Further Results on a Function Relevant for Conformal Blocks

We present further mathematical results on a function appearing in the conformal blocks of four-point correlation functions with arbitrary quasi-primary operators. The $H$-function was introduced in a previous article and it has several interesting properties. We prove explicitly the recurrence relation as well as the $D_6$-invariance presented previously. We also demonstrate the proper action of the differential operator used to construct the $H$-function.


Introduction
Conformal field theories (CFTs) play an important role in modern physics. The introduction of the full conformal algebra constrains non-trivially N -point correlation functions. For example, two-point correlation functions are completely determined by conformal invariance while threepoint correlation functions are settled in terms of a finite number of coefficients. This can be seen as originating from the existence of a convergent operator product expansion (OPE) [1].
Moreover, using the OPE twice in four-point correlation functions leads to conformal blocks which represent the contributions of exchanged quasi-primary operators to the four-point correlation functions. Using associativity in N -point correlation functions further constrains the OPE, leading to the crossing symmetry of the four-point correlation functions which can be used to restrict the unknown conformal dimensions and OPE coefficients [2]. Recent work in spacetime dimensions larger than two resulted in some explicit expressions for certain specific conformal blocks [3,4], and the conformal bootstrap in spacetime dimensions larger than two has also been implemented numerically with impressive results [5].
The computation of conformal blocks in spacetime dimensions larger than two is not straightforward. Although they are technically determined by conformal symmetry, they are better understood from the embedding space where the conformal generators act linearly [6]. For example, the OPE has been studied utilizing the embedding space formalism in [7][8][9].
In [8], it was shown how to employ the OPE in the embedding space formalism to compute the scalar conformal block. In [9], the results of [8] were used to find a very general function, the H-function, that appears in general conformal blocks containing fields in more complicated representations. Using a Rodrigues equation for the H-function, it was shown that it satisfies a recurrence relation and a specific symmetry property leading to invariance under the dihedral group of order 12.
In this paper, an explicit expression for the H-function is used to show directly that the recurrence relation and the symmetry property are indeed satisfied. After briefly reviewing the definition of the H-function in Section 2, several new expressions for this function are obtained in Section 3. These results use the properties of Pochhammer symbols listed in Appendix A as well as properties of hypergeometric-like polynomials listed in Appendix B. Section 4 gives a proof of the recurrence relation and the symmetry property using various expressions explored in the previous section. Finally, in Section 5 the action of the differential operator is found and it is shown how to use it to compute the H-function constructively.

Functions
In this section we review the G-and H-functions as well as the associated differential operators introduced in [9].

Power Series
It is a well-known fact that the conformal blocks for scalar exchange in four-point correlation functions of four scalar quasi-primary fields in arbitrary spacetime dimension d are related to the function G (q;r;t) d (u, v). This function can be expressed as a double sum over powers of the variables x = u/v and y = 1 − 1/v, where u and v are the conformal cross-ratios, Moreover, (2.1) can be expressed in terms of the hypergeometric function G(α, β, γ, δ; x, y) of Exton [4,9,10] as where q, r and t are related to the conformal dimensions of the five scalar quasi-primary fields appearing in the conformal blocks.
In [9], it was argued that more general conformal blocks are given by linear combinations of the following function,

(2.2)
The functional form ( In the equation above, the second-order differential operator D (u,v) , as well as two related firstorder differential operators D (u) and D (v) , are defined as and satisfy the algebra

Recurrence Relation and Symmetry
In [9], the recurrence relation

Several Expressions for H
In this section several equivalent but completely different expressions for the H-function are given. The first subsection lists the various expressions, while the proofs are left for the following subsections. The reader only interested in the different forms of H can certainly skip the proofs.

H-function
By trivially combining Pochhammer symbols together, the original solution (2.2) for the Hfunction can be rewritten as (3.1) Another expression for the H-function, which allows to show that P (p,q;r;s,t) d (m, n) is invariant under the interchange of r + 1 − d/2 and r + m + n, is given by

(3.2)
A slightly more complicated expression with one extra sum, useful to prove the symmetry The final rewriting of the H-function, relevant to prove the recurrence relation (2.5), is To prove (3.2) from (3.1), it is convenient to reorder the sums in the polynomial using which leads to after simplifying the pre-factors. With the help of

Proof of (3.3)
Now that the equivalence of (3.2) and (3.1) is established, the third form for P Reordering the sums as after a trivial simplification of the pre-factors. Using and separating the sum over i gives At this point, the symmetry property (B.3) with a = −t + j can be used for the sum over i leading to Combining the Pochhammer symbols in the last line yields which is equivalent to (3.3) once a few simplifications of the Pochhammer symbols are performed and the indices are changed as in i ↔ k.

Proof of (3.4)
The expression (3.4) can be obtained starting from the form (3.5) which can be written as where the sum over i was factored out and its pre-factor simplified. Using the symmetry property Combining the Pochhammer symbols together leads to P (p,q;r;s,t) d after straightforward simplifications of the pre-factors. Shifting i → i − j followed by reversing the order of the sums brings the results to which is exactly (3.4).

Recurrence Relation and Symmetry
This section proves the recurrence relation and the symmetry using suitable expressions for the H-function obtained in the previous section.

Proof of the Symmetry
In [9] it was argued from the definition of the H-function in terms of the differential operator where the pre-factors in the first equality have been simplified. This result for P (p,q;r;s,t) d (m, n) therefore shows that the H-function is invariant under the dihedral group of order 12.

Proof of the Recurrence Relation
The recurrence relation (2.5) can be verified directly starting from expression (3.4). It is actually simpler to introduce a generalization of (3.4) in order to prove (2.5). Defining Q (p,q,t,a,b,c,d,e,f ) d The new polynomial Q satisfies several contiguous relations. Two such relations are needed to prove (2.5). Using the fact that leads directly to the two following contiguous relations for (4.1), These two contiguous relations are not obeyed by the polynomial P

Differential Operators
In this section the differential operator D (u,v) is used to derive both the G and H-functions constructively. Generalizations to higher-point correlation functions will be discussed elsewhere.
originates directly from the action of D (u,v) , it is now clear why (3.4) is the appropriate form to prove the recurrence relation (2.5).

Conclusion
We used several identities for the Pochhammer symbols and hypergeometric-like polynomials in order to show that the H-function computed in [9] is the appropriate function appearing in conformal blocks. With the help of these identities, several different expressions for the H-function were presented. This allowed us to demonstrate explicitly that the H-function is invariant under the dihedral group of order 12 and that it satisfies the proper recurrence relation.
We also found the explicit action of the differential operator on simple products of the conformal cross-ratios. This differential form was used to give a constructive proof of the H-function, independent of the approach based on identities used before. As far as computing conformal blocks is concerned, the action of the differential operator is actually the most important result of this paper. Indeed, there exists a generalization of this expression that acts straightforwardly on higher N -point correlation functions. This result will be discussed elsewhere.
Finally, it is worth mentioning that the physical interpretation behind the D 6 -symmetry of the H-function remains unclear. Nevertheless, this symmetry might have implications for the analyticity properties in spin of the conformal blocks.

Acknowledgments
Two of the authors (JFF and WS) would like to thank the CERN Theory Group, where this work was conceived, for its hospitality. The work of VC and JFF is supported by NSERC and FRQNT.

A. Mathematical Properties of the Pochhammer Symbol
Pochhammer symbols satisfy several mathematical properties and some of those properties are necessary to show that the different representations of the H-function are equivalent. For completeness, this appendix presents several useful identities for the Pochhammer symbol.
First, the Pochhammer symbol (x) α is defined as and for α = n a non-negative integer, it satisfies (−x) n = (−1) n (x − n + 1) n , (A.2) as well as the binomial identity The Vandermonde's identity

B. Symmetry Properties of Hypergeomtric-like Sums
To demonstrate the identities relevant for the H-function, it is necessary to use symmetry properties of hypergeometric-like polynomials. The appropriate symmetry properties are presented in this appendix.
Defining the hypergeometric-like polynomial it is easy to show that it satisfies the following symmetry property,