . In this limit, the matrix integral (2.1) reduces to Since the orthogonal basis of this integral measure is the Hall-Littlewood function, for the two-point function of the Wilson line operators indexed by the two Young diagrams λ and µ, the half-BPS limit is shown to be given by [18] where K λν (q) is the Kostka-Foulkes polynomial [74] and m j (ν) is the multiplicity of the partition ν. For the antisymmetric representation, we have where P λ (σ; q) is the Hall-Littlewood function. Therefore if setting λ = µ = (1 m ) in (2.6), we obtain the exact result (2.9) In particular, for λ = (m), we have (2.10) For the symmetric representation, the Hall-Littlewood function are expanded by the sum of the Schur functions with the hook representations [74] (2.11) We can constrain relations of the correlators from the norm and the orthogonality of the Hall-Littlewood function. Using it, one finds the relations In the following sections, we will implicitly use these relations to check our evaluation.

Charged Wilson line operators
Let W n be the Wilson line operator of charge n which is described by the power sum symmetric function p n . In the large N limit, the normalized 2-point function of W n and W −n is given by [18] . (2.14) Turning off the flavored fugacity t, we get .
(2.15) In the half-BPS limit, we find (2.16) We have (2.18) For n = 1 they are the normalized 2-point functions of the Wilson line operators in the fundamental representation. One finds (2.21)

Factorization
It follows that in the large N limit, the Schur line defect even-point functions of the charged Wilson line operators have the following factorization [18] (2.24)

Holographic dual brane configuration
In later sections, we examine the large N limit of the Schur line defect correlation functions which play a significant role in the study of the AdS/CFT correspondence for the line operator. Let us briefly review the holographic dual brane configuration of the Wilson line operators in N = 4 U (N ) SYM theory. Suppose that N D3-branes are supported on (x 0 , x 1 , x 2 , x 3 ) in Type IIB string theory on R 1,9 . The low-energy effective description of the D3-branes is N = 4 U (N ) SYM theory on (x 0 , x 1 , x 2 , x 3 ). When a fundamental string on (x 0 , x 9 ) is added, one finds the half-BPS Wilson line operator in the fundamental representation for N = 4 U (N ) SYM theory [20,21]. When m fundamental strings end on the N D3-branes and on another D3-brane, we obtain the half-BPS Wilson line operator in the rank-m symmetric representation [22][23][24][25][26]. Without breaking further supersymmetry, the configuration also admit a D5-brane on (x 0 , x 4 , x 5 , x 6 , x 7 , x 8 ). When m fundamental strings ending on the N D3-branes and a D5-brane, the Wilson line operator in the rank-m antisymmetric representation is introduced [23,25,27,28]. The brane setup is summarized as follows: Here • denotes the directions in which branes are extended. The configuration preserves SO(1, 2) × SO(3) × SO(5) global symmetry, the bosonic subgroup of the 1d N = 8 superconformal group OSp(4 * |4). More generally, it is argued in [23] that the Wilson line operator in the representation labeled by the Young diagram λ = (λ 1 , · · · , λ r ) can be realized in the holographic dual gravitational description in terms of a configuration of r coincident D3-branes (D3 1 , · · · , D3 r ) in AdS 5 × S 5 where D3 i is the i-th D3-brane that carries λ i units of fundamental string charge dissolved in it. Alternatively, it is also shown to be described  (B.10) (B.10) in terms of a configuration of λ 1 coincident D5-branes (D5 1 , · · · , D5 λr ) in AdS 5 × S 5 where D5 j is the j-th D5-brane that carries λ ′ j units of fundamental string charge dissolved in it See Figure 1 for an example with the Young diagram (9, 8, 4, 4, 2, 1, 1).
In the near horizon limit of the brane configuration (2.25), the resulting Type IIB supergravity solution contains AdS 2 factor due to the one-dimensional conformal group SO(1, 2), S 2 factor corresponding to the SO(3) and S 4 factor associated to the SO(5). It takes the form [32] AdS 2 × S 2 × S 4 × Σ, (2.26) where Σ is a two-dimensional surface, over which the AdS 2 × S 2 × S 4 part is warped. It has the induced metric [32] ds 2 = f 2 1 ds 2 where f 1 , f 2 , f 4 and ds 2 Σ are real functions on Σ. When Σ is chosen as a hyperelliptic Riemann surface of genus g with boundary, the bubbling solutions are parametrized by two harmonic functions h 1 , h 2 on Σ as well as Σ in such a way that g is the number of parts of the Young diagram [32].
The fundamental strings wrapping AdS 2 in the global AdS 5 can meet the boundary of the AdS 5 at two points so that they should correspond to the superconformal lines which map to a pair of half-lines localized at the north pole and the south pole of S 3 . Accordingly, the fluctuation modes of the gravity dual configuration is expected to be calculated from the gauge theory side as the 2-point functions of the corresponding Wilson line operators in N = 4 U (N ) SYM theory. In fact, it is confirmed in [13,18] 1 that the large N normalized 2-point function of the Wilson line operator in the fundamental representation and that in the large (anti)symmetric representation precisely agree with the gravity indices encoding the corresponding gravity dual spectra [36,37].
The calculation of the fluctuation modes for more general geometries from the gravity side seems a rather non-trivial problem. To our knowledge, they have not yet been calculated in the literature. However, in the following sections, we will address them from gauge theory side by computing the large N Schur line defect correlators of the Wilson line operators in the representations labeled by the Young diagrams with hook shape and rectangular shape.

(Anti)Symmetric representations
In this section, we begin by examining the large N correlation functions of the Wilson line operators W (1 m ) (resp. W (m) ) transforming in the rank-m antisymmetric (resp. symmetric) representations labeled by the Young diagrams (1 m ) (resp. (m)). While the exact closed-form expression for the large N 2-point function of the two Wilson line operators W (1 m ) 's (resp. W (m) 's) is given in [18], several new properties and more general correlation functions are discussed.
The Wilson line operator W (1 m ) (resp. W (m) ) is associated with the elementary symmetric function e m (resp. complete homogeneous symmetric function h m ). According to Newton's identities

Rank-2 representations
The large N normalized 2-point functions of the Wilson line operators in the rank-2 (anti)symmetric representations are given by The two correlators (3.3) and (3.4) are related by

Unflavored limit
The large N unflavored Schur line defect 2-point functions of the Wilson line operators transforming in the rank-2 (anti)symmetric representations are (3.7) Here the coefficients and are the numbers congruent to 1 or 2 mod 4 and the numbers congruent to 2 or 3 mod 4 respectively.

Half-BPS limit
The half-BPS limit of the large N 2-point functions for the rank-2 (anti)symmetric representations is given by .

Rank-3 representations
The large N normalized 2-point functions of the Wilson line operators in the rank-3 (anti)symmetric representations are given by It is shown that

Unflavored limit
The large N unflavored 2-point functions of the Wilson line operators in the rank-3 (anti)symmetric representations are . (3.17) For the rank-3 (anti)symmetric representations the half-BPS limit of the large N normalized 2-point functions of the Wilson line operators is .

Rank-4 representations
The large N normalized 2-point function of the Wilson line operators in the rank-4 (anti)symmetric representations can be expanded with respect to the large N 2-point functions of the charged Wilson line operators We have

Unflavored limit
The unflavored 2-point functions of the Wilson line operators in the rank-4 (anti)symmetric representations are given by . (3.25)

Half-BPS limit
Also we find .
They satisfy the relation (3.28)

Rank-m representations
For general rank-m (anti)symmetric representations, the large N normalized 2-point functions of the Wilson line operators are given by [18] (3.29) Also we get We have

Unflavored limit
In the unflavored limit, the large N 2-point functions (3.29) and (3.30) reduce to They can be written as where G {(m)(m)} (q) is a polynomial in q with positive coefficients. So we have It follows that where the notation f (q) ≡ g(q) mod q m 2 implies that the coefficients of q k 2 in f (q) and g(q) are the same for k = 0, 1, · · · , m − 1.
In the large representation limit m → ∞, the large N normalized unflavored 2-point function of the symmetric Wilson line operators or equivalently that of the antisym-metric Wilson line operators becomes (3.38) The degeneracy a (3.40) As discussed in [18], the full large N unflavored 2-point function is given by . Therefore it does not allow for a non-trivial large representation limit The large N unflavored higher-point functions of (anti)symmetric Wilson line operators vanish in the large representation limit m → ∞ for |q| < 1. However, we find that the difference of the 4-point functions of the Wilson line operators with different ranks are (3.43)

Half-BPS limit
For general rank-m (anti)symmetric representations we find The expression (3.44) implies that the half-BPS local operators of dimension n which additionally appear at a junction of rank-m (anti)symmetric Wilson line operators are one-to-one with partitions of n into at most m parts. It follows that (3.46) In the large representation limit m → ∞, we find (3.48) The haif-BPS limit of the large N higher-point functions vanishes for |q| < 1, however, certain linear combinations of the correlation functions can be finite. For example the half-BPS limit of the large N 4-point functions of (anti)symmetric Wilson line operators obeys  Consider the Wilson line operator in the hook representation labeled by the partition λ = (m, 1) and its conjugate (2, 1 m−1 ) with m > 2.
For example, for m = 3 we have The Wilson line operators in the hook representations and also have correlators with those in the (anti)symmetric representations They are related by For m = 4 we have (4.11) It follows that (4.13)

Unflavored limit
For the partition and its conjugate , the large N unflavored 2-point functions are evaluated as , (4.14) . (4.15) The large N correlation functions with the rank-4 (anti)symmetric Wilson line operators are . (4.17) For the partition and its conjugate , we obtain . Also we find the correlators with the rank-5 (anti)symmetric Wilson line operators . (4.21) For general m, we can write where G {(m,1),(m,1)} (q) and G {(m,1),(m+1)} are polynomials in q with positive integer co-efficients. We find that (4.25) In the large m limit, we obtain (4.27)

Half-BPS limit
The half-BPS limit of the 2-point functions of the Wilson line operators associated with the partition and its conjugate is evaluated as .

Reconstructing flavored correlators
In the previous two subsections, we found analytic expressions in the large representation limit of the large N correlators for the unflavored case and the half-BPS case. To interpolate these two limits smoothly, we can guess analytic expressions for the flavored case. For example, from (4.26) and (4.35), we find Once we obtain such an analytic form, we can easily check its validity by comparing the q-series from the factorization property. Similarly, from (4.27) and (4.39), we find . For m = 4 we have (4.47) The large N normalized correlators obey the transformation law The Wilson line operators in these hook representations also have correlation func-tions with those in the (anti)symmetric representations. It follows that (4.50) We find that they are related by the transformation q → q −1 (4.51)

Unflavored limit
The large N unflavored 2-point functions of the Wilson line operators associated with the partitions and are Also we obtain For general m, the correlators can be written as (4.56) where G {(m,1 2 )(m,1 2 )} (q) and G {(m,1 2 )(m+2)} (q) are polynomials in q with positive integer coefficients. It follows that (4.59) In the large m limit, we find (4.61)

Reconstructing flavored correlators
We can also reconstruct the flavored correlators from special limits. From   For example, for m = 5 we have (4.82)
Under q → q −1 the (un)flavored correlator transforms as The Wilson line operators in the self-conjugate hook representations have correlation functions with those in the other hook representations including the (anti)symmetric representations. For example, (4.108) They obey

Unflavored limit
The unflavored large N 2-point functions of the Wilson line operators indexed by the Young diagrams and are .

Ramanujan's general theta functions
For general m the large N unflavored 2-point function of the Wilson line operator in the self-conjugate hook representation (m, 1 m−1 ) and that in the symmetric representation (2m − 1) can be expressed as Here G {(m,1 m−1 ),(2m−1)} (q) is a palindromic polynomial in q.

Rectangular representations
In this section, we study the large N correlation functions of the Wilson line operators in the representations indexed by the Young diagrams of rectangular shapes. According to the Jacobi-Trudi identities and Newton's identities (3.1)-(3.2), the Schur functions indexed by the rectangular Young diagrams are expressible in terms of the power sum symmetric functions.
Accordingly, we obtain the large N normalized 2-point functions of the Wilson line operators associated with the rectangular Young diagrams and They satisfy Similarly, for the rectangular diagrams and , the large N normalized 2-point functions are expressed as Under q → q −1 , they transform as More generally, we have The Wilson line operators in the rectangular representations (m 2 ) or (2 m ) also have correlators with those in the other representations. For example, the large N normalized 2-point functions of the Wilson line operators labeled by , and those in the (anti)symmetric representations can be calculated from the relation It follows that (5.14) More generally, we have where λ is the partition with |λ| = 2m.

Unflavored limit
The large N unflavored 2-point functions of the Wilson line operators in the rectangular representations and are evaluated as  We find that the large N unflavored 2-point function of the Wilson line operators in the rectangular representation labeled by the partitions (m 2 ) or (2 m ) can be expressed as . (5.22) In general, the large N unflavored 2-point function of the Wilson line operators in the rectangular representation (m 2 ) or (2 m ) and those in the (anti)symmetric representation takes the form Here G {(m 2 ),(2m)} (q) is a polynomial in q with positive coefficients.
We find that it obeys (5.24)

Half-BPS limit
In the half-BPS limit, the large N normalized 2-point functions of the Wilson line operators labeled by the partitions and are given by .
More generally, the half-BPS limit of the large N correlation functions of the Wilson line operators in the rectangular representations (m 2 ) and (2 m ) and those in the (anti)symmetric representations can be expressed as (5.32) One can check that the correlator transforms as under q → q −1 .

Unflavored limit
The large N unflavored 2-point functions of the Wilson line operators in the rectangular representations labeled by the partitions and are evaluated as For higher m the explicit form is rather expensive, however, for general m the large N unflavored 2-point function can be written as They obey More generally, we find that where λ is the partition with |λ| = m 2 .

Unflavored limit
In the unflavored limit, the large N normalized 2-point function of the Wilson line operators in the rectangular representations and are evaluated as  The finite m correction appears from the term with q m+1 2 . We find that Holographically, the degeneracy a {(∞ ∞ ),(∞ ∞ )} (n) will be understood as the fluctuation modes for the dual bubbling geometry with genus one surface. Curiously, the asymptotic behavior (5.68) grows much faster than that for the p-branes of the form exp(αn p p+1 ) [59][60][61][62][63] with some constant α.

q-MZVs
We have found that in the large representation limit m → ∞, the unflavored limit and the half-BPS limit of the large N normalized 2-point function of the Wilson line operators in the representation labeled by the m-square Young diagram becomes the generating functions (5.65) and (5.82) for the conjugacy classes of GL(n, 3) and GL(n, 2) respectively. It may be worth pointing out that they are expressible as [84] ∞ n=1 1 − q n 1 − cq n = k≥0 (c − 1) k ζ q ({1} k ), (5.87) where ζ q (k 1 , · · · , k r ) := is the q-analogues of multiple zeta values (q-MZVs) [45]. There are several versions of q-MZVs in the mathematical literature [39][40][41][42][43][44][45][46][47][48][49][50][51] where their algebraic structures have been examined. They are closely related to the multiple Kronecker theta series [18] which plays a role of the building blocks of the Schur line defect correlators. We leave it future work to investigate further connection between the Schur line defect correlators and the q-MZVs.

Conjectural properties
We have seen that the large N correlation functions satisfy certain transformation laws and that they can be expanded in specific forms by computing various examples. In this section, we summarize several conjectural properties for the large N Schur line defect correlation functions.

Conjugation
Consider the large N correlation function of the Wilson line operators in the representations labeled by a set of the Young diagrams {λ i } k i=1 associated with the positive powers of gauge fugacities and those labeled by the other set of diagrams {µ} l j=1 for the negative powers of gauge fugacities. The correlator is non-trivial when the total sums of the boxes are equal, k i=1 |λ i | = l j=1 |µ j |. Then we find that where λ ′ i and µ ′ j are conjugate to λ i and µ j respectively. For example, when k = l = 1 and λ 1 = µ 1 = (1 m ), the relation (6.1) results in the observation [18] that the large N 2-point function of the Wilson line operators in the rank-m antisymmetric representation is equal to that of the Wilson line operators in the rank-m symmetric representation 3 ⟨W (1 m ) W (1 m ) ⟩ U (∞) (t; q) = ⟨W (m) W (m) ⟩ U (∞) (t; q). (6.2)

Automorphy
While the large N correlation functions of the Wilson line operators are invariant when one replaces all the partitions labeling the representations of the Wilson line operators with their conjugate, they also have a nice transformation law under the replacement of one of the set of partitions with its conjugate. We find that ⟨W λ 1 · · · W λ k W µ 1 · · · W µ l ⟩ U (∞) (t; q −1 ) In other words, the replacement of one of the sets of partitions, e.g.

Hook-length exapansion
Suppose that λ and µ are the Young diagrams which describe the representations of the Wilson line operators have a finite number of boxes. Let h(b) be the hook-length of a box b in the Young diagram. It follows that the large N normalized unflavored 2-point functions of the Wilson line operators W λ and W µ can be factorized as where G {(λ),(µ)} (q) is some polynomial whose coefficients are positive integers. Similarly, we find that the half-BPS limit of the large N normalized 2-point functions of the Wilson line operators can be written as where H {(λ),(µ)} (q) is some polynomial whose coefficients are positive integers. Note that when either λ or µ is self-conjugate, the large N normalized 2-point functions have the automorphy in that they obey the relations (6.5) or (6.7). In that case, the polynomials G {(λ),(µ)} (q) and H {(λ),(µ)} (q) are palindromic.