Dimensional Reduction for Conformal Blocks

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.

In this note we develop a new representation of conformal blocks in d dimensions. This representation arises from the "dimensional reduction" of a CFT, i.e. the restriction of the conformal group SO(d + 1, 1) to a subgroup SO(d, 1) that preserves a hyperplane of codimension one. Although this is similar in spirit to a Kaluza-Klein reduction, we are not actually truncating the theory: rather, we simply organize all states in the Hilbert space of the CFT in representations of SO(d, 1) instead of the full conformal group. In particular, a d-dimensional conformal block will decompose into infinitely many (d−1)-dimensional conformal blocks with computable coefficients. As a corollary, this strategy provides an explicit formula for 3d and 5d conformal blocks in terms of 2 F 1 hypergeometric functions. This paper is organized as follows. Section 2 reviews basic facts about conformal blocks and develops the promised dimensional reduction. In section 2.3, we compare our expansion in d − 1 dimensional blocks to an expansion in 2d blocks. Finally section 3 discusses several directions for future work. Appendix A is a consistency check of the formalism developed in this note, applying it to the four-point function of the free scalar field.

Dimensional reduction
Let's start by recalling the definition of conformal blocks. For concreteness, consider a scalar operator φ of scaling dimension ∆ φ in a unitary d-dimensional CFT. Conformal invariance requires that its four-point function is of the following form: where the function G φ (u, v) depends only on two conformally invariant cross ratios The four-point function (2.1) can be computed using the operator product expansion (OPE): Here the sum runs over all primary operators O µ 1 ...µ (x) of even spin in the theory; with ∆ we denote their scaling dimension, and the OPE coefficient λ O is the constant of proportionality appearing in the three-point function φφO . The differential operator C ( ) ∆ (x, ∂) µ 1 ···µ depends only on the quantum numbers ∆ and . In passing, we note that unitary puts a lower bound on the possible values that ∆ can have: By applying the OPE twice to the four-point function (2.1), one can show that G φ (u, v) can be written as follows: The functions G ( ) ∆ (u, v; d) are conformal blocks, hence Eq. (2.5) is known as a conformal block (CB) decomposition. As the notation indicates, the blocks only depend on the quantum numbers ∆ and and the spacetime dimension d. In practice, they can be computed by solving a second-order PDE [18] while imposing the following asymptotic behaviour: is a rescaled Gegenbauer polynomial with parameter ν := (d − 2)/2: By construction, these functions obey C although we will leave c (d) arbitrary in the rest of this paper.
In even spacetime dimensions, simple expressions for the conformal blocks exist [17][18][19]. These are easiest to state in the Dolan-Osborn coordinates z,z, defined through u = zz, v = (1 − z)(1 −z). On the Euclidean section, z is a complex coordinate andz = z * its conjugate. After defining the 2d and 4d conformal blocks are: No similar formulas in odd d are known, although some simplifications occur when specializing to the "diagonal" line z =z [21,35].
The conformal block G ( ) ∆ has a representation-theoretical meaning: it is the contribution of a conformal multiplet of dimension ∆ and spin to the four-point function (2.1), containing a primary operator O µ 1 ···µ (x) and all of its derivatives. Such a multiplet can be described in a concrete fashion through the state-operator correspondence. The multiplet of O is built on top of the primary state |O µ 1 ···µ := lim x→0 O µ 1 ...µ (x)|0 , where |0 is the CFT vacuum. All other states in the multiplet are obtained by acting on |O with P µ , the generator of translations of the conformal algebra. A complete basis 1 of these descendants is spanned by the following states: It is understood that the µ indices must be symmetrized and made traceless. The state shown in (2.11) then has scaling dimension ∆ + 2k + m + p and spin + m − p. It follows that a descendant of level n -that is to say, with dimension ∆ + n -can have the following spins: For a suitable choice of coordinates, there is one-to-one correspondence between a descendant of level n and spin j and a term in the conformal block G ( ) ∆ . To make this concrete, we pass to the following coordinates: In the (s, ξ) coordinates, the contribution of a level-n spin-j descendant to the conformal block can be shown [20] to be proportional to P This is consistent with the fact that Gegenbauer polynomials are d-dimensional spherical harmonics. Consequently, conformal blocks admit an expansion of the form with j again restricted to the range (2.12). The coefficients a (d) n,j are fixed by conformal invariance, and are known in closed form as 4 F 3 hypergeometrics evaluated at unity [18].
As advertised, we will break the conformal group down to a subgroup of conformal transformations in d−1 dimensions, and we want to analyze the consequences of this dimensional reduction for conformal blocks. Let us first consider a toy example of what will happen, namely the restriction of the rotation group SO(d) to SO(d−1). If we think of SO(d) as the isometry group of the sphere S d−1 , this means that we take the subgroup of rotations that leave the equator invariant. Under this restriction, the spin-representation of SO(d), denoted as [ ] d , breaks up into SO(d−1) irreps as follows: The branching rule ( Since spherical harmonics form a representation of SO(d), the branching rule (2.16) applies in particular to the (rescaled) Gegenbauer polynomials. Concretely, the spin-Gegenbauer polynomial C (ν) can be written in the following form: are Gegenbauer polynomials in d − 1 dimensions. As a matter of fact, only spins j = , − 2, . . . , mod 2 appear in the RHS of Eq. (2.17), owing to the selection rule (2.18) The coefficients Z j in Eq. (2.17) can be computed using explicit expressions for the Gegenbauer polynomials [36] together with their orthogonality. This yields It will prove useful later in this work to have a bound on the coefficients Z j . It is easy to see that all Z j are positive, provided that d ≥ 2. 2 Moreover, the normalization condition C (ν) (1) = 1 implies that for fixed we have We conclude that 0 ≤ Z j ≤ 1 for all , j.
Having considered the restriction SO(d) → SO(d − 1), we now turn our attention to the conformal group SO(d + 1, 1). We will restrict the full group to a subgroup SO(d, 1) that preserves the hyperplane x d = 0. Doing so, a primary d-dimensional representation breaks up into infinitely many primary (d−1)-dimensional representations. The argument is the following. Recall that a state is a primary of SO(d + 1, 1) if and only if it is annihilated by all d generators of special conformal transformations, which we denote here by K µ . Therefore any state that is annihilated by K 1 , . . . , K d−1 but not by K d is a descendant of SO(d + 1, 1) but a primary of SO(d, 1). Among all descendants shown in Eq. (2.11), the following states fit that description: The state |O; j, m has SO(d − 1) spin j and scaling dimension ∆ + m. We arrive at the following branching rule: any SO(d + 1, 1) multiplet of dimension ∆ and spin splits up into infinitely many SO(d, 1) multiplets of spin 0 ≤ j ≤ and dimension ∆ + m with m ≥ 0. 3 Consequently, a conformal block G ( ) ∆ (u, v; d) can be written as an infinite sum over the conformal blocks G (j) ∆+m (u, v; d − 1) with 0 ≤ j ≤ and m ≥ 0. There are however some selection rules that apply, as was the case for the Gegenbauer polynomials. We will derive these in the ρ kinematics of [39], passing to the (r, η) coordinates defined as In the (r, η) coordinates, conformal blocks have an expansion where only descendants of even level appear [20]: with j restricted to the range (2.12). The SO(d + 1, 1) → SO(d, 1) branching rule described above must apply to any coordinate set, in particular to the (r, η) coordinates. By consistency with Eq.
with the sum running over j = , − 2 , . . . , mod 2 . (2.25) The coefficients A n,j (∆, ) are fixed by conformal invariance. In the following section, we will explain one method to compute them.

Recursion relation for coefficients
In this section, we will compute the coefficients A n,j (∆, ) appearing in Eq. (2.24). Our discussion will rely heavily on the representation (2.15) of conformal blocks in the (s, ξ) coordinates. In particular, we will use the fact that the coefficients a (d) n,j (∆, ) obey a three-term recursion relation: We can use this recurrence to compute the coefficients a (d) n,j (∆, ) to arbitrary order in n, starting from the initial condition a that is imposed by Eq. (2.6). A comprehensive discussion of this recursion relation is given in [20].
We will compute the A n,j (∆, ) coefficients by formulating a second recursion relation. As a starting point, remark that the conformal block G ∆+n,j (s, ξ). This expansion takes the following form: On the other hand, we can first apply the dimensional reduction formula (2.24) to the conformal block G n−2m,j (∆ + 2m, k) . (2.31) The sum over k is also restricted to j + 2m − n ≤ k ≤ j + n − 2m and k ≡ mod 2.
Now fix j ∈ { , − 2, . . . , mod 2} and n ≥ 0. Requiring that the two expressions for Y ( ) 2n,j agree, we obtain the following identity: where q * = ( + 2n − j)/2 and k is restricted to Notice that the RHS of (2.32) only involves coefficients A m,k (∆, ) with m < n. Moreover, the coefficients a 2n−2m,j (∆ + 2m, k) can be computed by means of the recursion relation (2.26). We can therefore use Eq. (2.32) to compute the coefficients A n,j (∆, ) recursively, up to arbitrary n, starting from n = 0. To be precise, Eq, (2.32) must be understood as a set of /2 coupled recursion relations, one for every allowed value of j. Finally, we remark that the above recursion relation is inhomogeneous, which means that the "initial condition" A 0,j (∆, ) is not arbitrary. Concretely, setting n = 0 in Eq. (2.32) yields which is consistent with the asymptotics imposed by Eq. (2.6).
Although the recursion relation (2.32) looks complicated, its solution can be written down in closed form: writing τ := ∆ − ( + d − 2) for the conformal twist. While we don't have a rigorous proof of this formula, we have checked that it satisfies (2.32) for , n ≤ 20 and we conjecture that it holds in general. Clearly Eq. (2.35) can be checked in other ways, e.g. using expansions of conformal blocks in the z or ρ coordinates and in the diagonal limit.
In passing, we notice that for the scalar ( = 0) block, only terms with j = 0 are allowed in (2.24), and the formula for the coefficients simplifies: A similar simplication occurs for = 1.

Convergence
Equations (2.24) and (2.35) are the main result of this note. At this stage, we want to point out two important properties of the coefficients A n,j . For convenience, we will set c (d) ≡ 1 in what follows. First, we note that all A n,j (∆, ) are positive, provided that ∆ satisfies the unitarity bound (2.4) and d ≥ 2. Second, we remark that A n,j decays exponentially fast with n. To prove this, let's consider the coefficient A n,j (∆, ) as a function of ∆, keeping , j and n fixed. We notice that A n,j (∆, ) is a rational function of ∆ of the form p(∆)/q(∆), where p and q are polynomials of equal degree. Furthermore p and q completely factorize over the reals, with all zeroes at values of ∆ at or below the unitarity bound. This means that above the unitarity bound, A n,j (∆, ) is a slowly varying function of ∆, and it is well approximated by its value in the limit ∆ → ∞: As promised, the coefficient A n,j (∆, ) decreases exponentially with n, as ∼ n −1/2 16 −n . Remarkably, this exponential behaviour holds not only asymptotically, but already starts at n = 1.
So far, we have encountered three different expansions for d-dimensional conformal blocks: the "z-series" from Eq. (2.15), the "ρ-series" from (2.23) and the expansion in terms of lowerdimensional blocks (2.24). In Fig. 1 we compare their convergence rates numerically, by truncating these expansions at finite order N and evaluating them at the crossing symmetric point u = v = 1/4. The results corroborate that the truncation error of (2.24) decreases exponentially with N . Rel. error, Δ = 1 For completeness, we can verify that the exponential decay with n also holds for ∆ close to the unitarity bound. For spinning operators ( ≥ 1), the limit τ → 0 is continuous, meaning that there are no important corrections to (2.37), and the exponential decay at large n persists. This is confirmed by an explicit expression for A n,j at τ = 0 shown in Appendix A. As is well known, the scalar ( = 0) block diverges at the unitarity bound ∆ = ν, where a level-two descendant becomes null. Using a conformal representation theory argument [40,41,22,30], we have omitting terms that are regular as ∆ → ν. Hence near the unitarity bound, G ∆ is dominated by a conformal block with ∆ = d/2 + 1, which itself is well above the unitarity bound. Therefore the estimate (2.37) applies, and we are done. The same conclusion can be reached by expanding Eq. (2.36) around ∆ = ν.
It may be interesting to systematically compute corrections to the leading-order behaviour (2.39). There is a group-theoretical approach to this problem, first discussed in appendix A of Ref. [46] (see also [56]). We will briefly review their argument here. The idea is to restrict SO(d, 2) -the conformal group in Minkowski signature -to SO(2, 2), the group of conformal transformations acting on the (z,z) plane. On the level of its Lie algebra, the latter splits into two copies of sl (2), spanned by three chiral generators L 0 , L ±1 and three anti-chiral generatorsL 0 ,L ±1 . Any SO(2, 2) primary state is therefore labeled by two numbers h,h, the eigenvalues of L 0 resp.L 0 ; such a state lifts to a local operator with scaling dimension h +h and spin |h −h|.
Under this restriction, any d-dimensional conformal multiplet breaks up into infinitely many "lightcone primaries". As with the dimensional reduction discussed in this paper, this implies that any d-dimensional conformal block can be decomposed into 2d blocks. Concretely, we have: for some coefficients P h,h (∆, ; d) fixed by conformal symmetry. Every term in the RHS corresponds to a different lightcone primary with quantum numbers (h,h). In the limit z → 0, the sum (2.40) is dominated by a single block with h = (∆ − )/2 andh = (∆ + )/2, all other terms being suppressed by powers of z.
In order to compute corrections to (2.39) it is therefore sufficient to determine the coefficients P h,h (∆, ; d). This has not been done so far, to our knowledge. We remark however that the expression (2.35) for the coefficients A n,j (∆, ) is sufficient to determine the P h,h (∆, ; d) for all integer d. For d = 3 this is obvious: after relabeling h,h in the RHS of (2.40) in terms of scaling dimensions and spins, the coefficients P h,h are identical to the coefficients A n,j with d → 3. The generalization to d > 3 is straightforward: in order to determine the coefficients P h,h (∆, ; d > 3) one has to "dimensionally reduce" d − 2 times.

Discussion
This note has presented a new method to compute conformal blocks in d-dimensional CFTs, by relating them to conformal blocks of CFTs in d−1 dimensions. In particular, Eqs. (2.24) and (2.35) together form an explicit formula for blocks in odd d: for d = 3 (resp. d = 5) our method leads to an expression in terms of 2d (resp. 4d) blocks shown in Eq. (2.10), which in turn are given by 2 F 1 hypergeometric functions. Moreover, the expansion in lower-dimensional blocks converges exponentially fast, which may prove to be useful for numerical applications.
Currently only two closed-form expressions are known for conformal blocks in odd d: the zseries expansion (2.15) and a formula that uses Mellin-Barnes integrals [57][58][59]19]. The latter involves so-called Mack polynomials that don't admit very compact expressions. The coefficients A n,j from Eq. (2.35) may therefore be easier to deal with in practice. In particular, they may be useful for the analytic bootstrap [60,61] in three dimensions, since the two-dimensional crossing kernel is known in closed form [62].
There are a few obvious ways to extend the results presented in this note. First, it is possible write down a similar expansion for conformal blocks with non-zero external dimensions. The resulting expressions are somewhat more complicated, as the selection rule described below (2.23) does not apply. Second, it is possible to dimensionally reduce more complicated representations of the Lorentz group. A starting point for this would be the "seed" conformal blocks in three and four dimensions [29,31]. An even further generalization consists of dimensionally reducing superconformal multiplets and the resulting superconformal blocks. We leave all of these issues for future work.