Conformal welding of quantum disks and multiple SLE: the non-simple case

TL;DR

This work advances the synthesis of conformal welding, SLE, and LCFT in two dimensions by extending the conformal welding of quantum disks to the non-simple regime κ∈(4,8) and by constructing global N-curve multiple SLE measures for arbitrary link patterns via a welding framework. The authors establish a precise coupling between generalized quantum surfaces (including forested/beaded quantum disks) and multiple SLE through LCFT partition functions, proving finiteness and conformal covariance of the partition functions and showing equivalence between global and local SLE constructions. The FD-disk welding result serves as a cornerstone for an inductive, cascade-based construction of mSLEκ,0,∞ along with a rigorous PDE/hypoellipticity argument ensuring smoothness of partition functions. The findings illuminate the LCFT interpretation of random moduli in conformal welding and provide a robust probabilistic realization of non-simple multiple SLEs, with potential implications for boundary observables and CLE/LQG integrability. Overall, the paper advances the understanding of how Liouville theory, SLE interfaces, and random planar maps interlock in finite-volume LQG settings, offering a concrete, mathematically rigorous bridge between field theory and stochastic Loewner evolutions in the physically relevant κ∈(4,8) regime.

Abstract

Two-pointed quantum disks with a weight parameter $W>0$ is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We extend the conformal welding of quantum disks in [AHS23] to the non-simple regime, and give a construction of the multiple SLE associated with any given link pattern for $κ\in(4,8)$. Our proof is based on connections between SLE and Liouville conformal field theory (LCFT), where we show that in the conformal welding of multiple forested quantum disks, the surface after welding can be described in terms of LCFT, and the random conformal moduli contains the SLE partition function for the interfaces as a multiplicative factor. As a corollary, for $κ\in(4,8)$, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance.

Figures (7)

• Figure 1: An illustration of Theorem \ref{['thm:fd-disk']} in the case $W_-\ge\frac{\gamma^2}{2}$ and $W_+\in(0,\frac{\gamma^2}{2})$. If we draw an independent $\mathrm{SLE}_\kappa(\rho_-;\rho_+)$ curve on top of a generalized quantum disk of weight $W_-+W_+$, then the curve-decorated quantum surface is equal to the welding of a pair of weight $W_1$ and $W_2$ generalized quantum disks conditioned on having the same generalized quantum length for the interface. If $W<\frac{\gamma^2}{2}$, then the interface $\eta$ is understood as the concatenation of $\mathrm{SLE}_\kappa(\rho_-;\rho_+)$ curves in each bead of the weight $W$ forested quantum disk.
• Figure 2: An illustration of Theorem \ref{['thm:main']}. Left: Under the link pattern $\alpha=\{\{1,6\},\{2,5\},\{3,4\} \}$, we are welding two samples from $\mathrm{GQD}_{2}$ (drawn in green and turquoise) with two samples from $\mathrm{GQD}_{4}$ (drawn in pink and yellow), restricted to the event $E$ where the welding output has the structure of a simply connected quantum surface glued to forested lines. Each generalized disk is composed of a countable number of (regular) disks, and the (regular) disks that are used to connect two of the marked boundary points of the generalized disk are shown in dark color, while the other disks are shown in light color. If we let $\ell_1,\ell_2,\ell_3$ be the interface lengths ordered from the left to the right, then the precise welding equation can be written as $\int_{\mathbb{R}_+^3}\mathrm{Weld}(\mathrm{GQD}_{2}(\ell_1),\mathrm{GQD}_{4}(\ell_1,\ell_2),\mathrm{GQD}_{4}(\ell_2,\ell_3),\mathrm{GQD}_{2}(\ell_3))|_Ed\ell_1\,d\ell_2\,d\ell_3$. Right: A similar setting where the link pattern $\alpha = \{\{1,6\},\{2,3\},\{4,5\} \}$, we are welding three samples from $\mathrm{GQD}_{2}$ (drawn in green, turquoise and yellow) with one sample from $\mathrm{GQD}_{6}$ (drawn in pink). The corresponding welding equation is given by $\int_{\mathbb{R}_+^3}\mathrm{Weld}(\mathrm{GQD}_{2}(\ell_1),\mathrm{GQD}_{2}(\ell_2),{\mathrm{GQD}_{2}}(\ell_3),\mathrm{GQD}_{6}(\ell_1,\ell_2,\ell_3))|_Ed\ell_1\,d\ell_2\,d\ell_3$. The forested line part of each generalized quantum disk is drawn in a lighter shade.
• Figure 3: An illustration of the case where the event $E$ in Theorem \ref{['thm:main']} fails to happen. The left two panels illustrate weldings of generalized quantum disks following the same link patterns as in Figure \ref{['fig:main']}. In these examples the output surface cannot be described by forested lines glued to a single simply connected surface. The right two panels indicate the corresponding topological gluing of trees, where in both cases there exists an edge which is not glued to any other edge.
• Figure 4: Left: The graph of the Lévy process $(X_t)_{t>0}$ with only upward jumps. We draw the blue curves for each of the jump, and identify the points that are on the same green horizontal line. Right: The Lévy tree of disks obtained from the left panel. For each topological disk we assign a quantum disk $\mathrm{QD}$ conditioned on having the same boundary length as the size of the jump, with the points on the red line in the left panel shrinked to a single point. The quantum length of the line segment between the root $o$ and the point $p_t$ is $t$, while the segment along the forested boundary between $o$ and $p_t$ has generalized quantum length $Y_t = \inf\{s>0:X_s\le -t\}$.
• Figure 5: An illustration of the proof of Proposition \ref{['prop:typical']}. Left: We sample a time $a>0$ from the Lebesgue measure. Right: The Lévy tree of disks obtained from the left panel with the marked point. The collection of green disks, which we shall prove to have law $c{\mathcal{M}}^\mathrm{disk}_2(\gamma^2-2)$, correspond to the jumps of $(X_{u-t})_{t \in [0,u]}$ hitting running infimum.

Theorems & Definitions (94)

• Definition 1.1: beffara2021uniquenessZhan23
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Thick quantum disk
• Definition 2.2: Thin quantum disk
• Lemma 2.3: Lemma 2.16 and Lemma 2.18 of AHS23
• Definition 2.4
• Definition 2.5