String equation for conformal loop ensemble on Liouville quantum gravity sphere

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 ${\gamma }^2\in (8\mathrm{/}3,4)$, 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 $\gamma \downarrow 0$.