A multipoint conformal block chain in $d$ dimensions

Conformal blocks play a central role in CFTs as the basic, theory-independent building blocks. However, only limited results are available concerning multipoint blocks associated with the global conformal group. In this paper, we systematically work out the $d$-dimensional $n$-point global conformal blocks (for arbitrary $d$ and $n$) for external and exchanged scalar operators in the so-called comb channel. We use kinematic aspects of holography and previously worked out higher-point AdS propagator identities to first obtain the geodesic diagram representation for the $(n+2)$-point block. Subsequently, upon taking a particular double-OPE limit, we obtain an explicit power series expansion for the $n$-point block expressed in terms of powers of conformal cross-ratios. Interestingly, the expansion coefficient is written entirely in terms of Pochhammer symbols and $(n-4)$ factors of the generalized hypergeometric function ${}_3F_2$, for which we provide a holographic explanation. This generalizes the results previously obtained in the literature for $n=4, 5$. We verify the results explicitly in embedding space using conformal Casimir equations.

symmetry as well as stringent consistency conditions such as the associativity of the OPE.
Here as well, conformal blocks are an essential ingredient needed for setting up the bootstrap equations involving conformal correlators.
Additionally, knowledge of higher-point scalar conformal blocks opens up the possibility of setting up an equivalent but possibly more efficient alternative to the conventional bootstrap program. In the conventional approach, typically one must study crossing equations of four-point correlators of all operators in the spectrum including those in non-trivial representations of the Lorentz group. As a potential alternative, one can instead aim to solve crossing equations for scalar n-point functions, but for all n [68,54]. Clearly, n-point blocks will play a crucial role here.
Motivated by these considerations, in this paper we will compute all higher-point ddimensional global conformal blocks in a channel which ref. [54] referred to as the comb channel, for external and exchanged scalar operators (see figure 1). This will generalize the series expansion for the d-dimensional five-point block computed in ref. [54] to n-point blocks for any n. The main techniques we will be employing are the AdS propagator identities and geodesic diagram techniques of refs. [44,49,56,57]. Our strategy will be to obtain the holographic geodesic diagram representation of an (n + 2)-point block in the comb channel with the help of various recently derived AdS propagator identities [57]. Higher-point geodesic diagrams, like the four-point case [44], are higher-point AdS diagrams where all bulk integrations are restricted to geodesic integrals, and they are related to higher-point conformal blocks [56,57]. For comb channel blocks, these geodesic diagram representations involve precisely two geodesic integrals. Taking a particular double-OPE limit gets rid of these geodesic integrals, producing an n(n − 3)/2-fold power series expansion of the n-point conformal block. We also verify our result via a proof by conformal Casimir equations.
This paper is organized as follows. In section 2 we illustrate the main computational strategy in the simplest non-trivial example. Particularly, in section 2.1 we derive the holographic geodesic diagram representation of the six-point comb channel block, and in section 2.2 we reproduce the well-known series expansion of the four-point block by taking a double-OPE limit. A second example is provided in appendix A, where we briefly discuss the holographic seven-point block and its double-OPE limit which recovers the series expansion of the five-point block. In section 3, with the help of these examples, we propose a holographic representation of the general (n + 2)-point block, whose double-OPE limit leads to the power series expansion of the n-point block.
is the Pochhammer symbol and we are using the notation for conformal dimensions ∆ i , whereas for the integral parameters k i , as well as the additional definitions 3 so that there are precisely (n − 3) independent k i parameters to be summed over in (1.1).
The other set of integral parameters is denoted j ⟨r|s⟩ , for 2 ≤ r < s ≤ n − 1, where we use the notation ⟨·|·⟩ in the subscript to index the n−2 2 independent j parameters. 4 Combined, this leads to an n(n − 3)/2-fold sum.
The coordinate dependence of the conformal block (1.1) is factorized into a "leg factor", which depends solely on external dimensions and is given by while the rest of the dependence is expressible as an n(n − 3)/2-fold power series expansion entirely in terms of a set of n(n − 3)/2 independent 5 cross-ratios 0 ≤ u i , w r;s ≤ 1 defined as follows, 6 The conformal block is uniquely determine based on the following conditions [22]. They satisfy the multipoint conformal Casimir eigenvalue equations AB are the generators of the Euclidean conformal group SO(d + 1, 1), realized as differential operators built out of and acting on the coordinate x i , with the quadratic Casimir operator defined as (L (i) ) 2 ≡ 1 2 L (i) AB L AB(i) (no sum over i). The eigenvalues are given by C 2 (∆) = ∆(∆ − d) for scalar exchange operators, which will be the case throughout this paper. Moreover, the blocks satisfy the following OPE limit 8) or pictorially, and symmetrically an analogous limit when x 2 → x 1 . Further, the blocks have been normalized such that for u i ≪ 1 for all 1 ≤ i ≤ n − 3, and w r;s ≈ 1 for all 2 ≤ r ≤ s ≤ n − 1, the n-point block has the leading behavior (1.10) The final claim follows trivially from an alternate, rapidly convergent power series expansion of the conformal blocks presented in section 4.

Low-point examples of holographic duals
In this section we provide the simplest non-trivial demonstration of the new techniques. First we will obtain the holographic dual of the six-point block in the comb channel, from which we shall recover the well-known purely boundary power series expansion of the four-point block.
A second non-trivial example is provided in appendix A -it focuses on the holographic dual for the comb channel seven-point block, from which a power series expansion is obtained for the five-point block. In the next section, we will generalize these results to obtain the n-point comb channel conformal block.

Holographic dual of the six-point block
Before discussing the six-point block, let us first establish some notation by reviewing recent results for the five-point block. In ref. [56] a holographic geodesic diagram representation was worked out for the d-dimensional global scalar five-point conformal block W ∆ 1 ,...,∆ 5 a block corresponds to external scalar insertions of dimensions ∆ 1 , . . . , ∆ 5 , and represents the contribution coming from the exchange of scalar representations (and their conformal families) labelled by dimensions ∆ δ 1 and ∆ δ 2 . The precise relation is, is the Euler Beta function, and W is a linear combination of five-point geodesic diagrams (see figure 2 for notation and definition), with the coefficients c given by, The geodesic bulk diagram in (2.2) is a generalization of the geodesic Witten diagrams first obtained in the case of the four-point block [44].
The (holographic representation of the) conformal block satisfies the Casimir equa- Figure 2: How to read comb-channel geodesic bulk diagrams. Geodesic bulk diagrams (also referred to as geodesic Witten diagrams) are AdS Feynman diagrams except with all bulk integrations restricted to boundary-anchored geodesics. Throughout this paper, boundaryanchored geodesics over which bulk points are to be integrated will be shown as red-dashed lines. In the diagram above, they represent the geodesics γ 12 and γ 56 joining x 1 to x 2 and x 5 to x 6 , respectively. Bulk-to-boundary propagatorsK ∆ (x, z) will be shown with solid blue lines, and whenever the conformal dimension ∆ associated with it is not clear from the figure, it will be mentioned explicitly. For example, in the six-point geodesic diagram above, the four bulkto-boundary propagators incident on bulk points to be integrated over boundary anchored geodesics are associated with the conformal dimensions ∆ i of the operator insertions O i as marked. The remaining four bulk-to-boundary propagators emanating from the coordinates x 3 and x 4 have conformal dimensions as displayed next to the blue lines. Unless stated otherwise, the operator O i is understood to be located at boundary coordinate x i . Solid black lines will refer to purely boundary contractions; for example in the diagram above the solid black line joining x 3 to x 4 corresponds to a factor of (x 2 34 ) −∆ D . Finally dotted black lines will stand for factors of chordal distance (ξ(w 1 , w 2 )/2) ∆ where ξ(w 1 , w 2 ) −1 = cosh σ(w 1 , w 2 ) where σ(w 1 , w 2 ) is the geodesic distance between bulk points w 1 and w 2 . We will be using the same propagator normalizations as in ref. [56]; see in particular [56, sec. 2] for the normalization of the bulk-to-boundary propagator as well as the relation between the bulkto-bulk propagator and the chordal distance factor above.
tions (1.7) (for n = 5 and K = 2, 3) and has the expected leading behavior in the two OPE limits. Later in this section we will obtain an alternate geodesic diagram representation for the five-point block involving a single geodesic integral. Moreover, in appendix A, we will provide a power series (purely CFT) representation of the five-point block obtained by taking a so-called double-OPE limit of the holographic dual of the seven-point block. In the rest of this section, we illustrate this procedure for the case of the six-point block where we recover the four-point block in the double-OPE limit.
We first briefly discuss how to obtain the holographic representation of the six-point comb channel block. This discussion is light on technical details; we refer to reader to refs. [56,57] where the systematic procedure is described and several examples are worked out in detail.
The first step involves partially evaluating a particular AdS diagram involving cubic scalar couplings, in this case the diagram where all internal (green) cubic vertices are to be integrated over all of AdS d+1 . In its directchannel conformal block decomposition, the term corresponding to the exchange of three single-trace primaries takes the form where C ∆ i ∆ j ∆ k are the OPE coefficients associated with the dual generalized free-field CFT. This is the six-point conformal block we are after. The trick to extracting the block is to use the split representation for the central bulk-to-bulk propagator as well as certain powerful two-propagator [44] and three-propagator identities [57]. These propagator identities are especially helpful in evaluating AdS diagrams in a way which makes their direct-channel conformal block decomposition manifest [44,57]. Performing the remaining the boundary and spectral integrals arising from the split representation, one manifestly isolates precisely the term shown above, proportional to the right combination of OPE coefficients. The object multiplying this factor of OPE coefficients produces a candidate for the holographic geodesic diagram representation of the block. Technically, the method outlined above is a heuristic derivation; the object obtained must be independently checked to be the claimed conformal block, for example via conformal Casimir equations. We will comment on certain interesting implications of this derivation in section 4 in the context of obtaining multipoint conformal blocks in arbitrary channels.
The procedure outlined above leads to the following geodesic diagram representation, which we claim to be the six-point conformal block in the comb channel, where the W is a linear combination of six-point geodesic diagrams (see figure 2), with the coefficients given by and It is worth pointing out the close structural similarity of the geodesic bulk diagram as well as the functional form of the coefficients with the five-point example. In particular, the five-point geodesic diagram had one "triangle" formed by two blue lines (bulk-to-boundary propagators) and a single dotted line (factor of chordal distance), and precisely one factor of the hypergeometric 3 F 2 function in the coefficient, while the six-point block has two triangles and two factors of 3 F 2 . 7 We will return to this observation in section 4.
To prove the claim above we need to show that the conformal block satisfies the right differential equations (1.7) (for n = 6, K = 2, 3, 4) with the right boundary conditions, expressed in terms of the OPE limits (1.8), reproduced below for convenience The conformal Casimir check is set up in embedding space and the calculation proceeds identically to the conformal Casimir checks for the holographic representations of the fivepoint block [56] and the six-point block in the OPE channel [57]. Since no new ingredients are required, we refrain from including the somewhat lengthy albeit straightforward proof and simply point the reader to Refs. [56,57] for reference. 8 In lieu of that, in the next section 7 The four-point case has no triangles and no factors of 3 F 2 in its double power series expansion. 8 The Casimir check relies on a particular non-trivial functional identity obeyed by the hypergeometric function 3 F 2 which was also previously employed in the five-point case [56], (2.11) The only difference is that due to two factors of the hypergeometric function in (2.8), this identity must now be applied twice, once for each factor.
we will provide a conformal Casimir proof for the power series expansion of the six-point block which will be obtained later.
In the remainder of this section, we would like to focus on the OPE limits (2.10) of the six-point block (2.6)-(2.7). If this is indeed the six-point conformal block as claimed, taking a single OPE limit should produce an (alternate) holographic representation for the five-point block involving a single geodesic integral, and taking a further OPE limit should reproduce the power series expansion of the four-point block purely in terms of boundary coordinates.
We will utilize this strategy in the next section to go from the holographic representation of the (n + 2)-point conformal block in the comb channel to produce an explicit power series expansion of the n-point comb channel block. For illustrative purposes, a seven-point to five-point block example is provided in appendix A.

OPE limit of the six-point block
Generically, all OPE limits in this paper will involve integrals of the following kind: which is straightforward to derive (assuming convergence of the integral). In this section we focus on the OPE limit x 2 → x 1 of the six-point block. Due to the symmetry of the six-point block, we need only check one of the OPE limits in (2.10); the other one follows immediately from a simple relabeling. Using (2.12), we have We are interested in the leading behavior as x 2 → x 1 , so we set the non-negative integral parameter k 1 = 0 above. Comparing with the first line of (2.10), it is sufficient to show that this holographic representation is indeed the five-point conformal block expected to be obtained in this limit, that is (2.14) This is easily checked by showing that the RHS above, which we call V , satisfies the differential equations and boundary conditions of a five-point block. Specifically, with V reducing to four-point blocks in the OPE limit, (2.16) Just as discussed near the end of section 2.1, the conformal Casimir check (2.15) is once again straightforward to show using directly the techniques of ref. [56] and the functional identity (2.11). In the interest of keeping the length of this paper reasonable, we will refrain from presenting the technical details.
The OPE limits (2.16) are also straightforward to work out. Interestingly, the first OPE limit in (2.16) leads to the (holographic) geodesic bulk diagram representation of the fourpoint block, and the second furnishes a (boundary) power series expansion. We discuss these limits next.

Recovering the four-point block from the six-point block
Let's first discuss the limit x 3 → x 1 . In this limit, using (2.12) we find that V is proportional to (x 2 13 ) ∆ δ 2 ,δ 1 3 +k 2 where k 2 is summed over non-negative integers. Thus the leading order contribution comes from setting k 2 = 0, which gives The free sum over j is easily performed to give Now let's consider the other OPE limit in (2.16). As x 6 → x 5 , the leading contribution is given by with the subleading contributions suppressed by higher positive powers of x 2 56 . Recalling from figure 2 that lines joining points on the boundary of the Poincaré disk represent factors of the form (x 2 34 ) j etc., this limit can be written explicitly as where the conformal cross-ratios are defined to be Up to the expected overall factor of (x 2 56 ) ∆ δ 3 ,56 , (2.20) is of the same form as (1.1) except for a different set of coordinate labels and conformal dimensions. More precisely, where the leg-factor W 0 was defined in (1.5). Up to the overall factor of (x 2 56 ) ∆ δ 3 ,56 , this is precisely the power series expansion of the appropriate global scalar four-point block [12], thus confirming the second line of (2.16).
To conclude, we emphasize the main results of this section: We obtained and verified a holographic representation of the six-point block in the comb channel (2.6)-(2.9), and in the double-OPE limit x 2 → x 1 , x n → x n−1 for n = 6, we recovered the explicit power series 9 The standard trick to do that is as follows: We expand Then extending the upper limit of the binomial sum above to infinity with impunity and switching the order of ℓ and j sums, perform the summation over j in (2.20) to obtain a power series expansion in powers of (1 − v).
expansion of the four-point block (2.19). In appendix A, we provide another example of this -the holographic dual of the seven-point block leading to the power series expansion of the five-point block. In the next section, we will generalize this result to obtain the power series expansion of the n-point comb channel block from a similar double-OPE limit of the holographic dual of the (n + 2)-point comb channel block.
3 Multipoint block in the comb channel 3.1 Holographic dual of the (n + 2)-point block and its double-

OPE limit
The low-point examples of five-and six-point blocks in the previous section, along with the seven-point example in appendix A allow us to make a guess for the holographic representation of any multipoint block in the comb channel. We conjecture the holographic representation of the (n + 2)-point comb channel block (n ≥ 3), labelled as follows, to be given by W, which is the following linear combination of geodesic bulk diagrams (see figure 2 for the graphical notation and below for more notes), where the indices ⟨r|s⟩ represent the n−2 2 values the subscript on j can take. The coefficients are given by (for n ≥ 3) , (3.4) and the additional definitions A few remarks are in order regarding the conjecture for the (n + 2)-point block above: • For clarity we have split up a single geodesic bulk diagram in (3.2) into a chain of constituent factors. The first factor should be familiar to the reader from the geodesic diagram representation of a four-point block [44] (see also (2.18)). The second factor, corresponding to n−2 2 contractions (in fact, a perfect graph) between boundary points x 2 , . . . , x n−1 , appears in the holographic dual for all n ≥ 4 (i.e. for the six-point block and higher). The third factor, which represents a product over (n − 2) pairs of bulk-toboundary propagators, is present in the holographic dual for all n ≥ 3 (i.e. five-point block and higher).
• All these constituent factors are to be merged together and should be understood as having been drawn on the same Poincaré disk. The bulk points w, w ′ , which are to be integrated over the two boundary anchored geodesics have been marked explicitly to emphasize that the bulk-to-boundary propagators are incident on precisely the same bulk points; thus there are only two geodesic integrals to be performed. If this conjecture is true, its double-OPE limit, x 1 ′ → x 1 , x n ′ → x n should lead to the n-point block (for n ≥ 4) described in figure 1, up to some expected overall scaling factors, where, using (2.12), 10 the explicit representation can be worked out to be (for n ≥ 4) x t+2 x 1 x n where we now additionally impose the identifications (1.4). Once again the chain of Poincaré disks above is interpreted as explained for the geodesic diagram representation. The coefficients themselves simplify to (for n ≥ 4), where λ n+2 is given in (3.4) and we employed (1.4) to write the coefficient above compactly. 11 The coordinate dependence of the putative conformal block is captured pictorially in (3.7).
Recalling the notation from figure 2, one can easily convert this to conformal cross-ratios as follows, where the "leg factor" W

Conformal Casimir check
Having established the OPE limits for the n-point conjecture, we now turn our attention to the multipoint Casimir equations (1.7), repeated below: where X r ∈ R d+1,1 are embedding space coordinates which upon taking an appropriate section give the corresponding Poincaré coordinates x r . In the rest of this section, x r will refer to a boundary coordinate so that X r will be a null coordinate in embedding space with X r · X r = 0 (no sum over r Expand the multipoint Casimir operator First, the following action of the quadratic Casimir on the conjectural conformal block (3.7) is easily checked: for all 1 ≤ i ≤ n and for all n. This follows from the explicit conformal dimension assignment of (3.7) and the following obvious identity where as explained in figure 2, the solid lines joining together boundary points are to be interpreted as a boundary contractions of the form (x 2 ij ) −∆ . The action of the cross-term in (3.18) on the putative conformal block (3.7) involves terms of the form with 1 ≤ r < s ≤ K fixed, and 1 ≤ a 1 < a 2 < . . . < a n−2 ≤ n with a 1 , a 2 , . . . , a n−2 / ∈ {r, s}.
Like in the previous subsections, the pictorial representation above is split into a product of Poincaré disks for clarity, but should be understood as merged into a single diagram. We are using the symbols ⟨a|b⟩ as indices labelling conformal dimensions with ∆ ⟨a|b⟩ = ∆ ⟨b|a⟩ .
Based on convenience, we will sometimes index a i using the subscripts i = 1, . . . , n − 2, and at other times directly as a i = 1, . . . , n with the restriction that a i ̸ = r, s.

Discussion
In this paper, we used the holographic principle, particularly its theory-independent kinematic aspects, to obtain for the first time explicit expressions for a class of multipoint con- There are various avenues for further investigation. In this paper, we restricted ourselves to studying n-point comb channel scalar conformal blocks involving solely scalar exchanges.
To be able to set up an alternative n-point conformal bootstrap involving scalar n-point functions for all n, one must also have in hand higher-point scalar blocks involving exchange of other non-trivial representations of the Lorentz group. It should be possible to generate n-point comb channel blocks involving both external and internal spin exchanges by, for example, operating on the blocks obtained in this paper via weight-shifting operators [27].
It should also be possible to generate higher-point spinning geodesic diagram representations using the AdS differential operators of ref. [34]. Various recursive techniques, when supplemented with the results of this paper, may also turn out to be fruitful.
For setting up the n-point conformal bootstrap, one also needs n-point blocks in channels other than the comb channel. The number of topologically distinct channels, not related via conformal transformation or simple relabeling, grows quickly with n. Thus it is likely inefficient to compute multipoint conformal blocks one specific channel at a time. On the other hand, it is conceivable there exist some version of "Feynman-like" rules for writing out conformal blocks, akin to Feynman rules for Mellin amplitudes [69,70,71], which can be worked out once and for all. We hope the explicit expressions for the n-point comb channel block we obtained in this paper will help elucidate these Feynman-like rules. This is not a coincidence, and is in fact reminiscent of Feynman rules for scalar Mellin amplitudes. Our work suggests a holographic origin for this. Given a conformal block, as argued previously [56,57], a general strategy for extracting its explicit holographic dual (and consequently a power series expansion via a double-OPE limit as discussed in this paper) is to start with a canonical tree-level AdS diagram. 17 The AdS diagram should have solely cubic couplings and its direct-channel conformal block decomposition should include the given block. Performing the bulk integrals in the diagram carefully using various two-and three-propagator AdS identities helps extract an explicit representation for the block [44,56,57]. Indeed for comb channel blocks, the (n − 4) factors of 3 F 2 arise due to the presence of (n − 4) cubic bulk integrals involving exactly one bulk-to-boundary propagator and two bulk-to-bulk propagators. This integral was fully worked out in ref. [57] and it involves precisely the right factor of the hypergeometric function 3 F 2 with precisely the right arguments. 18 In fact, this general argument should extend to any arbitrary-point scalar 16 One might be tempted to assert from (1.1) that there are in fact (n − 2) factors of the hypergeometric function, but the precise count is (n − 4) since k 0 and k n−2 vanish by definition (1.4). This is a simple generalization of the previously known cases for scalar blocks with scalar exchanges -the four-point block doesn't have any 3 F 2 functions in its double power series expansion [12], while the power series expansion for the d-dimensional five-point block obtained in ref. [54] (as well as its holographic representation [56]) carries precisely one factor of the hypergeometric 3 F 2 function. 17 The canonical choice for the AdS diagram for extracting the six-point comb channel block was provided in (2.4). 18 The remaining two bulk integrals involving integration over a product of precisely two bulk-to-boundary conformal block with scalar exchanges in an arbitrary channel. All the necessary threepropagator integrals appearing in such a derivation were worked out in ref. [57]. This should make it tractable to work out the putative Feynman rules for all scalar blocks.
It is interesting to compare our results with the parallel, albeit considerably simpler story in the framework of p-adic AdS/CFT, where the conformal group is PGL(2, Q p d ) [72,73]. 19 Due to the lack of descendant operators in p-adic CFTs [75], the conformal blocks are simply scaling blocks. Nevertheless, analogous to the real conformal blocks, the padic blocks admit holographic duals, written as geodesic bulk diagrams on the Bruhat-Tits tree [49,56,57]. In fact, all results presented in this paper also admit a p-adic counterpart -various recent accounts of comparison and translation between objects in the usual (real) and p-adic holographic settings can be found in refs. [72,73,76,77,49,78,79,80,81,82,83,84,85,86,87,88,89,90,56,57,91,92,93]. Essentially, to recover the p-adic result, one can truncate all power series expansions featured in this paper to their respective first terms, since the infinite multi-fold series expansions in real CFTs sum up descendant contributions which do not exist in p-adic CFTs. 20 Conversely, it is practical to work out holographic duals of blocks in the simpler p-adic setting first. This is because the conformal dimension dependence of propagators appearing in the associated p-adic geodesic diagrams is identical to that for geodesic diagrams in real CFTs (more precisely, the "primary contribution" is identical). So the simpler p-adic technology can be used to figure out in advance the expected primary contribution to conformal blocks in a real CFT; the full block, which sums up also the descendant contributions, can then in principle be determined from conformal invariance.
Finally, a potential practical concern about the power series expansion (1.1) could be that it is not rapidly convergent for operator insertions in the "OPE regime" of the comb channel, i.e. for cross-ratios u i ≪ 1 and w r;s ≈ 1 defined in (1.6) (for all allowed values of the subscripts). However, one can remedy this slow convergence by a simple transformation which re-expresses the series expansion in powers of w r;s as an expansion in powers of (1 − w r;s ). This is straightforward to work out, though increasingly tedious as n grows. More precisely, to transform (1.1) to a more efficient and rapidly convergent power series expansion propagators and one bulk-to-bulk propagator contribute simply factors of Gamma functions or Pochhammer symbols [57]. 19 Here Q p d is the unique unramified degree d field extension of the p-adic numbers Q p (see e.g. the book [74]). 20 For example, the holographic dual of the comb channel p-adic block is given by (3.1)-(3.2) except with all integral parameters being summed over set to zero.
for any fixed n, the following identity is useful (see e.g. footnote 9): For the four-, five-, six-, and seven-point blocks, we can explicitly check that a repeated application of this identity leads to the following alternate representation (for n = 4, 5, 6, 7) Recently, a similar power series expansion was obtained for the same five-point block in general spacetime dimensions [54]. There, a different set of conformal cross-ratios were used.
We expect it to be possible to establish analytically the equality between the result of ref. [54] and (4.2) for n = 5, but we have not found any simple transformations which achieve this.
However, we have verified numerically that the power series expansion (4.2) matches the one in ref. [54] to desired numerical precision in the mutual regime of convergence of the two series expansions, as expected. While we have not worked out the alternate representation for general n, we conjecture this to also hold for all n ≥ 8.

Acknowledgements
I thank Robert Clemenson for valuable discussions and collaboration in the early stages, and Christian B. Jepsen for useful discussions. I would also like to thank Steve Gubser, my Ph.D. advisor. Our joint work from about two years ago [49] served as an important stepping stone and a springboard for many subsequent ideas, including the present work.
Steve was an inspiring mentor, an extraordinary physicist and an exceptional human being.
I am deeply grateful to Steve Gubser for everything, and would like to dedicate this paper to his fond memory.

A Five-and seven-point examples
In this appendix we briefly discuss the holographic representation of the seven-point comb channel block and its double-OPE limit, which leads to a power series expansion for the five-point block. The holographic representations (2.2) and (2.7) for the five-and six-point cases inform the following conjecture for the seven-point block (which is consistent with the general conjecture (3.1)-(3.4)): where the linear combination of seven-point geodesic bulk diagrams W is given by with the coefficients Here we are using the subscript convention (1.3) for both k and j integral parameters. In drawing the geodesic diagram above, we have disobeyed the strict color-coding prescribed in To prove this conjecture, we must show that (A.1) satisfies the conformal Casimir equations (1.7) for n = 7 and K = 2, 3, 4, 5. Due to the symmetry of the object, we need only check the cases K = 2, 3; the remaining two cases follow trivially after relabelling. Further, we must show the OPE limit (1.8). As remarked in section 2.1, the conformal Casimir check is in fact straightforward though lengthy to work out, but the procedure is identical to the ones described in refs. [56,57] in the context of five-and six-point blocks. No new ingredients are needed, except for a triple-application of the hypergeometric identity (2.11), once for each factor of the 3 F 2 function in (A.3). Similarly, the object obtained in the OPE limit can be shown to be an alternate holographic representation of the six-point block involving a single geodesic integral, via a similar proof by Casimir. So to keep the paper to a reasonable length, we will refrain from providing the somewhat lengthy details here. In lieu of this, we provide a conformal Casimir check of the series expansion of the seven-point block in section 3. The OPE limits of the six-point block obtained above themselves lead to two different representations of the five-point block -one corresponding to a holographic representation involving a single geodesic integral similar to (2.14), with the other more interesting limit producing a power series expansion for the five-point block as discussed next.

A.1 Double-OPE limit and the five-point block
Consider the following double-OPE limit of the seven-point block, x 1 x 3 x 4 x 5 The claim is that V is a (power series expansion of the) five-point block. Explicitly, we may write it as where the leg factor W .
(A.8) 21 An alternate way to write the leg factor is as follows: Writing out the coefficients explicitly, we get where λ 7 was given in (A.4). It can be checked that this is just a rewriting of (1.1) for n = 5 in different variables. The proof that this is indeed a five-point block is given in section 3.

B Further technical details B.1 Proof of (3.15)
In this section we will prove (3.15). Consider first the j ⟨2|n−1⟩ sum on the right hand side.
We can use the following identity to evaluate this sum: which holds for integers ℓ 1 , ℓ 2 , and we assume convergence of the sum. Performing the sum using the identity above, the right hand side of (3.15) becomes where δ 0 = ∆ 1 , k 0 = 0 and k n−3 = 0 as before. Notice that (B.2) is of precisely the same form as the original sum on the right hand side of (3.15), except with one fewer sum. Thus one can iteratively apply (B.1) to systematically reduce RHS further and further. For instance, performing the j ⟨3|n−1⟩ sum next yields We will show that J = 0, which proves (3.31). First expand J as au,av / ∈{r,s} where we have broken the four-fold sums into the following constituent pieces: Thus, J vanishes identically.
Thus for the above expression to equal (3.35) for all choices of boundary insertion points, each individual term in the sum must equal the conformal Casimir eigenvalue, times the coefficient (3.8). That is, the proof of (3.36) boils down to demonstrating the following non-trivial identity involving the coefficients, we can simplify (B.9) to K+1≤au<av≤n c r,s;au,av (k s−1 −1,ks−1,...,k au−2 −1,j ⟨r|s⟩ −1,j ⟨au|av ⟩ −1,j ⟨s|au⟩ +1,j ⟨r|av ⟩ +1) ! = 0 . (B.12) Now, using the explicit expressions for the dimensions, one can further simplify (B.12) to (3.39). This is straightforward to work out for any given value of n and K (see the ancillary Mathematica notebook for calculational details).
To close this section, let's return to the as yet unproven symmetry transformations (B.10)-(B.11). Consider for example, the first identity in (B.11). To prove it, it is useful to take a closer look at the coefficients (3.8) and the dimensions ∆ ⟨i|j⟩ which can be visually read off of (3.7). Consider the left hand side first, where we need to study the effect of shifting the integral parameters j ⟨r|au⟩ → j ⟨r|au⟩ − 1, j ⟨s|au⟩ → j ⟨s|au⟩ + 1 in the scaled coefficient c r,s;au,n (·) , for fixed r, s, a u satisfying 1 ≤ r < s ≤ K and K + 1 ≤ a u ≤ n − 1. First assume r ≥ 2we will come back to the case r = 1 at the end: • The factor of ∆ 1n,2...(n−1) − 2≤ℓ 1 <ℓ 2 ≤n−1 j ⟨ℓ 1 |ℓ 2 ⟩ in the coefficient (3.8) remains unchanged since the sum over all j-parameters is invariant under this shift.
• Under the given shift, among other effects discussed below, the coefficient (3.8) picks up an overall factor of j ⟨r|au⟩ ! (j ⟨r|au⟩ − 1)! j ⟨s|au⟩ ! (j ⟨s|au⟩ + 1)! = j ⟨r|au⟩ j ⟨s|au⟩ + 1 = −∆ ⟨r|au⟩ j ⟨s|au⟩ + 1 , where we used ∆ ⟨r|au⟩ = −j ⟨r|au⟩ for r ≥ 2. Moreover note that the scaled coefficient (3.37) comes with additional factors of dimensions ∆ ⟨r|n⟩ ∆ ⟨s|au⟩ . We will address the first factor shortly, but the other one can be read off of (3.7) to be, ∆ ⟨s|au⟩ j ⟨r|au⟩ →j ⟨r|au⟩ −1 where we have appropriately shifted it as required by the left hand side being evaluated.
This will cancel the factor in the denominator of the previous displayed equation.
Combining all these observations, once again, the left hand side evaluates to the unshifted coefficient (3.8) times a factor of ∆ ⟨1|au⟩ ∆ ⟨s|n⟩ , which is precisely the expected right hand side for r = 1.
The other identities in (B.10)-(B.11) can be proven via very similar analyses. In particular, the proofs for the third identity in both (B.10) and (B.11) also go through without further difficulty, since the k i integer-shifts are uniform across both the left and right hand sides.