Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation

We present the computation of all the correlators of 1/2-BPS operators in $\mathcal{N} = 4$ SYM with weights up to 8 as well as some very high-weight correlation functions from the effective supergravity action. The computation is done by implementing the recently developed simplified algorithm in combination with the harmonic polynomial formalism. We provide a database of these results attached to this publication and additionally check for almost all of the functions in this database that they agree with the conjecture on their Mellin-space form.


Introduction
Recently the problem of computing the four-point functions of 1/2-BPS operators in planar N = 4 SYM at strong coupling has received new attention, due to the appearance of a surprisingly simple formula for the correlators of arbitrary weights in the Mellin space, conjectured by the authors of [1,2]. This conjecture, based on symmetry and physical assumptions, matched all the examples known then in the literature [3][4][5][6][7][8][9], including the infinite family studied in [9]. However, the fact that all these examples are in one way or another degenerate has required more tests. The first step in this direction was to prove that one of the assumptions on which the work [1,2] was based, namely that the four-derivative Lagrangian in the effective supergravity action vanishes in general, holds. This statement, which was already observed in all the concrete examples [3][4][5][6][7][8][9], was recently proved in [10]. The next test was to check the conjecture on less degenerate cases, which was recently initiated in [11] by computing the 2345 and 3456 correlators. These functions consist of all-different weight operators and are far from the extremality condition. It was shown that these correlators perfectly match with the Mellin formula. Apart from that, the algorithm of computing supergravity correlators from the compactified string action was significantly simplified, opening the possibility for calculating more complicated cases.
The knowledge of more complicated four-point functions could be used to further test the Mellin conjecture [1,2], but also to probe the non-planar spectrum of N = 4 SYM by computing 1/N corrections around the supergravity correlator [12][13][14][15].
In this work we combine the simplifications, obtained in [11], with the harmonic polynomial formalism [16,17]. The latter formalism provides a further significant simplification of the computational algorithm, allowing us to find the four-point functions of CPOs for practically any given weights. We discuss the explicit computation of all the correlation functions with weights up to 8 as well as 7 10 12 17 and 17 21 23 25 . This is a major improvement over the previously available set of correlators. We furthermore check whether these correlators match the Mellin conjecture. We make all the results available in a database attached to this publication. This paper, which follows the notation from [11], is organized as follows: in section 2 we recall how to compute four-point correlation functions from the type-IIB supergravity effective action. In section 3 we discuss how one can use the harmonic polynomial formalism to simplify the computation of the a, p and t tensors. In section 4 we discuss how we use this to compute new correlation functions. In particular, we discuss that all these new examples match the Mellin conjecture from [1,2]. In section 5 we conclude. In the appendix we discuss a way to obtain coordinate-space four-point functions from their Mellin-space form.

Four-point functions from type IIB supergravity
The central objects of our study are the four-point correlation functions in planar N = 4 SYM theory in the strong coupling limit 1 where the x i are the space-time coordinates and the t's are six-dimensional null vectors introduced to keep track of the R-symmetry. Here, the operators 3) where the φ are the scalar operators of N = 4 SYM, the κ k are normalization constants, the λ k,k 1 ,k 2 are fixed real numbers and we omit the total symmetrization over indices as this gets taken care of in the contraction with t vectors in (2.2).
According to the AdS/CFT correspondence [19][20][21] these operators are dual to the Kaluza-Klein modes of compactified type-IIB supergravity on AdS 5 ×S 5 . When computing the four-point function of such operators, the boundary values of the Kaluza-Klein modes act as sources with the generating functional being given by the on-shell value of the string partition function exp(−S IIB ). This approach warrants knowledge of the supergravity action up to fourth order in the fields, which was obtained in [22][23][24]. In these works explicit expressions were obtained for the coupling constants in the effective action in terms of so-called a, p and t tensors which are effectively Clebsch-Gordan coefficients for the representations of SO (6). Nevertheless, since the action, which splits into a contact and an exchange part, is still very complicated and finding the a, p and t tensors requires additional work executing the algorithm to compute four-point functions from this action is far from trivial.
Recently in [11] significant simplifications of this algorithm were obtained: the first simplification makes it possible to bypass having to write the full Lagrangian down, which becomes unpleasant very fast because of the growing number of descendants coupled to scalars in the cubic terms. Instead, the streamlined procedure allows one to obtain the four-point function for a given set of weights straight away. For example, the contribution to the non-normalized correlator, coming from the exchange terms, can be written as the following sum of the s, t and u channels: Here the exchange Witten diagrams, S, V and T are multiplied by the corresponding combination of quadratic couplings ζ and cubic couplings S, T, Φ, A, C and G that were derived in [22][23][24]. The exchange Witten diagrams can be expressed in terms of exchange integrals. A simple method to compute them was developed in [25] and further generalizations appeared in [5] and [7]. The appearing sums run over the possible set of exchange fields in the relevant channel restricted by the SU(4) selection rule.
Finding the contact part also forms no difficulty once all the quartic couplings are computed, as was discussed in [11]. Therefore the main remaining difficulty in computing the correlator sits in finding the a, p and t tensors that form the building blocks of the couplings. Their explicit expressions depend on the chosen weights and their computation becomes complicated quite quickly. In [11] we discussed how one can simplify the procedure outlined in [5] somewhat by carefully analyzing the sums over the symmetric group necessary in doing the tensor contractions.

Harmonic polynomial formalism
However, the computation of the a, p and t tensors simplifies incredibly in the harmonic polynomial formalism, developed in [16,17] and applied to compute some supergravity correlators in [8,9].
It turns out that, after an appropriate normalization, the a, p and t tensors can be expressed as harmonic polynomials Y which carry the non-trivial dependence on the null vectors t i . These functions are generalized eigenfunctions of the SO(6) Casimir operator L 2 , satisfying with C nm being the corresponding eigenvalue. Moreover, one can solve this equation [17] and find that the Y (a,b) nm can be expressed explicitly in terms of Jacobi polynomials P where (. . .) m is the usual Pochhammer symbol and which can be related to the original σ and τ variables via σ = 1 4 (y + 1)(ȳ + 1) and It was discussed in [9], that the product of C tensors appearing in the product of scalar a 125 a 345 , vector t 125 t 345 and tensor p 125 p 345 harmonics for arbitrary weights with fixed exchange leg k 5 are proportional to these Y where the t-dependent prefactor T is given by and k 5 satisfies for some nonnegative integer m respectively. The proportionality coefficients B were worked out in [9]: given a set of weights k 1 , k 2 , k 3 , k 4 ordered such that are nonnegative 3 and an intermediate weight k 5 satisfying (3.8) they take the following form: where α 123 = k1+k2−k3

2
. This now allows for a straightforward evaluation of a 125 a 345 , t 125 t 345 and p 125 p 345 for any weights as all the complicated tensor structure is captured by Jacobi polynomials. To obtain the corresponding tensors in the t and u channel, e.g. a 135 a 245 and a 145 a 235 , one simply reshuffles the t i .

Results
Computation. The aforementioned simplifications, obtained in [11], in combination with the harmonic polynomial formalism reviewed in the previous section allow one to compute any supergravity four-point function of the CPOs in (2.1) of given weights in very little time. We implement the entire algorithm in Mathematica and compute all the non-trivial connected four-point functions k 1 k 2 k 3 k 4 with 2 k 1 k 2 k 3 k 4 8 (94 in total and including 64 previously unknown correlators), which can be found in the database attached to this publication. Additionally, we compute two very high-weight cases, namely 7 10 12 17 and 17 21 23 25 . The computation of these latter correlation functions takes 1 minute and 40 minutes, respectively, on a standard computer.

Verification.
We have checked all of these four-point functions except for the high-weight case 17 21 23 25 for consistency with the structure predicted by superconformal symmetry, see e.g. [26], using the method described in [11]. In order to find this structure one should extract the free part of the correlator: in principle the free part can be computed straightforwardly by performing Wick contractions between extended CPOs. However, in our implementation we find the free part from requiring consistency with superconformal symmetry: after reducing the D-functions the difference between the four-point function and the prediction from superconformal symmetry splits naturally into four parts as and the F i are rational functions of the x i . The undetermined free part is contained completely in F 1 and its vanishing can be used to determine the free part. The vanishing of F 2,3,4 is independent of the free part and provides a non-trivial check that the computed correlator is indeed consistent with superconformal symmetry.
Mellin representation. We can go one step further by checking whether the correlator matches the conjectured formula from [1,2]. In order to do this we first reorder the full correlator k 1 k 2 k 3 k 4 such that the weights satisfy k 1 k 2 k 3 k 4 (which we will distinguish by writing G k1k2k3k4 instead) and then, following [1,2], write the correlator as where the overall prefactor A k1k2k3k4 depends on the weights as in (4.9) of [2]. One can decompose G k1k2k3k4 as G k1k2k3k4 = G free k1k2k3k4 + RH, (4.4) where The conjecture from [1,2] is a simple formula for the dynamical function H in Mellin space up to an overall weight-dependent normalization, which was later determined in [15].
Using the free part extracted in the verification process we can easily find an expression for RH. We can furthermore find H in terms ofD functions by solving a set of linear equations obtained from the decomposition into different tensor components. This expression can directly be Mellin-transformed and compared to the conjectured formulae and we find agreement for all checked correlation functions: these are all the 91 (of which 61 new) correlation functions with weights up to and including 8 except for the three with lowest weight k 1 7, 4 as well as 7 10 12 17 . In particular, we find agreement with the derived normalization function from [15], which in our notation becomes with L as in [2]: Database. With this publication 5 we include a database of all the non-trivial correlators k 1 . . . k 4 with 2 k 1 k 2 k 3 k 4 8. Moreover we include the results for the high weight cases 7 10 12 17 and 17 21 23 25 . For each correlation function there is a subfolder with the name k 1 k 2 k 3 k 4 containing up to five plain txt files: • Fullcorrelatork 1 k 2 k 3 k 4 .txt contains the full correlator as we compute it directly from the action, in the notation from [11], • Freepartk 1 k 2 k 3 k 4 .txt contains the free part as we extract it from consistency with superconformal symmetry (also in the notation from [11]), • Hk 1 k 2 k 3 k 4 .txt contains a coordinate-space expression for the dynamical function H in the notation of [1,2] as it follows from our direct computation, 6 • HfromMellink 1 k 2 k 3 k 4 .txt contains a much shorter coordinate-space expression for H in the notation of [1,2] and has been derived from its Mellin-space form (the construction of which we discuss in the appendix), • RZconjk 1 k 2 k 3 k 4 .txt contains two entries: if this four-point function coincides with the Mellin conjecture the first entry is yes and if not it would read no. The second entry is the value for the overall scaling function f (k 1 , k 2 , k 3 , k 4 ).

Conclusion
In this work we have demonstrated that the simplified algorithm obtained in [11] together with the harmonic polynomial formalism allow one to compute any four-point functions of CPOs of reasonable weights very fast. For example, now the computation of the 7 10 12 17 correlator takes only a minute. Attached to this publication we provide a database of all the correlators with weights up to and including 8. As an application of this new simplified algorithm we confirm for most of these correlation functions that they match the Mellin formula from [1,2], thereby further corroborating this hypothesis. These new correlators go far beyond the previously known set of four-point functions.
One could use the discussed simplifications to derive a closed formula for the coordinate space correlation function from the supergravity action, which could in turn be used to prove the Mellin conjecture in full generality. The main remaining obstruction sits in the fact that the correlation function cannot at present be written as a closed formula in terms of D-functions: the representations appearing in the tensor product decomposition, parametrizing both the couplings and the exchanged fields, depend on the external weights in an intricate way and the exchange Witten diagrams are found algorithmically.
If we manage to rewrite this expression as a linear combination of the Γ p1p2p3p4 , each summand in that sum gives rise to an integral of the form (A.2) after an appropriate identification of the ∆ i with the p i . Therefore, the problem of finding a coordinate-space expression is reduced to finding a linear combination of Γ p1p2p3p4 such that where the c are numbers and the sum is finite over the p i . The representation on the right-hand side of (A.7) is usually not unique, which reflects the fact that theD functions are not independent. The first step towards an expression as in the right-hand side of (A.7) is to rewrite its left-hand side as a pure product of gamma functions and linear factors. This can be done by applying the basic property xΓ(x) = Γ(x + 1) repeatedly to some of the gamma functions in the numerator, such that finally factors in the numerator cancel the denominator. For example This yields the intermediate form with C some constant and x 1,2 a linear factor in s, x 3,4 a linear factor in t and x 5,6 a linear factor inũ. Note that it could happen that each of the three sets of prefactors in s, t andũ might be empty. Suppose first for simplicity that there is only one factor −s+s1 and see that we have succeeded in our goal: by repeating the procedure described above recursively for the list of factors in (A.9) we can find a linear combination of products of gamma functions that are equal to (A.6), such that we have found a representation as in (A.7). Exchanging sum and integral we find that each summand is of the form (A.2) such that after matching the coefficients we find an expression for the inverse Mellin-transform in terms ofD functions. We have applied this algorithm to all the correlation functions in our database and included the result in the database. All cases have been checked explicitly with our coordinate-space results. In some cases it is possible that a more minimal representation exists, but due to the automatized nature of our application this is unavoidable. It is noteworthy that in exchanging the sum and integral during this procedure we do not run into any domain issues that exist for the full correlator as described in [2] that give rise to the free part upon inverse Mellin-transforming. After all, all we are doing is rewriting the integrand using a global property of the gamma function.