We demonstrate how to obtain integrability results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called . Our proof is built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper Ang, Holden and Sun (2023). Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating-of-trees based on these two connections.

中文翻译：

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通过随机表面保形焊接实现 SLE 的可集成性

我们演示了如何从刘维尔共形场理论 (LCFT) 和刘维尔量子引力 (LQG) 的树配对框架获得 Schramm-Loewner 演化 (SLE) 的可积性结果。特别是，我们证明了 SLE 经典变体的保形导数定律的精确公式，称为。我们的证明建立在 SLE、LCFT 和树交配之间的两个联系上。首先，LCFT 和树配对提供了等效但互补的方法来描述 LQG 中的自然随机表面。使用我们称之为LQG 表面均匀嵌入的新颖工具，我们通过允许更少的标记点和更通用的奇点来扩展早期的等价结果。其次，这些随机表面的保形焊接产生 SLE 曲线作为它们的界面。我们特别依赖我们的配套论文 Ang、Holden 和 Sun (2023) 中证明的保形焊接结果。我们的论文是证明 SLE、LCFT 和基于这两种连接的树配对的可积性结果的程序的重要组成部分。