论文标题
量子盘的共形焊接
Conformal welding of quantum disks
论文作者
论文摘要
具有重量参数$ W> 0 $的两点量子磁盘是一个有限区域随机表面的家族，在liouville量子重力中自然出现。在本文中，我们表明，根据它们的边界长度将两个量子磁盘焊接给了另一个量子磁盘，并用独立的和弦$ \ mathrm {sle}_κ（ρ_-;ρ_+）$曲线装饰。这是Sheffield（2010）的经典结果和Duplantier-Miller-Sheffield（2014）的有限体积对应物，它涉及Infinite-rator-rate两点量子表面的焊接，称为量子楔，这对于交配理论至关重要。我们的结果可用于给出量子磁盘和量子球的交配定理的统一证据，除了对重量的交配描述$ w = \ frac {γ^2} {2} {2} $量子磁盘。此外，它是我们同伴工作[AHS21]的关键要素，它证明了使用随机表面的保形焊接和共同焊接结果提供所谓的SLE循环的$ \ Mathrm {slerm {slerm {slerm {slerm {slerm {slerm {slerm {slerm {slerm {lect}_κ（ρ_-;ρ_+）$。
Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal $\mathrm{SLE}_κ(ρ_-;ρ_+)$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees theory. Our results can be used to give unified proofs of the mating-of-trees theorems for the quantum disk and the quantum sphere, in addition to a mating-of-trees description of the weight $W = \frac{γ^2}{2}$ quantum disk. Moreover, it serves as a key ingredient in our companion work [AHS21], which proves an exact formula for $\mathrm{SLE}_κ(ρ_-;ρ_+)$ using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.
