量子三角形与虚几何流线
Quantum triangles and imaginary geometry flow lines
AI总结：
本文定义了三参数量子三角形作为刘维尔量子引力中三角形的量子版本，证明其给出虚几何流线围成三角形的定律，并通过与双点量子圆盘的共形焊接推导出带三个力点的弦SLE可积性结果。
摘要：
我们在刘维尔量子引力（LQG）中定义了一个三参数随机曲面族，它可以被视为三角形的量子版本。这些量子三角形在两种意义下是自然的。首先，根据我们的定义，它们产生了圆盘上刘维尔共形场论的边界三点关联函数。其次，事实证明，由虚几何中与 LQG 耦合的流线所围成的三角形的定律正是由这些量子三角形给出的。在本文中，我们针对量子圆盘上的边界流线证明了第二点。我们的方法有潜力证明将量子三角形以任意方式粘合的一般共形焊接结果。量子三角形在通过共形焊接理解 SLE 和 LQG 的可积性方面起着基础性作用。在本文中，我们利用一个量子三角形与一个双点量子圆盘的共形焊接，推导出具有三个力点的弦 SLE 的可积性结果。在后续工作中，我们将探索它们在 LQG 的树配对框架中的应用，包括对斜布朗置换中逆序期望比例的精确计算。
英文摘要：
We define a three-parameter family of random surfaces in Liouville quantum gravity (LQG) which can be viewed as the quantum version of triangles. These quantum triangles are natural in two senses. First, by our definition they produce the boundary three-point correlation functions of Liouville conformal field theory on the disk. Second, it turns out that the laws of the triangles bounded by flow lines in imaginary geometry coupled with LQG are given by these quantum triangles. In this paper we demonstrate the second point for boundary flow lines on a quantum disk. Our method has the potential to prove general conformal welding results with quantum triangles glued in an arbitrary way. Quantum triangles play a basic role in understanding the integrability of SLE and LQG via conformal welding. In this paper, we deduce integrability results for chordal SLE with three force points, using the conformal welding of a quantum triangle and a two-pointed quantum disk. In a subsequent work we will explore their applications to the mating-of-trees framework of LQG, including the exact evaluation of the expected proportion of inversions in skew Brownian permutons.
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