The Virasoro fusion kernel and Ruijsenaars' hypergeometric function

TL;DR

This paper establishes a precise mathematical equivalence between the Virasoro fusion kernel and Ruijsenaars' hypergeometric function, revealing their shared eigenfunction properties and linking conformal field theory with integrable quantum systems.

Contribution

It proves that the Virasoro fusion kernel is a joint eigenfunction of certain difference operators and identifies it with the Ruijsenaars hypergeometric function through a renormalization process.

Findings

Virasoro fusion kernel equals Ruijsenaars' hypergeometric function up to normalization

The kernel is a joint eigenfunction of four difference operators

The renormalized kernel corresponds to the quantum relativistic hyperbolic Calogero-Moser Hamiltonian

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.