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The moduli of annuli in random conformal geometry

Morris Ang, Guillaume Remy, Xin Sun

TL;DR

This work provides exact laws for the modulus of annuli in three central random-geometry settings: the Brownian annulus, annuli bounded by CLE loops, and annuli arising from SLE_{8/3} loops. The authors implement a unifying framework based on the Polyakov 2D gravity decomposition into matter, Liouville, and ghost CFTs and a KPZ relation at the level of annulus partition functions, enabled by Liouville CFT bootstrap (Wu22) and Gaussian multiplicative chaos techniques. They derive explicit modulus laws, joint laws with conformal radius, and nesting statistics for CLE, as well as the Cardy-form annulus partition function for SLE_{8/3}, and they connect these results to Brownian annulus geometry via LQG limits. The results deepen the link between Liouville quantum gravity and moduli space measures, offering exact, computable formulas for observables tied to annular topology in random conformal geometries with broad implications for scaling limits of lattice models decorated by CLE/SLE/CFT structures.

Abstract

We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk and the disk boundary. The formula is as conjectured from the partition function of the O$(n)$ loop model on the annulus derived by Saleur-Bauer (1989) and Cardy (2006). The third is for the annulus partition function of the SLE$_{8/3}$ loop introduced by Werner (2008), {confirming another} prediction of Cardy (2006). The physics principle underlying our proofs is that 2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the Liouville CFT.

The moduli of annuli in random conformal geometry

TL;DR

This work provides exact laws for the modulus of annuli in three central random-geometry settings: the Brownian annulus, annuli bounded by CLE loops, and annuli arising from SLE_{8/3} loops. The authors implement a unifying framework based on the Polyakov 2D gravity decomposition into matter, Liouville, and ghost CFTs and a KPZ relation at the level of annulus partition functions, enabled by Liouville CFT bootstrap (Wu22) and Gaussian multiplicative chaos techniques. They derive explicit modulus laws, joint laws with conformal radius, and nesting statistics for CLE, as well as the Cardy-form annulus partition function for SLE_{8/3}, and they connect these results to Brownian annulus geometry via LQG limits. The results deepen the link between Liouville quantum gravity and moduli space measures, offering exact, computable formulas for observables tied to annular topology in random conformal geometries with broad implications for scaling limits of lattice models decorated by CLE/SLE/CFT structures.

Abstract

We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk and the disk boundary. The formula is as conjectured from the partition function of the O loop model on the annulus derived by Saleur-Bauer (1989) and Cardy (2006). The third is for the annulus partition function of the SLE loop introduced by Werner (2008), {confirming another} prediction of Cardy (2006). The physics principle underlying our proofs is that 2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the Liouville CFT.
Paper Structure (30 sections, 53 theorems, 113 equations, 3 figures)

This paper contains 30 sections, 53 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Random moduli for the Brownian annulus
  3. Matter-Liouville-ghost decomposition for 2D quantum gravity
  4. A KPZ relation for annulus partition functions
  5. Random moduli for simple CLE loops
  6. Annulus partition function of the SLE$_{8/3}$ loop
  7. Perspectives and outlook
  8. Liouville fields and their domain Markov property
  9. Gaussian free field
  10. Liouville fields on the annulus and the disk
  11. A resampling characterization of $\mathrm{LF}_\tau$
  12. The annulus boundary partition function for Liouville CFT
  13. Input from Wu22 on Liouville CFT on the annulus
  14. Proof of Theorems \ref{['thm:GMC-ratio']} and \ref{['thm:KPZ']}
  15. Quantum annulus: Liouville field description and the random modulus
  16. ...and 15 more sections

Key Result

Theorem 1.2

For $\tau>0$, let $X_\tau$ be a random variable such that $\mathbb{E}[X^{i t}_\tau]= \frac{2\pi t e^{ -\frac{2\pi \tau t^2}{3}}}{ 3\sinh(\frac{2}{3} \pi t )}$ for $t\in \mathbb{R}$. Let $\rho_\tau(\cdot)$ be the probability density function of $X_\tau$, which is supported on $(0,\infty)$. Then th

Figures (3)

  • Figure 1: Conformal embedding of the Brownian annulus to the annulus $\mathbb A_\tau = \{ z \: : \: |z| \in (e^{-2\pi \tau}, 1)\}$ with (random) modulus $\tau$.
  • Figure 2: Left: The cylinder $\mathcal{C}_\tau$ is obtained by identifying the top and bottom edges of the rectangle $[0,\tau] \times [0,1]$. Middle: With $\eta_j$ the $j^{\text{th}}$ nested loop around the origin, Theorem \ref{['thm:CLE-mod']} identifies the law of the modulus of the annulus bounded by $\partial \mathbb{D}$ and $\eta_j$. Right: For $\tau>0$, Corollary \ref{['cor:nesting']} gives the law of the number $\mathcal{N}$ of non-contractible loops for $\mathrm{CLE}$ in $\mathbb A_\tau$.
  • Figure 3: Illustration of objects in the proof of Lemma \ref{['lem:DMP1']}.

Theorems & Definitions (109)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1: Remy_annulus
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 99 more