标题： 一个来自Virasoro拓扑量子场论的非理性Verlinde公式 显示英文标题

标题： A non-rational Verlinde formula from Virasoro TQFT

摘要： 我们利用Virasoro拓扑量子场论（TQFT）推导出一个积分恒等式，我们认为它是在中心电荷为$c\geq 25$的Virasoro代数下对Verlinde公式的非有理推广。 该恒等式将Virasoro融合核表达为关于（挖去点的）环面模群S核比值的积分。 特别是，它表明单点S核对Virasoro$6j$符号进行对角化。 在仔细研究了这个“Virasoro-Verlinde公式”的解析性质后，我们给出了三个应用。 在边界Liouville共形场论（CFT）中，该公式确保了圆环上边界单点函数的开闭对偶性。 在纯三维引力中，它为计算超双曲三维流形在圆周上的配分函数提供了关键步骤。 最后，在AdS$_3$/CFT$_2$中，该公式计算了一个三边界环面虫洞，从而预测了对偶大-$c$CFT系综中态密度与两个算符积展开系数之间的统计相关性。 最后，我们讨论了我们的结果对一般非有理二维CFT中融合规则的启示。 显示英文摘要

摘要： We use the Virasoro TQFT to derive an integral identity that we view as a non-rational generalization of the Verlinde formula for the Virasoro algebra with central charge $c\geq 25$. The identity expresses the Virasoro fusion kernel as an integral over a ratio of modular S-kernels on the (punctured) torus. In particular, it shows that the one-point S-kernel diagonalizes the Virasoro $6j$ symbol. After carefully studying the analytic properties of this `Virasoro-Verlinde formula', we present three applications. In boundary Liouville CFT, the formula ensures the open-closed duality of the boundary one-point function on the annulus. In pure 3d gravity, it provides an essential step in computing the partition function on hyperbolic 3-manifolds that fiber over the circle. Lastly, in AdS$_3$/CFT$_2$, the formula computes a three-boundary torus wormhole, which leads to a prediction for the statistical correlation between the density of states and two OPE coefficients in the dual large-$c$ CFT ensemble. We conclude by discussing the implications of our result for the fusion rules in generic non-rational 2d CFTs.