Abstract
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 BC_1. We consequently prove that the renormalized Virasoro fusion kernel and the corresponding quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function, are equal.
Highlights
Two-dimensional conformal field theories (CFTs) have been intensively studied since the seminal work of Belavin et al [3]
We prove in Propositions 4.3 and 4.4 that the Virasoro fusion kernel is a joint eigenfunction of four difference operators
We have proved that the Virasoro fusion kernel is a joint eigenfunction of four difference operators
Summary
Two-dimensional conformal field theories (CFTs) have been intensively studied since the seminal work of Belavin et al [3]. The s- and t-channel conformal blocks are related by an integral transform called fusion transformation. 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 = eiπa−/a+ and q = eiπa+/a− , while γ is a set of four external couplings constants. Rren is proportional to a hyperbolic Barnes integral [5] It is a joint eigenfunction of four difference operators which are four versions of the rank one quantum hyperbolic RvD Hamiltonian. We show in Theorem 1 that they are equal: The proof is rather simple and follows from the identity (B.4) satisfied by the hyperbolic Barnes integral
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