The Virasoro fusion kernel and Ruijsenaars’ hypergeometric function

We show that the Virasoro fusion kernel is equal to Ruijsenaars’ hypergeometric function up to normalization. More precisely, we prove that the Virasoro fusion kernel is a joint eigenfunction of four difference operators. We find a renormalized version of this kernel for which the four difference operators are mapped to four versions of the quantum relativistic hyperbolic Calogero–Moser Hamiltonian tied with the root system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BC_1$$\end{document}BC1. We consequently prove that the renormalized Virasoro fusion kernel and the corresponding quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function, are equal.


Introduction
Two-dimensional conformal field theories (CFTs) have been intensively studied since the seminal work of Belavin et al. [3]. In addition to their broad range of physical applications in, for example, condensed matter theory and string theory, they possess rich underlying mathematical structures.
The infinite-dimensional symmetry algebra of 2D CFTs determines the structure of their correlation functions. The operator product expansion makes it possible to decompose N -point correlators on the Riemann sphere into combinations of threepoint structure constants and universal quantities called Virasoro conformal blocks. In particular, there are three ways to decompose the four-point correlation function. The universal parts of each decomposition are called s-, t-and u-channel conformal blocks. Since each decomposition must lead to the same four-point function, the three kinds of conformal blocks are related by duality transformations.
In particular, the s-and t-channel conformal blocks are related by an integral transform called fusion transformation. Such a transformation was conjectured in [21] and established in [36]. The corresponding kernel, the Virasoro fusion kernel, was constructed by Ponsot and Teschner [21,22] and revisited in [38] as b-6j symbols for the modular double of U q (sl 2 (R)). This terminology comes from the fact that b, which is related to the central charge of the Virasoro algebra by c = 1 + 6(b + b −1 ) 2 , is associated with two unimodular parameters q = e iπ b 2 andq = e iπ b −2 . The Virasoro fusion kernel also appears in quantum Teichmüller theory [20] and in 3d supersymmetric gauge theories [38,39], and recent physical applications were found in [6,7,12].
The Virasoro fusion kernel is denoted F (b, θ , σ s , σ t ). Here, θ is a set of four external conformal dimensions associated with the fields entering the four-point correlation function, while σ s and σ t are two internal conformal dimensions associated with the s-and t-channel decompositions of the four-point function.
The Ponsot-Teschner formula [21] for F is recalled in (4.1). It is given by a contour integral whose integrand involves a special function s b (z). The s b -function appears in several different contexts. It was introduced under the names of "Quantum Dilogarithm function," "Hyperbolic Gamma function" and "Quantum Exponential function" in [9,25,41], respectively. 1 In particular, F is proportional to a hyperbolic Barnes inte-gral which is a hyperbolic generalization of the Barnes representation for the Gauss hypergeometric function. Such an integral was constructed in [5] as a degeneration limit of the hyperbolic hypergeometric function, and it also arises as a limit of Spiridonov's elliptic hypergeometric V -function [32,38]. Finally, the Virasoro fusion kernel is a joint eigenfunction of four difference operators [17,21].
On the other hand, Calogero-Moser (CM) models are systems of particles living on the real line or the circle and interacting with a rational, trigonometric/hyperbolic or elliptic potential. Their relativistic deformations were found by Ruijsenaars and Schneider at the classical level [31]. Quantum relativistic CM systems tied with the root system A N −1 were found by Ruijsenaars [30], and their BC N generalization by van Diejen [40]. Therefore, we will refer to quantum relativistic BC N CM systems as quantum Ruijsenaars-van Diejen (RvD) systems. In this article, only the case N = 1 of the latter system will be considered.
The rank N quantum trigonometric RvD system is solved by the Koornwinder N -variable polynomials [13]. However, in the hyperbolic case the eigenfunctions are non-polynomial and are only known in the rank one. The corresponding eigenfunction, the Ruijsenaars hypergeometric function, was introduced in [24] and studied in greater detail in [26][27][28]. This function is denoted R(a − , a + , γ , v,v); here, a − and a + are associated with two unimodular quantum deformation parameters q = e iπa − /a + and q = e iπa + /a − , while γ is a set of four external couplings constants. Finally, v andv are viewed as geometric and spectral variables, respectively.
Of particular importance is the renormalized R-function, denoted R ren (a − , a + , γ , v,v), which was defined in [26]. The definition (2.3) and the properties of the function R ren resemble those of the Virasoro fusion kernel. First of all, R ren is defined as a contour integral whose integrand involves the hyperbolic gamma function G(a − , a + , z). The functions G and s b are simply related by R ren is also proportional to a hyperbolic Barnes integral [5]. Moreover, it is a joint eigenfunction of four difference operators which are four versions of the rank one quantum hyperbolic RvD Hamiltonian. Let us finally mention that the function R ren was related to the modular double of U q (sl 2 (R)) in [4].
The discussion above suggests that the Virasoro fusion kernel and the Ruijsenaars hypergeometric function are closely related. The aim of this article is to show that the two functions are equal up to normalization (see Theorem 1). The identification procedure can be summarized as follows. We prove in Propositions 4.3 and 4.4 that the Virasoro fusion kernel is a joint eigenfunction of four difference operators. We provide the Ruijsenaars/CFT parameters identification in Sect. 5.1. We define in (5.14) a renormalized version F ren of F, and we show in Proposition 5.3 that both R ren and F ren satisfy the same four difference equations. Therefore, the two functions are proportional. We finally show in Theorem 1 that they are actually equal: The proof is rather simple and follows from the identity (B.4) satisfied by the hyperbolic Barnes integral.

Organization of the paper
The Ruijsenaars hypergeometric function and various of its properties are recalled in Sect. 2. Section 3 reviews some properties of the four-point Virasoro conformal blocks and introduces the Virasoro fusion kernel in this context. In Sect. 4, we study various symmetry and eigenfunction properties of F. Finally, our main result-the identification of F with R ren -is presented in Sect. 5.

Ruijsenaars' hypergeometric function
This section is a brief overview of the (renormalized) Ruijsenaars hypergeometric function following [26]. We recall several of its properties which we will need to relate it to the Virasoro fusion kernel.

Definition
Define two sets of external couplings constants γ andγ by where J is a matrix satisfying J 2 = I 4 and is defined by The renormalized R-function denoted R ren (a − , a + , γ , v,v) is given by a contour integral whose integrand involves the hyperbolic gamma function G(a − , a + , z). The definition of G and its properties can be found in Appendix A. Its dependence on a − and a + will be omitted for simplicity. Following [26], the function R ren is defined for and where a = (a + +a − )/2. The function G has sequences of poles and zeros given in (A.4) and (A.5), respectively. Therefore, for instance, when a + , a − > 0 the integrand in (2.3) has eight semi-infinite lines of poles, four of them increasing and the other ones decreasing. The integration contour R runs from −∞ to +∞ separating the increasing and decreasing sequences of poles. Moreover, for fixed values of a − , a + and γ , the function R ren is meromorphic in v andv. Finally, it was shown in [5,Proposition 4.20] that the function R ren is proportional to a hyperbolic Barnes integral B h (a − , a + , u) whose definition is recalled in Appendix B. B h (a − , a + , u) is one possible degeneration of the hyperbolic hypergeometric function, and other types of degenerations led the authors of [5] to find other representations for R ren .

Symmetry properties
The function R ren possesses various discrete symmetries which follow from the representation (2.3). In view of the property (A.6) of the hyperbolic gamma function, R ren is scale invariant: Moreover, because of the fact that G(a − , a + , z) = G(a + , a − , z), the following identity holds: It can also be verified that R ren satisfies Finally, the function R ren is even in v,v and is symmetric under permutations of γ 1 , γ 2 , γ 3 . It was shown in [27, Theorem 1.1] that a similarity transformed version of R ren has an extended D 4 -symmetry in γ . More precisely, define the function E ren by where the function δ is given by The function E ren is invariant under all permutations of γ 0 , γ 1 , γ 2 , γ 3 and under flipping the sign of any pair of γ μ 's. The resulting invariance group is the Weyl group of the Lie algebra D 4 .

Eigenfunction properties
Define a translation operator e ±ia − ∂ x which formally acts on meromorphic functions The quantum relativistic Calogero-Moser Hamil-tonian tied with the root system BC 1 and with a hyperbolic interaction was defined in [40]. As mentioned in the introduction, it will be referred to as the Ruijsenaars-van Diejen (RvD) Hamiltonian. It is a difference operator of the form sinh 2π x a+ sinh 2π , (2.10) and where the potential V is defined by (2.11) The function R ren is a joint eigenfunction of four versions of this Hamiltonian [26, Theorem 3.1]. More precisely, for (a − , a + , γ , v,v) ∈ RHP 2 × C 6 the following eigenvalue equations hold: (2.12d) It can be noted that (2.12a) implies (2.12b), (2.12c) and (2.12d) thanks to the properties (2.5) and (2.6) of R ren .

Four-point Virasoro conformal blocks
In this section, we briefly review the four-point Virasoro conformal blocks and recall how the Virasoro fusion kernel arises in this context. A more complete overview of the subject can be found in [23,35,36].

Highest-weight representations of the Virasoro algebra
The Virasoro algebra is the symmetry algebra of two-dimensional conformal field theories. It has generators L n , n ∈ Z and relations where c is a central element in the algebra called central charge. Highest-weight representations V θ of the Virasoro algebra are generated from vectors |θ > which satisfy L 0 |θ >= (θ )|θ >, L n |θ >= 0, n > 0, (3.2) where a Liouville-type parameterization of and c is used: The representations V θ decompose as follows: It is defined by the following commutation relations: In particular, it admits the following formal series expansion: where v n ∈ V θ 2 and v 0 = |θ 2 >. We emphasize that the expansion (3.6) can be renormalized by an arbitrary factor N (θ 2 , θ 0 , θ 1 ). Such a factor will be introduced in Sect. 5.

Four-point Virasoro conformal blocks
We now define a set of external conformal dimensions θ by The four-point Virasoro conformal block is defined by the expectation value of a composition of two chiral vertex operators as follows: The parameters θ and σ s will be referred to as external and internal momenta. The block F admits a series expansion in z which can be computed recursively using (3.1), (3.5) and (3.6): (3.9) The discovery of the AGT relation [2] between two-dimensional conformal field theories and four-dimensional supersymmetric gauge theories led to a closed-form expression for all the coefficients c k . In our notation, the complete expansion of (3.9) can be found in [15,Eq. (3.1)]. Several conjectures exist for the analytic properties of four-point Virasoro conformal blocks. The series in (3.9) is believed to be convergent inside the unit disk |z| < 1. Moreover, the only singularities of the conformal blocks as a function of z are expected to be branch points at 0, 1, ∞ [10,42]. Under this assumption, conformal blocks are naturally defined for z ∈ C \ ((−∞, 0] ∪ [1, ∞)). Finally, they are believed to be analytic in θ and meromorphic in σ s , with the only possible poles located at ±σ (m,n) Let us finally mention that F admits another representation called Zamolodchikov's recursion [42]. This representation is a power series of the nome is the hypergeometric function. Whereas the representation (3.9) is expected to converge for |z| < 1, Zamolodchikov's recursion converges faster than (3.9) and for all z ∈ C\{1}.

Crossing transformations
The linear span (3.9) of four-point Virasoro conformal blocks with different internal dimensions σ s forms an infinite-dimensional representation of ( 0,4 ) = PSL 2 (Z), the mapping class group of the four-puncture Riemann sphere. It is generated by the braiding B and fusion F moves, such that F 2 = (B F) 3 = 1. The three ways of splitting four points on CP 1 into two pairs define the s-, t-and u-channel bases for the space of conformal blocks. The cross-ratio argument of conformal blocks in these channels is chosen from The braiding move B acts on the s-channel conformal blocks as follows: whereθ is obtained by performing the permutation θ 1 ↔ θ ∞ on θ . On the other hand, the fusion move was conjectured in [21] and established in [36]. It is represented by whereθ is obtained by performing the permutation θ 0 ↔ θ 1 on θ . It will be convenient to explicitly writeθ asθ Finally, the kernel F (b, θ , σ s , σ t ) of the fusion move is the Virasoro fusion kernel.

The Virasoro fusion kernel
In this section, we recall the Ponsot-Teschner formula [21] for the Virasoro fusion kernel. Let us also note that other representations for F were found in [38]. Moreover, we describe various symmetry and eigenfunction properties of F which will be useful in its identification with the Ruijsenaars hypergeometric function (2.3).

Definition
The Virasoro fusion kernel is defined by a contour integral involving two special respectively. The exact relations are given in (A.9). The Ponsot-Teschner formula reads (4.1) When b > 0, the integrand in (4.1) has eight vertical semi-infinite lines of poles, four of them increasing and the other four decreasing; the integration contour F runs from −∞ to +∞, separating the increasing and decreasing sequences of poles. More generally, the fusion kernel (4.1) can be extended to a meromorphic function of all of its parameters provided that c ∈ C \ R ≤1 , which corresponds to b / ∈ iR.

Symmetry properties
We now present various symmetry properties of the Virasoro fusion kernel. First, because the Virasoro conformal block (3.9) is a function of the conformal dimensions (x) = Q 2 4 + x 2 , it follows that it is even in θ and σ s . Therefore, the fusion transformation (3.11) implies that the Virasoro fusion kernel We now describe a duality transformation exchanging the internal momenta σ s and σ t .

Proposition 4.1
The Virasoro fusion kernel given in (4.1) satisfies whereθ is related to θ as in (3.12) and .
, it is easy to verify that the contour integral in (4.1) takes the form (4.5) where u ∈ G i Q and G k is defined by (B.1). Various choices for the eight parameters u lead to the same contour integral. Here, we choose (4.6) We are now going to apply the identity (B.4) satisfied by the hyperbolic Barnes integral. First, from (B.3) the action of ω on u ∈ G i Q is given by Second, a straightforward application of the identity (B.4) yields Recalling (4.5) and using where .
(4.12) Since the Virasoro fusion kernel is even in the external momenta, we have F b, Iθ , σ t , σ s = F b,θ , σ t , σ s . We deduce from (4.9) and (4.11) that the following identity holds: and a direct computation shows that where α is given in (4.4). Hence, the identity (4.3) is obtained.

Difference equations
We now show that the Virasoro fusion kernel is a joint eigenfunction of four difference operators, the first two acting on the internal momentum σ s and the remaining two acting on its dual σ t .

First pair of difference equations
Define the difference operator H F (b, θ , σ s ) by We now present a direct proof of the fact that the Virasoro fusion kernel (4.1) is an eigenfunction of H F . In order to provide rigorous proofs, in the remainder of this article we will need the following assumption on the parameters: Assumption 4.2 (Restriction on the parameters) We assume that

16) and
(4.18) Assumption 4.2 implies that the integration contour F in the Virasoro fusion kernel (4.1) is any curve going from −∞ to +∞ lying in the strip Im x ∈]− Q 2 , 0[. Moreover, Assumption 4.2 is made primarily for simplicity; we expect all the results of this article to admit an analytic continuation to more general values of the parameters, such as b ∈ C\iR, θ ∈ C 4 and (σ s , σ t ) ∈ C 2 .

Proposition 4.3
The Virasoro fusion kernel satisfies the following pair of difference equations: (2π bσ t ) F (b, θ , σ s , σ t ) , (4.19a) F (b, θ , σ s , σ t ) . (4.19b) Although the pair of difference Eq. (4.19) is known, we are not aware of a direct proof of it. More precisely, Eq. (4.19) are special cases of the pentagon equation which is one relation of the Moore-Seiberg groupoid [16,17,21]. They also arise from the fact that the fusion transformation (3.11) diagonalizes the Verlinde loop operator L γ t [37]. We now present a direct proof of (4.19). 2

Proof
We only need to prove (4.19a) thanks to the symmetry (4.2) of F. It will be convenient to rewrite the definition (4.1) of F as follows: where the dependence of X F (x, σ t ), Y F (x, σ s ) and Z F (x) on b and θ is omitted for simplicity, and (4.21) The proof relies on the following non-trivial identity for the action of the difference operator (4.15) on the block X F : .
(4.26) Hence, the proof of (4.22) is equivalent to proving the identity (4.27) We proceed as follows: Both f 1 (σ s ) and f 2 (σ s ) are meromorphic functions of σ s . It can also be easily be checked that they are even and ib −1 -periodic in σ s . Moreover, we have the following asymptotics: lim We now show that f 1 (σ s ) and f 2 (σ s ) have equal σ s -residues in a horizontal period strip Im σ s ∈ [0, b −1 ]. It can first be verified that f 2 (σ s ) has vanishing residues at σ s = ± ib 2 , 0. However, it has nonzero residues at σ s = ±(ib + θ ∞ + x) which are equal to

(4.30)
On the other hand, the only residues of f 1 (σ s ) are located at σ s = ±(ib + θ ∞ + x) and straightforward computations show that they are also equal to the right-hand side of (4.30). Hence, we have shown that the function f 1 (σ s ) − f 2 (σ s ) is bounded and holomorphic in a horizontal period strip Im σ s ∈ [0, b −1 ]. These properties extend to the whole complex plane by ib −1 -periodicity in σ s . Therefore, Liouville theorem ensures that the function f 1 (σ s ) − f 2 (σ s ) is constant everywhere. By evaluation at Re(σ s ) = +∞ using (4.29), we deduce that f 1 (σ s ) = f 2 (σ s ). Hence, the identity (4.22) is proved. We now let the difference operator H F defined in (4.15) act on the Virasoro fusion kernel (4.20). Using (4.22), we have (2π b(x + θ t )).
(4.31) The next step of the proof utilizes the following identity: We now perform a contour shift together with x → x − ib in the first line of (4.31), and we use (4.32). Assumption 4.2 ensures that the contour does not cross any pole. We obtain (4.33) Finally, the identity (4.34) implies that the last two lines in (4.33) cancel, and that the first line in (4.33) yields the desired result.

Second pair of difference equations
The duality transformation (4.3) exchanging the internal momenta σ s and σ t implies that the Virasoro fusion kernel (4.1) satisfies another pair of difference equations where the difference operators act on σ t . We introduce the dual operatorH F bỹ where H 0 F is given in (4.17),θ is defined by (3.12) and (4.36) We next show that the Virasoro fusion kernel is also an eigenfunction of this difference operator.

Proposition 4.4 The Virasoro fusion kernel satisfies the dual pair of difference equa-tionsH
(4.37a) Proof Again, we only need to prove (4.37a) thanks to the symmetry (4.2) of F. Let us first apply the duality transformation θ →θ and σ s ↔ σ t to Eq. (4.19a): where H F is given in (4.15). Substituting the identity (4.3) into (4.38), we obtain (4.39) where for the sake of brevity we denoted f (a, ±b) = f (a, b) f (a, −b). The operator on the left-hand side of (4.39) takes the form (4.40) A tedious but straightforward computation using the properties (A.10) and (A.11) of the functions g b and s b shows that for k = ±1, we have which finally implies the identity In view of (4.39), this completes the proof.
In summary, we have shown that the Virasoro fusion kernel satisfies two pairs of difference equations given in (4.19) and (4.37).

Main result
The renormalized Ruijsenaars hypergeometric function (2.3) and the Virasoro fusion kernel (4.1) were studied in Sects. 2 and 4, respectively. Although they appear in different contexts, the two functions resemble each other. Both of them are proportional to a hyperbolic Barnes integral (B.2) and are joint eigenfunctions of four difference operators. The main result of this section is Theorem 1. It shows that F and R ren are the same function up to normalization.
The key to the identification of F and R ren will be to compare and match the four difference equations that they satisfy. Let us first compare the difference operators H F defined by (4.15) and H Rv D given in (2.9). H F and H Rv D appear in the left-hand sides of the difference Eqs. (4.19a) and (2.12a) satisfied by F and R ren , respectively. Of particular importance are the "potentials" H 0 F defined by (4.17) and V given in (2.11), because they remain invariant under any change of normalization. The first step in the identification of F and R ren is to show that H 0 F and V are equal under a certain parameter correspondence.

Ruijsenaars/CFT parameter correspondence
We now provide a parameter correspondence between the Virasoro fusion kernel F (b, θ , σ s , σ t ) and the Ruijsenaars hypergeometric function R ren (a − , a + , γ , v,v).
The second relation between the pairs (v,v) and (σ s , σ t ) is simply given by Moreover, the couplings γ defined by (2.1) and the external momenta θ given in (3.7) are related by where the matrix L reads Finally, the dual sets of parametersγ andθ , respectively, given in (2.1) and (3.12) can be related as follows. It is straightforward to verify that the matrices J defined by (2.2), K given in (3.12) and L satisfy We emphasize that the parameter correspondence described above can be rescaled thanks to the property (2.4) of the function R ren . It can now be verified that the potentials H 0 F defined by (4.17) and V given in (2.11) are equal: This identification suggests that the difference operator H Rv D defined in (2.9) can be obtained from H F given in (4.15) after a proper change of normalization. It also suggests that the difference Eq. (4.19a) satisfied by F can be mapped to the difference Eq. (2.12a) satisfied by R ren .

Some requirements
The second step in the identification of F and R ren is to find a renormalized version F ren of F which satisfies a difference equation of the form (2.12a), but also three other equations of the form (2.12b), (2.12c), (2.12d). This requirement can be reformulated as follows. The difference Eq. (2.12a) satisfied by the function R ren implies the three remaining Eqs. (2.12b), (2.12c) and (2.12d) because R ren satisfies the identities (2.5) and (2.6). Moreover, using the parameter correspondence of Sect. 5.1, the self-duality (2.6) of R ren becomes This equation can be compared to the property (4.3) of the Virasoro fusion kernel: The function R ren is self-dual, while F is not. We deduce that the key to the identification of F and R ren is to find a renormalized version F ren of F which (i) satisfies a difference equation of the type (2.12a), (ii) is invariant under b → b −1 , and (iii) is self-dual.

Definition of F ren
We now present a renormalized version F ren of F which satisfies the three properties (i)-(iii). First, we define renormalized conformal blocks F ren by where F is given in (3.8), and where the normalization factor N is .
(5.10) We are now going to rewrite the fusion transformation (3.11) in terms of F ren . We first introduce a weight function 3 ω (b, θ , σ s ) = μ (b, θ , σ s ) μ (b, θ , −σ s ) , (5.11) where . (5.12) The fusion transformation (3.11) can now be rewritten as 13) where the renormalized Virasoro fusion kernel F ren is defined by (5.14) and where F is defined in (4.1). In the next three propositions, we show that F ren satisfies the desired three properties (i)-(iii).

Proposition 5.1
The following identity holds: We now prove that F ren is self-dual.

Proposition 5.2 The renormalized Virasoro fusion kernel F ren satisfies
Proof The proof consists of writing an explicit relation between F ren (b, θ , σ s , σ t ) and F ren b,θ , σ t , σ s using the definition (5.14) and the identity (4.3). First, from (5.14) we have (5.17) Substituting the identity (4.3) into (5.17), we have (5.18) where α is given in (4.4). Using once again the definition (5.14), we obtain shows that all the prefactors in the right-hand side of (5.19) cancel out. Therefore, the identity (5.16) is obtained.

Difference equations for F ren
We now show that the renormalized Virasoro fusion kernel and the renormalized Ruijsenaars hypergeometric function satisfy the same four difference equations.

Proposition 5.3
The renormalized fusion kernel F ren defined by (5.14) satisfies the following difference equations: where the matrix L is given in (5.4) and H Rv D is defined by (2.9).
Proof We only need to prove (5.20a) thanks to the symmetries (5.15) and (5.16) of F ren . Moreover, it will be convenient to rewrite (5.14) as F = (P(σ t )/Q(σ s ))F ren ,

(5.21)
Substituting F = (P(σ t )/Q(σ s ))F ren into the difference Eq. (4.19a) satisfied by F, we obtain (5.22) The factors P(σ t ) in the left-hand side of (5.22) cancel out, since the difference operator H F (b, θ , σ s ) acts on the variable σ s . Therefore, the proof of (5.20a) consists of verifying the following identity: where H F and H Rv D are, respectively, defined by (4.15) and (2.9). We have (5.24) where H + F is given in (4.16). Using the identity (A.10) satisfied by the g b -function, it is straightforward to verify that the following identities hold for k = ±1: . (5.25) We now substitute (5.25) into (5.24), and we use the reflection equation (1−z) (z) = π/ sin (π z). For k = ±1, we obtain where C is defined by (2.10). It finally remains to use the identification (5.7) of the potentials V and H 0 F to obtain (5.23).

F ren = R ren
We have shown that the renormalized Virasoro fusion kernel (5.14) and the renormalized Ruijsenaars hypergeometric function (2.3) satisfy the same four difference equations. Therefore, the two functions are proportional. We now present the main result of this article.

Theorem 1
The renormalized fusion kernel F ren given in (5.14) and the renormalized Ruijsenaars hypergeometric function defined by (2.3) are equal: (5.27) where the matrix L is given in (5.4).

Proof
The equality is not immediate and follows from the identity (B.4) satisfied by the hyperbolic Barnes integral. Let us first rewrite F ren using (5.14) and the representation (4.1). It will also be convenient to perform a shift x → x − i Q 2 in the integrand. This amounts to lifting up all poles of the integrand by i Q 2 . Because of Assumption 4.2, this shift maps the contour F to a contour F which runs from −∞ and +∞ and lies in the strip Im x ∈]0, Q 2 [. It is now straightforward to verify that F ren can be written as .
(5.28) On the other hand, the function R ren defined in (2.3) satisfies  G(b, b −1 , z), the contour integral in (5.28) takes the form (5.30) Several choices of v ∈ G i Q lead to the same contour integral. Here, we choose which maps the contour F to R, and using the identity s b (z) = s b (−z) −1 .

Conclusion and perspectives
In this article, we have proved that the Virasoro fusion kernel is a joint eigenfunction of four difference operators. We have found a normalization of the conformal blocks for which the four difference operators are mapped to four versions of the quantum relativistic hyperbolic BC 1 Calogero-Moser Hamiltonian. We have consequently proved that the Virasoro fusion kernel and the Ruijsenaars hypergeometric function coincide up to normalization and are the quantum eigenfunction of this integrable system. We now mention a list of perspectives related to this work.
(1) It would be interesting to understand the role played by the four-point Virasoro conformal blocks in the context of the present integrable system. In view of (5.13) and Theorem 1, the function R ren is the kernel of the fusion transformation relating the s-and t-channel renormalized conformal blocks. On the other hand, in [28] a unitary Hilbert space transform associated with the function R ren was constructed for special values of the couplings. We believe that this Hilbert space is in fact spanned by the four-point Virasoro conformal blocks. (2) The quantum eigenfunction of the A 1 relativistic hyperbolic CM system is a onecoupling specialization of the Ruijsenaars hypergeometric function [29]. It would be interesting to compare this limit to the transition limit from the Virasoro fusion kernel to the Virasoro modular kernel [11]. The latter describes how conformal blocks associated with the one-point torus transform under a mapping class group action [18]. (3) A natural question is to find higher rank generalizations of our result. The BC N , N > 1 generalization of the function R ren has not yet been constructed. Higher rank analogs of the Virasoro fusion kernel are associated with W-type algebras and also remain to be found. However, the framework developed in [33,34] allows us, in principle, to construct such a generalization from a quantum group perspective. (4) What is the meaning of the classical and/or non-relativistic limits in the conformal blocks setting? (5) The R-function reduces to the celebrated Askey-Wilson polynomials in a limit where one of v,v is discretized [26]. What does this limit mean from the conformal blocks viewpoint? (6) The Askey-Wilson polynomials can be studied using representation theory of the double affine Hecke algebra and the Askey-Wilson algebra [14,19]. An interest-ing program would be to generalize this algebraic study to the case of R ren . We believe that Virasoro conformal blocks and their quantum monodromies provide the correct framework for such a study. (7) Various confluent limits of the Virasoro fusion kernel were constructed in [15]. It would be interesting to understand these limits from an integrable system point of view. (8) An interesting project motivated by the WZW SL(2, R) model would be to construct the fusion kernel of the affine Lie algebraŝl 2 and to find a connection to integrable systems. (9) Due to the role played by the Virasoro fusion kernel in the bootstrap approach to Liouville theory [21,23], we hope that the present work can provide new insights on the role of integrability in two-dimensional CFTs.
where a = (a + + a − )/2. The functions G and E are obviously invariant under the exchange of a − and a + . Moreover, G satisfies the difference equations G(z + ia − /2) G(z − ia − /2) = 2 cosh (π z/a + ), G(z + ia + /2) G(z − ia + /2) = 2 cosh (π z/a − ). (A.3) As long as the ratio a − /a + stays in the subset C \ (−∞, 0], the function G extends to a meromorphic function with poles and zeros given by The function E satisfies the difference equation On the other hand, the Virasoro fusion kernel (4.1) is defined in terms of two special functions s b (z) and g b (z) such that s b (z) = g b (z)/g b (−z). They are related to the functions G and E as follows: . (A.9) The properties of s b and g b can be deduced from the ones of G and E. Both s b and g b are obviously invariant under the exchange of b and b −1 . Most importantly, the function g b (z) satisfies the difference equations (A.10) Finally, the s b function satisfies