# Integrability of the conformal loop ensemble

@inproceedings{Ang2021IntegrabilityOT, title={Integrability of the conformal loop ensemble}, author={Morris Ang and Xin Sun}, year={2021} }

We demonstrate that the conformal loop ensemble (CLE) has a rich integrable structure by establishing exact formulas for two CLE observables. The first describes the joint moments of the conformal radii of loops surrounding three points for CLE on the sphere. Up to normalization, our formula agrees with the imaginary DOZZ formula due to Zamolodchikov (2005) and KostovPetkova (2007), which is the three-point structure constant of certain conformal field theories that generalize the minimal…

## 14 Citations

### Integrability of SLE via conformal welding of random surfaces

- Mathematics
- 2021

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…

### FZZ formula of boundary Liouville CFT via conformal welding

- Mathematics
- 2021

Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev,…

### Conformal welding of quantum disks

- Physics
- 2020

Two-pointed quantum disks 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…

### Brownian half-plane excursion and critical Liouville quantum gravity

- Mathematics
- 2022

In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be obtained by gluing together a…

### Permutons, meanders, and SLE-decorated Liouville quantum gravity

- Mathematics
- 2022

We study a class of random permutons which can be constructed from a pair of space-ﬁlling Schramm-Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the…

### Three-point correlation functions in the $\mathfrak{sl}_3$ Toda theory I: Reflection coefficients

- Mathematics
- 2022

Toda Conformal Field Theories (CFTs) form a family of 2d CFTs indexed by semisimple and complex Lie algebras. They are natural generalizations of the Liouville CFT in that they enjoy an enhanced…

### Liouville quantum gravity from random matrix dynamics

- Mathematics
- 2022

We establish the ﬁrst connection between 2 d Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if ( U t ) is a Brownian motion on the unitary group at…

### The SLE loop via conformal welding of quantum disks

- Mathematics
- 2022

. We prove that the SLE loop measure arises naturally from the conformal welding of two Liouville quantum gravity (LQG) disks for γ 2 = κ ∈ (0 , 4). The proof relies on our companion work on…

### On Natural Measures of Several Random Fractals Induced from SLE and CLE

- Mathematics
- 2022

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal…

### On Natural Measures of SLE- and CLE-Related Random Fractals

- Mathematics
- 2022

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal…

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