Multipoint conformal blocks in the comb channel

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for n-point correlation functions. For conformal field theories in dimensions d = 1 and d = 2, we use the shadow formalism to compute n-point conformal blocks, for arbitrary n, in a particular channel which we refer to as the comb channel. The result is expressed in terms of a multivariable hypergeometric function, for which we give series, differential, and integral representations. In general dimension d we derive the 5-point conformal block, for external and exchanged scalar operators.


Introduction
Correlation functions of local operators are fundamental observables in quantum field theory. In computing and specifying correlation functions, it is useful to exploit the symmetries of the theory to the fullest extent possible. In particular, one would like to write the correlation functions in a way that separates the theory-dependent data from the universal pieces. In a d dimensional quantum field theory endowed with conformal symmetry, a CFT, the theory-dependent data are the dimensions and OPE coefficients of the primary operators, and the universal pieces are the conformal blocks. The conformal blocks provide a bridge JHEP02(2019)142 data are the conformal blocks. In the language of the OPE, in a CFT, the OPE the primary operator fully fix the OPE coe cients of the descendants -derivatives y. The conformal blocks sum all the descendants. iew a four-point conformal block as taking a four-point function and projecting the state, lying in between the second and third operator, onto the conformal family of intermediate operator O, as shown in Fig. 1(a). An n-point conformal block, such own in Fig. 1 higher-point blocks. There are now more cross ratios. In 2d the number is 2n.
nnels. In what order OPE is performed. We will compute the conformal blocks in nnel.
? . In Sec. 2 we consider two dimensions, and compute the conformal blocks in the l, for any number of points. In Sec. 3 we turn to d dimensions. In Sec. ?? we compute locks, .. We conclude in Sec. 4 with future directions. [7] more precisely cSYK [8], is a one-dimensional CFT. Appearance of blocks in solution 2 (a) ... between the observables: correlation functions that one measures, either experimentally or theoretically, and the CFT data: dimensions and OPE coefficients of operators. More technically, conformal symmetry fully fixes the OPE coefficients of the descendants in terms of the OPE coefficients of the primaries, and the conformal blocks sum all the descendants.
We can view a four-point conformal block as arising from taking a four-point function and adding a projector, in between the second and third operator, onto a particular intermediate operator O and its descendants, as shown in figure 1(a). An n-point conformal block, such as the one shown in figure 1(b), has n−3 intermediate operators.
Four-point blocks in dimensions d = 1 and d = 2 are simple and have been known since the 70's [1,2]. The modern study of four-point blocks in higher dimensions was initiated by Dolan and Osborn [3][4][5]. The goal of this work is to take initial steps towards the study of n-point conformal blocks, for any n. For fixed n ≥ 6, there are in fact multiple n-point blocks. What is perhaps the simplest channel is shown in figure 1(b), and we will refer to it as the comb channel.
In section 2 we compute the n-point conformal blocks in the comb channel, in dimensions d = 1 and d = 2. The result for d = 1, see eq. (2.7), is expressed in terms of an n−3 variable generalized hypergeometric function, which we call the comb function. In appendix A we derive some properties of the comb function. The answer for the blocks, while seemingly complicated, actually takes the simplest form one could have hoped for, and reflects the symmetry of the comb channel. In a technical sense, the statement of simplicity is that the conformal block is expressed as an n−3 fold sum, the smallest possible number, as there are n−3 independent cross-ratios. The n-point blocks for d = 2 immediately follow from those for d = 1, see eq. (2.3).
In higher dimensions, one expects the blocks to be significantly more involved. In section 3 we begin the study of higher-point blocks in d dimensions, focusing on the simplest case: the 5-point block with external and exchanged scalar operators. The relevant technical details are in appendices B and C. We end in section 4 with comments on future directions.

One and two dimensions
In this section we compute n-point blocks in the comb channel, for any n, in one and two dimensions. In section 2.1 we establish notation and summarize the result. In section 2.2 we review the 4-point block. In section 2.3 we derive the 5-point block. In section 2.4 we derive the n-point blocks in the comb channel. In section 2.5 we check that the blocks JHEP02(2019)142 have the correct leading OPE behavior and that they satisfy the appropriate Casimir differential equations.

Definitions
We first discuss conformal blocks in one dimension. An n-point conformal block with external operators of dimensions h i and exchanged operators of dimensions h i will be denoted by G h 1 ,...,h n h 1 ,...,h n−3 (z 1 , . . . , z n ). A n-point block can be decomposed into a function of the conformally invariant cross ratios χ i , of which there are n−3, and what we will refer to as the leg factor, The leg factor transforms as an n-point CFT correlation function and, in particular, has dimension n i=1 h i . The leg factor can of course be changed, by a function of the crossratios, at the expense of changing the bare conformal block. The above leg factor will emerge naturally in the derivation of the blocks, and leads to the bare blocks having a simple form.
The conformal Casimir in two dimensions (discussed later in section 2.5.2) factorizes into a sum of two SL 2 Casimirs, l 2 +l 2 , built out of z and z respectively, and the eigenvalues are a sum, h(h − 1) + h(h − 1). As a result, the two-dimensional conformal blocks are simply a product of two one-dimensional conformal blocks, one for the holomorphic sector and one for the antiholomorphic sector,  L h 1 ,...,h n (z 1 , . . . , z n ) = ✓ z 23 z 12 z 13 ◆ h 1 ✓ z n 2,n 1 z n 2,n z n 1,n ependence when writing latter on?

Conformal partial wave
The conformal partial wave, to be defined below, is denoted by Ψ h 1 ,...,h n h 1 ,...,h n−3 (z 1 , . . . , z n ). The 4-point partial wave is defined by the following integral, 3 The comb channel involves, at each iteration, gluing a three-point function to the end of the partial wave. In some different channel, one would, at some stage, glue the three point function somewhere else. The n+1 point partial wave involves n−2 integrals of a product of n−1 conformal 3-point functions. For instance, applying this equation for n = 4, and making use of (2.4), gives the 5-point partial wave, The conformal partial waves give a sum of a conformal block and shadow blocks. In particular, the 4-point partial wave is a sum of two terms: the conformal block for exchange of an operator of dimension h 1 and the conformal block for exchange of the shadow operator of dimension h 1 = 1 − h 1 . The n-point block is a sum of 2 n−3 terms, accounting for the blocks with exchanged operator dimensions (h 1 , . . . , h n−3 ) and all possible shadows.

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One can view the insertion of dz |O h (z) O h (z)| as a conformally invariant projector onto the exchange of the operator O h , or its shadow, O h . The shadow formalism [12][13][14] is a useful way of computing conformal blocks [15], and one we will exploit.

Result for n-point conformal blocks
Later in the section, we will compute the n-point conformal block in the comb channel, and find it to be, where F K is a multivariable hypergeometric function, which we will refer to as the comb function. The comb function is defined by the sum, where (a) n = Γ(a + n)/Γ(a) is the Pochhammer symbol. Some properties of the comb function are derived in appendix. A. 4 In the special case of two variables, the comb function reduces to the Appell function F 2 . It will be convenient to define the one variable comb function to be the standard hypergeometric function 2 F 1 . The conformal blocks (2.7) are for the case that all the cross ratios are between zero and one, 0 < χ i < 1. This occurs if one takes all the positions to be ordered or antiordered. We will assume that they are ordered, z 1 > z 2 > . . . > z n .
A nice property of the conformal blocks in the comb channel is that they have a Z 2 symmetry: figure 2 can be read either from left to right, or from right to left. 5 They also have a shift symmetry. Both of these symmetries are reflected in the formula.

Conformal blocks for small n
Let us write out the conformal blocks (2.7) for some small values of n. For the 4-point block, n = 4, we recover the usual conformal block, see figure 3(a), 6 The comb function recently appeared in [16] in the computation of SL 2 blocks for null polygon Wilson loops [17,18]. There may be a direct mapping between that result and ours, which would be interesting to understand. 5 For the partial waves, one should draw the diagram with arrows on the internal lines. The arrows would all point to the right, with the convention that an arrow leaves an operator and enter the shadow of an operator. The arrows break the symmetry, but only by trivial shadow transform factors that can be accounted for. 6 Our 4-point block is written in a form that is slight different from the standard form, due to our choice of leg factor. If one applies the identity for 2 F 1 in (2.21), then this gives the standard form of the block.
dependence when writing latter on? 4 scaling in terms of dimension P h i , for the correlator.
. , z n+1 ) = h n n 2 ✓ z n 1,n z n 1,n+1 z n,n+1 Discuss that the form we wrote is time ordered. Trivial to do anti-time ordrered. Discuss regiemes of cross ratios (for 4-pt), and how flipping times transforms the cross-ratios. fixing pionts. But want the right cross ratios and need to produce function in a form in which recognizable.

n-point block
For n-point, use block for n 1 point, but only need to act on last cross-ratio. We can us following formula for F K (the splitting formula).

Evaluating the integral
Should call the integration variables z instead of ⌧ , and let the cross-ratio be and not x?

Conformal Blocks and Conformal Partial Waves
Need notation for partial wave and for block. Use wh Discuss conformal symmetry. 2) Simmons-Du n defined monodromy conditions to p Say how it should behave in the OPE limit.
3) Write down di↵ eq. and explain why they are correct. Point out that there writing the same equation. For instance, for 5pt C(1, 2) or C(1, 2, 3).
the form we wrote is time ordered. Trivial to do anti-time ordrered. Discuss other ratios (for 4-pt), and how flipping times transforms the cross-ratios.
lock tly, by change of variables, and explain. That can restrict integration to regions.
exactly, by change of variables. Redo by adding onto 4-pt. (becomes harder to change t using previous result). Of course, trivial to change variables to cross-ratio, by just t want the right cross ratios and need to produce function in a form in which it is use block for n 1 point, but only need to act on last cross-ratio. We can use the for F K (the splitting formula).
he integral he integration variables z instead of ⌧ , and let the cross-ratio be and not x?
actly. To make use of SL 2 symmetry, perform variable changes 3 times. Of course, 5 We want higher-point blocks. There are now more cross ratios. In 2d the nu channels. In what order OPE is performed. We will compute the conformal block In Sec. 2 . In Sec. 3 we consider two dimensions, and compute the conform channel, for any number of points. In Sec. 4 we turn to d dimensions. In Sec. ?? w blocks, .. We conclude in Sec. 5 with future directions.

Conformal Blocks and Conformal Partial Waves
3) Write down di↵ eq. and explain why they are correct. Point out that the writing the same equation. For instance, for 5pt C(1, 2) or C(1, 2, 3).
3 out of the 4 points. Get conformal blocks by doing OPE on both pairs of operators. 4pt block in Figure. Consistency of OPE in two di↵erent channel is basis of bootstrap. e want higher-point blocks. There are now more cross ratios. In 2d the number is 2n. Choice of els. In what order OPE is performed. We will compute the conformal blocks in the comb channel.
Sec. 2 . In Sec. 3 we consider two dimensions, and compute the conformal blocks in the comb el, for any number of points. In Sec. 4 we turn to d dimensions. In Sec. ?? we compute the 5-point , .. We conclude in Sec. 5 with future directions.
nformal Blocks and Conformal Partial Waves hen discuss shadow formalism. Why is this correct? How to map between the two? ways of getting blocks. 1) OPE and doing the sum. 2) shadow formalism. 3) solving the di↵erential ion )Define OPE operator, and apply. We just want an explicit form for this expression (of di↵erential tors acting on the 3-pt function).
) Simmons-Du n defined monodromy conditions to pick out block. We just look at the expression. ay how it should behave in the OPE limit.
) Write down di↵ eq. and explain why they are correct. Point out that there are di↵erent way of g the same equation. For instance, for 5pt C(1, 2) or C(1, 2, 3).   2) Simmons-Du n defined monodromy conditions to pick out block. We just Say how it should behave in the OPE limit.

Conformal Blocks and Conformal Partial Waves
3) Write down di↵ eq. and explain why they are correct. Point out that th writing the same equation. For instance, for 5pt C(1, 2) or C(1, 2, 3). want higher-point blocks. There are now more cross ratios. In 2d the number is 2n. Choice of s. In what order OPE is performed. We will compute the conformal blocks in the comb channel.
Sec. 2 . In Sec. 3 we consider two dimensions, and compute the conformal blocks in the comb , for any number of points. In Sec. 4 we turn to d dimensions. In Sec. ?? we compute the 5-point .. We conclude in Sec. 5 with future directions.
formal Blocks and Conformal Partial Waves Simmons-Du n defined monodromy conditions to pick out block. We just look at the expression.
how it should behave in the OPE limit.
Write down di↵ eq. and explain why they are correct. Point out that there are di↵erent way of the same equation. For instance, for 5pt C(1, 2) or C(1, 2, 3).  For the 5-point block, see figure 3 and for the 6-point block, see figure 3(c), we have, (2.11) where the cross-ratios are (2.1), Let us also write out the leg factors (2.2) for some small values of n. For n = 3 the leg factor is just a conformal three-point function, (2.15) From the definition of the leg factors (2.2) one can trivially relate the n+1 point leg factor to the n point leg factor, This completes our summary of the results. The rest of the section is devoted to deriving the conformal blocks (2.7).

Four-point block
We start by recalling the standard four-point conformal block, see figure 3(a). We will find it by computing the integral defining the conformal partial wave. The definition of the conformal partial wave is, (2.4), We do a change of variables, z 1 → z 1 The integral has now simplified: instead of z 1 colliding with four points, it now only collides with three points. In particular, after the change of variables we do not have a term in the integrand involving |z 1 | to some power. This is because the sum of the exponents in the denominator of the integrand in (2.17) is equal to two. This is a result of conformal invariance. Namely, for (2.17) to transform correctly under inversion of the points z i → z −1 i , it is necessary that the sum of the exponents in the denominator of (2.17) be equal to two (as one can see by inverting the integration point z 1 → z −1 1 ). Performing a further change of variables on (2.18), Let us now assume that 0 < χ 1 < 1. The integrand is analytic in four separate regions of z 1 , which are: (−∞, 0), (0, χ 1 ), (χ 1 , 1), (1, ∞). We should do the integral in each of the four regions. In the region 0 < z 1 < χ 1 , the integral is proportional to, as one can see from the definition of the hypergeometric function, (A.23) in appendix. A, if one sends z 1 → χ 1 z 1 (recall that we have extended notation to let F K in the one variable case denote 2 F 1 ). By a hypergeometric identity, and so we have obtained the 4-point conformal block, (2.9), (2.14).

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In the other three regions, the integral can also be evaluated by performing simple variable changes. The result for each of the regions will be a superposition of the conformal block and the shadow block. In fact, this is guaranteed: once we evaluate the integral in the region 0 < z 1 < χ 1 and obtain (2.20), we know that the conformal partial wave must be a solution of the second order differential equation obeyed by the hypergeometric function. This has two solutions, corresponding to the conformal block and the shadow block. Thus, the partial wave must be a superposition of these two solutions. We have learned a lesson that will be useful later on: to find the conformal block, it is sufficient to evaluate the integral for the conformal partial wave (2.19) in any of the the four regions.
To go from the definition of the partial wave (2.17) to the form in (2.19), we did two simple changes of integration variables. Combining them, the transformation is, This transformation sends the four points z 1 = (z 4 , z 3 , z 2 , z 1 ) ← z 1 = (0, χ 1 , 1, ±∞). Of course, the reason that this change of variables simplified the integral is SL 2 symmetry. We could have equivalently taken the original integral (2.17) and set z 4 = 0, z 3 = χ 1 , z 2 = 1, z 1 = ∞.
To summarize: one method for computing the conformal block is to start with the definition of the partial wave (2.4) and to evaluate the integral. One can simplify the problem by restricting the integration range to lie in any of the four regions in between the singularities where points collide: The integral from any one of these four regions gives some combination of the block and the shadow block. Distinguishing between the two is trivial, and so this is enough to find the block. 7

Five-point block
Direct evaluation. We start with the definition of the 5-point conformal partial wave, (2.6), and change integration variables, z 1 → z 3 −z 1 −1 and z 2 → z 3 −z 2 −1 to obtain, Performing a further change of variables, z 1 → z 1 (z −1 If one is interested in the conformal partial wave, then this can easily be established from knowledge of the conformal block. The partial wave is a sum of a block and a shadow block, with some coefficients which can be established by evaluating the integral in the OPE limit, see e.g. [19,20]. The same applies to higher point partial waves as well.

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The portion of the integral from the region of integration 0 < z 1 < 1 and 0 < z 2 < 1 is an Appell function (A.24), Taking the shadow with respect to h 1 , by sending h 1 → 1 − h 1 , we obtain the 5-point conformal block (2.10). The derivation we just gave of the 5-point conformal block relied on starting with the definition of the conformal partial wave, and finding an appropriate change of integration variables that gives an integrand depending only on the conformal cross-ratios χ 1 , χ 2 . Obtaining an integrand that depends only on the cross-ratios is not difficult: one can, without loss of generality, take the definition of the conformal partial wave (2.6) and write it as the leg factor L h 1 ,...,h 5 times some function of the two cross-ratios, and then make some choice for the external points, such as z 1 = ∞, z 2 = 1, z 5 = 0, and express the remaining two points z 3 , z 4 appearing in the partial wave in terms of χ 1 , χ 2 . What is slightly more challenging, if one does not make the best choice, is recognizing the resulting integral as some special function. In the case of the 5-point block, the special function one is looking for is a two variable function, the simplest ones are Appell functions, and there are only four kinds of Appell functions. As a result, one has a fairly good sense of what the answer might be.
For higher point blocks, it will be harder to find a good change of variables that turns the integral into something we can recognize. In part, this is because there are a large number of multivariable hypergeometric functions, when the number of variables is greater than two. The key step in our derivation of the n-point block will be to make use of the result for the n−1 point block, as input in computing the n-point block.
In order to illustrate the method, we will now redo the computation of the 5-point block, in a way that makes use of the 4-point block.

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We write the 4-point conformal block appearing above in terms of the leg factor and the bare conformal block, , (2.28) where the leg factor is (2.16), while the bare conformal block is (2.9), The restriction on the integration range we made in (2.27), that z 2 < z 3 , implies 0 < ρ < 1, and allows us to use the 4-point conformal block in the form above. Inserting these terms gives, where the first three-point function in the integral emerged from the powers of z 2 − z 3 and z 2 − z 2 that occurred. We recognize the integral here as the same one defining the conformal partial wave, Ψ h 1 +m,h 3 ,h 4 ,h 5 h 2 (z 2 , z 3 , z 4 , z 5 ). As discussed in section 2.2, restricting the integration range to z 4 < z 2 < z 3 will still give rise to a linear combination of the block and its shadow. We may therefore replace the integral above with the conformal block G h 1 +m,h 3 ,h 4 ,h 5 h 2 (z 2 , z 3 , z 4 , z 5 ), which we write in terms of the leg factor and the bare block, to get Writing out the leg factors, we reorganize this as,

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We identify the sum as the bare 5-point conformal block. Inserting the 4-point conformal block (2.9), written as a sum, we thus have, (2.33) sum is just the Appell function F 2 , and we thus get the 5-point conformal block stated earlier (2.10).

n-point block
To obtain the n+1 point block from the n point block we use a similar procedure as the one used in the previous section for getting the 5-point block from the 4-point block. Namely, we start with the definition of the n+1 point conformal partial wave as an n-point partial wave glued to a 3-point function. We use this expression, with the range of integration restricted, and insert the n-point block, rewritten as a sum involving an n−1 point block (so as to strip off the last cross-ratio), and recognize the integral to be the one that gives rise to a 4-point conformal block (involving the last 4 points). The sum then defines the n+1 point block. After doing this for the 6-point and 7-point blocks, the pattern becomes clear, allowing us to guess the n point block, (2.7). We will show by induction that this guess is correct. We start with the definition of the conformal partial wave in the comb channel,  We insert the conformal block appearing on the right, written as a leg factor times the bare block, We recognize the integral on the right side is the same one that appears in the definition of the 4-point partial wave, Ψ h n−3 +m,h n−1 ,h n ,h n+1 h n−2 (z n−2 , z n−1 , z n , z n+1 ), and so we replace this integral by the conformal block G h n−3 +m,h n−1 ,h n ,h n+1 h n−2 (z n−2 , z n−1 , z n , z n+1 ), which we write as a leg factor times the bare block, We identify the right hand side with the n+1 point block. Simplifying gives,  (2.40) where the dots are the descendants of O h 1 . This gives for the four-point function, Let us check that our conformal blocks behave correctly in the OPE limit. For the 4-point block we take z 4 → z 3 , as above. The conformal cross ratio χ 1 goes to zero in this limit, and the behavior of the bare block and the leg factor in this limit is, where in the denominator of χ 1 in g we replaced z 24 with z 23 . Combining the bare block and the leg factor, we have that the block behaves as, which is the correct behavior (2.41). Now, consider performing the OPE on the last two operators in a n-point function, (2.44) Let us check that our n-point conformal block has this behavior in the z n → z n−1 limit. The cross ratio χ n−3 → 0, while all other cross ratios remain finite. Thus, from (2.7), we see that the bare n point block reduces to a bare n−1 point block, Using this, and taking the limit z n → z n−1 , we get,

Casimir equations
Here we check that the conformal blocks behave correctly as eigenfunctions of the Casimir of the conformal group. In particular, in one dimension the conformal group is SL 2 (R), with generators l a and Casimir l 2 , Let us denote the SL 2 (R) generator acting on point z i by l (i) a . We define the Casimir acting on two points, z i and z j , to be, We make an analogous definition for more points: for instance, C(i, j, k) = (l (i) +l (j) +l (k) ) 2 . A defining property of the 4-point conformal block is that it is an eigenfunction of the Casimir acting on two points [4], We may insert into the these equations the decomposition of the conformal block into the leg factor and the bare block (2.1), to then obtain sets of differential equations in terms of the cross-ratios. This is simple to do for low values of n, and to then recognize the equations, and to see that the solutions are what we wrote down before for the conformal blocks.

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This is a good approach for checking that one has the correct conformal blocks, but is less good for actually finding the blocks, for n ≥ 5, due to the large number of choices one needs to make: the choice of the leg factors, as well as the choice of the cross-ratios. Any (correct) choice will give correct equations, but they may not be in a form in which one can recognize the solution. In appendix A.2 we derive differential equations for the comb function, which are what these Casimir equations become. 9

d dimensions
In this section we study conformal blocks in general dimension d. This is more involved than in one or two dimensions, and we restrict to external operators that are scalars and exchanged operators that are scalars. In section 3.1 we review the computation of the 4-point blocks. In section 3.2 we compute the 5-point blocks.

Four-point block, scalar exchange
The four-point conformal partial wave is, where ∆ i are the dimensions of the external operators, and ∆ is the dimension of the exchanged operator O. We take all operators to be scalars. 10 Here O refers to the shadow of O, which has dimension ∆ = d − ∆.
Writing out the three-point functions, the integral we must evaluate is,

40
, This is a 4-point integral, of the form studied in appendix. B. Applying (B.8) the result is a sum of the conformal block and the shadow block , where the prefactor is, , (3.4) 9 We checked that the Casimir equations for n-point blocks are the same as the differential equations for the comb function, for n up to 6. It is straightforward to check for larger n, but it is not obvious how to make the match manifest. 10 It is common to denote the partial wave by Ψ ∆ i ∆,J with J denoting the spin of the exchanged operator. We have not written our partial wave in this notation, as Ψ    and the conformal block is [3], where the bare conformal block is, where u and v are the two conformal cross ratios, (3.7) One can write these cross-ratios as u = χχ and v = (1 − χ)(1 − χ). In d = 2, these are the χ and χ that we used in the previous section. In d = 1, there is only one independent cross ratio, and χ = χ.

Five-point block, scalar exchange
In this section we compute the five-point conformal block, with scalar exchange. As in the other cases, we do this by evaluating the integral expression for the conformal partial wave, and picking out the conformal block in it. The five-point conformal partial wave is, .
We now evaluate the integral over x a , representing it as a five-fold Mellin-Barnes integral, using the result in appendix. B, eq. (B.19). Combined with the s and t integrals, this gives us a seven-fold Mellin-Barnes integral, , (3.12) and 2 +s 1 +s 2 +s 3 Γ ∆ a −∆ 12 2 +s 1 +s 4 Γ (∆ a +2s 1 +s 2 +s 3 +s 4 ) 2 +s+s 2 +s 3 +s 4 +2s 5 , (3.14) We have written the 5-point block in a form that makes the left-right symmetry of figure 4 manifest: one can exchange (1,2,3,4,5) The form of the block is, as would be expected, more complicated than the 4-point block. It is also qualitatively different, in that the coefficients are not just Pochhammer symbol, but also involve a 3 F 2 . We can not exclude the possibility that there is a different choice of cross-ratios that gives a simpler answer than the one we found.
For the 4-point block, there were two independent conformal cross-ratio in any d ≥ 2. However, for the 5-point block, there are 4 independent conformal cross-ratios in d = 2, but 5 in d ≥ 3: the cross-ratio we called w is the new one in d ≥ 3.

Discussion
We computed the n-point conformal blocks, for arbitrary n, in d = 1, 2, in the comb channel. The comb channel corresponds to a particular order in which one does the OPE, and for n ≥ 6, there are other channels (which are not related by symmetry). For instance, for n = 6, a different channel is one in which one does the OPE between the first and the second operator, the third and the fourth, and the fifth and the sixth. It is possible to compute the conformal blocks in these other channels, using the same methods applied here. However, it is not obvious that the answer will take as simple a form as the one for the comb channel.
We derived the 5-point conformal block in arbitrary dimensions, for external and exchanged operators that are scalars. In d = 4 and d = 6, for external scalars, the 4-point JHEP02(2019)142 Figure 5. The six-point conformal partial wave in the comb channel, represented as a Feynman diagram. Each line is a particular propagator, 1/x 2 to some power involving the operator dimensions, as specified by the definition of the partial wave, and the three internal points are integrated over. There are six external points which are held fixed. The n-point partial wave will look like this figure, but with n−2 triangles. conformal block is known to simplify, taking a similar form as the d = 2 block [3,4]. It is conceivable that the 5-point block could also be simplified further in these dimensions; however, the degree of simplification is limited, since the number of conformal cross-ratios built out of 5 points is 4 in d = 2 but 5 in d = 4. For the 4-point blocks, obtaining the blocks with spinning operators is nontrivial. One generally uses recursion relations to relate spinning blocks to scalar blocks [4,5,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Weight-shifting operators [34] can be used to get blocks with either external or internal operators with spin from those with scalars. One could apply weight-shifting operators to the 5-point block we found. Finally, although we only computed the 5-point block in d dimensions, one could use the same Mellin-Barnes technique to compute n-point blocks; the result will get progressively more complicated with increasing n. It is conceivable, however, that, at least for the comb channel, there is a pattern which would allow one to eventually guess the n-point answer, much as we were able to do in d = 1, 2.
We computed the conformal blocks by computing the conformal partial waves, using the shadow formalism. This method turns a conformal block computation into a multi-loop integral. For instance, evaluating the partial waves in the comb channel is equivalent to evaluating the Feynman diagrams shown in figure 5. There have been many studies of multi-loop Feynman integrals. An interesting and tractable class are those with iterative structure, such as ladder diagrams [54,55]. One can view our result as a computation of a new class of Feynman diagrams. A number of methods have recently been developed for the evaluation of multi-loop Feynman integrals, for example [56][57][58][59][60][61][62][63]. Many of these focus on propagators that are 1/x 2 . It would be useful to try to extend these methods to cases in which the propagators contain 1/x 2 to a non-integer power; this is what occurs in the computation of conformal partial waves.
It is sometimes useful to translate CFT results into AdS statements. Recently, the AdS duals of 4-point conformal blocks were found to be geodesic Witten diagrams [64]. It would be interesting to find the AdS duals of n-point conformal blocks; the explicit form for the blocks that we gave in d = 1, 2 should make this tractable.

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The extensive recent studies of 4-point conformal blocks have in part been motivated by their application to the conformal bootstrap program [65][66][67][68][69][70][71][72][73][74][75][76][77]. It may be useful to study the bootstrap using n-point blocks. Although crossing relations for the four-point function are sufficient for the bootstrap if one incorporates all operators, including spinning operators, it may be that if one uses n-point functions, then external scalars are sufficient. 11 Finally, even though two-point and three-point functions of all operators contain all the CFT data, it may happen that n-point correlation functions of simple operators are more simple and more natural than 3-point correlation functions of complicated operators. A Properties of the comb function F K

A.1 Definition
The standard one variable hypergeometric function 2 F 1 (x) is defined as the infinite sum, It is common to encounter the single variable generalized hypergeometric function, p F q (x), defined as, p F q a 1 , . . . , a p c 1 , . . . , c q ; x = ∞ n=0 (a 1 ) n · · · (a p ) n (c 1 ) n · · · (c q ) n x n n! . (A.2) We will encounter multivariable hypergeometric functions. The number of simple hypergeometric functions of k variables grows rapidly with k, and there does not seem to be a general classification of these beyond three variables. However, for any particular multivariable hypergeometric function, it is straightforward to work out its properties, following standard techniques in the literature for the two variable case, see e.g. [78]. We define the following function of k variables, which we will refer to as the comb function, 11 We thank D. Gross for this suggestion.

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In the slightly degenerate case of one variable, k = 1, it will be convenient to interpret the definition of F K to be that of the one variable hypergeometric function, In the case of two variables, the comb function is, which is the Appell function F 2 .
Splitting identities. From the definition of the comb function, combined with the trivial identity (a) n+m = (a + n) m (a) n , one can derive "splitting" identities, such as, Here we have split off the last variable. We may also split off the last two variables, Since the conformal blocks are expressed in terms of the comb function (2.7), these two identities give the following two identities for the blocks, which we use in the main text: from (A.6) we get,

Differential equation representation
Starting with the definition of the k variable comb function as a sum (A.3), we derive the set of k differential equations that it satisfies. We write the comb function as, where the coefficients are given in (A.3). It is simple to find recursion relations for the coefficients. For instance, the recursion relation which iterates n 1 is, A n 1 +1,n 2 ,...,n k = A n 1 ,...,n k (a 1 + n 1 )(b 1 + n 1 + n 2 ) (c 1 + n 1 ) 1 (n 1 + 1) .

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In total we have a set of k partial differential equations, (A.13), (A.15), (A.18). In the special case that k = 2, we only have (A.13), (A.15), which, written out explicitly, are, which are the correct system of differential equations for the Appell function F 2 .

A.3 Integral representation
In this section, we derive an integral representation of the comb function, by using its definition as a series. We start by observing that the ratio of Pochhammer symbols can be written so as to involve a beta function. The using the integral representation of the beta function, we get, Let us take the series definition of the one variable comb function, in which case it is the standard hypergeometric function 2 F 1 , and make use of (A.21), and then sum the series, to get, which reproduces the integral representation of the Appell function F 2 . For the three variable comb function, we write the series as, (1−t 1 ) c 1 −a 1 −1 .
For the k variable comb function, for k ≥ 4, we write the series in a way that involves a hypergeometric function of x 1 , and a hypergeometric function of x k , analogous to what we did in the three variable case (A.25), insert the integral representation of the hypergeometric function, and sum the series, to get, (1−t 1 ) If k = 4, then the comb function on the right only has the first and the last arguments.

B Mellin-Barnes integrals
In the computation of the conformal blocks in d dimensions, we encounter integrals of the form, (B.1) We will refer to these as n-point integrals. For general n and general dimension, the best one can do towards evaluating these integrals is turning them into Mellin-Barnes integrals, as we review in this appendix. We write each of the factors 1/X a i 0i in the integrand as an integral, 1 X a i 0i and then evaluate the x 0 integral, by completing the square. This gives, where Λ = n i=1 λ i . The n-point integral is conformally invariant if n i=1 a i = d. In this case, we may replace Λ by Λ = n i=1 α i λ i , provided α i > 0 [79].

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where the first term came from the poles of Γ(−s), located at s = m, and the second term came from the poles of Γ(a 3 + a 4 − d 2 − s), located at s = a 3 + a 4 − d 2 + m. Both terms picked up the poles from Γ(−t) at t = n. We may move terms between the numerator and denominator by using the identity,

B.2 5-point integral
Next, we consider the 5-point integral. We would like to maintain symmetry between (1,2) and (4,5), so in (B.3) we take Λ = λ 3 . We apply the integral representation (B.6) to some of the exponentials, to write I 5 as,