Projectors and seed conformal blocks for traceless mixed-symmetry tensors

In this paper we derive the projectors to all irreducible SO(d) representations (traceless mixed-symmetry tensors) that appear in the partial wave decomposition of a conformal correlator of four stress-tensors in d dimensions. These projectors are given in a closed form for arbitrary length $l_1$ of the first row of the Young diagram. The appearance of Gegenbauer polynomials leads directly to recursion relations in $l_1$ for seed conformal blocks. Further results include a differential operator that generates the projectors to traceless mixed-symmetry tensors and the general normalization constant of the shadow operator.

One missing ingredient for the conformal bootstrap program  for correlation functions with operators with spin in d dimensions is the explicit knowledge of seed conformal blocks exchanging mixed-symmetry tensors. Seed conformal blocks are the conformal blocks exchanging a given irreducible representation (irrep) of SO(d), while having a minimal amount of spin in the external operators, such that the conformal block is unique. They are seeds in the sense that conformal blocks for the exchange of the same representation, but for external operators with higher spin, can be derived from them by acting with differential operators that generate the required extra tensor structures, as described in [26,35,36].
Much of the structure of a seed conformal block is encoded by the projector to the SO(d) irrep which labels the exchanged operator. This can be seen by considering the integral expressions of the conformal blocks in the shadow formalism [37][38][39]. Indeed, the lack of explicit expressions for the projectors is the main reason why the seed conformal blocks are still unknown. So far, the projectors and seed conformal blocks in d dimensions are only known for the exchanged operators in the irreps (l 1 ) = ... [38,40,41] and (l 1 , 1) = ... [42]. Expressions for the projectors to the irreps (l 1 , 1, 1) and (l 1 , 2) were given in [43,44]. Table 1 shows all irreps that appear in a correlator of four stress-tensors, and for each irrep the correlator where it appears in a seed conformal block.
In section 2 the projectors for all irreps appearing in table 1 will be derived in a compact form. The length of the first row of the Young diagram l 1 is left unspecified and only appears in the final results as a parameter of Gegenbauer polynomials and in the overall normalization. A consequence of this are universal recursion relations in l 1 for the seed conformal blocks. These recursion relations are shown to hold for the seed conformal blocks of all the correlators in table 1 and conjectured to hold for any seed conformal block of bosonic operators. These relations are derived in section 3, making use of the integral representations of the conformal blocks in the shadow formalism, where the projector appears explicitly. Section 4 presents final remarks.
The appendices contain several other general results related to projectors and seed conformal blocks. In appendix A we derive a differential operator that generates projectors to traceless mixed-symmetry tensors for Young diagrams of two rows. This operator is a generalization of a well known operator for traceless symmetric tensors. Appendix B deals with a relation between projectors of different irreps that arise from certain index contractions. In appendix C we state some of the longer explicit results for projectors. Appendix D computes the normalization constant that arises when a shadow transformation is performed on an operator in any three-point function that can appear in a seed conformal block. In appendix E the OPE limit of general seed conformal blocks in the shadow formalism is computed to facilitate comparisons to other results. Finally, in appendix F we explain the relation between projectors to traceless mixed-symmetry tensors and tensor harmonics on the sphere. Included with this work is a Mathematica notebook containing the derived projectors.

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Correlator Exchanged SO(d) irreps as seed conformal blocks Table 1. Exchanged irreps in correlators of currents and stress-tensors. Each line shows only the irreps exchanged as a seed block. For a correlator in a given line, the irreps in the lines above can also be exchanged, but those conformal blocks can be constructed by acting with derivatives.
2 Projectors to traceless mixed-symmetry tensors

Review of projectors to traceless symmetric tensors
As an inspiration for the ideas ahead let us briefly review how to quickly derive the projector to traceless symmetric tensors encoded in a simple polynomial, following [38]. The projector is defined by its symmetry . (2.3) Due to the first property the projector can be implemented as a function of a single variable by contracting with two auxiliary vectors z a ,z b ∈ R d , π (l) (z,z) = z a 1 . . . z a l π (l)a1...al,b1.

Projectors for Young diagrams with two rows
Next let us consider the irreps which are labeled by Young diagrams with two rows λ = (l 1 , l 2 ). The total number of boxes in the Young diagram λ will be denoted by |λ|, so in the case at hand we have |λ| = l 1 + l 2 . Again we want to construct these projectors as polynomials and then impose differential equations. In this case the projector can be encoded in a polynomial depending on four vectors Our strategy is now to first implement the mixed-symmetry property and then the tracelessness. Along the way we will always make sure to keep the construction symmetric under exchange of z i andz i , This means it is enough to impose conditions on one side of the projector. The mixed symmetry or Young symmetrization of a tensor f in the symmetric representation of the irrep (l 1 , l 2 ) amounts to the following two conditions 1 (2.10)

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This can be easily implemented by constructing π λ (z 1 , z 2 ,z 1 ,z 2 ) out of structures Q s that have this property (2.14) The structures Q s have weight l 2 in z 2 andz 2 , and zero weight in z 1 andz 1 . We are already familiar with such transverse structures from the construction of conformally invariant correlators in embedding space. These structures can be built out of transverse building blocks. In this case one can use JHEP07(2016)018 respectively. Combining these three conditions yields for this case Furthermore, both functions have to be even to avoid a square root in z 2 1z 2 1 . These polynomials are determined by solving a coupled system of second order differential equations arising from the tracelessness conditions (2.20) After discussing a first example, we will describe the algorithm that is used to solve these equations in section 2.2.3. Then the structures of the individual families of projectors will be presented. Finally, the overall normalization constants c λ appearing in (2.14) will be computed in section 2.4.

Birdtrack notation
To construct Young symmetrized structures Q i it is convenient to use birdtrack notation, with lines denoting index contractions. Using multiple copies of the same vector results in a group of symmetric indices, which is denoted by a white bar while a black bar denotes antisymmetrization The building blocks transverse to z a 1 and toz b 1 that were defined in (2.15) will be denoted by the following symbols (2.23) The δ ab was defined to indicate that short lines connecting and do not stand for H ab . The notation should be clear after the first examples which are given both in birdtrack and explicit notation.

Projectors to the irreps (l 1 , 1)
The projectors to SO(d) irreps with Young diagrams of shape ...
were already derived in [42]. We include them here for completeness. The structures are (2.24)

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Imposing the tracelessness conditions (2.20) results in many differential equations for the functions f 1 (t) and f 2 (t). For example the term proportional to (z 1 · z 2 )(z 1 ·z 2 ) in the condition ∂ ∂z 1 · ∂ ∂z 1 π λ (z 1 , z 2 ,z 1 ,z 2 ) = 0 is with l 1 -independent coefficients w s,n (d, t). That is, we write This assumption turns out to be true for all computed projectors, with the sum ranging from n = l 2 up to n = 2l 2 . The overall normalization of all functions f s (t) in a projector will be chosen such that 2 The independence on l 1 of the coefficients is relevant for the derivation of recursion relations for conformal blocks. We will use the following relation, which is a version of the Gegenbauer differential equation for the Gegenbauer polynomial appearing explicitly in (2.27), (2.30) Using this relation repeatedly on a polynomial of the form (2.28) of order l 1 or lower, one can remove all but the two highest derivatives and write it in the following way

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where N is the highest n appearing in the sum in (2.28). Plugging this ansatz into the differential equations that arise from demanding tracelessness results in a system of linear equations. In all cases discussed below, these linear systems have a unique nontrivial solution for u s,i (d, l 1 ) and v s,j (d, l 1 ), as long as one chooses N large enough. After finding the solution, (2.30) can again be used to bring the solution into a form with l 1 -independent coefficients.
The mixed-symmetry property amounts to relations such as (2.10), but now such relations hold between any two of the three sets of symmetrized indices. With the same argument as in the case of two-row Young diagrams, this property can be imposed on π λ by requiring The first two conditions can again be implemented by using the building blocks defined in (2.15), now also allowing z 3 ,z 3 in the place of z 2 ,z 2 . The third condition can be implemented by Young symmetrizing in z 2 and z 3 (andz 2 ,z 3 ). As before, this leads to the following ansatz separating structures depending on z 2 , z 3 ,z 2 ,z 3 and functions of t Tracelessness now amounts to the old equations (2.20) and to the three new ones there are only two possible structures, due to the antisymmetry between z 2 , z 3 andz 2 ,z 3 , The resulting functions f i (t) are similar as in the case of irreps  Starting with this example, whose Young diagrams have shape ... , it becomes helpful to consider some tensor products to make sure that one finds the correct number of structures. As explained above, the allowed structures are constructed from the building blocks in (2.15), including also the dependence on z 3 ,z 3 in the place of z 2 ,z 2 . In order to satisfy the condition z 2 · ∂ ∂z 3 Q i = 0 of (2.36) it is enough to Young symmetrize in z 2 and z 3 according to the Young diagram (l 2 , l 3 ) = (2, 1). In the case at hand that means to consider Although this expression is a mixed-symmetry tensor, it is not traceless, hence it is in the reducible representation ⊕ . (2.42)

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The only way to contract the building block T a 1 a 2 to expression (2.41) is This term contains a contraction with the primitive SO(d) invariant δ a 1 a 2 . Thus, the number of possible contractions with T a 1 a 2 is the same as with δ a 1 a 2 . We conclude that the number of independent contractions of (2.41) to the corresponding expression withz 2 , z 3 , with H and T building blocks in the middle, is given by the multiplicity of the scalar representation • in the SO(d) tensor product 4 Since one can also use multiple copies of the building blocks V andV , which form a symmetric representation, one can then take a further tensor product with a one-row Young diagram of any length. Thus, the total number of birdtracks that one has to consider is the multiplicity of the scalar representation in the tensor product Two of these eight birdtracks are combined into a single structure (Q 7 below), when requiring the building blocks to respect the left-right symmetry z i ↔z i . The resulting structures are

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The dots here indicate Young diagrams with more than one row, which cannot contribute to the multiplicity of the scalar representation in (2.45). The solution of the tracelessness conditions can then be written as Possible contractions with T a 1 a 2 are (2.52) Hence the number of birdtracks to consider is given by The individual contributions are The projector to the family of irreps (l 1 , 3, 1) is given in appendix C.

Normalization of the projectors
The normalizations can be computed by using that the projectors have a term with a known normalization, namely the term projecting to generic mixed-symmetry tensors, from which the traces are subtracted by further terms. This is the only term in π λ (z 1 , z 2 , z 3 ,z 1 ,z 2 ,z 3 ) containing a factor (z 1 ·z 1 ) l 1 (z 2 ·z 2 ) l 2 (z 3 ·z 3 ) l 3 . For concreteness let us consider the Young . In this section all lines in birdtracks denote just simple contractions, without any of the definitions of (2.23) (2.58) The antisymmetrizations appearing here are defined as (2.59) and the normalization of the first term is [46] where h i is the height of the i th column of the Young diagram λ. Using the identity on both antisymmetrizations in (2.58), one can isolate the terms containing (z 1 ·z 1 ) l 1 , (2.62) In general this factor 3 · 2 will be h i and the part of the birdtrack with z 2 , z 3 ,z 2 ,z 3 always matches the z 1 andz 1 independent part of Q 1 (z 2 , z 3 ,z 2 ,z 3 ). Comparing to (2.37) this means that the normalization constant appearing in the projectors is

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and can be easily computed in each case using Doing this for each family of projectors one finds

Recursion relations for seed conformal blocks
In this section recursion relations in l 1 will be read off from the integral representation of conformal blocks in the shadow formalism. The recursion relations are based on the observation that the projectors π λ , as computed in the previous section, are linear combinations of just a few (at most l 2 + 1) different Gegenbauer polynomials, does not depend on l 1 , and the sum stops at min(l 1 , 2l 2 ) since C (n) l 1 (t) = 0 for n > l 1 . We showed this for the projectors to the irreps that can appear in conformal blocks of four stress-tensors by explicit computation. This fact can be used to turn the recursion relation for the Gegenbauer polynomials into recursion relations for the conformal blocks.

Classification of seed conformal blocks
We will focus on conformal blocks that are unique given the irreps of the external and exchanged operators. These will be called seed conformal blocks. The origin of this terminology is the fact that conformal blocks involving three-point functions with multiple tensor structures can be derived from these seed blocks by acting with differential operators, using the method of [35]. Uniqueness of a conformal block exchanging the irrep λ = (l 1 , l 2 , . . .) in the channel λ 1 λ 2 → λ 3 λ 4 , means that the three-point functions λ 1 λ 2 λ and λλ 3 λ 4 both have a single OPE coefficient. For such combinations of irreps the tensor product contains only one symmetric tensor (or as special cases one scalar or vector), with multiplicity one and similar for λ 3 , λ 4 . The most trivial case is when λ 1 and λ 2 are scalars, and λ is a symmetric tensor. We will consider the case when λ 1 and λ 2 are symmetric tensors, and λ is a mixed-symmetry tensor. In this case the lower rows of λ, which will be denoted by the Young diagram λ − = (l 2 , l 3 , . . .), must be removed by contraction to λ 1 and λ 2 . To understand this better let us consider three cases.
If λ − appears in the tensor product of λ 1 and λ 2 with multiplicity one, then (3.3) is satisfied and the symmetric tensor on the right hand side is of rank l 1 .
If there are symmetric tensors in the tensor product (3.3), then there are more than one. Consider for example |λ 1 | + |λ 2 | = |λ − | + 1. After removing the lower rows of λ, there is still a tensor product of and the remaining first row of λ, hence there will be symmetric tensors of rank l 1 − 1 and l 1 + 1.
Hence the second case is the only relevant one, leading to the necessary condition for seed conformal blocks (for λ 1 , λ 2 , λ 3 , λ 4 being symmetric tensors) To see this condition in action one can consider the OPE. For example, the OPE of two vector operators has only a single term in the irreps λ = (l 1 , 2) or λ = (l 1 , 1, 1), both of the form

Three-point functions
We will use the embedding formalism and the methods to construct correlators of [45,47]. The embedding space is d + 2 dimensional Minkowski space R d+1,1 , where the conformal group SO(d + 1, 1) acts linearly. Capital letters will denote vectors in this space P i , Z ij ∈ R d+1,1 . The second label on Z ij labels the different vectors required to encode mixed-symmetry tensors, e.g. (z 1 , z 2 , z 3 ) from the previous section could be replaced by (Z 01 , Z 02 , Z 03 ) in embedding space. Contractions of tensors can be written using derivatives acting on vectors Furthermore, boldface letters indicate a set of vectors that are used to encode a mixedsymmetry tensor and the corresponding sets of derivatives, normalized to include the factorial appearing in (3.6), , It is convenient to consider polynomials that encode the correlators without fully implementing the symmetry and tracelessness of the operators. The full correlators can then be obtained by projecting

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Throughout the paper, the external operators will mostly be symmetric traceless tensors, which we will encode by a single null vector instead of projecting to a traceless tensor. Instead of (3.8) we will thus use 9) where Z 2 1 = Z 2 2 = 0 will always be implied. When considering conformal blocks for external operators that are either scalars, currents or stress-tensors, the three-point functions with a single tensor structure are encoded by

Recursion relations from the shadow formalism
Next we recall the formula for the conformal partial wave in the shadow formalism (an overview of the formalism in the case of mixed-symmetry tensors can be found in section 5 of [45]). When using the three-point functions defined above, the exchanged representation JHEP07(2016)018 must be projected to an irreducible representation Here∆ = d − ∆ is the conformal dimension of the shadow operatorÕ and S λ ∆∆ 34 = S λ ∆∆ 34 | ∆→∆ is the constant that occurs when an operator in a three-point function is replaced by its shadow 14) It is computed in appendix D for any three-point function that can appear in a seed conformal block, with the result .
(3.15) We start by inserting the expressions for the three-point functions (3.10) into (3.13), to find Next we wish to insert the expression (3.1) for the projector π λ . The factor z 2 appearing in (3.16) and c λ in (3.1) depend on l 1 , it is convenient to use the following normalization for the conformal blocks 5 By inserting (3.1) into (3.16) we can now write the conformal block as a sum of functions that only depend on a single Gegenbauer polynomial each, The Gegenbauer recursion relation (3.2) implies the following recursion relations for each of these functions where u = P 12 P 34 P 13 P 24 , v = P 14 P 23 P 13 P 24 are the conformal cross-ratios. To use the recursion relations, it is only necessary to know the conformal blocks for l 1 = l 2 up to l 1 = 2l 2 as seeds. Once these are known, one can forget about the conformal integrals that were used for the derivation of the recursion relations and even the precise form of the functions K λ n . Any family of conformal blocks can be mapped to the functions F λ 1 λ 2 λ 3 λ 4 ∆,λ,i using that Hence the conformal blocks for l 1 = l 2 , . . . , 2l 2 can be used in conjunction with the recursion relation to compute the seeds For example, the conformal blocks for exchange of and act as seeds for the family ...
in the following way The recursion relation (3.21) allows us to move down along the arrows. Note that in order for the conformal blocks to satisfy the stated recursion relations, it is crucial that their normalization depends on l 1 , ∆ 12 and ∆ 34 as defined in (3.17). A good method to compare normalizations for conformal blocks obtained via different methods is to consider the OPE limit, which is done for the partial wave (3.13) in appendix E.

Solution of the recursion relation in terms of scalar conformal blocks
For n = 0 the recursion relation (3.21) is equivalent to the one for scalar conformal blocks that was solved for d = 2, 4 in [40]. The new parameter n can be removed from the prefactors by using the variables l 1 = l 1 − n and d = d + 2n,

(3.25)
This implies that the scalar conformal block in d dimensions is a solution of the recurrence relation for seed conformal blocks. Maybe this can be used to solve the recursion relations in terms of known functions. While we do not know if this is actually possible, assume for a moment that the seeds for the recursion relation can be written in terms of scalar conformal blocks as We allowed for arbitrary shifts i, j, k in the three parameters ∆ 12 , ∆ 34 and ∆ and functions f i,j,k that do not influence the recursion relation, i.e. f i,j,k can depend on ∆, u, v and the polarizations of external operators, but not on l 1 , d, ∆ 12 or ∆ 34 . Then the result for arbitrary l 1 would be the same linear combination of scalar conformal blocks with appropriately increased spin l 1 , In [39,45] the two initial blocks of the family (l 1 , 1) were computed explicitly in terms of conformal blocks with l 1 = 0 in higher dimensions, however the block G • • ∆,(1,1) = F ∆,(1,1),1 was given in terms of scalar blocks in 8 dimensions instead of the d = 6 required by (3.27). While the simple relation to scalar conformal blocks assumed in (3.27) might not be true, it is likely that the recursion relations should be solvable in terms of functions similar to scalar conformal blocks in higher dimensions. This correspondence was also observed in the recent paper [48], where the mixed-symmetry seed conformal blocks in four dimensions were computed using a twistor formalism.

Recursion relation for radial coordinates
Here we will discuss how to use these recursion relations for expansions of conformal blocks in the radial coordinates (r, η) of [49][50][51]. These coordinates are related to the cross-ratios (u, v) by and one typically considers expansions of the conformal blocks up to some order m in r The identification of the different parts of the conformal blocks can be performed as illustrated in the diagram (3.24), by starting at the lowest allowed l 1 and using the recursion to find the term that has to be subtracted from the next conformal block to get the seed for the next recursion relation. To derive the recursion relations in (r, η) one has to expand u − 1 2 and v in r Unfortunately the appearance of the term of order r −1 means that the recursion relations decrease the order of the expansion in r. They read (l 1 − n)F λ1λ2λ3λ4 ∆,(l1,l2,l3),n (∆ 12 , ∆ 34 ; r, m) = , which is depicted in the diagram (3.24). To this end the normalizations were matched in the OPE limit.

Concluding remarks
In this paper the projectors to traceless mixed-symmetry tensors that appear in the correlator of four stress tensors were derived in terms of Gegenbauer polynomials. Knowledge of the explicit form of the projectors led us to a single universal recursion relation in l 1 for seed conformal blocks, given by (3.21). Interestingly, the existence of the recursion JHEP07(2016)018 relation does not rely on the complete expressions for the projectors, but only requires that their dependence on l 1 is according to (3.1). Of course this implies that, to show that the recursion relation is truly universal (i.e. holds for any seed conformal block of bosonic operators), it is enough to prove that all projectors to traceless mixed-symmetry tensors can be written as in (3.1).
In order for the conformal blocks to obey the recursion relation, it is required that they are normalized in a particular way. In particular, the normalization constant of the projector as well as the normalization of the shadow operator need to be canceled from the integral expression of the conformal partial wave. To this end the normalization of the shadow was computed. Furthermore, the OPE limit of the shadow integral was analyzed to allow for comparisons to other results.
Another remark concerns the solution of the recursion relations. The new recursion relations (3.21) are generalizations of the recursion relation for scalar conformal blocks of [38] with a new parameter n. This new parameter can be absorbed into l 1 and d by using shifted parameters As a result the recursion relations for general seed conformal blocks are solved by scalar conformal blocks in higher dimensions, suggesting that seed conformal blocks can generally be expressed in terms of such scalar blocks. A similar correspondence was also observed in the recent paper [48] for the case d = 4. Beside this application, we want to stress that the projectors actually play a very important role in CFTs. The most basic example in which they appear is the two point functions of mixed-symmetry operators, Moreover, the projector π λ is a necessary ingredient to compute a conformal block for the exchange of an irrep λ. For example, the OPE limit of any seed conformal block with external operators O i of spins i is always written (here without being precise on the placement of the indices on the π i ) in terms of the two-point function of the exchanged operator as therefore it involves the projector π λ (see appendix E). Of course, also the leading OPE of any other conformal block is written in terms of the projectors since it can be generated by acting with some derivatives on the seed block. From this remark it is clear that the knowledge of π λ is needed to compute conformal blocks from their radial coordinate expansion [49][50][51], since the leading term of the expansion is the leading OPE.
To compute conformal blocks with the shadow formalism it is crucial to know the form of π λ as well, since it appears explicitly in the shadow integral. In this paper we expressed π λ in terms of derivatives of the Gegenbauer polynomial, which encodes the JHEP07(2016)018 projector to traceless symmetric irreps. Therefore, it should be possible to compute any family of seed conformal blocks (for generic l 1 ) in terms of scalar conformal blocks by a direct computation. To achieve this, it is enough to rewrite the integrand of the shadow integral of the seed conformal block (3.16) in terms of derivatives of integrands of scalar conformal blocks. The general result for seed conformal blocks of the family λ = (l 1 , 1) was obtained in this way in [42].

A A mixed-symmetry differential operator
In this section we find an alternative way to generate the projectors into traceless mixedsymmetry tensors for Young diagrams with two rows. The main idea is to generalize the result of [52], where the differential operator was defined in order to generate projectors to traceless symmetric tensor representations One can construct this operator by looking for an operator of weight −1 in z that preserves the space defined by z 2 = 0, i.e.

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Clearly this expression is Young symmetrized (in the antisymmetric representation) and traceless, provided the differential operators D 1 and D 2 preserve the space defined by z 2 1 = 0, z 2 2 = 0 and z 1 · z 2 = 0, namely Furthermore, D 1 must have weight −1 in z 1 and 0 in z 2 , and D 2 must have weight 0 in z 1 and −1 in z 2 . These requirements fix the operators completely as 6 where d mn are differential operators with weight m in the variable z 1 and n in the variable z 2 , defined by Moreover, these operators automatically satisfy for i ∈ {1, 2}. The general normalization factor appearing in (A.4) is fixed asking for the idempotence of the projector, As an example we construct the projector into the representations The projector that is generated in this way is in the antisymmetric representation. It is related to the other expressions derived in the main text by the contraction π λ (z 1 , z 2 ,z 1 ,z 2 ) = n λ z a 1 1 z a 2 2 . . . z , (A.11) 6 There exist higher order differential operators that satisfy the same requirements. The ones that we found are the lowest order ones.

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where the factor n λ appears due to the change from the antisymmetric to the symmetric representation and is defined in (2.60).
It is in principle possible to generalize this method in order to study Young diagrams with more than two rows. However this requires to find a new set of differential operators, which we expect to be lengthy and therefore not very efficient.

B Relating different projectors
This section introduces a useful relation between similar projectors. The main observation is that a contraction between corresponding indices on the left and right of a projector leads to an object that is still traceless and has mixed-symmetry. Hence it should be proportional to another projector, e.g.
The proportionality factor can be found by using that the full trace of a projector is given by the dimension d λ of the SO(d) irrep, which is given by [53] where H(λ) was defined in (2.60). Since both sides in (B.2) can be reduced to the corresponding dimensions by doing full contractions, the missing constant is given by the ratios of dimensions Of course the same relation holds also for the other rows in the Young diagram, e.g.

C More projectors
In this appendix we state the remaining projectors to irreducible representations of SO(d) that can appear in a three-point function with two stress-tensors.

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C.1 Projectors to the irreps (l 1 , 3) For Young diagrams of shape ... the possible combinations of the proposed building blocks are Imposing the tracelessness conditions (2.20) one finds that the functions f i (t) that appear in the projector (2.14) can then be written as In this case the functions f i (t) that appear in the projector (2.14) can then be written as

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C.3 Projectors to the irreps (l 1 , 3, 1) To construct the tensor structures for projectors to irreps ... we need to consider the reducible representation for the shape , given by with the building blocks Note that the second expression here is not symmetric in its indices. However, the third expression is antisymmetric and the second one has a symmetric part, so this will suffice as a basis. From the tensor product ... we conclude that in this case there will be 21 allowed structures, which can be identified with different terms in the tensor product (C.10) D Computation of the constants S λ ∆∆ 12 from the shadow formalism In this section the constant  [42]. In this appendix, the constant is computed for any three-point function that has a single tensor structure and obeys |λ 1 | + |λ 2 | = |λ| − l 1 (as discussed in section 3.1). We start in subsection D.1 by showing that S λ ∆∆ 12 does not depend on the SO(d) irreps λ 1 and λ 2 , by constructing a differential operator that transfers spin between the operators O 1 and O 2 without violating the condition |λ 1 | + |λ 2 | = |λ| − l 1 . In the following subsections S λ ∆∆ 12 is computed for the case of an arbitrary irrep λ = (l 1 , l 2 , l 3 , . . .) and λ 1 = •, λ 2 = (l 2 , l 3 , . . .).

D.1 Spin transfer operator
The shadow operator is hence the three-point function of the shadow is Our strategy will be to find a differential operator which transforms a three-point function with operators O, O 1 and O 2 in the representations λ = (l 1 , l 2 , . . . , l h 1 ) and λ 1 = •, λ 2 = (l 2 , . . . , l h 1 ) , (D.4) into another three-point function with a single tensor structure by transferring spin from the operator O 2 to O 1 . This operator must decrease the homogeneity of a polynomial in Z 2i and increase it in Z 1j by one each, while not changing the homogeneity in P 1 and P 2 . Furthermore, it must preserve the transverseness of the functions, that is This last requirement means that the operator should transfer terms of order into terms of the same kind. Such an operator is given in terms of the differential operator derived in appendix A in d + 2 dimensions (i.e. each d in its definition must be replaced by d + 2), where c is an unspecified constant. The term containing V (Z 3(i+1) ) (3,12) vanishes upon Young symmetrization with V (Z 31 ) (3,12) l 1 . By repeated use of such operators, any three-point function with a single tensor structure can be generated whereÔ 1 (P 1 ) is in the scalar representationλ 1 = • andÔ 2 (P 2 ,Z 2 ) in an irrepλ 2 satisfying |λ 2 | = |λ 1 | + |λ 2 |. In order to compute S λ ∆∆ 12 one can insert (D.9) into (D.1). The action of the operator on both three-point functions is obviously the same, and the resulting S λ ∆∆ 12 is the same that one gets when using the three-point function given in (D.4). In the next two subsections, S λ ∆∆ 12 is computed using this three-point function.

D.2 Young diagrams with two rows
The computations here are done using the conventions of [42] to allow reusing some of their computations. We start with the case of λ = (l 1 , l 2 ). To compute the constant we will consider a three-point function of an operator O in this representation with a scalar O 1 and a symmetric tensor O 2 of spin l 2 . This correlator can be written as where the building blocks m (ij) and k (ijk) are physical space variants of the building blocks H ij and V i,jk defined in (3.12), given by We will also use the notation To compute the ratio in (D.1) it is not necessary to keep track of terms containing z 2 31 , z 2 32 or z 31 · z 32 . Those will be collectively denoted by O(z 3i · z 3j ). Furthermore, it is enough to consider the term of order l 2 in z 2 · z 32 , so terms of order O (z 2 · z 32 ) l 2 −1 can be dropped.

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We need to compute the three-point function of the shadow operator. This is given by where we defined 14) The reason why the trace subtracting terms of the projector π λ (z 0 ; ∂z 0 ) in (D.13) do not contribute is that m (30) ac m (30) bc = δ ab , so these terms are annihilated by the other projector. Not keeping track of terms containing z 2 31 , z 2 32 or z 31 · z 32 means that the trace subtracting terms of the other projector can be ignored as well and we can use instead the Young symmetrizer Y λ (z 1 , z 2 ;z 1 ,z 2 ) = π λ (z 1 , z 2 ;z 1 ,z 2 ) z 2 One needs to compute the conformal integral which is done in subsection D.4 below. Next we do a trinomial expansion of y(z 31 ) and a binomial expansion of y 2 (z 32 , z 2 ) in (D. 16). Note that we consider only the term proportional to (z 2 · z 32 ) l 2 , for which the action of the Young symmetrizer results only in the JHEP07(2016)018 factor S l 1 ,k+b,i , which will be explained below The sums were evaluated by using first that , (D. 20) and finally The factor S l 1 ,k+b,i appearing in (D.18) is given by where we used the following notation for the falling factorial As a next example we will compute the constant for the representations λ = (l 1 , l 2 , l 3 ), λ 1 = •, λ 2 = (l 2 , l 3 ). From the derivation it will also be clear what the result is for arbitrary λ. The three-point function is in this case (D.26) The three-point function of the shadow operator is then (D. 27) To compute this expression one needs the following integral, which can be immediately read off from the result (D.36) (D.28)

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By now it is clear that the computation for Young diagrams with more rows will work out analogously, and that the constant relating a three-point functions of an operator and the one of its shadow for a Young diagram with row lengths (l 1 , l 2 , . . .) and column heights (h 1 , h 2 , . . .) is .

D.4 Conformal integrals
We will use the following conformal integral (for α + β + γ = d) to compute the following integral (for α + β + γ = d) The computation of the derivative also simplifies significantly when considering only terms of order O (z 2 · z 32 ) m . It is enough to consider the contribution of (

E OPE limit of conformal blocks in the shadow formalism
In order for the conformal blocks to satisfy the recursion relation derived above, it is crucial that they have the correct normalization. To compare to other results it is a good idea to consider the OPE limit x a 12 → 0, x a 34 → 0. This can be done in physical space by generalizing a trick from [38]. To this end let us work again in physical space, as in the previous appendix. The shadow operator of an operator O in the irrep (∆, λ) is given bỹ (E.1) For a lighter notation we will choose for the remainder of this appendix to consider Young diagrams with at most three rows λ = (l 1 , l 2 , l 3 ). Furthermore, we will compute the OPE limit for conformal blocks with λ 1 = λ 3 = • and λ 2 = λ 4 = (l 2 , l 3 ). Other configurations can be treated analogously by replacing some of the vectors z 2i by z 1j or z 4i by z 3j .
We want to compute the OPE limit of the conformal partial wave

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where the three-point functions are given by 7 To perform the OPE limit one can use and hence Inserting this into (E.2), and using the definition of the shadow operator (E.1), one finds Doing the limit x a 34 → 0 in a similar way leads to .
(E.7) In order to extract the normalization it is convenient to consider only the term which is of leading order in the building blocks m (24) (x 12 , x 34 ), m (24) (z 21 , x 41 ) and m (24) (z 22 , x 42 ). This term has a prefactor that can be found by considering the birdtrack diagram in (2.58) and removing all antisymmetrizations, resulting in a factor of n λ Finally one can use that (E.10) 7 We are omitting the projector π λ 2 which has no impact on this computation.

JHEP07(2016)018 F Spherical tensor harmonics
In this appendix the relation between projectors to SO(d) irreps and spherical tensor harmonics on the sphere S d−1 is explained. In the case of projectors to traceless symmetric tensors this relation is just the fact that the projectors are encoded by Gegenbauer polynomials (2.7), which are scalar spherical harmonics. In the radial coordinates of [49], the conformal blocks are naturally written in terms of spherical tensor harmonics [51]. Consider a tensor field on S d−1 ∈ R d . We shall work in the embedding space R d and impose transversality x a i t a 1 ...a k (x) = 0 , (F.1) for all i = 1, . . . , k and x a x a = 1. As shown in subsection F.1 below, covariant derivatives on the sphere are just partial derivatives ∂ a = ∂ ∂x a on R d projected to the sphere (over all indices of the resulting tensor). Therefore, the laplacian on the sphere can be written as where P ab = δ ab − x a x b is a projector onto the unit sphere. Using (F.1) on the sphere x a x a = 1, one can simplify this expression to Another interesting differential operator to consider is the quadratic Casimir of the symmetry group SO(d), generated by with M ab c e = i(δ a e δ bc − δ b e δ ac ). The quadratic Casimir is then given by Acting on a tensor obeying (F.1) on the sphere x a x a = 1, it gives where we used expression (F.3) for the laplacian on the sphere. Now consider a tensor field defined by the following contraction with a traceless mixedsymmetry tensor

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Notice that tracelessness and property (2.10) of f implies transversality (F.1) and ∇ a i t a 1 ...a k = 0 , (F.9) where k = i=2 l i and a i can be any of the indices of the tensor. Using formula (F.3) it is easy to obtain ∇ 2 t a 1 ...a k = − l 1 (l 1 + d − 2) − k t a 1 ...a k . (F.10) Moreover, because Σ ab only rotates the indices, and therefore 1 2 Σ ab Σ ab just measures the Casimir of the irreducible tensor. This statement can also be checked explicitly using the definition (F.5). One can also check that (F.7) leads to the expected result Rotational and permutation symmetry imply that I a 1 ...a k = 0 for odd k and I a 1 ...a 2k = q k δ (a 1 a 2 . . . δ a 2k−1 a 2k ) , (F. 16) for some constant q k . This constant can be determined by computing the full contraction 8 δ a 1 a 2 . . . δ a 2k−1 a 2k I a 1 ...a 2k = q k 4 k k! d 2 k = Vol(S d−1 ) .

(F.17)
This is sufficient to verify the orthogonality relation (F.14) and to determine the normalization constant c λ = 1 l 1 ! q l 1 = 4 l 1 d

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We will now show that the last line vanishes and therefore covariant derivatives on the sphere are equal to partial derivatives on R d projected to the sphere. It is sufficient to show that ∂ 2 x a ∂y β ∂y α − Γ γ βα ∂x a ∂y γ ∝ x a , (F. 27) because the last line in (F.26) vanishes due to the transversality condition (F.1). Equation (F.27) is equivalent to ∂ 2 x a ∂y β ∂y α ∂x a ∂y µ − Γ γ βα ∂x a ∂y γ ∂x a ∂y µ = 0 , (F. 28) which can be easily verified using the sphere metric g αβ = ∂x a ∂y α ∂x a ∂y β , (F. 29) and the expression for the Levi-Civita connection Γ γ βα = 1 2 g γµ (g µβ,α + g µα,β − g βα,µ ) .

(F.30)
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