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 $$BC_1$$ B C 1 . We consequently prove that the renormalized Virasoro fusion kernel and the corresponding quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function, are equal.

中文翻译：

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Virasoro 融合核和 Ruijsenaars 的超几何函数

我们证明 Virasoro 融合核在归一化之前等于 Ruijsenaars 的超几何函数。更准确地说，我们证明了 Virasoro 融合核是四个差分算子的联合特征函数。我们找到了该内核的重整化版本，其中四个差分算子映射到与根系统 $$BC_1$$BC 1 相关联的量子相对论双曲 Calogero-Moser Hamiltonian 的四个版本。我们因此证明了重整化的 Virasoro 融合核和相应的量子本征函数，（重整化的）Ruijsenaars 超几何函数，是相等的。