A Young diagram expansion of the hexagonal Wilson loop (amplitude) in ${\cal N}=4$ SYM

We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D ${\cal N}=4$ Super Yang-Mills, or, precisely, an integral representation of its matrix part, via an ADHM-like instanton construction. In this way, we can apply localisation techniques to obtain combinatorial expressions in terms of Young diagrams. Then, we use our general formula to obtain explicit expressions in several explicit cases. In particular, we discuss those already available in the literature and find exact agreement. Moreover, we are capable to determine explicitly the denominator (poles) of the matrix part, and find some interesting recursion properties for the residues, as well.

Needless to say that we are interested in gauge theories for phenomenological reasons and in this particular one because it is conformal, albeit there are at least two other connected motivations. One is that this theory forms one side of the most known example of AdS/CFT correspondence [1][2][3], as it lives on the 4d Minkowski boundary of its gravitational dual, the type IIB superstring theory on AdS 5 × S 5 . The second reason is that the latter appears to be classically integrable in the sense that it can be written in Lax form [4], and moreover quantum integrable at least in the planar limit, N c → ∞ with g Y M → 0 so that the 't Hooft coupling λ ≡ N c g 2 Y M = 16π 2 g 2 stays fixed, in that N = 4 SYM shows remarkable connections with 1 + 1 dimensional integrable models [5][6][7][8][9][10][11][12][13][14][15][16]. The presence of integrability has opened the way to a better comprehension and partial proof of the aforementioned correspondence.
More precisely, quantum integrability was discovered for the spectral problem of N = 4, i.e. the computation of the anomalous dimensions of local (single trace) gauge invariant operators. But, more recently, it played an important rôle also in the evaluation of null polygonal Wilson loops (Wls), which are dual to and the same as gluon scattering amplitudes [17][18][19]; but in a different, though connected, guise.
In a nutshell, N = 4 SYM, as any conformal quantum field theory, enjoys an Operator Product Expansion (OPE) of the (renormalised) null polygonal Wilson loops [20], the simplest ones of non-local nature. This method has an intrinsic non-perturbative nature, given by the collinearity expansion of the null edges which gives rise to an infinite series over the number of particles (cf. infra). This OPE series was developed for the Wls in N = 4 SYM by employing the underlying integrability of the theory, which manifests itself in the flux-tube, dual to the Gubser, Klebanov and Polyakov (GKP) [21] string. The connection with the spin chain picture was initiated in [22,23], but the OPE series for the Wilson loop was developed in a series of papers [24][25][26], through the so-called pentagon approach. Briefly, the proposal was to write the expectation value of the Wl as an infinite sum over intermediate excitations on the GKP string vacuum: gluons and their bound states, fermions, antifermions and, finally, scalars. In this way, it is reminiscent of the Form Factor (FF) spectral (large distance) expansion of the correlation functions in integrable 2d quantum field theories, and the pentagonal operator has been identified [24,[27][28][29] with a specific conical twist field [30,31]. Therefore, this methodology needs to know the dispersion relations of the GKP string excitations [32] and the 2d scattering factors between them [33][34][35]: these quantities can be gained by filling up (with infinite Bethe roots) the original BPS vacuum of the single trace operators (as ferromagnetic vacuum) to construct the GKP (or anti-ferromagnetic) vacuum and the excitations thereof.
The validity of the OPE series has been checked in the weak coupling (e.g. [25,[36][37][38]) and the strong coupling regimes (e.g. [26,[39][40][41][42][43][44]), with comparisons against gauge and string theory, respectively. In the latter regime, string theory has so far given only the leading order as minimization of the world-sheet area living in the AdS 5 space and insisting on the polygon in 4d Minkowski [20,[45][46][47]. This contribution is given by gluons and (bound states of) fermions in the OPE series [41,44,48]. Although scalars elude the minimal area argument, nevertheless they are even more important to understand as they provide a comparable non-perturbative contribution, heuristically derivable from the O(6) non-linear sigma model and corrections as emerging from the low energy action of the string on S 5 : for the hexagon this has been proven in the FF set-up by [39,40,[49][50][51]. In general, their OPE contribution to the N -sided polygon is a form factors series of a (N − 4)-point correlation function of a string model, which reduces to the O(6) non-linear sigma-model in the strong coupling [39,40,49,50]. In this regime the dynamically generated mass is exponentially suppressed m ∼ e − √ λ 4 and entails the short-distance regime for the correlation function and thus a power-law behavior [39]. Thanks to the specific form of this contribution at all couplings, it has been possible to expand it with exact computations in this regime by using integrability (only, so far) derived ideas and methods, so proving it to be of the same order as the one coming from the classical string computation [40,49,50]. In this respect, the all coupling form of the contribution of scalars assumes a particular relevance and it will be given a new interpretation in this paper. In general, if u i are the rapidities of the GKP string (flux tube), the hexagonal Wilson loops can be represented as the OPE (1.1) over all admissible n particle states. The explicit form of Π dyn and Π F F can be found e.g. in [53] and S n is the symmetry factor: given n = n 1 + n 2 + ... + n k , where n j are the number of identical particles of a given kind j, then S n = n 1 !n 2 !...n k !. In this paper, we will be interested only in Π mat , which admits an integral representation as multi-integrals over the nested Bethe (or isotopic) roots [35,44] (see below (4.1) and notice that it depends only on the rapidities of scalars, fermions and antifermions {u i }, {v i }, {v i }). For large number of particles the evaluation of these integrals soon becomes formidable. In fact, our aim here is to represent this integral as a combinatorial sum, which instead allows for explicitly calculation also for a large number of particles.
In papers [49,50] one of the authors (D.F.) and collaborators have focused on the matrix part with only scalars, cf. (2.1), or only fermions respectivey, cf. (3.1), namely as multi-integrals over the nested Bethe roots of SO (6) or SU (4), respectively. They systematically evaluated by residues the result and encoded it in some Young diagram combinatorics [40,49,50]. This method gives rather explicit final formulae in terms of simple rational functions and is reminiscent of the pole contributions to the instanton partition function of N = 2 SYM [54]. But still the diagrams are rather different and comparison is not so strict, neither physically motivated. This is why in the present paper, we want to draw a more refined correspondence between the two methods by making explicit and adequate reference to instantons, namely the equivariant localization technique and its related Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction [55]. Importantly, we can in this way treat the most general situation with any number of scalars and fermions and reach the simple form of sum over multiple Young diagrams of certain shapes, where each term is the inverse of a nice factorized polynomial expression.
In this perspective, the paper is organized as follows: in section 2, we consider the matrix part of MHV amplitudes where one has only 2n scalars. Upon taking into account the structure of the integral representation we construct an appropriate complex manifold defined by a system of matrix equations. Then we apply equivariant localization to get a combinatorial representation of the matrix part of Wl. Several applications and consequences of our formula are given as well. In section 3, we extend the above analysis to the case with an equal number of fermions and antifermions, without scalars. Then, in section 4, we consider the matrix part of the Wl with N φ scalars, N ψ fermions and Nψ antifermions. In this general case too, we suggest for the integral representation (4.1) the matrix equations (4.8)-(4.10), which, via localization, lead to the combinatorial formula presented in subsection 4.3. Eventually, in section 5 we use our general formula (4.29) to establish the result (5.2) for the denominator of Π (N ψ N φ Nψ) mat . We also have found two interesting residue formulae (5.3) and (5.4). At the end of this section for several specific values of scalars, fermion, antifermions and the bottom R-charge r b we derive the matrix parts explicitly.

Scalars
In this section we rewrite the integral representation of the matrix part of the hexagonal bosonic Wl in such a way to make the connection with the integral representation of instanton partition function in Ω background more obvious. The structure of integral representation suggests how to construct an ADHM-like moduli space which reproduces the same integral via equivariant localization. As usual the denominator encodes the required set of algebraic data (matrices) and the numerator reflects the symmetry properties of equations to be satisfied by these data. Thus, we introduce the appropriate ADHM-like moduli space defined in terms of six matrices subject to three quadratic equations. It is shown that this moduli space admits a U (1) 2n+1 symmetry which is used to apply equivariant localization technique. Subsection 2.3 is devoted to the classification of fixed points of moduli space under aforementioned U (1) 2n+1 action. It is shown that each fixed point can be described by an array of 2n Young diagrams. Only six types of diagrams (an empty one, a 1-box, two 2-box, a 3-box and a 4-box diagrams shown in Fig.2) are allowed. We investigate the tangent space of an arbitrary fixed point and find a closed character formula which encodes the pattern of how a tangent space decomposes into one (complex) dimensional invariant subspaces of the U (1) 2n+1 action. Finally, we apply localization technique to present the matrix part as a sum over fixed points, i.e. 2n-tuples of above young diagrams constructed in subsection 2.3. We use our formula to derive explicitly the first 2 and 4 particles expressions, Π (2) mat and Π (4) mat respectively and find complete agreement with all previous results available in literature.

Connexion to the N = 2 (Ω background) SYM partition function
As for the hexagonal bosonic Wl in N = 4 SYM, the matrix factor accounting for the 2D scattering of 2n scalars (of GKP string or flux tube) does not depend on the 't Hooft coupling constant λ and can be written as a multi-integral over the three kinds of nested Bethe Ansatz roots of a SO(6) spin-chain [35,44], a k , b j and c k [39,40]: where the functions f (x) and g(x) are defined as Inserting this in (2.1) we get , where the prime on the product symbol indicates that all the vanishing factors (i.e. the factors a ii , c ii , b α,α ) are omitted. After generalizing the last expression by the substitution i 2 → and shifting the variables of integration by this amount we get . A similar integral arises in a completely different context, the N = 2 SYM theory with gauge group U (n) in Ω-background parameterized by 1 and 2 . In the latter case, the k-instanton contribution to the partition function can be written in the form [54,56] , (2.5) where a are the gauge expectation values and the prime on the product symbol again means that the vanishing factors should be suppressed. In the case 1 = 2 = this expression is suspiciously similar to (2.4). On the other hand, upon applying localization techniques for the moduli space of instantons, according to the ADHM construction, the full instanton partition function can be arranged as a sum over Young diagrams [57] in the following way where Y is an array of n Young diagrams Y = {Y 1 , Y 2 , . . . , Y n } and | Y | is the total number of boxes. q is the instanton counting parameter, related to the gauge coupling constant in the standard manner: q = exp 2πiτ , with τ = i g 2 + θ 2π the complexified coupling. The coefficients f Y are factorized as

Matrix equations for the hexagonal Wilson loop
A nice exposition on how ADHM construction leads to the integral representation can be found in [58]. Here we consider the opposite problem to obtain an ADHM like construction corresponding to (2.4). The ADHM analogy suggests that we have to introduce four vector spaces • u: a 2n complex dimensional space related to the parameters u 1 , u 2 , . . . , u 2n • b: a 2n complex dimensional space related to the parameters b 1 , b 2 , . . . , b 2n • a and c: each a n complex dimensional space related to the parameters a 1 , a 2 , . . . , a n and c 1 , c 2 , . . As usual in order to guaranty smoothness of moduli space these equations are supplemented by stability condition [59]. We found that the suitable Stability condition reads: There in no proper subspace of b which contains the image M bu and is invariant under two operators M ba M ab and M bc M cb . In addition it is required that M ab M bu and M cb M bu are onto maps on a, c respectively. The first condition means that if one acts in all possible ways on the image of M bu the entire space b will be recovered.
Let us introduce also transformations T a , T b , T c , T u acting in respective spaces (e.g T a : a → a is a n × n matrix). While T u is a genuine symmetry (which is the analog of global gauge transformations in N = 2 SYM), the remaining T a , T b , T c are auxiliary transformations (the moduli space is found by factorizing space of solutions over these auxiliary transformations). In addition we introduce also transformation T ∈ C * (C * is the group of multiplication by nonzero complex numbers). Let us quote below the transformation lows of matrices M Notice that the above rules follow the pattern of factors in denominator of (2.4) (e.g. the first transformation rule matches with the factor (a i − b α + )).
It is straightforward to check that the matrix equations (2.9)-(2.11) respect these transformations. The left hand sides of matrix equations get transform as The equations (2.9)-(2.11) are designed so that their transformation lows exactly much with corresponding factors in the numerators of (2.4). For example the equation (2.9) transforms as (2.16) which obviously agrees with the factor a ij + 2 in (2.4).

The set of all admissible diagrams
To apply localisation we need to find all fixed points under transformations T , T u . Thus a fixed point is a set of matrices {M ab , M ba , M bc , M cb , M bu , M ub } satisfying the equations (2.9)-(2.11) and stability condition, invariant under T , T u up to auxiliary transformations T a , T b , T c . It is possible to show that at fixed points the matrix M ub = 0, so that due to (2.11) Let us choose the basis vectors e 1 , e 2 , . . . , e 2n in space u to be eigenvectors of transformation T u : The we introduce a useful graphical interpretation as follows: • To each e i such that M bu e i = 0 we associate a dot (empty diagram).
• To each e i such that M bu e i = 0 ≡ v we associate a box v . Thus a box represents a non zero basis vector in space b.
• If M ab v i = 0 then we can add a extra box (NE direction) v , notice this new box represents a vector in the space a.
• Because the spaces a and c enter our construction in a symmetric way so obviously we have in addition the diagram v .
Due to the matrix equations (2.9) and (2.10) it is not possible to enlarge these diagrams further on. So we conclude that the diagrams listed in Fig.2 are the only possible ones. Since the dimension of u is 2n we should have 2n diagrams chosen from the list depicted in Fig.2. Let n i be the number of diagrams of type i, i = 0, 1, 2,2, 3, 4, then The dimensions of b, a and c are 2n, n and n respectively, so The last four conditions lead to n 4 = n 0 , n 1 = n 3 , n 2 = n2 , n 0 + n 1 + n 2 = n . (2.23)

Tangent space decomposition
To apply localization formula we have to describe the tangent spaces of M n at the fixed points. The procedure is standard: from the total space where the unconstrained variation δM lives, one "subtracts" subspaces corresponding to equations (2.9)-(2.11) and auxiliary transformations. Notice that (cf. transformation laws (2.12)) Obviously the subspace corresponding to the three equations (2.9)-(2.11) is the direct sum of three terms Finally the remaining part corresponding to auxiliary transformations T a ∈ a ⊗ a * , Combining all together for the tangent space character we get where (without changing notation) we have replaced the spaces by their character (with respect to T , T u transformations) as follows: the character of space u can be written as (see (2.20)) as for the spaces a, b and c we will have Notice that since the spaces a and c are n dimensional exactly n therms in each (2.31) must be zero. The characters a * , b * , c * can be found by replacing i . For a given fixed point specified by diagrams Y = {Y 1 , Y 2 , ..., Y 2n } the character has the following structure The summands X (u i , Y i |u j , Y j ) can be easily read off from (2.29): Table 1. It is essential that though (2.29) besides positive terms includes also negative ones, nevertheless in final formula (2.33) and Table 1 all terms are positive, which is required by self consistency (the initial space contains all the sub spaces to be factored out due to matrix equations and equivalence relation (2.15)). Though we did not provide a mathematically rigorous proof, the above property strongly suggests that the moduli space M n is a smooth algebraic manifold with dimension 4n 2 .

Matrix part as a sum over diagrams
For given n the matrix part of the hexagonal Wl can be represented as Fig. 2 subject to constraints: • The total number of boxes in middle line (related to the space b) is equal to 2n.
• Both the upper and lower lines (related to spaces a and c) contain n boxes each.
A little consideration ensures that above conditions are equivalent to the requirements (2.23) where n i (i = 0, 1, 2,2, 3, 4) is the number of diagrams of type i (see Fig.2) in Y .
We can also deduce that if a diagram of type 0 (1 and 2) enters in Y then 4 (3 and2 respectively) must be present too.
The summands F Y are given by .
Here P (a, λ|b, µ) is a product of factors obtained from the terms of where m and n are integers. The final result (alternatively represented in Table 2) is To derive P (a, 2|b,2) we can use both the second and third lines of (2.37) because they give the same result.
Here the e i (i = 1, 2, 3, 4) are elementary symmetric polynomials in four variables Next let us consider the case when n = 3 then Π  Table 3.
polynomial. Below we present the result for the case when all u i are very large so the in (2.37) can be neglected. We have found Here e i , i = 1, 2, ..., 6 are elementary symmetric polynomials from six variables. In [49] it was shown that the denominator of Π As one can see from (2.37) or Table 2, poles of the form u ij cancel out. In order to demonstrate the advantage of our approach let us compare it with the method invented in [49]. If we take into consideration the constraints on the diagrams then for fixed n the number of summands (possible Y ) in (2.34) is equal to whereas it can be shown that the number of summands when the method of [49] is applied equals to n k=0 (2n)!(2n − 2k)! For the first few numbers of scalars these gives Table 3. So we have a good reason to think that our formula is substantially more efficient for both symbolic and numerical computations.

Asymptotic factorisation and recursion property of the residue
Using integral representation (2.1) it was shown in [49] that if one shifts 2k of 2n rapidities in Π Here we will demonstrate that this property follows from (2.34) in a straightforward way. For a diagram Y we denote byȲ the unique diagram Y =Ȳ which has 4 − |Y | boxes. It is not difficult to see that the arrays of the form and those which can be obtained by permutations not mixing the first 2k diagrams with the last 2n − 2k (i.e. those arrays which contain any diagram together with its partner separately in both groups) give the most relevant contributions in above described clustering limit Λ → ∞ as seen from Table 2. We observe from (2.37) that the both cases λ = µ and λ = µ though seem different, in fact give the same outcome: ( P (u i + Λ, λ|u n , µ)P (u i + Λ, λ|u m ,μ)P (u n , µ|u i + Λ, λ)P (u m ,μ|u i + Λ, λ) ) −1 × (2.53) P (u j + Λ,λ|u n , µ)P (u j + Λ,λ|u m ,μ)P (u n , µ|u j + Λ,λ)P (u m ,μ|u j + Λ,λ) Hence it is quite straightforward to observe that contributions of arrays of type (2.52) are given by Due to such a nice factorization, summing over all allowed choices of diagrams we arrive at the desired formula (2.51).
In [49] it was demonstrated that .

Fermions
This section is analogous to the one before, with the only difference that we consider fermions instead of scalars. More precisely, we write down the analogues of ADHM equations that upon localization lead to (3.1). As a result, we recover also a combinatorial representation for Π (n) mat (see (3.2) together with (3.12)). By using our formula, we can perform explicit computations and study different properties of Π (n) mat .

Matrix equations and their consequent combinatorial expression
In this section, we will consider the MHV case with no scalars, N φ = 0, but equal number of fermions and antifermions N ψ = Nψ = n: where, like for scalars, we have a multi-integral over the three kinds of nested Bethe Ansatz roots of a SU (4) spin-chain [35,44], a k , b j and c k and the same functions g(x) and f (x) (2.2) [26,50]. In this case, too, we want to apply localization in order to obtain a combinatorial representation. It turns out that from the point of view of localization it is more natural to consider a slightly different quantity With simple manipulations one gets As earlier ≡ i/2 and the vanishing factors should be suppressed in the product with a prime. For this case the ADHM analogy suggests that we have to introduce five vector spaces: • v andv: each one is a n complex dimensional space associated with the parameters v 1 , v 2 , . . . , v n andv 1 ,v 2 , . . . ,v n respectively • a, b, and c: each one is a n complex dimensional space associated with the parameters a 1 , a 2 , . . . , a  We introduce five transformations, three of which T a , T b , T c are auxiliary while the remaining two T v , Tv are diagonal matrices corresponding to genuine symmetries. As before the additional transformation T ∈ C * related to the parameter is introduced. The denominator of Π It is easy to see that the matrix equations Proceeding as in the case of scalars we find that the set of admissible diagrams are those depicted in Fig.5. Our final result for fermionic case can be formulated as 1 .., Yv n }. Y v (Yv) are chosen from the first (second) row of diagrams listed in Fig.5. In addition it is required that the total number of boxes in upper, middle and lower lines (related to the spaces a, b and c) contain precisely n boxes each. In terms of Young diagrams the summands in (3.12) can be represented as The factors P (a, λ|b, µ) are listed in Table 4, which also can be represented by the formula where the upper indices m and n take the values v andv.

If n = 1 we have four fixed points given by pairs of diagrams
Inserting these expressions into (3.15) we will find F {0v,3v} , the contributions of remaining three fixed points are found similarly: Inserting these into (3.12) and recalling (3.2) we obtain which coincides with the result in [50].
In [50] it was shown that Π (n) mat can be represented as where the numerator is a polynomial in v i andv i . For n = 2 we have 28 fixed points: {0 v , 0 v , 3v, 3v} and those obtained by the permutations of their first two and last two entries. We used (3.12) with (3.13) and the Table 4 to derive Π 0v, 0v, 1v}. Using mathematica we were able to compute the polynomial N (3) explicitly. Unfortunately the result is too lengthy to be presented here 2 . For n = 4 and n = 5 we have 2716 and 31504 fixed points and mathematica gives the expression as a huge sum over the fixed points.

Asymptotic factorisation and recursion property of the residue
Let λ be any of the diagrams Y v 1 , ..., Y v k andλ a diagram from Yv 1 , ..., Yv k having 3 − |λ| boxes. Similarly let µ ∈ {Y v k+1 , ..., Y v n } andμ ∈ {Yv k+1 , ..., Yv n } is such that 3−|µ|. Using Table 4 one can check that where i, j ∈ {1, 2, ..., k} and a, b ∈ {k + 1, k + 2, ..., n}. We will denote by Y Λ a particular choice of Y such that for any i = 1, ..., k, | Y v i | = 3 − | Yv p(i) | with p(i) being some permutation of 1, ..., k. It is easy to see that for such a choice of Y Λ an analogous property for k + 1, ..., n holds automatically. So from (3.23) and (3.13) we get One can see from (3.12) that In [50] it was demonstrated that . (3.26) This result too can be easily obtained by using our formula for Π (n) mat . Indeed, as can bee seen from Table 4, the arrays Y with nonzero residues Table 4 for the residue we get . (3.28) Summing over the diagrams (see (3.12)) we recover (3.26).

Young diagram representation for MHV, NMHV and N 2 MHV amplitudes
According to [52], inside the hexagonal Wilson loop in N = 4 SYM the factor accounting for the matrix structure can be again written as a multi-integral over the three kinds of nested Bethe Ansatz roots of a spin-chain a i , b m , c j , where i = 1, . . . , K 1 , m = 1, . . . , K 2 , j = 1, . . . , K 3 [35,44]: with the same functions f (x) and g(x) (2.2). After the usual specification ≡ i 2 and shifting the integration variables a i , b m and c j by the last expression becomes Nψ, the prime on the product symbol again indicates that all the vanishing factors must be ignored. Here u are the rapidities of the scalars and v α andv β are the rapidities of the fermions and anti-fermions, respectively. N φ , N ψ and Nψ are the numbers of scalars, fermions and anti-fermions. K j j = 1, 2, 3 satisfy the following conditions (see [52]) where 0 ≤ r b ≤ 4 is the R charge carried by the bottom pentagon. For MHV and N 2 MHV the parameter r b is fixed and takes the values r b = 0 and r b = 4 respectively.
In NMHV case r b may assume any of the five allowed values. The MHV case with 2K 1 = 2K 3 = K 2 = 2n, N ψ = Nψ = 0, N φ = 2n is considered in grate details in section 2 while the MHV amplitudes with K 1 = K 2 = K 3 = n, N ψ = Nψ = n, N φ = 0 are considered in section 3. Here we treat the general case.

Matrix equations
Let us start by constructing a more general system of matrix equations corresponding to the integral representation of Π (N ψ N φ Nψ) mat (4.2). As a result, by means of localization, we will get an efficient combinatorial procedure for the evaluation of these integrals. Similar to the case considered in section 3 here too it is more natural to consider a slightly different quantity defined as where As the integrand of (4.2) suggests, one needs to introduce six spaces: • u, v,v 1 that are N φ , N ψ and Nψ dimensional complex vector spaces respectively. They are connected to parameters u 1 , u 2 , ..., Construction of fixed points goes parallel to the cases discussed in previous cases (see section 2.3). In fact the general case under consideration is a simple combination of the purely bosonic and the case with several fermion antifermion pairs. The complete set of allowed diagrams is depicted in Fig.7. Remind that a box of a diagram located on the dotted lines a, b or c represents a basis vector in the respective space. A fixed point is represented by N ψ + N φ + Nψ diagrams from Fig.7. Since the dimension of u is N φ we must have N φ diagrams from the second row. The dimension of v andv are N ψ and Nψ respectively hence we need N ψ diagrams from the first and Nψ diagrams from the third row. In addition to match the dimensions of spaces a, b and c we should have in total K 1 boxes on first dotted line, K 2 boxes on the second and K 3 on the third lines.
To apply localisation we need the tangent space of M at the fixed points. As usual we start with the total unconstrained space δM then "subtract" subspaces corresponding to equations (4.8)-(4.10) and auxiliary transformations.
The characters of the spaces v, u andv are (the same latter is used for both the space and its character) The dual characters u * , v * ,v * are obtained by substitution and the conjugates are obtained by replacing the summands by their inverses. Examining the structure of the diagrams we get convinced that the summands in equations (4.24) explicitly are given by In view of above decompositions the detailed structure of the character (4.22) takes the form Nψ k,l=1 where the summands, derived from (4.22) and (4.25)-(4.27), are listed in Table 5.

Representation of Π
as a sum over diagrams Due to localization for given N φ , N ψ and Nψ the matrix part of the hexagonal Wl is a sum over fixed points: Specifically the sum is over all collections of N φ + N ψ + Nψ diagrams of the form .., Yv Nψ } with entries taken from the list given in Fig. 7. As the notation indicates Y v i , Y u i and Yv i are diagrams taken from the first second or third row in Fig.7 respectively. In Y the total numbers of boxes on the upper, middle and lower lines are K 1 , K 2 and K 3 respectively.
The summands F Y are given by where P (a, λ|b, µ) are displayed in Table 6. Table 6.
Examining numerous cases (some of them are presented in subsections 5.1-5.4) we got convinced that the denominator of has the form We have succeeded to generalize the recursion properties (2.55) and (3.26): where m 1,2 are defined in (4.7) for pairs Π mat related by above recursions. We do not present proofs of these recursion relations, since structurally they are similar to the proofs of simpler cases considered in sections 2 and 3.
In the upcoming subsections we will use our combinatorial formula to compute Π (N ψ N φ Nψ) mat explicitly for some fixed values of r b , N ψ , N φ , Nψ. For some cases 3 we have checked our results against (4.1) by performing numerical integrations.
Here there are 12 fixed points where e 1 and e 2 are the elementary symmetric polynomials in u 1 and u 2 .
• The case with N ψ = 2, N φ = 1 and Nψ = 1 thus K 1 = K 2 = 2 and K 3 = 1. We have 27 fixed points and the final result is where s 1 and s 2 are the elementary symmetric polynomials in v 1 and v 2 . The denominator is given by (5.2).
Here e 1 and e 2 (s 1 and s 2 ) are the elementary symmetric polynomials in u 1 and u 2 (v 1 and v 2 ). The denominator is given by (5.2).

Conclusions and perspectives
It would be interesting to find a physical interpretation of the ADHM-like moduli space we have constructed. Perhaps the identification of the real ADHM equation counterpart of the matrix equations (2.9-2.11) would have some significance as well.
Another achievement of this paper may be consider that of a general approach for passing from an integral representation with some group-theoretical structure to a combinatorial sum over Young diagrams. Thanks to the peculiar two SU (4) matrix structure of the fermions and antifermions contributions, the papers [35,41,50] have re-summed the leading contributions to those from gluons (and their bound states) at strong coupling so that to give the same Thermodynamic Bethe Ansatz results as (classical) string theory ( [50] furnishes by the same method also subleading corrections, waiting for one-loop confirmation). This computation resembles the poles contributions of the instanton partition function of N = 2 SYM [54] in the so-called Nekrasov-Shatashvili (NS) limit [60] (and similarly for the subleading correction to the NS limit, computed in [61,62]). If this represents a second way to TBA (with respect to the ordinary one [63])), which surprisingly stems from FFs, a third one can be counted as the massive Ordinary Differential Equation/ Integrable Model (ODE/IM) correspondence which [64] has recently shown to 'solve' the dual (classical) string theory thanks to the full-fledged quantum integrability structures: not only T -, Y -systems and TBA [45][46][47], but also the more fundamental Q-functions with relative functional and integral equations. These structures have been derived from the discrete (Ω-and Λ-) symmetries acting on the gauge (or differential equation) moduli space (with the extension of a twist or angular momentum for the string solution). In this perspective, the FF series is incorporated in a full integrability structure in its strong coupling and hence the (exact) all coupling expressions, we are dealing with here, will acquire even more importance as a possible (second) quantization of the massive ODE/IM correspondence. In a nutshell, the extension (at all couplings) and rôle of the Q-functions shall be punctually scrutinized as these are the most fundamental objects on the integrability side and the closest to the ODE wave function according to [65].
Yet, the OPE series is much more complicated than the NS one and, in particular, its strong coupling seems insensible to many details of the weak or all coupling regime (the presence of the scalars, for instance). Maybe this is a positive point in favor of the quantization of massive ODE/IM correspondence as could be argued by looking at the simple and elegant structure of the next to leading expression in the NS regime [61,62]. In fact, how to quantize the TBA is a long-standing question, but gauge or string theory may know the answer.