Flavour singlets in gauge theory as Permutations

Gauge-invariant operators can be specified by equivalence classes of permutations. We develop this idea concretely for the singlets of the flavour group $SO(N_f)$ in $U(N_c)$ gauge theory by using Gelfand pairs and Schur-Weyl duality. The singlet operators, when specialised at $N_f =6$, belong to the scalar sector of ${\cal N}=4$ SYM. A simple formula is given for the two-point functions in the free field limit of $g_{YM}^2 =0$. The free two-point functions are shown to be equal to the partition function on a 2-complex with boundaries and a defect, in a topological field theory of permutations. The permutation equivalence classes are Fourier transformed to a representation basis which is orthogonal for the two-point functions at finite $N_c , N_f$. Counting formulae for the gauge-invariant operators are described. The one-loop mixing matrix is derived as a linear operator on the permutation equivalence classes.


Introduction
The AdS/CFT [1,2,3] correspondence has led to detailed studies of local operators in gauge theories. A remarkable success has been the discovery of integrability in planar N = 4 SYM, which allows the computation of conformal dimensions and other quantities at any value of the 't Hooft coupling [4]. In parallel, the study of AdS/CFT at finite N c , the rank of the gauge group, is making progress. The CFT duals of giant gravitons [5] are local BPS operators, which have been investigated using techniques based on permutation groups and Fourier transformation in representation theory [6,7,8,9,10,11,12,13]. A class of quarter BPS operators, showing finite N c cutoffs related to Brauer algebras, were constructed [14]. The general quarter BPS for the case where the dimension of the operators is less than N c were constructed using permutation algebras in [9,12,15,16]. Integrability of excitations around large half-BPS operators has been established [17,18,19]. A natural direction of investigation is the application of the permutation-based methods to problems in the 1/N c expansion and far from planarity. In this paper we will take a further step in this direction, studying singlet operators in the SO(6) sector made from six scalars. The study of non-planar corrections in this sector was initiated in [20].
We will focus on the sector of hermitian scalar fields in N = 4 SYM with U (N c ) gauge group, and consider "mesonic" gauge-invariant operators. The mesonic SO(6)-invariant operators form a simple closed subsector under the action of one-loop dilatation operator; namely the explicit form of the one-loop dilatation tells that there is no mixing between mesonic operators and other operators like non-scalar singlets with derivatives or "baryonic" operators. 1 It is in fact convenient to replace SO (6) by SO(N f ) and discuss the general N f theory. Various arguments simplify at large N f thanks to the absence of finite N f constraints. We obtain identities relating different ways of counting gauge-invariant operators.
There is an essential distinction between N c ≥ 2n and N c < 2n, where 2n is the operator length. For example the planar limit is a limit in the former regime. The latter regime contains interesting limits like N c 1, n ∼ O(N c ), which is related to the description of giant gravitons. Conventionally the former is called large N c and the latter is called finite N c . Likewise we mean large N f and finite N f by N f ≥ 2n and N f < 2n.
In section 2 we show that operators O α of length 2n can be parametrised by permutations α in S 2n , the symmetric group of all permutations of 2n distinct objects. Different permutations giving rise to the same mesonic operator are related by conjugation with an element γ in the wreath product subgroup S n [S 2 ] ⊂ S 2n . This group, of dimension 2 n n!, contains n copies of S 2 as well as the symmetric group S n consisting of permutations of n pairs. The two-point function of operators O α 1 O α 2 is expressed in terms of a sum of permutations in S 2n . The all-orders expansion in N f and N c is given in terms of symmetric group data such as cycle types of appropriate permutations. This wreath product group is also used in [21,22] to organise the colour structure in SO(N c ) and Sp(N c ) gauge theories.
In section 3, we show how to take a linear combination of the operators labelled by permutations to form a new basis of operators labelled by representations of S 2n , along with a group-theoretical multiplicity label. This procedure can be thought of as the Fourier transform on finite groups, which replaces permutation labels by representation labels. The two-point function for these representation-labelled operators is diagonal, as given in (3.9). The diagonal operators become null when the constraints of finite N c or N f are violated. The construction in this section is an explicit realisation for the mesonic sector of a general construction in [12]. In section 4 we take a careful look at the counting of the mesonic operators constructed. We obtain the exact counting formula using representation theory, in particular Schur-Weyl duality. The representation labels are equipped with finite N c and finite N f cut-offs, which give the correct counting for general n. We return to the language of permutations, and analyze the large N c , N f limits. In this way, we make contact with counting of the graphs via the Burnside Lemma.
In section 5 we consider a topological lattice gauge theory with a discrete gauge group S 2n of the type discussed in [23,24,25]. The counting of mesonic operators in the permutation basis, and their free-field two-point functions can be interpreted as the partition functions of this topological field theory, defined on 2-complexes with boundaries and a defect.
Finally in section 6, we derive formulae for the action of the one-loop dilatation operator acting on the operators constructed above.
In appendices we will explain our notation, collect mathematical statements and details of computation.

Key technical results
Here is a brief summary of the key technical results of this paper.
Mesonic gauge-invariant operators, composite fields made of 2n scalars (Φ a ) i j , are associated to permutations α ∈ S 2n . They are written as O α . Different permutations α related through conjugation by a permutation γ ∈ S n [S 2 ] are identical. (1.1) We denote the two-point functions of normal-ordered operators O α of length 2n by where the bracket O α 1 O α 2 contains the sum of all Wick contractions between the two operators. The two-point functions O α 1 O α 2 in the free theory will be described by an elegant formula expressed in terms of cycle structures of permutations (2.31). However, O α 1 O α 2 viewed as a function of the permutation equivalence classes, is not diagonal.
To find a basis of operators with diagonal two-point functions in the free theory, we use Fourier transformation on finite groups. Ordinary Fourier transformation can be written as where D K (θ) is the action of the U (1) group element e iθ in the representation of charge K. Fourier transformation on finite groups is given by where D R ij (g) is the matrix element of g in the representation R and i, j run over some basis of the representation, which we will choose to be an orthogonal basis. A similar transformation in the group algebra C(G), formed by linear combinations of group elements with complex coefficients, is Specializing G to the symmetric group, the irreducible representations R will correspond to Young diagrams.
In this paper, we find a new diagonal basis for the free-field two-point functions, labelled by two irreducible representations R, Λ 1 of the permutation group S 2n and a multiplicity label τ , as in (3.1). Here R corresponds to a Young diagrams with 2n boxes, Λ 1 to an even Young diagram with 2n boxes, i.e. all the row lengths are even numbers, and C(R, R, Λ 1 ) is the number of times Λ 1 appears in the irreducible decomposition of R ⊗ R. P R,Λ 1 ,τ is a linear combination of the sums over the equivalence classes of permutations which live in the group algebra C[S 2n ]. The equation (3.1) for P R,Λ 1 ,τ is a generalisation of (1.5), and the equation (1.6) for O R,Λ 1 ,τ is a generalisation of (1.4), where f (g) has been replaced by gauge-invariant polynomials in (Φ a ) i j parametrised by permutations. The coefficients in P R,Λ 1 ,τ do not depend explicitly on N c or N f , though R, Λ 1 are required to have a bound on the number of rows when N f , N c < 2n.
The number of diagonal operators (1.6) is given by (1.8) The two-point functions of (1.6) are given by (3.9), and they vanish unless the corresponding representation labels on the two operators are identical. The normalisation factor ω Λ 1 /2 (Ω (f ) 2n ) is a polynomial in N f . The wreath product group S n [S 2 ] in (1.7) also appears as a symmetry of the Kronecker delta's used for the contraction of flavour indices. The groups S 2n and S n [S 2 ] form what is called a Gelfand pair (S 2n , S n [S 2 ]). A notable property of the Gelfand pair is that the reduction of an irreducible representation of S 2n into the singlet representation of S n [S 2 ] is multiplicity-free. This property plays an essential role in our construction of the diagonal operators (1.6), which is in the trivial representation of S n [S 2 ].
The free-field two-point functions and the number of operators (1.8) are closely related, which become manifest in the large N c , N f limit. Indeed, this agreement opens up a novel interpretation as the partition function of a topological field theory with a discrete gauge group.
The representation basis highly constrains the one-loop mixing, shown in (6.11). We then establish that the elements of the mixing matrix are non-zero for operators having a pair of representation labels R, R related by the move of at most one box. This is a familiar fact from previous studies of one-loop mixing in representation bases.

Singlet operators 2.1 Mesonic operators and wreath-product permutation group
We denote a hermitian scalar field of gauge theory by Φ = (Φ a ) i j with a = 1, 2, . . . , N f and i, j = 1, 2, . . . N c . The N = 4 SYM corresponds to N f = 6. The upper gauge indices are identified with the lower gauge indices, up to an ordering parametrised by a permutation α Regarding (Φ a ) i j as matrix elements of operators Φ a : V N → V N and defining with α acting in the standard way on the tensor product V ⊗2n N . In the second equality, the trace has been abbreviated as tr 2n .
Mesonic operators are defined as gauge-invariant operators whose flavour indices are pairwise contracted. The general mesonic operator can be written by a permutation α ∈ S 2n , Generally this is a multi-trace operator, whose trace structure is given by the cycle type of α. If the cycle type of α is p, then the number of traces of O α is equal to the number of cycles in α, The flavour contractions appear in pairs (2k − 1, 2k), which is invariant under the permutation Σ 0 = (1, 2)(3, 4) · · · (2n − 1, 2n) consisting of pairwise swops. The permutation Σ 0 is invariant when conjugated by another set of permutations γ belonging to a subgroup S n [S 2 ] of S 2n This wreath product group S n [S 2 ] 2 has order 2 n n! where δ 2n (g) = 1 (g = identity ∈ S 2n ) 0 (otherwise). (2.14) Consider O α with fixed trace structure, where the cycle type of α is p in (2.5). The number of such gauge-invariants is where T p consists of permutations of cycle type p. This is also equal to where the elements of H p commute with a fixed permutation of cycle type p. A derivation of the equality of these formulae, along with counting at finite N f is in section 4.3. When N c < 2n or N f < 2n, there exist a number of linearly dependent relations among operators. They are called finite N c constraints or finite N f constraints. These constraints can be expressed in terms of Young diagrams. 3 In section 3, we will construct a set of operators with representation labels, where the finite N c and finite N f constraints are manifest.

Two-point functions
Consider the free two-point functions of the mesonic operators (2.4). Using the Wick contraction rule (A.1), we obtain Here is a simple example of a finite N c constraint. For a 2 × 2 matrix X, we have the identity We can rewrite this identity in terms of the projection operator associated with the anti-symmetric representation as In short, we cannot anti-symmetrise more than N c = 2 indices. = Figure 1: The left figure shows C 12 C 34 · · · C 2n−1,2n σ acting on V ⊗2n F . The upper and lower horizontal lines are identified when taking the trace, as in the right figure. where σ represents all possible Wick contractions, and δ a = n i=1 δ a 2i−1 a 2i . The results can be expressed by permutations, 4 (2.20) Here C(σ) is (2.6), and W (σ) is the flavour factor We have W (1) = N n f . The σ in (2.21) acts on V ⊗2n F , with matrix elements equal to Kronecker deltas. We can also write W (σ) by introducing contraction operators C 12 , C 34 · · · , where C 12 acts on the first and second tensor factors of V ⊗2n Then W (σ) = tr V ⊗2n F (C 12 C 34 C 56 · · · C 2n−1,2n σ) . (2.23) The contractions form part of the Brauer algebra which is the commutant of O(N f ) acting on V ⊗2n F , the 2n-fold tensor product of the fundamental representations. The contraction operators obey as can be seen from Figure 2. It follows that Let us denote by z(σ) the number of cycles W (σ) represented in Figure 1. Then W (σ) is given by Indeed, the function z(σ) satisfies We will show in Appendix B that it can be expressed as Since permutations can be multiplied efficiently using group theory software such as GAP or with Mathematica, this is a very useful expression for practical calculations. It does not involve explicitly doing sums over indices ranging from 1 to N f . The quantity (2.29) also has a nice mathematical meaning. It is equal to the number of cycles in the coset type of σ as explained in Appendix A.3.
For later purposes, let us define In terms of these quantities the two-point functions (2.20) have the form of

Orthogonal two-point functions
We now propose a set of operators labelled by representations, where R, Λ 1 are irreducible representations of S 2n , τ runs over 1, . . . , C(R, R, Λ 1 ) is the Clebsch-Gordan multiplicity for S 2n tensor products (see (A.21)), and the S τ,Λ1 k R i R j is the Clebsch-Gordan coefficient (A.20). The use of this group theory data in the construction of covariant bases was developed in [9,12]. The B Λ 1 k , called a branching coefficient, is defined in terms of the reduction of the irreducible representation Λ 1 of S 2n in terms of a direct sum of irreducible representations of S n [S 2 ] ⊂ S 2n . A generic state |Λ 1 , k can be expanded in terms of irreducible representations of S n [S 2 ]. The one-dimensional irreducible representation of S n [S 2 ] is known to appear in the decomposition of Λ 1 with unit multiplicity if Λ 1 is even, i.e. the partition of Λ has the form (2λ 1 , 2λ 2 , · · · ) with integers λ i . Denoting this one-dimensional subspace of Λ 1 as |Λ 1 → 1 Sn[S 2 ] , we have the branching coefficients where k runs over 1, · · · , d Λ 1 . 5 The branching coefficient is related to the singlet projector The quantity P R,Λ 1 ,τ has the symmetry for γ ∈ S n [S 2 ]. From the inverse of (3.1), [α] is expressed by Let us introduce where Λ is an even Young diagram (A.27). The function ω Λ/2 (σ) is called the zonal polynomial of the Gelfand pair (S 2n , S n [S 2 ]) [26]. We have other expressions in terms of the projection operator associated with the singlet representation of S n [S 2 ], The zonal polynomial has the property The prominent property of the operators is that they have diagonal two-point functions where Dim(R) is the dimension of U (N c ) associated with the representation R, and d R is the dimension of S 2n associated with the representation R. ω Λ 1 /2 (Ω It is a polynomial in N f , which was also encountered in [21,22]. This equation is derived in Appendix C. 1. Concrete examples of the operators O R,Λ 1 ,τ are given in Appendix D. In section 4.1 we count the number of mesonic operators using Schur-Weyl duality. It will turn out that the representation-labelled operator (3.1) should vanish if where c 1 (R) denotes the length of the first column of the Young diagram R. This expectation is consistent with the normalisation of our two-point functions (3.9). We can find the vanishing properties from the formulae and where (i, j) labels the row and column of the boxes of the Young diagram, and h(i, j) is the hook-length (A.11). The formula (3.12) was derived in [26,21]. Both (3.11) and (3.12) depend on N c and N f through the product of factors, e.g.
In [12] a general formula of diagonal operators is presented, which works for any group G and representation V . However, the formula requires the calculation of the CG coefficients arising from the decomposition of V ⊗2n F by G × S 2n . It turns out that our operator O R,Λ 1 ,τ is a concrete realisation of the general formulae of [12]. The relevant CG coefficients can be obtained explicitly as explained in Appendix E. The branching coefficients associated with the Gelfand pair (S 2n , S n [S 2 ]) provide a neat description of these Clebsch-Gordan coefficients, and hence of the space of O(N f ) singlet operators of length 2n.

Operator counting
In this section we discuss several aspects of the counting formula, taking care of essential distinction between large N c and finite N c ( i.e. N c ≥ 2n and N c < 2n ) as well as large N f and finite N f ( i.e. N f ≥ 2n and N f < 2n). First we employ Schur-Weyl duality and to obtain group theoretic expressions for the dimension of the space of operators at finite N c , N f . Then we rewrite the counting formula in terms of delta functions of permutations in the large N c and large (N c , N f ) limits. The latter formulae are recognised as the counting formulae for graphs based on Burnside's Lemma. We will rediscover various ways of associating permutation labels to gauge-invariant operators at large N c , N f .

Schur-Weyl duality and counting for finite N c , N f
The construction of our operators can be explained by Schur-Weyl duality, along the lines of [9,12,27,28]. The scalar field (Φ a ) j i has one flavour and two colour indices, and belongs There is a natural action of permutations σ ∈ S 2n on the tensor product, The RHS is a polynomial of bosonic variables (Φ a ) j i . Since they commute with each other, this polynomial is invariant under σ.
Let us count the number of gauge-invariant scalar operators. We regard V F , V C , V C as U (N f ), U (N c )-modules and apply Schur-Weyl (SW) duality. We find and the operators should also be S 2n -invariant . (4.5) The number of gauge-invariant scalar operators is given by taking the dimension of both sides. Now we count the number of mesonic operators (2.4), which contains the Kronecker delta n i=1 δ a 2i−1 a 2i . The mesonic operator is invariant under the following action of permutations γ in the wreath product group S n [S 2 ], This γ is not part of σ in (4.1), because γ does not change the colour indices. Recall that the wreath product subgroup S n [S 2 ] is the set of permutations which leaves Σ 0 = (12)(34) . . . (2n − 1, 2n) ∈ S 2n invariant under action by conjugation  Table 1: The number of all mesonic operators of length 2n for N c ≥ 12 and N f ≥ 6.
The contraction of the flavour indices breaks U (N f ) to O(N f ) and projects to the invariant representation of O(N f ). So we can count the number of mesonic operators by restricting (4.8) ) are Gelfand pairs [26], which has the multiplicityfree property. In particular Λ of the parent group contains the trivial of the subgroup with unit multiplicity if Λ 1 is an even partition (A.27). Now we repeat the above argument, taking into account that the two factors in (4.8) are one-dimensional, to obtain The number I 2n (N c , N f ) of mesonic singlet operators of length 2n at finite N c , N f is thus This result agrees with the number of diagonal operators (3.1). Table 1 shows some values of I 2n (N c , N f ) at large N c , N f . The formula (4.10) counts only the O(N f ) singlets. The method to count the SO(N f ) singlets or other representations are described in [12]. The difference between O(N f ) and SO(N f ) lies in the existence of "baryonic" operators. We will explain more about these points in Appendix E.2.

Large N c
The number of singlet mesonic operators for finite N c and finite N f is given by (4.10). We consider simplifications at large N c ; more precisely N c ≥ 2n. There we can convert the sum over R into a sum over permutations. and apply the formula (A.12), valid for N c ≥ 2n, to the counting formula (4.10). It simplifies as

Let us define
where δ 2n (g) is defined by (2.14). When the cycle type of σ is (4.14) Note that H p (σ), H p (σ ) are conjugate with each other if σ, σ ∈ T p . We define H p as H p (σ * ) for a fixed σ * ∈ T p . The order of H p , namely the number of elements that commute with any permutation of cycle type p, is given by The two expressions give the large N c equivalence This equivalence can also be derived from the following group theory identity The expressions (4.17) and (4.18) will be used in the next subsection.

Large N c and N f
We take large N f and large N c limits to convert all representations to permutations, and obtain several expressions of the counting. We define where we used the formula (A.26) saying that (S 2n , S n [S 2 ]) is a Gelfand pair. We also have Let us define which is also written as From (4.17) the total number of singlets with 2n fields is We will derive three expressions for Z p below. 7 The quantity Z p is the number of equivalence classes in the double coset S n [S 2 ]\S 2n /H p , The double coset space is the set of equivalence classes of permutations in S 2n , generated by the left and right multiplications by the subgroups S n [S 2 ] and H p respectively.
This description is redundant in the following way, The permutation γ 1 comes from the re-ordering (A.6), and γ 2 is the symmetry of the Kronecker delta's. By using this redundancy we can gauge-fix either α or ρ. If we fix ρ, we obtain the operator O α in (2.4). The corresponding counting formula is (4.34). If we fix α, then the cycle structure of α, denoted by p 2n, determines the colour (or multi-trace) structure. It corresponds to the counting formula is (4.29). Finally, we use the contraction operators (2.22) and write We find that the transformation rule of the quantities The expression (4.28) is related to the last counting formula (4.33).
indices of the corresponding fields being contracted. The connection between double cosets and graph counting is explained in a physics context in [29]. By going to large N c , N f , we see that counting SO(N f ) invariants is simply counting the graphs. Now observe that the last line in (4.22) can be rewritten as 8 From (4.12), (4.25) and (4.32), This can be recognised as the counting of pairs (σ, τ ) in conjugacy classes (T p , Such equivalence classes of pairs form another way of encoding bi-partite graphs. It amounts to choosing a labelling of the edges using integers {1, · · · , 2n} and reading off the labels of the edges around the black and white vertices. This is an alternative way to encode graphs, which differs from the encoding by a permutation σ ∈ S 2n which links directly with the counting by double cosets. This way of encoding graphs, in the context of Feynman graphs (which have symmetric rather than the cyclic vertices here) is illustrated in Figure 7 of [29]). The way that links directly with double cosets is shown in Figure 10 there. Some further manipulation of (4.33) gives The equation (4.34) establishes the equivalence between (2.15) and (2.16). We can reproduce this result also by applying the Burnside Lemma directly to the equivalence class of (4.27).
Note however thatZ q = Z q . 9 We now have some physical insight into the two ways of writing I 2n as sums over partitions in (4.20). (4.19) In one way we have Z p . In another, we have ϕ(γ) with γ ∈ T q . The partition p is the trace structure. The partition q is the cycle structure of γ which commutes with τ ∈ [2 n ]. Note that we have arrived at the sums over a product of two delta functions in this section by taking large N c , N f . It is instructive to reconsider in reverse what we did in this section. Start from gaugeinvariant operators parametrised by permutation equivalence classes. Graph counting associated with gauge-invariant operators can be expressed in terms of permutation sums with a product of delta functions. The Young diagrams R, Λ come from Fourier transforming these two delta functions. In the present context, we have seen that the numbers of rows of R, Λ are cut off by the rank of gauge and global symmetry groups. In a wider context, we may wonder about the physical meaning of introducing extra integers to cut off the numbers of columns.

Permutation topological field theory
We explore the connection to a two-dimensional topological field theory (TFT) of permutation groups. This TFT is a topological lattice gauge theory whose gauge group is S 2n , defined on a 2-complex (collection of 0-, 1-and 2-cells glued together). The computation of observables of the TFT involves a sum over the group elements of S 2n for every edge (1-cell), with a weight equal to a product of delta functions, one for every face (2-cell). The delta function weight ensures that the sum is invariant under refinement of the cell decomposition, so that a continuum limit can be reached. This type of TFT is discussed in the physics literature in e.g. [23,24,25]. A review of the TFT of permutations and application to a large class of observables in quiver gauge theories is given in [30].
As a first step, we reconsider the number of mesonic states as a partition function in TFT. By summing over p 2n in (4.33), we find This formula gives the number of mesonic singlets as a partition function for S 2n TFT on the 2-complex shown in the upper left of Figure 3. The 2-complex consists of two tori (drawn as cylinders with top and bottom boundaries identified) joined along a circle, associated with permutation σ. One of the tori has a cycle with permutation Σ 0 , which is fixed rather than being summed. This cycle with constrained permutation is a defect. Note that the delta function δ 2n (Σ 0 γΣ 0 γ −1 ) can be solved explicitly as in (4.34). Next we rewrite the free two-point functions (2.31) to make contact with (5.1), Note that the free two-point functions (5.2) become diagonal in the large N f , N c limit owing to the symmetry (4.27). The formula (5.2) can be recognised as the partition function of the S 2n TFT on a 2complex M with boundaries and a defect, as we now describe. The first delta function is associated to a 2-torus with the disc removed. The boundary of the disc has permutation (this forms the Ω N f factor). One of the cycles of the 2-torus is constrained to be the permutation Σ 0 . This constraint can be viewed as a defect. The second delta function is associated to a topological quotient of a cylinder with a disc removed, related to β 1 , which is summed with weight N C(β 2 ) c to give Ω Nc . The permutations α 1 , α 2 correspond to the two boundary circles of the cylinder, and a point from each of the two boundary circles is identified by the quotienting. The 2-torus and the quotiented cylinder are glued along a circle, to form the 2-complex which we call M. This is illustrated in Figure  3.
The 2-complex M cannot be the cell decomposition of a 2-manifold (which should locally be R 2 ), because a 1-cell (the red-dashed circle denoted by σ) is incident on four 2-cells. Still, it should be possible to realise it as the 2-skeleton of a higher dimensional manifold. In that case higher dimensional TFT would be the natural setting for the interpretation of the 2-point function. TFT3 has arisen in the context of refined counting formulae for graphs in [31].

One-loop operator mixing
In this section we will compute the mixing matrix under the action of the one-loop dilatation operator [32] On the representation basis, the mixing matrix is almost diagonal, where the non-zero components are explained by the repositioning of boxes.
On the permutation basis, the mixing matrix is given by where Σ 0 = (12)(34) · · · (2n − 1, 2n), and we have defined The sum (i,j) is over (1,2), (3,4), · · · , while the sum i,j is over the other pairs. We do not distinguish (i, j) and (j, i). The first line of (6.3) comes from the first term of (6.1), and the remaining lines come from the second term. The derivation of this mixing matrix is presented in Appendix F.
Given that we have expressed the mixing matrix purely in terms of permutations, we expect that there will be an interpretation in terms of permutation topological field theory. The construction of 2-complex will be analogous to our interpretation for the free field twopoint function given in Figure 3, but will involve some new features given the additional complexity apparent here. We will return to this problem in the near future.
In the latter part of this section we study the mixing matrix on the representation basis. We denote the change of basis by Let us take one term in the first line of (6.3), and simplify the mixing matrix on the representation basis, Expand the Kronecker delta's using (A.13). We now use the following formula to remove the sum over β Here we have introduced the branching coefficient where the r is a Young diagram with 2n − 1 boxes, and m runs over 1, · · · , d r . The quantity I RR ,r il is the intertwiner map of [17,19], here expressed in terms of branching coefficients. In the formula, the RHS is non-zero when both g([1], r; R) and g([1], r; R ) are non-zero. 10 In other words, the RHS of (6.8) is non-zero only when the R is obtained from the S by moving a single box. Performing the sum over σ, ρ in (6.7), This mixing matrix is non-zero if R and R become identical after the move of a single box. This kind of mixing, re-positioning of boxes, is common for representation bases, which has been studied concretely in [33,34,35,12,17,36]. The diagonalisation of one-loop mixing is still not trivial, which has been achieved in the SU (2) sector in special cases [18,19]. This will be an interesting avenue for future investigation.

Conclusion and Discussion
We summarise the results of this paper. We studied mesonic operators, that is the O(N f )singlet scalar operators in U (N c ) gauge theory, and computed free field two-point functions. The two-point functions are expressed them in terms of permutations. We performed a Fourier transform from the permutation basis to the representation basis, which made the two-point functions diagonal. Our mesonic operator provided a concrete realisation of the formula for diagonal operators [12]. We counted the number of operators in both bases by applying inverse Fourier transform. It was important to remove the redundancy of the wreath product group S n [S 2 ], noting that (S 2n , S n [S 2 ]) is a Gelfand pair. We computed the one-loop mixing matrix in both bases.
Our expression for the 2-point function on the permutation basis was used to give an interpretation in terms of TFT, based on topological lattice gauge theory of permutations equipped with defects. This extends the connection between TFT and quiver gauge theories [30]. An interesting problem is to connect these results with axiomatic TFT as discussed in [37,38,39,40].
An important point is that the R-symmetry group of N = 4 SYM is SO(6) rather than O(6). The SO(N f )-singlet operators include  (6), the two-point functions remain orthogonal to all orders of perturbation theory. A systematic generalisation of our study of correlation functions to baryonic operators is left for the future.
Our explicit construction of operators and free-field 2-point functions has been given for U (N c ) theories. Generalisations to other gauge groups, e.g. SU (N c ) along the lines of [7,41,42], and SO(N c ) or Sp(N c ) following [21,22] will be interesting.
Our results provide a foundation for the systematic studies of planar zero modes (PZM) in the SO(6) singlet sector. Group-theoretical counting methods for the PZM's can be developed, analogously to the construction of general mesonic singlets here. Often the counting formula implies the existence of a basis of PZM's labelled by permutations or representations. An interesting application of such a formalism is to compute the non-planar anomalous dimension of the PZM's [20]. The PZM's are expected to have negative anomalous dimensions by 1/N c corrections, according to the results at strong coupling [43,44,45,46,47] and those of conformal bootstrap [48,49,50]. Following this paper, the general N f setup can provide a tractable approach to finite N f = 6.
Another application of our results is to determine physical quantities of N = 4 SYM at this level of generality, such as non-planar correlation functions [51,52], partition functions [53,54,55,56], and statistical properties of one-point functions via matrix product states [57].
The sector of SO(6) singlets offers a rich and interesting setting to explore non-planar effects in a non-supersymmetric sector of N = 4 SYM.

Acknowledgements
The work of YK is supported by JSPS KAKENHI Grant Number 15K17673. SR is supported by STFC consolidated grant ST/L000415/1 "String Theory, Gauge Theory & Duality". RS is supported by FAPESP grants 2011/11973-4 and 2015/04030-7. We thank Robert de Mello Koch for discussions.

A.1 Notation
We denote by Φ = (Φ a ) i j a hermitian scalar field, a = 1, 2, . . . , N f and i, j = 1, 2, . . . N c . The flavour group is SO(N f ) and the colour group is U (N c ). Φ belongs to the fundamental representation V F of SO(N f ), and to the adjoint representation V C ⊗ V C of U (N c ). The case of N f = 6 describes the six scalars of N = 4 SYM in four dimensions.
We introduce a gauge-covariant operator Φ a , which is related to the component fields by Permutations act on the bases as Thus, permutations act on SYM fields by The order of a finite group G is denoted by |G|. The words "representations of S L ", "Young diagrams", and "partitions of L" are used interchangeably. A partition of L is expressed in two ways. The first expression is If we collect the same q's together, we obtain the second expression (2.5). The symbol c 1 (Y ) is the length of the first column of the Young diagram Y . In (A.7), Clearly c 1 (Q) ≥ c 2 (Q) ≥ · · · ≥ c q1 (Q).

A.2 Formulae
In this subsection the definition of group theory quantities, and group theory formulas are collected. The matrix elements (i, j) of the group element σ ∈ S L in the representation R are denoted by D R ij (σ). We assume all representations of S L are real and unitary, and thus D R ij (σ) = D R ji (σ −1 ). They satisfy the so-called grand orthogonality relation where d R is the dimension of R of symmetric group S L , The product is over the boxes of the Young diagram R with i, j labelling the rows and columns. The quantity h(i, j) is the hook-length associated with the box at (i, j), namely the number of boxes intersecting the hook which extends from (i, j) toward the right and bottom. For example, The character of σ in the representation R is denoted by χ R (σ). The characters satisfy the orthogonality where the delta function is defined by Consider the tensor space V ⊗L , where V is the fundamental representation of the unitary group U (N ). The symmetric group acts on the tensor space by permuting the L factors. From the fact that these two actions commute each other, the tensor space can be decomposed as where the sum is over the Young diagrams with at most N rows, which is expressed in terms of the length of the first column c 1 (R) by c 1 (R) ≤ N . This equation is the Schur-Weyl duality between U (N ) and S L . A trace over this tensor space is denoted by tr L , where we have defined the quantity According to the Schur-Weyl duality, a trace of an element σ ∈ S L can be written as where Dim(R) is the dimension of R of Lie group U (N ) Setting σ = 1 in (A.17) gives the identity i 2 is the Clebsch-Gordan (CG) coefficients, defined by the irreducible decomposition of the tensor product R 1 ⊗ R 2 = ⊕S, with the multiplicity label τ , (A.20) The indices (k, i 1 , i 2 ) specifies the elements of (R, R 1 , R 2 ). The multiplicity label τ runs over 1, · · · , C(R, R, Λ), where C(R, S, T ) is called the CG number (also known as CG multiplicity or Kronecker coefficient) The CG coefficients satisfy the following properties Let R 1 , R 2 , R be the irreducible representations of S m , S n , S m+n , respectively. The Littlewood-Richardson coefficient g(R 1 , R 2 ; R) counts the number of R 1 ⊗ R 2 appearing in the decomposition of R under S m × S n ,

A.3 Gelfand pair and coset type
A pair of finite groups (G, H) with G ⊃ H is called Gelfand pair if they satisfy either of the following conditions: (i) Any irreducible representation of G contains at most one singlet representation of H.
(ii) Consider a set of functions on the double coset H\G/H, where the multiplication is defined by convolution. The algebra C(G, H) is commutative.
The two conditions are equivalent [26]. The Gelfand pair for Lie groups is defined similarly. An important example of the Gelfand pair is (S 2n , S n [S 2 ]). From (i), an irreducible representation R of S 2n satisfies An even representation R corresponds to even Young diagram The double coset function in the condition (ii) corresponds to W (σ) defined in (2.21), which satisfies the equivalence relation, (A.28) The function W (σ) defines the unique coset type for each permutation σ,

B Powers of N f
We denote the number of loops in the graph of W (σ) by z(σ). We will show the identity where Σ 0 = (12)(34) . . . (2n − 1, 2n) and C(σ) counts the number of cycles in σ ∈ S 2n . The structure of W (σ) is depicted in Figure 4. Consider the labelled points {1, 2, · · · , 2n} in that diagram. If these points move along the lines toward the upper arcs, they will return to the labelled points {1, 2, · · · , 2n} after a permutation Σ 0 = (12)(34) · · · (2n − 1 2n) (B.2) = Figure 4: This is identical to Figure 1. The left figure shows C 12 C 34 · · · C 2n−1,2n σ acting on V ⊗2n F . The upper and lower horizontal lines are identified when taking the trace, as in the right figure. If the points went down through σ, the lower arcs, and back up σ again, they will undergo the permutation The graph of W (σ) contains flavour loops. A flavour loop is a sequence of transitions of the form or their inverses. Note that Σ 2 0 = Σ 2 1 = 1. We cannot return to the original point by after an odd number of Σ's, because both Σ 0 and Σ 1 have cycle type [2 n ]. In other words, Σ 0 and Σ 1 acting on i behaves as a permutation of odd signature for any i. The number of flavour loops, i.e. the power of N f denoted by z(σ), is same as the number of orbits in the subgroup of S 2n generated by Σ 0 , Σ 1 . This is also the number of connected components in the ribbon graph determined by the permutations Σ 0 , Σ 1 . Let us introduce the notation Recall that S n [S 2 ] is the stabiliser of Σ 0 , By using this "gauge degree of freedom" of S n [S 2 ], we may transform Σ 1 to Σ 1 = ξ −1 Σ 1 ξ without changing C(Σ 0 Σ 1 ). There exists a useful gauge: Lemma 1. By a gauge transformation in S n [S 2 ], we can transform Σ 1 to the form (B.7) Figure 5: Another graph of W (σ). The dashed edges represent the elements of Σ 0 , and the sold ones those of Σ 1 . This graph contains two loops, (N Σ 0 ) and (HΣ 0 N Σ 0 HΣ 0 N Σ 0 ).
Proof of 1. We draw another graph of W (σ) emphasising the structure of loops, with 1 + , 2 + , . . . along the upper line and 1 − , 2 − , . . . along the lower line. We connect the points i ± and j ± when (i ± , j ± ) belong to Σ 0 or Σ 1 as shown in Figure 5. The horizontal edges of Σ 1 , namely those connecting (+, +) or (−, −), will be called H-edges. The other edges of Σ 1 , (+, −) or (−, +), will be called N-edges. Let us prove that every loop has to have an even number of H-edges. As discussed in (B.4), every loop consists of even number of edges. Within the loop, only Σ 0 and N-edges change the parity ±. If we circle around the loop, then there should be no change in parity. Thus, every loop ( Since #(H) + #(N ) + #(Σ 0 ) is even for each loop, the number of H-edges is also even. The statement (B.7) is equivalent to saying that we can remove all H-edges by gauge transformations S n [S 2 ]. Consider how the flip (i − i + ) ∈ S n [S 2 ] acts on the Σ 1 edges connected to the points i ± . Inspecting Figure 5, we find (B.9) The flip cannot change HΣ 0 N into N Σ 0 N or HΣ 0 H, because it violates the parity rule (B.8).
The same is true for N Σ 0 H. Now, if we have a flavour loop (HΣ 0 N · · · Σ 0 H . . . ), flipping all the edges in Σ 0 between two H-edges will convert the loop to (N Σ 0 N · · · Σ 0 N . . . ). Since the number of H-edges is even, applying this process repeatedly will remove all H-edges. This means that there is a gauge transformation which converts Σ 1 to Σ 1 of the form (B.7).
The permutation τ is itself defined up to conjugation in S n . This is in fact a way to understand the correspondence between partitions of n and the double coset space S n [S 2 ] \ S 2n /S n [S 2 ], as we will explain subsequently.
Let us denote by ( 1 , 2 , . . . ) the number of edges in the loops of the graph W (σ). These { i } are all even, and satisfy i i = 2n. Thus, λ i ≡ i /2 defines a partition of n. This partition λ is same as the cycle decomposition of τ . We are going to relate the number of loops with the number of cycles in Σ tot ≡ Σ 0 Σ 1 .
Corollary 2. Σ tot maps minus variables to minus variables, plus to plus.
Thus, Σ tot splits into two disjoint actions Σ − tot × Σ + tot , where Σ ± tot acts on the set V ± n = (1 ± , . . . , n ± ). Then, the number of cycles is equal to Lemma 3. The number of loops in Σ − tot acting on V − n is equal to C(τ ), and similarly for Σ + tot acting on V + n .
Proof of 3. Let us define We can express the number of loops in Σ ± tot as showing that #(loops) = C(τ ). It can also be derived graphically as in Figure 6.
The identity (B.1) follows from (B.12) and (B.10), The above discussion gives a concrete insight into the coset type (A.29); A coset type is a partition of n which parametrises the elements of double coset S n [S 2 ]\S 2n /S n [S 2 ]. Lemma 1 says that Σ 1 = σ −1 Σ 0 σ can be gauge transformed by ξ ∈ S n [S 2 ] to the form (B.7). We takeτ to be a permutation in S 2n which leaves i − fixed and acts nontrivially on i + (i = 1, 2, . . . n). The gauge transformation of ξ can be written as Soτ ξ −1 σ −1 is in the stabiliser of Σ 0 , andτ ξ −1 σ −1 = η ∈ S n [S 2 ] for some η. Hence any σ can be written asτ Therefore, the elements of the double coset S n [S 2 ]\S 2n /S n [S 2 ] correspond to the permutationsτ ∈ S 2n , or the permutations τ ∈ S n . The condition (B.14) does not completely fix the gauge. The residual gauge freedom is conjugation of τ by ξ ∈ S n ⊂ S n [S 2 ], which should not change the double coset element. As a result, the double coset elements are in 1-1 correspondence with the conjugacy classes in S n , i.e. partitions of n called coset types.

C Diagonal two-point functions
In this Appendix, we will derive (3.9) and compute the normalisation factor.

C.1 Proof of diagonality
Let us first rewrite the two-point functions of the permutation basis The colour factor can be expanded using (A.17) as where Dim(R) is the dimension of R associated with U (N c ). The flavour factor can be written from (2.33) as The sum over σ in (C.1) can be removed using (A.22) to obtain We now compute the two-point functions of the representation basis Substituting (C.4) into this and using the relation (A. 23), The last factor has several expressions The second equality comes from γ 1 Ω . It is non-zero only for the case Λ 1 , Λ 1 are even Young diagrams.

C.2 Twisting Wick-contraction rules
We derive the formula (3.12) by developing the connection with [26]. The function ω Λ 1 /2 (Ω (f ) 2n ) which appears in the normalisation of the diagonal two-point functions (C.7) is equal to the zonal spherical function of the Gelfand pair (GL(N f ), O(N f )), introduced in [26]. 13 In developing this connection, it is instructive to introduce a twist of the two-point functions parametrised by matrices T ab ∈ GL(N f ) and Y i j ∈ GL(N c ). Since P R,Λ 1 ,τ does not depend explicitly on N c or N f , our construction of diagonal operators can be readily generalised to the case where the Wick-contraction rules are twisted, The two-point functions (2.19) become Figure 7: Generalisation of W (σ) in Figure 1 including the twist T .
where W T collects the flavour factors and P Y the colour factor. They can be written as the power sum, where p 2n is the cycle type of ρ = α 1 σα 2 σ −1 , andp n is the coset type of σ. When we untwist as T = Y = 1, both factors reduce to A graphical representation of W T (σ) is shown in Figure 7.
Owing to (C.10), the operator Ω The flavour factor in the diagonal two-point functions in (3.9) becomes where we used χ Λ 1 (σ −1 ) = χ Λ 1 (σ). It turns out that the function ω Λ 1 /2 (Ω 2n ) is identical, up to normalisation, to the zonal spherical function Z Λ (T T T ) of the Gelfand pair (GL(N f ), O(N f )) introduced in [26]. In [26] it is shown that where (i, j) specifies the position of the Young tableau Λ. From (C.15), we reproduce the formula (3.12). The overall normalisation of ω Λ 1 /2 (Ω 2n ) is determined as follows. If we take T = 1 and N f 1 in (C.14), the leading terms come from σ ∈ S n [S 2 ], whose coset type is [2 n ]. Since (S 2n , S n [S 2 ]) is a Gelfand pair, the restriction to S n [S 2 ]-invariant subspace is multiplicity-free;

D Examples of diagonal operators
We will explain how to construct the diagonal operators (3.1) and give explicit examples at 2n = 2, 4.

D.1 Generality
Let us explain our strategy. First, we classify all irreducible representations R such that R ⊗ R contains an even partition Λ 1 . The irreducible decomposition of R ⊗ R can be computed from the character table by using (A.21). Second, we specify an orthonormal basis of the irreducible representations of S 2n explicitly. We use the Young-Yamanouchi orthonormal form for this purpose [58,59]. The Young-Yamanouchi basis is labelled by the standard Young tableaux λ of shape R, and the transposition (j, j + 1) acts on them as where ρ λ (j + 1, j) is the axial distance from j + 1 to j in the standard Young tableau λ. 14 The matrix representation of all other elements follows from (D.2). It is straightforward to compute the branching coefficient from (3.3).
The final ingredient to obtain the diagonal operators (D.1) is the CG coefficient defined by (A.20). To extract S τ,ΛRR kij , we apply σ ∈ S 2n to (A.20) and compare both sides. Let us explain the computation of the CG coefficients in detail. We rewrite the matrix representation of the product D R ⊗ D R (σ) into a block diagonal form by a similarity 14 The axial distance between a and b is defined by counting the number of boxes we need to pass through from a to b in the Young tableau. We add +1 by going left or down, and −1 by going right or up. For example, ρ transformation, The CG coefficients are equal to the elements of the rotation matrix U R . We compute each column of U R by using the Young symmetriser, where p λ (σ) is a coefficient, N λ is a normalisation constant, and A k (or S ) is anti-symmetric (or symmetric) combination of the entries in the k-th column (or -th row) of the standard Young tableau λ, respectively. For example, The Young symmetriser projects D R⊗R onto the state corresponding to {ẽ λ }. 15 Thus, the combination σ p λ (σ) D R⊗R (σ) becomes a rank-one matrix corresponding to a single eigenvectorẽ λ . By collecting all eigenvectors (and orthogonalizing them appropriately), we obtain the rotation matrix U R .
E Relation to covariant approach In [12] a general construction of free-field diagonal operators was given, where the operators are built out of fields transforming in a general representation V of a general global symmetry group G. This general construction requires the explicit computation of the Clebsch-Gordan coefficients, which decompose the tensor products V ⊗m in terms of irreducible representations of G × S m . Our approach to the free-field diagonal operators in the current paper is similar to [12] since both diagonal operators carry the same representation labels. However, the two operators look slightly different, since the mesonic operators discussed here do not involve the CG coefficients. We will show that the two operators are identical, by giving an explicit formula for the relevant CG coefficients. Recall that in [12] the colour and flavour indices are treated separately. A gauge-covariant operators turn into gauge-invariant operators by combining indices appropriately. Instead, we may fix a gauge in the gauge-covariant form, and then combine them into gauge-invariant operators. In this way we can reproduce the mesonic operators. The CG coefficients are related to the product of branching coefficients and Kronecker delta's.
Let us just focus on the CG problem for the 2n-fold tensor product of V F with G = O(N f ). In particular, we are interested in the one-dimensional representation of both S 2n and SO(N f ). Let v a be basis vectors in A vector v ρ , parametrised by a permutation ρ ∈ S 2n which controls the pairwise contractions, Following a theme we have seen repeatedly, whenever we have some invariants parametrised by permutations, in the present case tensor products of vectors invariant under O(N f ), we must ask about the redundancy in the description. Here the redundancy is Again, following a familiar theme, disentangle these equivalence classes by using representation theory. The first step is to define Exploiting the invariance in the Fourier transformed basis, The branching coefficient is zero if Λ 1 is not even. The invariant vectors are therefore 16 This can be written in terms of a CG coefficient coupling V ⊗2n Multiplicity labels are not needed. It is tempting to identify the CG coefficients as K δ a ρ(1) a ρ(2) · · · δ a ρ(2n−1) a ρ(2n) (E.10) There is an important subtlety of normalisation which has to be considered when comparing to [12]. The key point is that the normalisation of the above-defined C a Λ 1 ,I is 16 See e.g. Appendix B of [9] of the relevant fact from linear algebra.
Since C a Λ 1 ,I = C σ( a) Λ 1 ,I for any σ ∈ S 2n , the first line is also written as (E.14) Note we assumed, as in [12], that the vectors in The norm of the permutation-parametrised vectors is If we substitute the CG coefficients (E.10) to the general diagonal operator of [12], we obtain gauge-invariant operators involving a sum over permutations ρ, α. After doing the sum over ρ, our diagonal operators in section 3 can be recovered.
Let us next focus on the mesonic singlet operators We decompose the sum over i, j into where (i, j) run over (1,2), (3,4), · · · , and i, j over the other pairs. 20 We have