AI 中文总结
本文在Liouville量子引力球面上构造量子n孔球面，计算其面积拉普拉斯变换，证明其满足超几何弦方程，并揭示其与Weil-Petersson体积的联系。
AI 中文摘要
矩阵模型通过弦方程表达了具有指定边界长度的随机曲面的配分函数。我们在Liouville量子引力（LQG）中给出了这一结构的概率实现。对于$\gamma^2\in(8/3,4)$，我们从由独立共形环系综（CLE）装饰的量子圆盘构造了量子$n$孔球面，并计算了其在固定边界长度下的面积拉普拉斯变换。该变换除了一个简单的前因子外，仅依赖于总边界长度，并由一个超几何弦方程决定。我们利用LQG/CLE耦合从稳定Lévy游荡推导出该方程。共形焊接方程产生了表示积分LQG/CLE耦合可观测量的对称多项式。当$\gamma\downarrow0$时，这些多项式收敛到具有$n$条测地边界的球面的Weil-Petersson体积。
英文摘要
Matrix models express partition functions of random surfaces with prescribed boundary lengths through string equations. We give a probabilistic realization of this structure in Liouville quantum gravity (LQG). For $γ^2\in(8/3,4)$, we construct the quantum $n$-hole sphere from a quantum disk decorated by an independent conformal loop ensemble (CLE) and compute its area Laplace transform at fixed boundary lengths. The transform depends only on the total boundary length apart from a simple prefactor and is determined by a hypergeometric string equation. We derive this equation from stable Lévy excursions using LQG/CLE coupling. The conformal welding equation yields symmetric polynomials that represent integrated LQG/CLE coupling observables. These polynomials converge to Weil--Petersson volumes of spheres with $n$ geodesic boundaries as $γ\downarrow0$.
Comments39 pages, 1 figure
