Superconformal block from holographic geometry

We explicitly construct the holographic dual configuration for the four dimensional N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 superconformal block containing half-BPS scalar primary operators by considering its full AdS5× S5 dual geometry. We extend the embedding space formalism and the related Harmonic analysis to general d-dimensional sphere Sd, and obtain precisely the R-symmetry contribution to the half-BPS scalar superconformal blocks, which we refer as “R-symmetry block”. We also observe that the R-symmetry quadratic Casimir operator can be mapped to BC2 Calogero-Sutherland system Hamiltonian, such that R-symmetry block is in terms identified as its bound state wave function.


Introduction and summary
Among the various developments in studying AdS/CFT correspondence, one of the most common approaches has been reformulating various field theoretical observables into geometrical objects in their dual gravitational theories. This approach is particularly powerful when the class of observables considered are constrained by their underlying symmetries,

JHEP07(2020)028
and often provides more efficient and intuitive ways to compute them. Moreover, such a geometrization of field theoretical observables usually yields interesting alternative representations which lead to new physical insights or non-trivial mathematical connections.
The so-called "conformal blocks" or the closely related "conformal partial waves", 1 which are the universal building ingredients of the conformal correlation functions, precisely belong to this class of observables. They play the role of harmonic functions in conformal field theories, and their explicit form is governed kinematically by the wellknown conformal Casimir equation [1,2]. Finding the holographic dual configuration of d-dimensional conformal partial wave in the d + 1 dimensional Anti-de Sitter space (AdS) can be regarded as geometrizing the conformal Casimir equation, indeed it has close relation with the AdS harmonic equation. This task was performed in an elegant paper [3], and the resultant configuration is now known as "geodesic Witten diagram". As its name suggests, the interaction vertices of geodesic Witten diagrams are restricted to move along the AdS geodesics connecting the boundary insertion points of the CFT primary operators. In a somewhat parallel but equally exciting development, it was discovered that conformal Casimir equation can be directly identified with the eigenvalue equation of BC 2 Calogero-Sutherland quantum integrable system [4][5][6], hence the conformal block and the eigenfunction. The BC 2 Calogero-Sutherland eigenvalue equation can be further mapped to so-called Heckman-Opdam hypergeometric equation, whose solution can be constructed in terms of the Harish-Chandra series. Using this chain of relations, the conformal blocks can be constructed as the scattering solutions to the BC 2 Calogero-Sutherland system and their properties such as monodromy can be systematically studied. We will review some of these details in the later section.
It is interesting to ask if we can extend the story we summarize so far about the conformal block/partial wave into full string theoretic constructions, in other words we need to consider the compact manifolds as well as the non-compact AdS spaces. This also requires us to generalize the conformal symmetries to much richer superconformal symmetries, and consider the holographic dual configurations which reproduce the superconformal blocks/partial waves. In this work, we focus on the prototype of AdS/CFT correspondence, namely the exact correspondence between four dimensional N = 4 super Yang-Mills theory in the planar limit and Type-IIB superstring theory on AdS 5 × S 5 background. In the nascent days of AdS/CFT, various Witten diagram computations have been performed and matched with the appropriate correlation functions, see e.g. [7], [8] and [9] for review. Here we would like to fill the gap in the literature by explicitly constructing the holographic dual configuration that reproduce the superconformal blocks involving exclusively half-BPS operators. This is the simplest type of the four dimensional N = 4 superconformal blocks, whose corresponding superconformal Casimir equation has been constructed in [10,11]. For this particularly simple choice, the superconformal Casimir equation can be separated into two parts: one for global conformal symmetries, and the other one for R-symmetries. To geometrize this equation, we first extend the embedding formalism for AdS space [12] JHEP07(2020)028 to general d-dimensional sphere, while some of the details are quite similar there remain various interesting subtleties as we will discuss. 2 In particular, we will generalize the split integral representation of AdS harmonic function to the spherical one for arbitrary spin, and demonstrate it satisfies the appropriate equation of motion. Using this and combining with the AdS contribution to explicitly construct the holographic dual of the desired superconformal partial waves. Moreover, as an interesting observation, we also notice that the quadratic Casimir operator for the R-symmetries can also be mapped to BC 2 Calogero-Sutherland system Hamiltonian. This times due to the compact R-symmetry group, the eigenfunctions correspond to the bound state solutions instead of the scattering ones.
This work is organized as follows: In section 2, we review the necessary details about the superconformal partial waves/blocks for half-BPS scalar primary operators in four dimensional N = 4 SYM, and their superconformal Casimir equations. This also allows us to fix the notations used throughout this work. In section 3, we systematically construct the holographic dual of the half-BPS superconformal block by extending the embedding space formalism to general d-dimensional sphere. Using the split representation of spherical Harmonic function, we obtain what we called the R-symmetry block which is a solution to the R-symmetry part of superconformal Casimir equation. In section 4, we discuss how superconformal Casimir equation for half-BPS primary operators can be mapped to two copies of BC 2 Calogero-Sutherland system. We also review related details about Heckman-Opdam systems and Harish-Chandra hypergeometric functions. We have streamlined our main text, and relegated various computational details in several appendices.

Four-point functions with half-BPS operators in N = 4 SYM
In this section, we will summarize the essential details about four-point correlation functions in four dimensional N = 4 Super Yang-Mills (SYM) with SU(N ) gauge group in the so-called planar limit such that N → ∞ and 't Hooft coupling λ = g 2 YM N is kept fixed and arbitrary.

Four-point functions of half-BPS operators
Let us focus on the single-trace half-BPS operators in four dimensional N = 4 SU(N ) SYM, which are the gauge invariant operators constructed from the elementary scalar fields φÂ(x) (Â = 1, . . . , 6) transforming in the adjoint representation of SU(N ): Here TÂ ∈ C 6 is an auxiliary complex null vector for performing the tensor contraction and ensuring the traceless condition. Since the scalar fields φÂ(x) transform as the vectors of SO(6) R ∼ = SU(4) R , the single trace operator (2.1) describes a rank-p SO(6) R symmetric JHEP07(2020)028 traceless tensor. In terms of the Dynkin labels of SU(4) R , (2.1) belongs to [0, p, 0] representation. This operator (2.1) is annihilated by half of the Poincaré supercharges, so it has the scaling dimension ∆ = p which is protected from any quantum correction.
This is the lowest component of a four-point correlation function of superfields considered in [14,15]. If we expand the four-point function (2.2) in the s-channel OPE, the conformal and the R-symmetries allow us to fix the form of G( The conformal cross ratios u and v are defined as Similarly, we can introduce the R-symmetry cross ratios σ , τ as Here for the R-symmetry cross ratios, we follow the notation of [14]. 4 The cross-ratio dependent function G a,b (u, v, σ, τ ) can be expanded in terms of the lowest component G a,b ∆,J,l,s (u, v, σ, τ ) of the superconformal block associated with the superconformal primary operator with scaling dimension ∆ , spin J and the SU(4) Cartan numbers [s, l − s, s]. More explicitly, the expansion of four point correlation function G a,b (u, v, σ, τ ) into conformal blocks G a,b ∆,J,l,s (u, v, σ, τ ) associated with the exchanged operator labeled by (∆, J, l, s) has the following form: ∆,J,l,s (u, v, σ, τ ) .
(2.7) 3 We can also use the Poincare embedding to express T 2 ij as y 2 ij (A.19). 4 Note that the definitions of σ, τ are slightly different from the conventional one introduced in [10]. The two different R-symmetry cross ratios are related by In (α1, α2) , this relation is translated to The advantage of the definition is that the conformal block and the R-symmetry block can be treated on the equal footing.

JHEP07(2020)028
The exchanged operators should be in the irreducible representations of SU (4) The coefficient c(∆, J, l, s) is the product of the three-point function coefficients of two external half-BPS operators and one intermediate superconformal primary operator.

Superconformal Casimir equation for the lowest component
The superconformal block is defined as an eigenfunction of the suitable quadratic Casimir equation derived from PSU(2, 2|4) superconformal group. As explained in [10,11,14], the superconformal Ward identities can then relate the lowest component G a,b ∆,J,l,s (u, v, σ, τ ) of the superconformal block with its superconformal descendants. By combining the superconformal Ward identities and the quadratic superconformal Casimir equation, we obtain the quadratic differential equation for G a,b ∆,J,l,s (u, v, σ, τ ) [10,11,14], where C ∆,J,l,s is the eigenvalue of the quadratic Casimir operator C for PSU(2, 2|4) given by 5 The eigenvalue given above consists of two contributions: the first two terms come from the eigenvalue for SO (2,4) Casimir operator with the shift: ∆ → ∆ + 4 ; the remaining two terms come from the eigenvalue of SO(6) Casimir operator. Let us give explicit expressions of the differential operators D and D f . For convenience, we will decompose the differential operator D into the contributions from SO(2, 4) and SO(6) subgroups, are the quadratic Casimir equations for SO(2, d) and SO(d + 2) given by:Ď JHEP07(2020)028 The differential operator D f consists of the first order derivatives with respect to z i and α i , (2.14) This operator comes from the fermionic part of the quadratic superconformal Casimir equation and is anti-symmetric under the exchange of variables z i ↔ α i .

Superconformal block for the long multiplets
As described in [10,11] (see also [14,16]), the solutions to the quadratic superconformal Casimir equation are classified by the representations of the exchanged operators under the superconformal symmetry group PSU(2, 2|4). In this paper, we focus on the case that the exchanged state belongs to the so-called long multiplet with ∆ , J and the SU(4) representation [s, l − s, s] . The solution to the superconformal Casimir equation associated with a long multiplet has the following form [10,11,14,16]: where the explicit overall factor F (z i , α i ) is given by If we perform the similarity transformation on D and D f using F (z i , α i ), we can simplify the Casimir operator in (2.9) into: (2.17) Therefore, the reduced superconformal block H a,b ∆,J,l,s (z i , α i ) satisfies the quadratic Casimir equation for SO(2, 4) × SO(6) , This fact implies that H a,b ∆,J,l,s (z i , α i ) can be further decomposed into The conformal symmetry dependent partǦ a,b ∆+4,J (z i ) is the usual four dimensional conformal block given by

JHEP07(2020)028
On the other hand, the R-symmetry dependent part:Ĝ a,b l,s (α i ) is called R-symmetry block, and the explicit expression is [10,14,16] Here, P (α,β) m,n (ŵ 1 ,ŵ 2 ) can be expressed in terms of the Jacobi polynomial whereŵ 1 andŵ 2 are defined through: The solution (2.22) to the quadratic Casimir equation is obtained by specifying the boundary condition,Ĝâ n (x) is the Gegenbauer polynomial. The holographic dual to the global conformal block has been constructed by considering the so-called geodesic Witten diagram in five dimensional AdS space [3], our main goal in this paper is to extend this result and construct a holographic dual configuration along the spherical space which reproduce to the Rsymmetry block (2.22), hence the full superconformal block H a,b ∆,J,l,s (z i , α i ) (2.19).

R-symmetry block from harmonic analysis on S d+1
In this section, we would like to generalize the widely used embedding space formalism for AdS space to the spherical geometry to construct the holographic dual to the R-symmetry block, for an introduction, see [17]. In particular, we will present an generalization of the AdS split representations to the spherical harmonic function and the formulas of threepoint functions on S d+1 . As we will see, these are ingredients for constructing a holographic dual to the R-symmetry block. Moreover, we should stress here that while our main focus will be AdS 5 × S 5 geometry, our analysis however is valid for the sphere of arbitrary dimensions S d+1 .

Embedding formalism for S d+1
Similar to the AdS case, it is convenient to employ the embedding formalism to describe functions on S d+1 for the construction of a holographic dual to the R-symmetry block. The embedding formalism for sphere can mostly be constructed by performing analytical continuations of the physical quantities in AdS one. Therefore, here we only summarize the major differences between the AdS and the spherical embedding formalism. For more details of the spherical case, please see appendix A.
On the other hand, the sphere does not have the boundary, but let us recall that in the previous section, the R-symmetry cross ratios are constructed from the complex null vectors TÂ associated with the half-BPS operators. Therefore, we will regard the complex null cone as the spherical counterpart of the AdS boundary [18]: Note the complex null cone is invariant under the (complex) dilatation transformation, as TÂ is now complex. We should note that the main difference between the AdS-space and the spherical embedding formalisms is that there does not exist a notion of a geodesic interpolating between the bulk space S d+1 to the complex null cone in contrast with the AdS case. This distinction plays an important role to construct a holographic dual of the R-symmetry block.
Bulk-to-boundary propagator. Having clarified what we meant by the "boundary" and "bulk' coordinates in the spherical embedding space formalism, let us next consider the counterparts of the AdS bulk-to-boundary propagators. The AdS scalar bulk-to-boundary propagator is given by whereČ ∆,0 is an overall normalization constant. This function satisfies the quadratic Casimir equation for SO(2, d) with the eigenvalue ∆(∆−d) . The eigenvalue can be mapped to the spherical one by a direct replacement ∆ ↔ −l. Therefore, through replacements of embedding space coordinatesXǍ ↔XÂ , PǍ ↔ TÂ and ∆ ↔ −l, a natural counterpart of the AdS scalar bulk-to-boundary propagator is obtained as whereĈ l,0 is an overall normalization constant remained to be fixed. The null condition (3.4) is required in order to describe the symmetric traceless tensors which belong JHEP07(2020)028 to the irreducible representations of SO(d + 2) . This polynomial is known as the scalar spherical harmonic (SSH) function. The space of such homogeneous polynomials has been extensively discussed in [18]. When d = 4 , the SSH function corresponds to the SU(4) symmetric traceless representation labelled by the Dynkin label [0, l, 0]. The generalization to the spinning case can easily be done. In the AdS case, if we introduce the additional null vectors WǍ , ZǍ ∈ R 2,d satisfing the AdS spinning bulk-to-boundary propagator is given by whereČ ∆,J is an overall normalization constant. As in the AdS case, we can further introduce the additional auxiliary complex null vectors UÂ , RÂ ∈ C d+2 satisfying sphere : The spherical spinning bulk-to-boundary propagators are given by where the parameter s now plays the role of "spin" andĈ l,s is an overall normalization constant. When d = 4 , the tensor harmonic function corresponds to the SU(4) representation labelled by the Dynkin label [s, l − s, s].

Split representations of harmonic functions
Now we would like to construct the spherical harmonic functions using the spherical bulk to boundary propagators we just listed, this can be regarded as a generalization of split representation of AdS harmonic function [12]. First of all, let us consider a harmonic function on the sphere S d+1 which depends on two bulk pointsX 1 ,X 2 and two polarization null vectors U 1 , U 2 . The function satisfies where∇ 1,Â is the covariant derivative with respect toX 1 , U 1 defined in (A.14). For general s, a solution to the equation (3.12) can be constructed by generalizing the AdS split representation [12] toΩ l,s (X 1 , U 1 ;X 2 , U 2 ), as we will do in this subsection.
Scalar case. As a warm up, let us consider the scalar s = 0 case. In this case, the harmonic functionΩ l,0 (X 1 ;X 2 ) only depends on z ≡X 12 ≡X 1 ·X 2 from the SO(d + 2) symmetry. The equation of motion (3.12) becomes

JHEP07(2020)028
and a solution to this equation is given by the Gegenbauer polynomial (3.14) This solution can be reproduced by a split representation ofΩ l,0 (X 1 ;X 2 ), Here the integration measure is defined by where the symbol Vol(C × ) expresses the volume for the complexified dilatation transformation (3.5), and T r and T i are the real and imaginary parts of T , respectively. Here,l is a negative integer defined as:l which is a counterpart of the shadow scaling dimension∆ = d−∆. If we define a conformal integral in analogy to the AdS case, the definition (3.17) ofl can be fixed by requiring the invariance of the integral (3.16) under the complexified dilatation transformation (3.5) . 6 In fact, the eigenvalue of the quadratic Casimir operator is invariant under the replacement l →l . Furthermore, the use of the bulk-to-boundary propagator withl in the split representation guarantees that the integral (3.15) is invariant under the action of SO(d + 2) . The integral (3.15) is performed in appendix D, and the final result is again proportional to the Gegenbauer polynomial where the symbol Vol(S d ) expresses the volume of the d-dimensional sphere S d . In this way, the split representation (3.16) reproduces (3.14).
Tensor case. The extension to the tensor case is straightforward. By using the tensor bulk-to-boundary propagator (3.11), we can construct the split representation ofΩ l,s : Another motivation to usel is that the Gegenbauer function satisfies the following relation [19]:

JHEP07(2020)028
where (x) n is the pochhammer symbol and the differential operatorD R responsible for index contraction is defined in (A.23). As shown in appendix D, the harmonic function (3.19) after the integration takes the form: 20) and by construction, satisfies the equation of motion (3.12), as the bulk to boundary propagators in the integrand do.
Let us see the s = 1 case in more details. The equation of motion (3.12) can be rewritten as where the action of∇ 2 1 on any function f (z) is given by: The integral (3.19) can be performed, as done in appendix D following the procedure similar to the one described in appendix C of [12]. By performing the integral (3.19), we obtain where g (0) l,1 (z) and g (1) l,1 (z) are given by We can check that (3.23) and (3.24) together satisfy the equation of motion for s = 1 (3.21).
In general, performing the integral in (3.19) requires rather cumbersome calculations. However, when we take a special choice of the polarization vectors U 1 , U 2 such thatX 2 ·U 1 = 0 =X 1 · U 2 , we can easily perform the integral and obtain (3.26)

JHEP07(2020)028
The details of the derivation for (3.26) are given in appendix D . As a consistency check, we can see that the expression (3.26) reproduces the s = 0 case (3.18) and g (0) l,1 (z) in (3.24). Note that the expression (3.26) does not solve the equations of motion (3.21), forΩ l,s (X 1 , U 1 ;X 2 , U 2 ) to satisfy the equation of motion (3.12), not only a leading term g (0) l,s (z) but also a higher-order terms g (r) l,s (z) (1 ≤ r ≤ s) are needed (For the s = 1 case, see (3.21)).

Three-point functions
Next, we will consider the simplest three-point functions involving only three spherical bulkto-boundary propagators, this is the spherical analogue of three-point contact Witten diagram in AdS space, and serves as the building block for higher point correlation functions.

Scalar-scalar-scalar case
Let us first see three-point functions involving three scalar bulk-to-boundary propagators, which are given bŷ where the integration measure is defined in (B.4). Temporarily, we assume that all quantum numbers l i (i = 1, 2, 3) are positive integers, and we will analytically continue them in the final expression (3.32) obtained after the integration. As shown in appendix B, the above integral can be evaluated asÂ where the overall constantÂ l 1 l 2 l 3 and α ijk are given bŷ The above formula (3.28) can also be expressed in terms of the anti-symmetric C-tensor [7,8]. From the overall coefficientÂ l 1 l 2 l 3 , the integration (3.27) with positive integers l i does not vanish only if each α ijk satisfies In the next subsection, we will evaluate an integral representation of the R-symmetry block by employing the split representation (3.19) of the spherical harmonic function. For this purpose, it is useful to rewrite the overall constant (3.29) in terms of gamma functions aŝ (3.32)

JHEP07(2020)028
This expression can be analytically continued to complex values l i except for the poles l i = −n − 1 , n ∈ Z ≥0 of the gamma functions appearing in the numerator. Note that the Gamma functions appearing in (3.32) are consistent with (127) in [12] with the replacement −l i = ∆ i by using the reflection formula Γ(z)Γ(1 − z) = π sin πz , we obtain:

Scalar-scalar-tensor case
Next, let us consider the three-point functions involving tensor spherical bulk-to-boundary propagator. Here we will consider the scalar-scalar-tensor case given bŷ where Y 3 ≡ ∂ U 3 · ∂X and we assume l 2 ≥ l 1 ≥ s 3 . We can also consider other threepoint functions that the derivatives with respect toXÂ act on either of K s l 1 ,0 (X; T 1 ) and K s l 2 ,0 (X; T 2 ) . By performing partial integrals, these vertices are reduced to the form is divergence free, and the sphere does not have the boundary hence the additional boundary contributions. The s = 1, 2 cases have been computed in [8].
This integral can be easily computed by using the relation (B.24), and the final expression for (3.34) can be written as: where the differential operatorD T 3 (T 1 , R 3 ) is defined in (B.25), and we have introduced In the second equation, we have used the fact The overall constantÂ l 1 l 2 l 3 ;0,0,s 3 is given bŷ .
The Gamma function factors of the three-point function above are also consistent with (131) in [12]. JHEP07(2020)028

Holographic dual to the R-symmetry block
Finally, let us use the three-point functions we just obtained to construct a holographic dual to the R-symmetry block. For this purpose, we first give a holographic dual to the R-symmetry partial waves. Suppose that the external and exchanged states transform in the R-symmetry representations [s i , l i − s i , s i ] and [s, l − s, s], respectively. Then a possible holographic dual of the R-symmetry partial waves is the following four-point function of K s For l i = s i = 0, this four-point function is an eigenfunction of the quadratic Casimir equation for SO(d + 2) by construction. In particular, we do not restrict the interaction vertices to move along the geodesics on the sphere as the integration regions to describe the R-symmetry block holographically. This follows from the fact that there does not exist a notion of geodesics interpolating the bulk space S d+1 to the complex null cone (3.4), hence integrating over the entire S d+1 generates the only SO(d + 2)-preserving configuration. As we will see, after performing integrals in (3.39), the R-symmetry partial wave (3.39) has a form which is a linear combination of the R-symmetry block (3.48) and its shadow (3.49), as in the AdS case. Remarkably, we will observe that the shadow part vanishes after imposing the selection rule (3.56) on the exchanged operator after taking an appropriate normalization (3.57). In this way, our R-symmetry partial wave (3.39) also describes the R-symmetry block. This situation is different from the AdS case.
Let us give a brief sketch on how we perform the integrals of (3.39) with s i = 0 which is the same method applied in [1]. For this purpose, we first use the split representation (3.19) of the harmonic functionΩ l,s and the formula (3.35) of three-point functions. Then the R-symmetry partial wave becomes:  integers k ij = k ji (i, j = 1, . . . , 4) are defined as We defined the parameters κ 1,2 l and κ 3,4 l as which ensures the conformality of the integral. Therefore, the integral in (3.41) may be regarded as the conformal integral except that the integration region is now over a complex null cone. In fact, as performed in [1], we can also evaluate it through a Mellin-Barnes transform by assuming Here we will skip the details of the calculations, as the steps are somewhat similar to the derivation of AdS Mellin amplitudes (See e.g. [20,21]) and only present the final result which is expressed as a linear combination of the R-symmetry block and its shadow, and can be regarded as one of the main results in our work: where the coefficients c l,s and cl ,s are given by Here we also introduced the notation Γ(x ± y) = Γ(x + y)Γ(x − y) . The R-symmetry block Gâ ,b l,s (σ, τ ) and its shadowĜâ ,b l,s (σ, τ ) are:

JHEP07(2020)028
where the two variable function G(α, β, γ, δ; x, y) is defined as the double power series expansion This function was originally introduced by Exton in [22] and we will call it the Exton function in this paper. Its properties are summarized in appendix C. When s = 0 , the R-symmetry block (3.48) has a simple form We can explicitly check that this function is an eigenfunction of the quadratic Casimir equation for SO(d + 2) (see appendix C for details). In the case of the conformal block, the same expression has been derived in [1].
In addition to the selection rule (3.56), we have to consider the boundary condition of the R-symmetry partial wave (3.46). Here we further require that the R-symmetry partial wave (3.46) has the same boundary condition (2.26) of the R-symmetry block after imposing the selection rule (3.56). To this end, we normalize the R-symmetry partial wave (3.46) by the factor 1/c l,s . We next impose the selection rule (3.56) leads to R-symmetry partial wave (3.46), 1 c l,sŴ because the ratio cl ,s /c l,s vanishes due to the condition (3.56). In this sense, the Rsymmetry partial wave (3.39) also encodes the R-symmetry block. It is important to note the similarity between the double trace operator condition τ = ∆ − J = ∆ 1 + ∆ 2 + 2n , n ∈ Z ≥0 and the selection rule (3.56). An equivalent condition to (3.56) is −l − s = −(l 1 + l 2 ), . . . , −(l 1 + l 2 ) + 2(l 1 − s), which is analogous to the double trace operators, if we identify −l and s with ∆ and J, also −l 1,2 with ∆ 1,2 etc., but with a cutoff on the descendants due to the compactness of the group SO(d + 2). If we take s = 0 , we can show that the R-symmetry block (3.51) reduces to the known result (2.22). In order to see the equivalence between this expression and (2.22), we assumê a ≤b . By using the formula (C.8), (3.51) can be rewritten aŝ where the overall constant is given by (3.60) Therefore, the R-symmetry block (3.51) becomeŝ

JHEP07(2020)028
where we assumedâ ≤b again. While we have not been able to show analytically the equivalence between (3.48), (3.49) and (3.62) for arbitrary l and s, we have explicitly checked their equivalence in appendix E for s ≤ l ≤ 4 , 0 ≤ s ≤ 2 , up to the overall factors. This explicit check is somewhat non-trivial as we also need to perform the summation over the four fold partition of integers for given l (For the explicit expressions, see (E.2)-(E.39)). Note thatâ,b are taken to satisfy the selection rule (3.56).
The d = 3 case. Finally, let us comment on the connection of our result with superconformal field theories in other dimensions. We focus here on a six dimensional (2, 0) superconformal theory which is holographically dual to eleven dimensional supergravity on AdS 7 × S 4 . The associated superconformal algebra is osp(8 * |4) that contains a bosonic subgroup usp(4) so(5) as the R-symmetry.
The four-point functions of the half-BPS operators, that transform in the [2, 0] representation of so(5) , and its OPE structures have been considered by many authors [11,[23][24][25][26]. As in the previous case, the four-point functions are expanded in terms of the so(5) spherical harmonic functions. For a consistency check, we confirmed that our R-symmetry block (3.48) reproduces the known results listed in (B.14) of [10] up to overall constants: (3.63) The R-symmetry blockĜ 0,0 l,s (σ, τ ) corresponds to the so(5) spherical harmonic function Y l+s 2 l−s 2 (1/σ, τ /σ) associated with the usp(4) Dynkin labels [l − s, 2s] . In this way, the R-symmetry block (3.48) works well for the odd d dimension cases.

Superconformal Casimir equation and the Heckman-Opdam systems
In this section, we will establish the relation between the quadratic superconformal Casimir equation (2.18) and the Heckman-Opdam (HO) hypergeometric systems associated with the BC 2 root system. This is a generalization of the similar relation for the conformal Casimir equation discovered in [4,5,27].

JHEP07(2020)028
where the differential operator L BC 2 (k) is The equation (4.1) is characterized by three complex parameters k s , k m , and k l which are associated with the short, middle, and long positive roots of BC 2 root system, respectively. The bracket ·, · is the inner product in the two-dimensional Euclidean space, and λ and ρ BC 2 (k) are defined as where e i are orthonormal vectors satisfying e i , e j = δ ij . As observed in [4,5], any d-dimensional scalar conformal block can be described in the HO system (4.1) associated with the BC 2 root system. This observation can easily be extended to the case of the superconformal block for the long multiplets. In the following discussions, we will explain it.

Superconformal blocks as HO hypergeometric function
We will consider the reduced part of the superconformal blocks H a,b ∆,J,l,s (z i , α i ) for the long multiplets with the SU(4) Dynkin index [s, l − s, s] satisfying (2.18).
The differential operatorsĎ defined in (2.12) and (2.13) are related to the differential operator L BC 2 by performing the following similarity transformations, 7 where variables z i and α i are related tow i andŵ i inĽ BC 2 andL BC 2 by In these HO systems, the each multiplicity functionsǩ ,k take valueš From the similarity transformations (4.6), let us defině Then, the quadratic superconformal Casimir equation (2.18) can be rewritten aš By comparing the HO systems (4.1) with the above equations, the spectral parameterš λ ,λ for each system are taken aš (4.14) In this way, the superconformal Casimir equation (2.18) can be translated to two copies of BC 2 HO hypergeometric equations.
Finally, let us comment on the relation between HO system and Calogero-Sutherland (CS) model. It is well known that a given HO system is related to a CS model associated with the same root system by a similarity transformation [28]. We can show that the eigenvalues of the CS Hamiltonians corresponding toĽ BC 2 andL BC 2 are given by Since the eigenvalue of the conformal Casimir operator takes continuous values, the conformal case corresponds to the scattering problem of the CS model. On the other hand, the R-symmetry case describes a bound state problem because its eigenvalue (4.17) takes discrete values.

R-symmetry part
The relation between the conformal block and the HO hypergeometric system has been discussed in it and focus on a solution of the HO hypergeometric system (4.13) for the R-symmetry part. As we will show, the generalized Jacobi polynomial (2.24) or equivalently the Rsymmetry block can also be described in the HO hypergeometric system associated with the BC 2 root system.
The solutions of the HO hypergeometric system are constructed from the liner combinations of the Harish-Chandra series Φ BC 2 (λ,k;ŵ) and its BC 2 Weyl transformed ones [28] (See also [5]). The BC 2 Weyl group is generated bŷ where we used the relation (4.13). 8 As in the conformal case, we consider a linear combination of Φ BC 2 (λ,k;ŵ) and the Weyl transformed HO series Φ BC 2 (λ,k;ω 1ŵ ) withk m = 1, Since the BC 1 Harish-Chandra series Φ BC 1 (λ i ,k;ŵ i ) is proportional to the Jacobi polynomial, after requiring that l and s satisfy the selection rule (3.56). In this way, the R-symmetry blocks (2.22) for the long multiplets can be described in the HO hypergeometric system associated with the BC 2 root system.

Acknowledgments
We would like to thank P. Heslop

JHEP07(2020)028
A The details of embedding formalism of S d+1 In this appendix, we will present more details of the embedding formalism on S d+1 used in the main text.

A.1 Bulk side
Let us first consider a symmetric traceless rank-s tensor field on R d+2 with the components hÂ 1 ···Âs (X) . The tensor field can be restricted to S d+1 by using the projection operator: whereĜÂB is the induced metricĜÂB Such that a null vector with respect to the induced metric: describes a unit sphere. In fact, the projection operator guarantees that (Ph)Â 1 ···Âs (X) is transverse to the hypersurfaceX 2 = 1 i.e.
In other words, (Ph)Â 1 ···Âs (X) is the spherical analogue of so-called symmetric, traceless and transverse (STT) tensor. This condition also implies the tensor field hÂ 1 ···Âs (X) has a gauge symmetry where ΨÂ 1 ...Â s−1 (X) is any rank-(s − 1) tensor field on S d+1 and {. . . } denotes traceless symmetrization of indices, as they project to the same symmetric traceless tensor on S d+1 . In this sense, the term containing sub-leading ΨÂ 1 ...Â s−1 (X) is regarded as an un-physical mode of hÂ 1 ···Âs (X) . In the direct analogy with the AdS space tensor, there is an efficient way to implement index contraction on tensor fields on S d+1 . That is to express such a field as a polynomial of the auxiliary null vector UÂ: Since S d+1 is embedded in the Euclidean space R d+2 , the null vector UÂ should be complex i.e. UÂ ∈ C d+2 . Furthermore, in order to restrict the tensor field on S d+1 , we assume JHEP07(2020)028 The rank of the tensor field hÂ 1 ···Âs (X) is then translated into the homogeneity of the polynomial h s (X, U ), as encoded in the equation: The (projected) components (Ph)Â 1 ···Âs (X) can be reproduced by using the differential operator:KÂ We can check that the above differential operator (A.9) satisfieŝ where (x) n is defined as (x) n = Γ(x+n) Γ(x) . Therefore, we obtain Note that if the operatorKÂ acts on such a polynomial h s (X, U ), this operator is effectively simplified toKÂ Finally, let us introduce the covariant derivative on S d+1 defined bŷ (A.14) HereΣÂB are Lorentz generators in the differential representation acting on null vectors UÂ . The covariant derivative acting on the projected tensor field (Ph)Â 1 ···Âs (X) can be written as∇B where we used the transversality (A.4) of (Ph)Â 1 ···Âs (X) .

JHEP07(2020)028
A.2 "Boundary" side Next, we will discuss the embedding formalism for the "boundary" side of the sphere, which is described by the complex null cone (3.3) with complex coordinates {TÂ}. Let FÂ 1 ...Âs (T ) be an arbitrary tensor field on the complex plane C d+2 . The physical tensor fields on the complex null cone surface (3.3) can be obtained by restricting the tensor fields FÂ 1 ...Âs (T ) on C d+2 to the complex null cone surface (3.3). This is performed by using the projection operator ΠÂ 1 ...ÂsB In fact, from the fact TÂ ∂TÂ ∂y i = 0, the projected tensor field satisfies the transversality condition Here for convenience, we introduced a parameterization of the null vector This parameterization is regarded as the counterpart of the Poincaré boundary coordinates for AdS d+1 , but notice that we now have complex entries instead. The tensor fields on the complex null cone can be obtained by using the projection: The tensor fields on the embedding space can also be described as a homogeneous polynomial of the null vector RÂ satisfying T · R = 0, As in the previous subsection, we can construct a differential operator similar toKÂ defined in (A.9)D Therefore, the (projected) components (ΠF s )Â 1 ...Âs can be reproduced from F s (T, R) (A.26)

B Spherical harmonic functions
In this appendix, we will give some properties of the scalar and tensor spherical harmonic functions.

B.1 Scalar case
Let us first see the scalar spherical harmonic (SSH) function K s l,0 (X; T ) given by (3.7). By using the expression of the covariant derivative (A.14), we can check that (3.7) is a solution to the equation∇ or equivalently satisfies the quadratic Casimir equation for SO(d + 2) with the eigenvalue l(l+d) . In the following discussion, we will show the orthogonality relation and three-point functions involving only the SSH functions.
Orthogonality relation. The SSH function (3.7) satisfies the orthogonality relation, where the overall constant Z l is given by The integration measure is symmetric under SO(d + 2) transformations and defined as The relation (B.2) is regarded as the spherical counterpart of joining two bulk to boundary AdS propagators to obtain the two-point CFT correlation functions. The orthogonality of the SSH functions easily follows from the SO(d + 2) symmetry. In fact, if we let g be an element of SO(d + 2) , I ll (T 1 , T 2 ) transform as I ll (g · T 1 , g · T 2 ) = I ll (T 1 , T 2 ) . (B.5)

JHEP07(2020)028
Now let us use the formula (B.15) to evaluate (B.10). Note that the term in the expansion of s l (|∂ y |) gives non-vanishing contribution only if j = L 123 = l 1 +l 2 +l 3 2 . Therefore, A l 1 l 2 l 3 (T i ) can be rewritten aŝ Note that this expression does not vanish when α 231 is zero or positive integers. Since ∂ 2 y K s l 2 ,0 (y; T 2 ) = ∂ 2 y K s l 3 ,0 (y; T 3 ) = 0 for y ∈ R d+2 , we obtain Since this formula is symmetric under the permutations of l i , when l i ∈ Z ≥0 ,Â l 1 l 2 l 3 (T i ) does not vanish only if α 231 , α 312 , α 123 ∈ Z ≥0 . The above proof is a higher dimensional generalization of the proof given in [32].

B.2 Spinning case
Next, let us see the tensor spherical harmonic function K s l,s (X, U ; T, R) given in (3.11), and the homogeneity and the transversality condition The eigenvalue (B.19) of the Laplacian precisely matches with the one of the rank-s tensor spherical harmonic function on S d+1 [33]. The harmonic function (3.11) In the final equation, we used (B.27) and

C Exton function
In this appendix, we will summarize some properties of the so called Exton function [22] (see also [35]), which recently becomes increasingly relevant in constructing conformal blocks [36]. This function is defined as a double power series This double power series can be expressed as a linear combination of two Appell hypergeometric function F 4 , where Appell hypergeometric function F 4 is defined as This hypergeometric function satisfies the following partial differential equations: D α,β,γ 1 F 4 (α, β; γ, γ ; x, y) = 0 , D α,β,γ 2 F 4 (α, β; γ, γ ; x, y) = 0 , (C.4)

JHEP07(2020)028
where t =X 1 ·X 2 + 2 −2t andt is defined as This integral has a similar form as (202) in [12]. Next, let us use the finite series form of the Gegenbauer polynomial Then, the integral (D. 3) becomeŝ where we defined the integral This integral can be rewritten as where the integral W l,r is defined as As we will show later, the integral (D. l,s (z) , (D.10)

JHEP07(2020)028
where the coefficients g (r) l,s (z) are functions of z . When s = 1 , the explicit expressions of g (0) l,1 (z) and g (1) l,1 (z) are presented in (3.24), and we can show that the resulting harmonic functionΩ l,1 satisfies the equation of motion (3.21).

(D.17)
A proof of (D.14). Finally, we will show the formula (D.14). First of all, let us note that (D.14) can be expressed in terms of the most simplest case Indeed, by differentiating I(Q) with respect to QÂ , we obtain C d+2 Therefore, in the following discussion, we focus on a proof of the formula (D.18).