Multi-trace Correlators from Permutations as Moduli Space

We study the $n$-point functions of scalar multi-trace operators in the $U(N_c)$ gauge theory with adjacent scalars, such as ${\cal N}=4$ super Yang-Mills, at tree-level by using finite group methods. We derive a set of formulae of the general $n$-point functions, valid for general $n$ and to all orders of $1/N_c$. In one formula, the sum over Feynman graphs becomes a topological partition function on $\Sigma_{0,n}$ with a discrete gauge group, which resembles closed string interactions. In another formula, a new skeleton reduction of Feynman graphs generates connected ribbon graphs, which resembles open string interaction. We define the moduli space ${\cal M}_{g,n}^{\rm gauge}$ from the space of skeleton-reduced graphs in the connected $n$-point function of gauge theory. This moduli space is a proper subset of ${\cal M}_{g,n}$ stratified by the genus, and its top component gives a simple triangulation of $\Sigma_{g,n}$.

In the large N c gauge theories, Riemann surfaces are carved out by the Feynman diagrams in the double-line notation [1]. This observation was a precursor of the AdS/CFT correspondence, that is the conjectured duality between gauge and string theories [2]. The AdS/CFT correspondence can be checked explicitly at any values of the 't Hooft coupling λ = N c g 2 YM in the integrable setup, such as N = 4 super Yang-Mills (SYM) in four dimensions and superstring on AdS 5 × S 5 [3,4].
The AdS/CFT correspondence beyond the large N c limit is not well-understood, including the question of integrability. It is known that the non-planar integrability, even if exists, does not protect the spectrum, because the 1/N c corrections lift the spectral degeneracy of the planar N = 4 SYM [5]. One of the promising ideas in this direction is based on the hexagon form factor [6], which can capture 1/N c corrections [7][8][9].
Integrability predictions are often formulated and refined with the help of extensive perturbative data. At weak coupling, the 1/N c corrections can be computed by summing over the graphs at a fixed genus [10]. However, this method is tedious. The finite group method is more efficient. There one expresses multi-trace operators in terms of permutations, and applies finite-group Fouriertransform to obtain a new basis of gauge-invariant operators labeled by Young tableaux [11]. This representation basis diagonalizes tree-level two-point functions at any N c , and solve finite N c constraints [12,13]. The finite group methods are used to compute various quantities, such as partition functions, two-point functions, and recently extremal n-point functions [14].
We ask two questions in this paper. How do we find general non-extremal n-point functions in the finite group methods, at any n and to any orders of 1/N c ? And how do these correlators describe Riemann surfaces?
A similar problem was studied in the Hermitian matrix model, which is a simpler version of large N c gauge theory [15]. This matrix model describes two-dimensional gravity in the continuum limit, and its exact free energy is given by the τ -function of the KdV hierarchy [16,17].
We will study the tree-level n-point functions of scalar operators in N = 4 SYM with U (N c ) gauge group. We express Feynman graphs in terms of permutations, and describe the space of Wick-contractions in an algebraic manner. We obtain a formula which naturally factorizes into the product of pairs of pants, i.e. three-point functions. This formula is invariant different pants decomposition, and resembles the interaction of n closed strings.
Then we perform a skeleton reduction to the Feynman graphs in the n-point functions. Under the skeleton reduction, Feynman graphs become connected metric ribbon graphs. It is known that there is an isomorphism between the space of connected metric ribbon graphs and the decorated moduli space of Riemann surfaces [18][19][20][21]. Thus we define the moduli space of Riemann surfaces in gauge theory M gauge g,n as the space of Wick-contractions in the skeleton-reduced Feynman graphs. These graphs resemble the interaction of open strings which triangulate Σ g,n .
The gauge theory moduli space M gauge g,n exhibit two properties. First, it is a proper subset of the moduli space of the decorated arithmetic Riemann surfaces, equivalent to the connected integral ribbon graphs [22,23]. Second, our definition of the skeleton reduction stratifies M gauge g,n by genus, meaning that the diagrams with smaller genera contribute to higher powers of 1/N c . 1 In [27] the correlators of gauge theory are used to define an effective two-dimensional worldsheet theory, by rewriting propagators in the Schwinger parameterization. This "CFT" has been studied in detail, and its unusual properties have been found [28][29][30][31][32][33][34], including M gauge g,n M g,n mentioned above. 2 Another idea is to interpret the correlators of Hermitian matrix model as the counting of holomorphic maps [36]. It is argued that the Gaussian matrix model is dual to the A-model topological string on P 1 [37][38][39], similar to the Eguchi-Yang model [40].
Our results may be regarded as a variation of the open/closed duality, exploring the n-point functions of a gauge theory at tree-level from various aspects. Correspondingly, this paper consists of two parts. In Sections 2 and 3, closed-string-like formulae of the n-point functions are discussed, where one of the formulae has natural pants decomposition. In Sections 4 and 5, open-string-like formulae of the n-point functions are discussed, where the skeleton-reduced graphs define the moduli space M gauge g,n through triangulation.

Outline of ideas
Our objectives in this paper are to compute n-point functions by using permutations, and to define M gauge g,n by using permutations.
Let O i be a general multi-trace scalar operator of length L i in N = 4 SYM, 3 whose color structure is specified by the permutation α i ∈ S L i and flavor structure unspecified. The tree-level n-point function is schematically written as the sum over all possible sets of Wick-contractions, (1.1) The space of Wick-contractions W decomposes as where ij is the number of all Wick-contractions between O i and O j , also called bridge length. The bridge lengths satisfy 3) The space of Wick-contractions is equivalent to the sum over Feynman diagrams. In order to study graphs, the permutation is a powerful tool. Feynman diagram can be interpreted as the Cayley graph generated by a finite group. The graph data (vertices, edges and faces) are translated into a triple of permutations. Hence, there should be permutation-based formulae of G n . Once we start looking for such formulae, we encounter the following questions.
i) How to multiply the elements of different permutation groups ? ii) How to sum over { ij } ?
Our answer to both is by embedding, i) to embed {α i } into a big permutation group to make the group multiplication well-defined. ii) to embed W into a larger but simpler space by adding unphysical Wick-contractions.
We will obtain three different expressions of G n in Section 2, which we call vertex-based, edgebased and face-based formulae. Let us explain the edge-based formula, which reads whereα i specifies the color structure of the i-th multi-trace operator, and h A 1 A 2 ...An is related to the flavor inner-product. This formula follows from the idea of extending the operators by adding identity fields, where ΦÂ We decompose the formula (2.28) into the product of pairs of pants in Section 3 G n ∼ Glue (X [S 1 ], X [S 2 ], . . . , X [S n−2 ]) , Σ 0,n = S 1 S 2 · · · S n−2 . (3.1) where {S f } is a local three-point interaction. The RHS of (3.1) depends only on the topology of Σ 0,n . We introduce a defect on Σ 0,n which carries all information on the powers of N c . As a result, no higher-genus surfaces show up.
In Section 4, we study G n in the opposite way by reduction, and take the sum over { ij } literally. This sum is related to the moduli space of Riemann surfaces, which becomes transparent after a skeleton reduction. The skeleton reduction can be defined by taking the Wick-contractions in two steps. Let us define a partition of L i by i ≡ ( i|1 , i|2 , . . . , i|L i ) L i . We denote a sequence of consecutive i|r fields in O i by Φ (i) r , and define the reduced single-trace operator by There are many ways to create O i from O i , which we call partitions of O i . We define the space of the external Wick-contractions by all possible pairings of {(i, r), (j, s)} in Φ (i) r and Φ (j) s . Then we impose the internal Wick-contraction rule between Φ (i) r and Φ (j) s , which is non-zero only if i|r = j|s and expressed as a sum over the internal pairing map τ ∈ S i|r . Since τ contains non-planar Wickcontractions, the topology of the skeleton-reduced graphs is greatly simplified.
We will obtain two more permutation-based formulae for G n , which we call vertex-based and face-based skeleton formulae. The face-based skeleton formula for the n-point function reads constraints (4.79) whereL = C( ) and the elements of S ×× 2L ⊂ S 2L have no one-and two-cycles. The permutation ν ∈ S ×× 2L defines a skeleton graph. The function F maps a set of skeleton graphs to a polynomial of color and flavor factors, and takes a complicated form z · h τ μ 1,sμ1,s . . .μ n,sμn,s . (4.78) The set (S ×× 2L ) phys is defined so that it corresponds to a skeleton graph Skeleton graphs with 2L unlabeled vertices ↔ ν ∈ S ×× 2L /Aut V phys . (4.61) From this correspondence, we translate the formula (4.79) into a sum over graphs as where F (τ |Γ) is rewriting of F(τ | ν) in the language of graphs.
We call M gauge g,n ({L i }) the moduli space of gauge theory. This is a proper subset of the connected metric ribbon graphs, or equivalently of the decorated moduli space of Riemann surfaces. We give its alternative definition in Section 5, and study examples.
In Section 6 we briefly discuss how metric ribbon graphs appear in string theory.

Correlators from permutations
We express tree-level n-point functions by permutations in various ways, and call them vertex-based, edge-based and face-based formulae. Basic techniques to study Wick-contractions by permutations will be explained below.
We define the permutation basis of scalar multi-trace operators by and impose the U (N c ) Wick rule (A.18) between scalar fields. 4 Then we evaluate G n in (1.1) by the repeated application of the pairwise Wick-contractions as where W ij runs over the Wick-contractions at fixed bridge lengths. We rewrite RHS by permutations.

S 2L (or vertex-based) formula
Let us label the fields by P = 1, 2, . . . , 2L so that the n-point function looks like a one-point function, Here α = n i=1 α i ∈ S 2L is the permutation for all external operators, parameterized by Denote the cycle type of α by λ = [1 λ 1 2 λ 2 . . . ] (2L), where λ K is the number of single-trace operator of length K in O all . Next we introduce a pairing map This W 0 can be regarded as the top element of Z ⊗L 2 , and we write a 2k−1 = W 0 (a 2k ) with W 2 0 = 1. By using (A.4), we permute W 0 by the adjoint action of γ ∈ S 2L to create the general element of Z ⊗L 2 ⊂ S 2L as 5 (2.7) 4 Our notation will be explained further in Appendix A. 5 In the literature on graph theory and combinatorics, the adjoint action of γ on edges is called an edge permutation.  (6,4,4). The left figure corresponds to a planar term, and the right figure to a non-planar term.
The above definition of W is redundant, because it remains invariant under some relabeling of {a i }. This redundancy is generated by the group S L [Z 2 ] as explained in Appendix A. In contrast, our parameterization of α ∈ S 2L has no redundancy, because they carry flavor indices. We can identify Feynman graphs with the Cayley graph generated by α ∈ S 2L and W ∈ Z ⊗L 2 , as shown in Figure 1. 6 In the above notation, the original n-point formula (2.2) becomes where we take the square root of 2L factors, and used W 2 = 1. The δ-functions can be simplified as where C(ω) is defined in (A.8). It follows that This expression looks like a one-point formula on the permutation basis for any n. Suppose W 0 is one such pairing, namely All other pairings are generated by the group action It is easy to show the equivalence. First, we see that RHS does not change { ij }. Recall that the elements in RHS take the form (2.7) under the constraint γ ∈ n i=1 S L i . Since any elements of S L is a product of transpositions, we assume γ = (aa ) without loss of generality. Since γ ∈ n i=1 S L i , both a and a must come from the same operator, say O a . Now which does not change any of ( ab , bc , ca ). Secondly, we argue that any γ which preserves { ij } should belong to n i=1 S L i . To see this, suppose that O i and O j are paired by W 0 and W as (2.14) The two permutations are related by By repeating this argument to all (i, j), one finds that . For this reason, it is difficult to compute the sum over W in (2.10).
In order to simplify the above formula further, we introduce the new inner-product, (2.16) The space of all pairing maps W becomes is any set of L pairs out of 2L numbers without overlap like (2.5). Now the space W becomes a group coset. It follows that where |Aut W 0 | = |S L [Z 2 ]| = L! 2 L . The sum over { ij } has been successfully removed.

S L (or edge-based) formula
In the previous section, we labeled fields Φ A P by P = 1, 2, . . . , 2L and derived formulae as a sum over S 2L . Here we label Wick-contractions by 1, 2, . . . , L, and give another formula in terms of S L .
First, we extend the operator O i by adding identity fields, When ΦÂ (i) p = 1, the permutationα i acts as the identity on p.
Second, we introduce n-tuple Wick-contraction by We demand that hÂ 1Â2 ...Ân is equal to the two-point inner-product g AB , if (n − 2) of the flavor indices {Â 1 ,Â 2 , . . . ,Â n } are the identity field 1, and otherwise hÂ 1Â2 ...Ân = 0. For example, triple contraction is given by, 7 and in general ( where δ L and Ω are defined in (A.5) and (A.24), respectively. The symbol means that the sums over W 12 , W 23 , W 31 are constrained by (2.25) We can solve this condition formally by introducing (2.27) We introduced the factor |S L | = L! because the simultaneous transformation U k → U k γ with ∀γ ∈ S L does not change W ij in (2.26).
The formula for general n-point functions ofÔ i is This quantity is proportional to the original correlator of {O i }. The proportionality constant comes from the permutations of the identity fields in (2.19). SinceÔ i has (L − L i ) identity fields, we find (2.29)

Symmetry of the S L formula
We summarize the symmetry of the formula (2.28).
First, the n-point formula is invariant under the simultaneous redefinition {U i } = {U iγ }. This corresponds to relabeling the Wick-contractions p →γ(p) withγ ∈ S L . We can use this redundancy to set e.g. U 1 = 1.
Second, the extended operator (2.19) has the gauge symmetry generalizing (A.17). Our n-point formula is also invariant under the gauge transformations (2.30) for each fixed i, because the change (α i ,Â γ i (p) ) can be absorbed by the redefinition Third, the n-pt formula is invariant under any S n permutations, This follows from S n invariance of the n-tuple metric h A 1 A 2 ...An in (2.22), and the redefinition In other words, two conjugacy classes commute. Recall that we introduced an artificial cyclic ordering (12 . . . n) in (2.20), apparently breaking S n to Z n . We could use the S n -invariant Wickcontraction rule, where Der n is the derangement of n elements (permutations without fixed points). The two definitions give the same result, because h A 1 A 2 ...An is locally a two-point inner-product, and the order of the color indices does not matter.

Cayley graph
We discuss how to draw a Cayley graph based on the S L formula. The main idea is to define an n × L Wick-contraction matrix In generalα k permutes the k-th row ofμ in a non-trivial way. We wrote the matrixμ in the form whereα 1 acts trivially. The p-th column ofμ corresponds to the p-th flavor factor U 2 (p) ...Â Since W 11 = 1, we can think ofμ as an element of Z ⊗L n consisting of L columns. 8 The two groupŝ can be used to draw a Cayley graph of the n-point function the S L formula, as in Figure 2. This graph looks different from the Cayley graph in the S 2L formula. In particular, planar graphs in the S 2L formula may not be realized as a planar graph in the S L formula, even though the number of color cycles is same for both.
Intuitively, the n-tuple Wick-contraction can be regarded as a hoop with two hooks. There are L hoops in total. We hook the hoop at a pair of operators (O i , O j ), where each operator O i has L i holes where one can hook the hoop. The position of the holes is permuted by U . 8 The matrixμ has the symmetry S ⊗L n if we sum over {U k }. Here we consider a gauge-fixed version.
We can slightly generalize the same matrix so that it has double columns, 9    Let us assign dummy indices {Â 4 ,Â 5 ,B 5 ,Ĉ 4 ,Ĉ 5 } to empty entries as (2.39) We reconstruct the original Wick-contraction (2.37) by remembering the label of the dummy indices.

Face-based formulae
From a Cayley graph generated by (α, W ), we define the dual graph by Since the flavor inner-product g AB is symmetric, we do not distinguish two ways to double the columns, i.e.
We want to express the n-point functions as a sum over the face permutation ω, instead of a sum over W , where F computes the color and flavor factors associated to ω.
In order to define F, we need to parametrize the faces of the graph. The Feynman graphs in the double-line notation have two types of edges, those from Wick-contractions and the others from external operators. We call the edge from Wick-contractions the Wick-edge, and the edge from external operators the rim. Each Wick-edge is connected to a pair of rims, and each rim is oriented by α = n i=1 α i ∈ S 2L . There are two ways to label rims, depending on whether one simplifies α or W . In the first method, we label the rims by {1, 2, . . . , 2L} as in Figure 3. These numbers are ordered in the same way as α explained in (2.4). 10 In the second method, we label the Wick-edges by {1, 1 , 2, 2 , . . . , L, L }. The Wick-contraction is given by W = p (p p ). Since the whole graph is oriented, each rim starts from a Wick-edge and ends at another Wick-edge. We label the rim by using the ending Wick-edge as in Figure 4.

Rim-labels by W
We consider n-point functions of single-trace operators by using the rims labeled by W . The rimlabels by α will be studied in Appendix B. For simplicity, we assume that all external operators are single-traces.
To begin with, we show the equivalence between the set of unlabeled graphs and face permutations, In RHS, we define ω as the permutation of {1, 1 , 2, 2 , . . . , L, L }. The face permutation should not have one-cycles, because the one-cycles come from self-contractions. Owing to the redundancy of relabeling, ω is in one-to-one correspondence with the coset S × 2L /S L [Z 2 ], where the denominator has the adjoint action like (2.17). We have already explained how to construct a face permutation from a given graph in the double-line notation; label the Wick edges using {1, 1 , 2, 2 , . . . , L, L } and export them to rims. Conversely, we can construct a graph from the face permutation as follows. Consider the permutation (2.44) If we regard α • as the permutation for the external single-trace operators, we can make all rims oriented. By connecting the end-points of the rim pairs p, p , we obtain a Feynman diagram in the double-line notation. Hence the relation (2.42) is proven.
The same graph whose rim between aP and a α(P ) is labeled as P . We make two more comments. First, the relation (2.42) tells nothing about the cycle type of α • . To compute G n we need to restrict the sum over ω further. Second, in this construction ω produces an unlabeled graph as exemplified in Figure 4. In other words, the rim labels {1, 1 , 2, 2 , . . . , L, L } are not related to the flavor indices of the external operators, and the identification is not unique. Generally, an unlabeled graph is an equivalence class of the relabeling group acting on labeled graphs. Thus, the function F in (2.41) is a map from a set of graphs to a polynomial of (N c , g AB ), where the graph set is a certain group orbit.
We define F in the following way. To begin with, we interpret a permutation as a set of ordered The ambiguity comes from the translation of each cycle and permuting the cycles of the same length.
In other words, CL(α • ) is isomorphic to the stabilizer subgroup, 11 Given the permutation α • , we pick up one element from CL(α • ) and write it as β ≡ (β 1 , . . . , β n ). 13 In the example of Figure The ordered list β ∈ CL(α • ) relates the rim labels to the flavor indices as is uniquely defined if and only if x = 0. 14 We parameterize α as In the literature, Stab(α • ) is also called the automorphism group Aut(α • ). 12 Since all operators are assumed to be single-traces, λ k is the number of single-trace operators of length k. 13 At this point n is a free parameter. We will equate n with the number of external single-trace operators in (2.59).
We should sum over the choices β ∈ CL(α • ) entering inμ β . This procedure corresponds to labeling unlabeled graphs. Then we define the function on faces by h z·µ 1,s z·µ 1,s ... z·µn,s z·µ n,s . (2.58) Using the correspondence (2.42), we can write the n-point function of single-trace operators as We inserted the δ-function in order to guarantee that α • = ωW • has the same cycle type of α ex specified by the external operators. We rewrite (2.59) in terms of the conjugacy class (A.6) of S 2L as We checked this formula by Mathematica for simple cases.

Pants decomposition
Let us discuss the geometric interpretation of the n-point function in the S L formalism. We rewrite the formula (2.28) as a product of three-point functions, or pairs of pants, schematically as where {S f } defines a pants decomposition of CP 1 with n punctures, The last equation of (3.1) will be made precise in (3.11). This expression implies that the tree-level n-point function is a certain topological partition function on CP 1 with n boundaries and defects. We call it topological, because it is invariant under different pants decompositions, and does not depend on the complex structure of Σ 0,n . 15 The punctures represent external operators {Ô i }, and each factor X [S f ] represents a local three-point interaction. The defects carry all the information about N c . This is why higher-genus Riemann surfaces do not show up. The sum over U 's in (3.1) can be thought of as a sum over different twisted sectors, or the partition function of a topological gauge theory with finite gauge groups [41][42][43][44]. 16 To derive (3.1), consider the color and flavor factors separately.
The color factor decomposes as follows. Thanks to the permutation symmetry (2.31), without loss of generality we may assume S 1 contains {α 1 , α 2 }, S k contains α k+1 for 2 ≤ k ≤ n − 3, and S n−2 has {α n−1 , α n }. Then we insert the resolution of identity on the permutation basis 17 Now we rewrite the δ-function as where [ω k ] is the conjugacy class defined in (A.6). Since two conjugacy classes commute, the first line is manifestly invariant under the permutation of different ω's. This decomposition is similar to Although this figure has a fixed number of bridge lengths, the sum over { ij } has already been performed in our formula.
the following set of OPE's, 18 This structure is also depicted in Figure 5.
In order to split the flavor part of the formula (3.1), we introduce the functions where φ is an extra parameter to count the scalars (i.e. non-identity fields), and g AB is the inverse metric g AB g BC g CD = g AD , According to (2.21), h A 1 A 2 ...An = 0 when it has precisely two scalars. Thus H A 1 A 2 A 3 = 0 when it has zero or two scalar indices.
By using (3.6) and (3.7), we find the identity To prove it, recall that precisely two indices in {Ǎ p } on LHS are scalars for each p. On RHS, there are (3n − 6) flavor indices for each p 18 Strictly speaking, this operation is not OPE. Generally, an OPE produces several terms, while inserting the resolution of identity produces one term. and the factors H or ζ are trivial if they consist of the identity fields only. We look for the term of O(φ −1 p ). This comes from the case Since g BC is a diagonal metric, this condition is satisfied precisely when two of {Ǎ p } are scalars. This completes the proof.
Let us combine the pants decomposition of the color and flavor factors (3.4), (3.8), and substitute them into the n-point formula (3.1). The result is The equation (3.11) gives the pants decomposition of the n-point function addressed in (3.1). The gluing operation consists of the sum over the defect factors {ω f }, the residue operation and inverse metrics {ζ BC }.
Note that the above discussion is different from TFT interpretation discussed in [45]. There we required the invariance of the topological partition function under any cell decompositions of the two-dimensional surface, where each 2-cell is a strip. In other words, the n-point functions should decompose into a collection of strips, where each strip collects the local Wick-contractions between O i and O j . However, it turns out that G n in the S L formalism is compatible with such rewriting only if n ≤ 4, or equivalently as long as the complete graph K n is planar. For n ≥ 5, we find obstruction for rewriting, unless we assume certain commutation relation between the permutations for external operators.

Skeleton reduction
We will perform a skeleton reduction of Feynman graphs.
Let us first clarify what we mean by skeleton reduction. One possible definition of the skeleton reduction is to make a new graph by assembling all edges connecting the same pair of vertices. However, this definition is ambiguous for us, because we want to keep the color structure of Feynman graphs in the double-line notation. We define skeleton as a type of graphs which does not have faces made of two or fewer edges. This definition is convenient because we will eventually relate the skeleton-reduced graph with the complex structure of a Riemann surface. There are a few ways to reduce the original Feynman graphs satisfying this condition. The skeleton reduction used in [8,9,27] assembles the planar, ladder-type internal Wick-contractions, which we call ladder-skeleton. Our skeleton reduction includes non-planar internal Wick-contractions, which simplifies the topology of the reduced graphs. Figure 6 shows a simple example of the skeleton reduction.
For simplicity, we consider only the n-point functions of single-traces in this section. Our notation is summarized in Appendix A.4, and the implementation of the reduction procedure is explained in Appendix D.

Open two-point functions
Let us denote a sequence of numbers by and a sequence of consecutive fields by We define open two-point functions by taking all possible Wick-contractions between a pair of sequences, where k = τ (k) for some τ ∈ S . We sumW over the space of "local" Wick-contractions We inverted the argument of τ by k → − k to keep track of the orientation of the whole graph. We assign the counterclockwise ordering to all external operators. This means that α i and α j rotates the color indices in the opposite directions, as explained graphically in Figure 7.
Here Z(ω|a, b) counts the number of cycles inω without a or b . To understand Z(ω|a, b), recall that the permutationω = α iW α jW −1 represents the faces of the part of the original Feynman graph. 19 Since all Wick-contractions are locally taken between O i and O j , repeated applications of ω send the upper indices (a p+1 , b q+1 ) to the lower indices (a p+ +1 , b q+ +1 ). If we writẽ By substituting (4.5) into (4.3), we obtain 19 SinceW is a set of local Wick-contractions as in (4.4),ω knows only the faces inside O i and O j .
Powers of N c and the color R-matrix As a linear operator, the color matrix R ab a b is just a product of two δ-functions. Its index structure is correlated with the powers of N c as (for some n ∈ Z ≥1 ). (4.8) We can prove this selection rule as follows. If we multiply δ , the equation (4.8) becomes part of the two-point function of single-traces, written as Since α i and α j have the same cycle type and belong to the same conjugacy class, we may redefinē W to rewrite This is an element of the commutator subgroup of S +1 , namely A +1 , consisting of an even number of transpositions. Thus the power of N c changes only by an even number whenW changes, which translates into (4.8).
The powers of N c are also determined by τ ∈ S in (4.4), and we can rewrite Some examples are shown in Figure 8.Z is maximal if τ is identity, which corresponds to the planar case. The O(1/N c ) terms come from cyclic permutations, The O(1/N 2 c ) terms come from cyclic permutations of lower order with consecutive numbers, (4.14)

Relabeling Wick-contractions
The skeleton reduction reorganizes the structure of Wick-contractions. This structure can be explained in two ways.
In the first explanation, we introduce {  which generalizes (1.3). Note that planar graphs may have r ij > 1. In this notation, G n becomes Each W ij,ρ takes the form as in (4.4).
The second explanation was given in Introduction. We split O i into the sequential fields {Φ (i) r } in (4.2), and call it a partition of O i , We take open two-point functions as the internal Wick-contractions between Φ (i) p and Φ (j) q . Now the n-point functions contain three sums.
iii) Sum over the internal Wick-contractions, for each Φ (i) p Φ (j) q as in (4.7). Figure 9: Rewriting the open two-point function in Figure 7 in terms of (V, τ ). The color indices at the four end-points are denoted by a p , b q , aᾱ i (p) , bᾱ j (q) .
The choice of τ ij,ρ in (4.18) is equivalent to the internal Wick-contraction.
For a given partition {O i }, the equivalence of the two explanations can be understood as follows. We relabel the color indices of O i as where p labels the open end-points andp labels the internal indices. The equation (4.19) can be written as Here the reduced color permutationᾱ i ∈ SL i shifts the open end-points {p}, and thus aᾱ i (p) = a p+ p . We write the Wick-contractions in the new notation as The new permutation V is equivalent to the external Wick-contraction. The relation between W ij,ρ and (V ij,ρ , τ ij,ρ ) is also depicted in Figure 9.
The open two-point function (4.7) can be rewritten as whereZ(τ ) is given in (4.11), and we replaced g AB by h AB in (2.16) for a later purpose. In this notation, G n is given by (2) 23 (1) 23

Vertex-based skeleton formula
We express G n by gluing open two-points, following the vertex-based method in Section 2.1.

Gluing open two-point
Let us unify the color indices of {O i } in (4.26) as We also unify the reduced permutations and the reduced Wick-contractions as just like (2.7). The Cayley graph of (V,ᾱ) is the skeleton-reduced Feynman graph, and a positive integer E is associated to the E-th edge. Figure 10 shows an example of the skeleton reduction applied to a Feynman graph in a four-point function.
In the new notation, the bridge lengths { (ρ) ij } in (4.15) simply become { P }, and the number of non-zero Note that the range of P is twice as big as that of (ij, ρ), which means that The same remark applies to the color R-matrix, which has the symmetry R ab a b = R ba b a . Let us rewrite G n in (4.24) as 20 where we introduced new symbols We can think ofω as a face permutation, because R is a pair of δ-functions as discussed in (4.8).
The equation (4.33) makes sense because RHS has 4L color indices, and the same index always appears as a superscript and a subscript, once for each. Furthermore, by comparing (2.10) and (4.31), we find where we used (4.11). This means thatZ counts the number of cycles without open end-points, and C(ω) counts those with open end-points.

Face permutations of the skeleton graph
There exist two face permutations for the skeleton graph. The first one is given by which generates the faces of the skeleton graph. Since the skeleton graph triangulates a Riemann surface, ν does not have one-and two-cycles. The second one isω ∈ S 2L introduced in (4.33), 20 By abuse of notation, we use (a P a Q ) and (P Q) in V interchangeably. The two face permutations ν andω agree if the matrix R ab a b in (4.5) takes the planar form, For non-planar cases, we can define another involution θ ∈ ZL 2 such that ν ≡ω θ, θ 2 = 1. This θ also follows from τ in (4.4) thanks to the selection rule (4.8). An example of the relation (4.37) is illustrated in Figure 11.

Space of the reduced Wick-contractions
Our next goal is to construct the space of the reduced Wick-contractions in G n , namely the range of three sums in (4.31).
First, we generate partitions of the operator O i in (4.19) as follows: choose a partition of L i , divide O i according to that partition, and apply Z L i translations. A partition of L i is denoted by We group the sequences of i|q consecutive fields together as in (4.2) to define the reduced operator withᾱ i ∈ SL i . The reduced operators also have the gauge symmetry (A.17).
For simplicity, we assume thatᾱ i ∈ ZL i and O i are single-traces. The gauge symmetry is equivalent to the identification of cyclically translated operators, The original Z L i symmetry generates different reduced operators. For example, we regard the following two reductions as inequivalent: We introduce an ordered set of L i numbers i as a function of z ∈ Z L i , and denote its equivalence class by [ i (z)] modulo the translation byᾱ i , In general, the number of equivalence classes is given by the orbit-stabilizer theorem. In our problem, it means where z 0 is a reference point, e.g. identity. In summary, the space of the partitions of O i can be written as where r k is the number of consecutive Wick-contractions with k fields. This partition is related to Third, we study the space of external Wick-contractions. One definition of this space has already been given in (4.27) as is the external Wick-contractions of length k. We sum V k over the space (4.50) The restricted space of external Wick-contractions is defined as the union of {V k }, This definition of V still contains two types of unphysical Wick-contractions. The first type is self-contractions, which can be removed by h AB . The second type is mutually adjacent pairs of open two-points. As discussed at the beginning of this section, a skeleton graph should not have faces made of two or fewer edges. As a result, the symbol Φ (i) p represents the maximal sequence of consecutive fields appearing in the original Wick-contractions. This condition is equivalent to the exclusion of taking external Wick-contractions from the mutually adjacent pairs. For example, when (Φ (i) p , Φ (j) q ) are contracted, then the four possibilities (Φ ) are not allowed. We define V phys as the subspace of V in (4.51) having no mutually adjacent pairs. A necessary but not sufficient condition for getting V phys ⊂ V is to require that the faces has no two-cycles, 21 RHS does not have adjacent ladder-type skeleton edges, but still contains adjacent skeleton edges which mutually cross and do not form a two-cycle. We need to throw away external Wickcontractions of this type.
As discussed in Section 4.2, there is a bijection between the original Wick-contraction and W and the triple {O i }, V, τ . The spaces (4.45), (4.52) and (4.48) offer an alternative definition of the space of (physical) Wick-contractions given in (2.17). We write the n-point functions (4.31) as The symbol G n in (4.53) means that the disconnected parts involving a two-point function are removed. This is because the skeleton reduction of a two-point function gives a face permutation consisting of a two-cycle. We excluded such terms in (4.52).

Face-based skeleton formula
We discuss how to reconstruct G n or G n of single-trace operators from the graph data, following our discussion in Section 2.3. In (4.53) we fixedᾱ and take a sum over V . Below we fix V and sum over the face permutation ν of a skeleton graph. We also need the set of internal Wick-contractions τ to recover the full n-point function.

Reconstruction of correlators
To begin with, we choose a partition = ( 1 , 2 , . . . , L) L. (4.55) We needL ≥ n/2 for general n-point functions, andL ≥ n − 1 for connected n-point. We impose the ordering  Since there are r k skeleton edges with length k from (4.46), the symmetry of relabeling is . These graphs are same as in Figure 10.
Due to the redundancy of relabeling, ν is in one-to-one correspondence with the coset S ×× 2L /Aut V . The correspondence (4.59) can be proven as in Section 2.3. Following (4.52) we define the space (S ×× 2L ) phys as the generalization of (4.59) Skeleton graphs with 2L unlabeled vertices ↔ ν ∈ S ×× 2L /Aut V phys (4.61) where LHS involve non-planar consecutive Wick-contractions as. Unfortunately, we do not find a simple algebraic characterization of RHS. The terms ν / ∈ (S ×× 2L ) phys should be excluded in order to avoid double-counting.
We setᾱ • ≡ νV • ∈ S 2L and useᾱ • to make all skeleton-rims oriented. We denote the cycle decomposition ofᾱ • byᾱ The ladder-skeleton graph is made by assembling the planar, ladder-type consecutive Wick-contractions.
which should count the number of scalars in O i . We impose the conditions that graph data should match those of physical data, These conditions determine the cycle type of α = i α i ∈ S 2L . Thus, the equations (4.64) solved the constraint δ cycle-type (α −1 ex , ωW • ) used in (2.59). Figure 13 shows an example of the skeleton reduction using the notation introduced here.
Next, we generate a set of ordered lists from the permutation α • like (2.45). In (2.46) we defined the map CL as the composition of the cyclic translations and the permutations of the cycles of the same length. Here we consider the cyclic translations and permutations separately.
Let us define the map from a permutation to a set of the reduced lists, denoted by CR. This consists of the permutations of the operators with the same dimensions ∆(ᾱ i ) = ∆(ᾱ j ). For example, (4.65) The map CR depends only on the operator dimensions ∆(ᾱ i ) = ∆(ᾱ j ), and they may haveL i =L j . If we denote the cycle type of α by λ = [1 λ 1 2 λ 2 . . . ] 2L, then The cyclic translations will be discussed later.
Choose an elementβ = (β 1 , . . . ,β n ) from CR(ᾱ • ), and use it to relate the rim labels to the flavor-like indicesβ We can better understand this notation by introducing the extended reduced operator where the length of the sequential fields Φ (i) r is found fromβ −1 i (r) ∈ {1 1 , 1 1 , . . . ,L L ,L L }. The total number of scalar fields inÔ i is given by The reduced Wick-contraction matrixμβ is one-to-one correspondence with the external Wickcontraction of the reduced operators. For example, we find We introduce the flavor factor for the E-th edge by which agrees with the flavor factor in (4.23). We generalize h as (4.73) like (2.55). We define the flavor function ofμβ by where RHS is generalization of (2.22). Now we consider the cyclic translations Z α = i Z L i . In fact, although the formula (4.74) computes the flavor factor for a set of reduced operators O i , we did not specify the relation between This symmetry was also part of the stabilizer (2.48). We write F(τ |z ·μβ) ≡L s=1 z · h τ μ 1,sμ1,s . . .μ n,sμn,s , (z ∈ Z α ) (4.77) and take a sum over the entire orbit of Z α .
Finally we define the function on the skeleton-faces by We count the powers of N c in the same way as (4.34) in Section 4.3. Namely, we compute θ = θ(τ ) in (4.37) and count the number of cycles inω = νθ. The non-planarity inside open two-points also contribute throughZ(τ ) in (4.54).
The n-point function of single-trace operators is written as where the sums over and τ are given in (4.46) and (4.48), respectively. We inserted two types of δ-functions from (4.64). We sum the face permutation ν over the range (4.61). We used G n as in (4.53) because the last procedure removes two-point functions from G n .
The fact that the powers of N c in (4.78) depend on the internal data τ causes stratification, namely the skeleton graph of smaller genera contribute to the 1/N c corrections to the n-point functions at genus g * [8,9,26].

Sum over graphs
We simplify (4.79) further in two steps. The first step is to take the connected part of n-point functions. This makes the subtle difference between G n and G n unimportant, The second step is to interchange the order of summation in (4.79). Previously, we fixed the bridge length distribution L first, and summed over the skeleton graph with the edge lengths { 1 , 2 , . . . , L}. Now we fix the skeleton graph first, and distribute the bridge lengths later. Let us define a set of skeleton graphs with a given set of dimensions by SG(L 1 , L 2 , . . . , L n ) = Skeleton graphs with n vertices ∆(i-th vertex) = L i . The n-point function (4.79) becomes z · h τ μ 1,sμ1,s . . .μ n,sμn,s (4.84) whereω = Face(Γ) · θ(τ ). The second sum of (4.83) is taken over the space of assigning a length to each edge of Γ, The first sum of (4.83) is taken over labeled skeleton graphs. 23 Note that we do distinguish the vertices having the same dimensions, and hence removed the sum overβ ∈ CR in (4.84). Recall that the sum over CR was needed in (4.66), because in the group-theory language a permutation α • cannot specify an ordered list uniquely. In the graph-theory language, we may distinguish the i-th and j-th vertices even if L i = L j .
Let us take the connected part of the n-point function and rewrite (G n ) connected further. We decompose the space SG({L i }) by the genus of a graph Γ, where Γ( ) is the skeleton graph whose bridge lengths are specified. Note that g is a parameter of Γ( ) which is not the powers of N c in (G n ) connected due to the stratification.
The skeleton graph with specific bridge lengths describing the connected n-point function can be identified as the so-called connected metric ribbon graphs. Using this connection, we will study the properties of M gauge g,n ({L i }) in Section 5.

Geometry from permutations
We start our argument in an abstract way by looking at the equivalence between the space of connected metric ribbon graphs and the decorated moduli space. Then we define a subset of these spaces by imposing three conditions. We conjecture that this subspace is equal to the gauge theory moduli space M gauge g,n ({L i }) introduced above, and discuss some examples.

Ribbon graph and Riemann surface
As explained in Introduction and Appendix C.3, there is one-to-one correspondence between the space of connected metric ribbon graphs (CMRG) with the decorated moduli space of Riemann surfaces, CMRG g,n = M g,n × R n + . (5.1) In this equality, a graph defines a cell decomposition of a Riemann surface, Σ g,n = n i=1 R i . Each region R i contains a puncture of Σ g,n , and two regions are separated by the edges of the graph. The edge lengths of the graph { E } specify the complex structure of Σ g,n .
We denote the sum of edge lengths around the i-th puncture by On Σ g,n , LHS is a cycle integral of a differential around the i-th puncture, and RHS is the residue. In fact, for any {p 1 , p 2 , . . . , p n } ∈ R n + , there exists a unique quadratic differential ϕ having the residue p i at the i-th puncture, so-called the Jenkins-Strebel (JS) differential. The critical graph of ϕ is isomorphic to the metric ribbon graph Γ g,n . See reviews [46][47][48] for details.
Let us inspect the properties of ribbon graphs, constrained by the corresponding Riemann surfaces. If a ribbon graph completely triangulates a Riemann surface as Σ g,n = n i=1 R i , then the number of vertices, edges and faces are given by Incomplete triangulations also constrains the graph data, as discussed in Appendix C.2. If we focus on all possible ways of distributing lengths (a positive number) to each edge of the graph, we find CMRG g,n R 3n−6+6g which is consistent with dim R M g,n = 2n − 6 + 6g. (5.5) This equation has been made precise in [17], where the integration measure of M g,n is expressed by the edge length differentials {d 1 , d 2 , . . . , d 3n−6+6g } under the constraints (5.2).
By definition, the space CMRG g,n splits into the metric part and the ribbon graph part as CMRG g,n = Choice of { E > 0} × Choice of ribbon graph Γ g,n .

Constraints from gauge theory
We impose three constraints on M g,n to select the gauge theory moduli space M gauge g,n M g,n .
The first condition is integrality, E ∈ Z + . The residue condition (5.2) becomes which is equivalent to (4.15).
The second condition is the absence of self-contractions, ii = 0. As a corollary, we obtain the extremality condition j =i L j ≥ L i for any i. This can be shown by which is non-negative if ii = 0. Indeed, the correlator of scalar operators in gauge theory vanishes when the extremality condition is violated. The situation was different in matrix models, because self-contractions are allowed and ii can be non-zero.
The above conditions also constrain the complex structure, which can be argued in two ways. One argument goes as follows. Since the complex structure is induced by decoration in the metric ribbon graph method, any inequality on the decoration parameters selects a proper subset of the moduli space. Thus we find CMRG g,n = M g,n × Z n + = M g,n × (Z n + ) (5.10) where (Z n + ) is the subspace of Z n + satisfying the generalized triangular inequality, (Z n + ) = (L 1 , L 2 , . . . , L n ) ∈ Z n + j =i L j ≥ L i (i = 1, 2, . . . , n) .

(5.11)
The moduli space without self-contractions M g,n is a subspace of M g,n according to (5.9). Another argument is based on the fact that Feynman graphs are dual to the critical graphs on Riemann surfaces, as discussed in Appendix C.5. In general, the critical graph of Riemann surface may contain self-edges [17]. The self-edge is a boundary of the same face as shown in Figure 14. The gauge theory moduli space M gauge g,n should not contain self-edges, because they correspond to the self-contractions in the dual Feynman graph [29].
The third condition is discrete homotopy, which is stronger than the usual homotopy in classifying edges. In classical geometry, we call two edges are homotopic if they end on the same vertices and if they wind various one-cycles on Σ g,n in the same way. In our skeleton reduction, two edges are discrete-homotopic as long as they are consecutive and connect the same pair of operators, even if they cross each other. This procedure selects a subspace of the moduli space as M ,disc g,n ⊂ M g,n ⊂ M g,n .
(5.12) Figure 14: Examples of the connected metric ribbon graphs with (g, n) = (0, 3) (above) and (0, 4) (below). The diagrams on the right half have self-edges, and some diagrams on the left half are extremal.
Importantly, the discrete homotopy induces a special type of triangulation of Riemann surface.
Since the discrete homotopy is stronger than the usual homotopy, our triangulation is simpler than those used in [8,9]. 24 We conjecture that the restricted moduli space is equal to the gauge theory moduli space defined by the skeleton-reduced n-point formula When some of E become zero, then vertices collide and the graph topology changes. In the literature, the classification of M gauge g,n by graph topology is called cell decomposition.
Consider the second factor of (5.14) in detail. Let us denote the valency set of the skeleton graph by d = [3 d 3 4 d 4 . . . ], where d k is the number of k-valent vertices. From Appendix C.2, we find that trivalent skeleton graphs have d = [3 2n−4+4g ], and the extremal skeleton graph with one vertex has d = [(2n − 2 + 4g) 1 ]. The valency set of general skeleton graphs at (g, n) is in between, This pattern also constrains the number of edges in the skeleton graphs, denoted byL in Section 4. The trivalent graphs haveL = 3n − 6 + 6g and the extremal graph has n − 1 + 2g. The number of edges in general skeleton graphs is L ∈ L g,n = 3n − 6 + 6g, 3n − 7 + 6g, . . . , n + 2g, n − 1 + 2g .
Generally there is no correspondence between the entries in (5.16) and (5.15). As shown in Figure  14, the edges of a non-trivalent graph may end on the same vertex or different vertices, which changes the number of edges.
Classical limit of M gauge g,n The classical limit is defined by taking all bridge lengths to be large E 1 with g, n fixed, for the ribbon graphs whose faces are all triangles. We can neglect the quantization condition (5.8) in this limit. The second line of (5.14) is still a finite set, because the graph data are constrained by (g, n) as in ( We expect that the generalized triangular inequality (5.11) disappears in more general setups. For example, when we include the covariant derivatives, then there are non-zero three-point functions with ∆ 1 > ∆ 2 + ∆ 3 due to the Laplacian n . If we write ∆ = τ * s + s + 2n and study the minimal twist τ min , then a triangular inequality was found in [49].

Examples
We compute the connected n-point functions (G n ) connected ({L i }) for small L = 1 2 i L i and recast them into the form of (4.87). This procedure explicitly determines the space of reduced Wickcontractions M gauge g,n ({L i }) for simple cases. We expand (G n ) connected in 1/N c and use ξ = 0, 1, 2, . . . to count the powers of N c , where ξ = 0 corresponds to the planar term. For a given {L i }, we define N • ξ,λ ({L i }) = Feynman (or ribbon) graphs of genus ξ whose skeleton reduction has the face permutation and of cycle type λ (5.18) Different elements in N • ξ,λ have different cyclic orderings in (5.14), or different τ . We introduce which roughly counts the number of skeleton graphs whose face permutation of cycle type λ. This counting is not precise, because the action of Z α on Feynman graphs may have non-trivial fixed points, as discussed in Appendix D. We write

(5.20)
It will turn out that the lists N ξ are perfectly consistent with our earlier classification (5.15). If g represents the genus of the ribbon graph, ν ξ may contain graphs of smaller genera, ν g with g < ξ, owing to the stratification.     In general, the planar three-point functions have the unique skeleton graph. This is because the bridge lengths are completely fixed by the external operators {i, j, k} ∈ {1, 2, 3} (5.24) and the cyclic orderings are fixed uniquely. Thus we have up to the change of graph topology in the extremal limit.
Another interesting observation is that the skeleton graph of (G 3 ) connected (L, L, 2) is unique irrespective of the genus. By taking first the Wick-contraction of O 3 with O 1 , O 2 , we find that the remaining Wick-contractions are same as the two-point function of length L − 1, whose graph topology is trivial.
Four-point functions. We consider three examples, (L 1 , L 2 , L 3 , L 4 ) = (3, 3, 3, 3), (6, 2, 2, 2) and (6,4,2,2). For the first example, the list of the cycle type of face permutations is N 0 = 6 [4 2 ] + 2 [3 4  The planar four-point functions have several skeleton graphs. The moduli space can be written as The first factor comes from the sum over bridge lengths given by where p, q can be any number owing to the cyclic symmetry. Then, the subsequent color index

Metric ribbon graphs in string theory
We briefly discuss how metric ribbon graphs show up in string theory.
In the light-cone SFT, closed string worldsheet is characterized by the Giddings-Wolpert quadratic differential [50], whose analytic structure is classified by Nakamura graphs [51]. This situation is in parallel with the JS differential and its critical graph. The moduli space of light-cone worldsheet theory can be regarded as the space of Wick-contractions in the Hermitian matrix model [52], described by permutations [53]. The light-cone and conformal methods differ in many ways. For example, Nakamura graphs consist of quadrivalent vertices (four-point interactions) only. Furthermore, the Giddings-Wolpert differential introduces a global time coordinate on Σ g,n , whereas the JS differential does not have such quantity.
In conformal gauge, we argue that the metric ribbon graphs arise from the analytic properties of the classical energy-momentum tensor, and characterize the permutation of Stokes sectors of string worldsheet on AdS 5 × S 5 .
As discussed in Appendix C, the JS quadratic differential dual to a Feynman diagram of n-point functions has the analytic structure It also has 2n − 4 + 4g simple zeroes somewhere on Σ g,n . This behavior is similar to that of the classical worldsheet energy-momentum tensor Here {κ i } is related to the conformal dimensions of vertex operators. If we interpret T (z) as the "JS" differential, then the parameters {κ i } determine the complex structure of the underlying Riemann surface. Physically, we may replace the Strebel condition by other conditions like the equations of motion, as long as the external parameters uniquely fix the quadratic differential and its critical graph. 26 In AdS/CFT, the three-point functions of heavy operators in N = 4 SYM at strong coupling have been studied by Pohlmeyer reduction [55][56][57][58], borrowing the techniques developed for the gluon scattering [59][60][61][62]. The Pohlmeyer reduction on S 3 gives us the Hitchin system, The SL(2, C) flat connections (B z , Bz) can be written as [57] 26 Non-Strebel critical graphs are considered in [54].
where T (z),T (z) are the worldsheet energy-momentum tensors, ζ = 1−x 1+x is the spectral parameter, and (γ, ρ,ρ) are the SO(4)-invariant degrees of freedom after the Pohlmeyer reduction. The flatness condition (6.3) is equivalent to the compatibility of the auxiliary linear problem (∂ + B z ) ψ = ∂ + Bz ψ = 0. (6.5) These have two linearly independent solutions, called small and large depending on the asymptotics of |z| → ∞ with arg (z) fixed. When we rotate z the asymptotic behavior of the solutions changes. The region of the common asymptotics is called the Stokes sector.
In the semi-classical analysis, we uniformize the worldsheet by dw = T (z) dz, which introduces branch points at the simple zeroes of T (z). 27 If we regard T (z) as a quadratic differential on Σ g,n , then the number of simple zeroes in T (z) is given by (C.6). We can draw a metric ribbon graph by connecting the zeroes and the poles of T (z). This graph knows how Stokes sectors are permuted around the branch points. A similar argument has been done in the light-cone string theory in [52,53].

Conclusion and Outlook
We studied the general n-point functions of scalar multi-trace operators in N = 4 SYM with U (N c ) gauge group at tree-level. We obtained permutation-based formulae valid for any n and to all orders of 1/N c . The edge-based formula is interpreted as the topological partition function on Σ 0,n with defects, which naturally decomposes into pairs of pants. We applied a new skeleton reduction to find another set of formulae. The skeleton-reduced Feynman graphs generate metric ribbon graphs, which form a subset of the moduli space of complex structure of Riemann surfaces. Our skeleton reduction stratifies the moduli space with respect to the genus, whose top component is simpler than the usual skeleton reduction.
We find open/closed duality from the n-point functions of gauge-invariant operators. The pants decomposition resembles closed-string interaction, while the triangulation of Σ g,n through the skeleton reduction resembles open-string interaction.
This work hopefully initiates several future directions of research, which we want to revisit in the near future.
The first direction is to generalize our formulae into the full sector of N = 4 SYM, and to include the g YM loop corrections. Perturbatively, the loop corrections are computed by taking the OPE with the insertion of interaction terms in the Lagrangian, and integrating over the insertion point in spacetime. The integrand of n-point functions at -loop can be rewritten roughly as the (n + )-point function at tree-level. Pants decomposition in Section 3 may be useful to deal with such quantities.
The second direction is to compare our results with the BPS correlators derived by the S 4 localization [63][64][65]. Whether one can recast the generating function of BPS correlators as the τ -function of an integrable system is an interesting question.
The third direction is to apply our formalism to the determinant-like operators, or equivalently the LLM geometry [66]. It has recently been conjectured that certain states on the LLM backgrounds are isomorphic to those of N = 4 SYM, and the correlators are related by the redefinition of N c [67]. This conjecture may be checked by rewriting our results in an appropriate representation basis.
The last direction is to employ our Wick-contractions techniques for studying integrable systems in the free-field representation [68,69]. The skeleton reduction in Section 4 is analogous to the hexagon program [7][8][9]. Then, the pants decomposition in Section 3 may give a hint on the threepoint functions or related form-factors in terms of Quantum Spectral Curve along the line of [70,71].

A Notation
We denote an indexed set α 1 , α 2 , . . . collectively by {α i }. A multiple sum or product is written as (A.1)

A.1 Permutations
A cyclic permutation is denoted by (i 1 i 2 . . . i ), and (i)(j) . . . represents an identity element. 28 We define the permutation action by as represented in Figure 22. We find the identity In Mathematica, the group multiplication α · β is implemented by PermutationProduct[β,α].
The δ-function on the symmetric group is defined by (A.5) 28 Usually we do not put commas inside the bracket for permutation cycles. Often we use [α] to signify the cycle type of α.
The wreath product group is denoted by S L [G], which is a composition of G ⊗L and S L permuting the order of the copies of G's. The order of this group is given by (A.10) For example, the action of S L [Z 2 ] on L pairs is generated by The stabilizer subgroup is denoted by Here g(x) can be any group action, such as g(x) = gxg −1 . By the orbit-stabilizer theorem, the number of inequivalent elements of the group coset is given by |G|/|Stab G (x)|.

A.2 Permutation basis of operators
We express a multi-trace operator of length L as the equivalence class in the permutation group S L . Let O A α be a scalar multi-trace operator, Here α ∈ S L defines the color structure of O A α . If α ∈ Z L , then O A α becomes a single-trace operator, usually denoted by For later purposes, we introduce another notation for the same operator by The δ-function prefactor can be represented by another permutation We can reorder the flavor indices in RHS of (A.13), which leads to the gauge symmetry, If we neglect the spacetime dependence, the scalar fields obey the U (N c ) Wick-contraction rule, whereg AB = δ AB if Φ A is a real scalar of N = 4 SYM. If the gauge group is SU (N c ), we replace the scalar fields of SU (N c ) theory by those of U (N c ) theory as Let us write the two-point functions of scalar operators of N = 4 SYM as The RHS is given by the sum over all possible Wick-contractions, . (A.21) The product over the color indices can be simplified by relabeling, where we used α 2 σ −1 (p) = σ −1 (r). This gives where the δ-function is given in (A.5). The symbol Ω is an element of the group algebra C[S L ] where C(ω) is in (A.8). Thanks to the property C(ω) = C(γ ω γ −1 ), the Ω is a class function

A.3 Feynman diagrams in the double-line notation
The space of Wick-contractions W represents the sum over all Feynman diagrams. In the doubleline notation, Feynman diagrams become ribbon graphs owing to the cyclic ordering at each vertex. The graph data of vertices, edges, and faces uniquely specifies a ribbon graph. Thus we introduce a triple of permutations to represent the graph data: 29 (Vertices, Edges, Faces) ↔ (α, W, ω) ∈ (S 2L , Z ⊗L 2 , S 2L ), ω = αW. (A.26) Here the vertex permutation α knows the color structure of external operators {O i }, and the face permutation ω knows the color-index loops. By using permutations, we can represent the Feynman graph as the Cayley graph of the composition of (α, W ). 30 The Feynman graphs in the double-line notation can be understood as a Cayley graph in the following way. We rewrite the permutation basis of operators in the double-index notation as in (A.15). Now a scalar field carries one flavor half-edge and two color half-edges. All half-edges are paired by the pair of involutions (ι α , W ) with ι α = n i=1 ι α i in (A.16). The Cayley graph of the composition of (ι α , W ) is the usual Feynman graph in the double-line notation; see Figure 23.
The Feynman graphs in the double-line notation have the ribbon graph structure. A ribbon graph is a graph with cyclic ordering at each vertex, which is represented by α or ω in our setup.

A.4 Summary of skeleton reduction
Let us present an overview of our notation in Section 4, where we introduced various different symbols for making our argument straightforward. 29 The permutations (α, W, ω) are denoted by (σ 0 , σ 1 , σ ∞ ) in [36]. 30 The Cayley graph Γ(G) is a colored directed graph, defined canonically from the generators of a finite group G. Figure 23: (Left) Usual Feynman graph as the Cayley graph of (α, W ). (Right) The Feynman graph in the double-line notation as the Cayley graph of (ι α , W ), representing the Wick-contraction The number of all Wick-contractions between O i and O j is denoted by { ij } in (1.3). We introduce two more symbols as the refinement of { ij } in (4.15) and (4.38). The first one gives a partition of ij where each segment represents the consecutive sets of Wick-edges between O i and O j , The second one defines a partition of L i and of the operator O i , ij is illustrated in Figure 10, and another example of i is given in Figure 12.
We call O i a reduced operator, and Φ (i) p a sequence or sequential fields. Each Φ (i) p consists of p consecutive fields. There are many ways to create a reduced operator from O i . We call {O i } a partition of O i .
We impose the following rule for a pair of sequential fields, which we call an open two-point function. Using the δ-functions of color indices in the open twopoint function, we define two functions R and Z as The properties of these functions are studied at the end of Section 4.1.
After the skeleton reduction, the Wick-contractions are taken in two steps. The first step is to take all pairs of sequential fields (Φ (i) p , Φ (j) q ), and the second step is to take the sum inside open two-points. Correspondingly, the pairing map W ∈ Z ⊗L The skeleton-edges can be represented in two notations, P = 1, 2, . . . , 2L and E = 1, 2, . . . ,L. The symbol E is rewriting of (ij, ρ), The symbol P is a double copy of E. For example, we find the identity

B Another face-based formula
In Section 2.3 we derived a face-based formula of n-point functions. Below we discuss another formula by using the rims labeled by α, as illustrated in Figure 3. It turns out that the constraint δ 2L (α −1 ωW ) disappears in this labeling, but another constraint must be imposed.
Let us denote the cycle type of ω by [ω] = [1 w 1 2 w 2 . . . ] 2L, The Feynman graph has w f faces with f edges, and C(ω) color-index loops. We have w 1 = 0 because the self-contractions are forbidden. Each face defines a cyclic translation by going around the boundary of the face counterclockwise. We can parametrize ω as a product of cyclic permutations, or using {w f } as The face permutation ω written as sequences of rims define Wick-contractions. For example, In general, we introduce the function on faces by and write n-point functions as G n = ω F(ω).
Let us sum over ω in two steps. The first sum is over the cycle type w 2L and the second sum over ω's at a fixed cycle type. The latter sum can be generated by where ω 0 is a fixed (or representative) permutation with the cycle type w. The division by Aut(ω 0 ) removes the redundancy of relabeling in (B.3).
the square-root on h AB in (B.5) remains unresolved. These two sequences produce the same flavor factor √ h ApAq . Thus, the consistency condition (B.7) is equivalent to the square-root-free condition; namely, the total product Based on the above arguments, we conjecture that the n-point function is given by the sum over ω ∈ W(w) which does not contain √ h's: (B.9) We checked that this equation is consistent with the original S 2L formula (2.18), by computing some simple cases by Mathematica.
C More on geometry and graphs C.1 Moduli space In the literature there are various definitions of the moduli space of Riemann surfaces denoted by M g,n ; • Space of the complex structure on Σ g,n • Space of the conformal class of Riemannian metric g ∼ e 2ρ g on Σ g,n • Space of hyperbolic metrics modulo mapping class groups for g ≥ 2 [72] • Space of smooth or stable curves up to isomorphism [73] • · · · The moduli space M g,n has the dimensions dim C M g,n = n − 3 + 3g (C.1) which is parametrized locally by One way to identify {µ a } is to take pants decomposition by cutting a genus g curve (3g − 3) times, and use Fenchel-Nielsen coordinates to fix the moduli [25]. The Riemann surface degenerates on the boundary of M g,n .
In general, it is not easy to construct the whole M g,n explicitly, because M g,n is not compact. When one approaches the boundary of M g,n , the Riemann surface Σ g,n degenerates, either by pinching the handle or by colliding punctures. Deligne and Mumford considered compactification of M g,n by adding boundary components in the language of algebraic geometry [24].
Harer, Mumford and Thurston suggested to study the moduli space of decorated Riemann surfaces M g,n × R n + instead of M g,n [20]. By decorated we mean that the complex structure is induced by the quadratic differential having the double pole with the specific residue r i ∈ R + at the puncture p i . This idea works if there is a bijection between the complex structure and the quadratic differential on Σ g,n . This is indeed true for the Jenkins-Strebel quadratic differential discussed in Appendix C.3.

C.2 Ribbon graphs
A graph is defined as a set of vertices and edges, and we usually talk of connected unoriented graphs whose vertices are labeled. A metric ribbon graph is a graph with the following property: 1. Each vertex has valency at least three, 2. Edges connecting to a vertex are cyclically ordered, 3. A positive real number called length is assigned to each edge.
Any graphs drawn on a Riemann surface without intersection can be ribbonized [47]. 31 For this purpose, we replace vertices by disks, edges by strips, and glue them together while keeping the cyclic ordering.
A topological cell decomposition of Σ g,n defines a ribbon graph, by identifying the punctures as the vertices of the graph. The cell decomposition is called complete (or ideal) triangulation if all faces of the graph are triangles. If some faces are squares or higher polygons, then the triangulation is called incomplete.
Consider a graph which completely triangulates Σ g,n . The number of vertices, edges and faces of the graph is given by V g,n = 2n − 4 + 4g, E g,n = 3n − 6 + 6g, F g,n = n, χ = V g,n − E g,n + F g,n = 2 − 2g. (C.3) For example, a planar graph can have at most 3n − 6 non-zero edges, and a toric (genus-one) graph can have 3n non-zero edges. A graph is called complete if any pairs of faces must be connected by a single edge. A complete graph exists when namely, n ≤ 4 at g = 0 and n ≤ 7 at g = 1. 31 A ribbon graph is also called a fat graph in the literature.
2. φ has a double pole at {p i } 3. All horizontal trajectories are compact, except for measure zero subsets of Σ g,n 4. Each compact trajectory surrounds one of the poles, and the residue at p j satisfies The branch of the square root is chosen such that r j > 0 for all j. 32 By definition, a JS differential does not have poles of order > 2. We define (Order of ϕ) = It is known that the order of ϕ for Σ g,n is given by 33 (Order of ϕ) = 4g − 4. (C.14) In particular, if ϕ has only simple zeroes and double poles, For a given JS differential ϕ, we define its critical graph Γ as the collection of all horizontal trajectories. Each zero of ϕ is a vertex of Γ, and each pole of ϕ is a face of Γ. Suppose w is a local coordinate around the zero or pole of ϕ. From 16) we find that a simple zero of ϕ gives a trivalent vertex of Γ, while a face of Γ can be surrounded by any number of edges.
The critical graph Γ has the structure of metric ribbon graphs. Γ is a ribbon graph because the graph is drawn on an oriented Riemann surface, and Γ is a metric graph because we measure the length of the edges by (C.10). The structure of Γ as a metric ribbon graph is fixed by ϕ, which implies that a metric ribbon graph specifies a point of the moduli space M g,n . The metric ribbon graphs classify other structures as well, such as the (decorated) Teichmüller space [75] and the projective structure [76].
Example. The JS differential of Σ 0,3 takes the form Figure 24 shows the horizontal trajectories for some (r 0 , r 1 , r ∞ ). When (r 0 , r 1 , r ∞ ) are not identical, the distance between two zeroes (trivalent vertices) depends on the path one chooses. In particular, there must be a branch cut somewhere on Σ g,n .  Critical graphs may have self-edges as shown in Figure 24. If we denote the length of the edges between zeroes of φ on Σ 0,3 as ( 1 , 2 , 3 ), then the self-edges show up if three edges do not satisfy the triangular inequality i + j > k for any (i, j, k).
Some examples of the JS differential of genus-one surfaces are given in [22]. In general, we need to solve transcendental equations to find the explicit JS differential of given lengths [77].

C.4 Drawing bipartite ribbon graphs
According to (5.3), any graph on Σ g,n (or ideal triangulation of Σ g,n ) has a fixed number of vertices, edges and faces when all vertices are trivalent. From such a graph, we derive a bipartite graph, by drawing a white circle for the face and a black dot for the vertex of the original graph. The white circles and black dots should correspond to the double poles and simple zeroes of the Jenkins-Strebel differential on Σ g,n .
Consider examples of the ribbon graph obtained by a triangulation of Riemann surface. Planar bipartite graphs are shown in Figures 25 -26, and non-planar graphs in Figures 27 -28. Figure 27 shows the bipartite graph from a triangulation of Σ 1,3 . This is a toric graph with periodic boundary conditions in both horizontal and vertical directions. The fundamental region of the bipartite graph contains 6 black dots, 9 edges and 3 white circles, in agreement with (5.3). Other examples were given in Figure 18. These graphs have the shape of a honeycomb lattice. We can draw similar toric honeycomb lattices for general n, satisfying the constraints (5.3).     Beware that the case of (g, n) = (1, 2) is singular. The equation (5.3) says that we should have four trivalent vertices. However, we have only "bivalent" vertices because there are only two punctures. One solution is to introduce fictitious punctures [7]. Another solution is to use two quadrivalent vertices.

C.5 Feynman graphs on a Riemann surface
Given a bipartite graph in Appendix C.4, we can draw two more ribbon graphs, called black and white. The black graphs are defined by adding edges between the pair of black nodes for each square of the bipartite graph, and then removing all white nodes and the edges connected to them. The white graphs are defined by the same procedure, interchanging black and white. The black and white graphs are dual to each other, as shown in Figure 29.
The black graph represents the complex structure of a Riemann surface. The white graph represents the Feynman diagram appearing in the connected n-point function of single-trace operators. Thus, the critical graph on a Riemann surface is the dual graph of a Feynman diagram of a gauge theory like N = 4 SYM. As a Feynman graph, the white circles represent external operators and black dots represent effective three-point interactions. 34 The equation (5.3) constrains the graph data of a white graph as V • g,n = n, E • g,n = 3n − 6 + 6g, F • g,n = 2n − 4 + 4g.
(C. 19) Recall that any Feynman diagrams in the double-line notation can be drawn on a Riemann surface. 35 where the last equation comes from (C.6). These equations have the unique solution This value of C(ω) remains invariant even if we consider more general [ω] made by gluing triangles under the constraint (C.6). Hence, the formula (C.21) suggests that higher genus corrections to any n-point functions must come in powers of 1/N 2 c , rather than 1/N c . 36 34 They are not three-point interactions in the N = 4 SYM Lagrangian, because we talk of tree-level correlators. 35 In other words, the skeleton reduction is unimportant if we are interested in Σ g,n rather than M g,n . 36 The 1/N c terms appear when the gauge group is SO or Sp [78][79][80][81]. Also, if the gauge theory has the object in the fundamental representation such as quarks and Wilson loops, the Riemann surface has boundaries [82]. Suppose that a ribbon graph completely triangulates Σ g,n . Then, any edge is the boundary of two triangles. Let us make a square by gluing the two triangles, and study the four edges { a , b , c , d } of the square. We define the shift map s E on the E-th edge as which is also shown in Figure 30. Since there are no more relations among s E , the number of linearly-independent simultaneous shift maps is equal to 2n − 6 + 6g. This is equal to dim R M g,n in (5.5), which implies (5.17).

D Details of the skeleton reduction
We explain how to compute the skeleton reduction of G n in order to find M gauge g,n ({L i }). The attached Mathematica files implement the procedures given below.
We need a few preliminary steps. To begin with, we fix {O i } and compute the connected n-point functions of single-trace operators in the standard way, e.g. [83]. Let us write the result as a function acting on a set of Feynman graphs {Γ}, (D.1) Since (G n ) connected is long and complicated, we study its cyclic decomposition, Here the action z · Γ 0 is given in (4.76). The division by |Aut Zα Γ 0 | is needed to avoid double counting. For each term of G n , we can extract W ∈ Z ⊗L 2 in the notation of Section 2.1. Then we change the labeling so that W becomes W = L p=1 (p p ). Using the new labeling we compute ω = αW for each set of Wick-contractions.
In order to apply the skeleton reduction, we need to determine the end-points of open two-point functions. Once we know the open end-points, we can decompose W into (V, τ ). We also obtain There are two methods to determine the open end-points. The first method is intuitive but slow. The second method needs detailed case-studies, but efficient particularly when L is small.

(D.4)
A pair of edges can be multiply adjacent. Let us call a pair of sequences of consecutive fields between two operators an Wick-edge group. A Wick-edge group of length consists of a pair of consecutive indices which are Wick-contracted within themselves, E = p, α(p), α 2 (p), . . . , α −1 (p) , q, α(q), α 2 (q), . . . , α −1 (q) any α m (p) for 0 ≤ m < is Wick-contracted with some α n (q) for 0 ≤ n < . (D.5) We also use the notation E = [p 1 p 2 . . . p ] using edge labels. We look for the maximal Wick-edge group containing p, assuming that p connects O ip and O jp . Figure 31 shows examples. We study the other side of Wick-contractions starting from n ∈ R 1 . Since both p and W α n (p ) belong to O ip , we can define an integer u n such that W α n (p ) = α un (p). Similarly, we can define W α n (p) = α u n (p ) for n ∈ R 1 . They are summarized as W α n (p ), α n (p ) ≡ α un (p), α n (p ) = α un (p), W α un (p) , (n ∈ R 1 ) α n (p), W α n (p) ≡ α n (p), α u n (p ) = W α u n (p ), α u n (p ) , (n ∈ R 1 ).

(D.8)
Clearly R 2 ⊂ R 1 and R 2 ⊂ R 1 . If R 2 , R 2 are the two maximal Wick-edge groups (D.5), the two sets should be identical. We call the elements with (i p j p ) = (i q j q ) diagonal parts, and the rest off-diagonal parts. From adjacency relations, we draw the associated adjacency graph. The graph topology of the diagonal part J (ij) (ij) knows how the adjacency inside the Wick-contractions between O i and O j in a cryptic way.
If J (ij) (ij) is block-diagonal with more than one blocks, then the associated adjacency graph has more than one connected components. Consider the examples. Assuming that the left and right ends of the open two-points are not periodically identified as in Figure 32 Figure 32: Open two-points and edge-adjacency matrices represented as adjacency graphs, corresponding to (D.11). Here p represents a Wick-edge. The edge (or edges) between p and q means they are (multiply) adjacent.  Figure 33: Open two-points and edge-adjacency matrices represented as adjacency graphs for periodic cases.
In contrast, if we identify the left and right ends of the above operator as in Figure  In the periodic cases, we need to identify both ends of either O i or O j as in Figure 33. In non-periodic cases, there are at most four external adjacent Wick-edges. In periodic cases, this number decreases by two (or more).
Notice that two edges go outside the open two-point in the right graphs of Figures 32 -33. We call them external Wick-edges, because they are connected to the other operators. We can count the       as follows. Suppose there is a single Wick-edge group. We divide this group by inserting an external Wick-edge. This reduces the adjacent multiplicity of two edges which belonged to the single Wick-edge group. Therefore, the sum of J  38 There are two underlying assumptions in writing this table. First, we consider only those matrices which correspond to a set (or sets) of physically possible Wick-contractions W . Second, if several sets of Wick-contractions have the same J and the same periodicity, then they have the same Wick-edge group E. If these assumptions are correct (which seems to be the case), an edge-adjacency matrix J gives the unique corresponding Wick-edge group.