Commutation relations for two-sided radial SLE

TL;DR

This work classifies interchangeable locally commuting 2-radial SLE$_\kappa$ in the unit disk for $\kappa\in(0,8)$ by deriving and solving the radial BPZ equations for a partition function $\mathcal{Z}$. It shows that the only admissible partition functions under the interchangeability constraint are two one-parameter families: $\mathcal{G}_{\mu}$, yielding two-sided radial SLE$_\kappa$ with spiral rate $\mu$, and $\mathcal{Z}_{\alpha}$, yielding a chordal SLE$_\kappa$ weighted by CR$(\mathbb{D}\setminus\gamma)^{-\alpha}$. The paper further proves a resampling property for the spiral case when $\kappa\le4$, connects the second family to a variational problem involving the chordal Loewner energy, and discusses semiclassical limits as $\kappa\to0$, linking partition-function limits to energy-minimizing chordal curves. These results unify radial and chordal SLE constructions via partition functions, provide explicit formulas, and illuminate connections to SLE/GFF couplings and conformal field theory.

Abstract

We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points. We follow the framework introduced by Dubédat. Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_κ$ for $κ\in (0,8)$: it is either a two-sided radial SLE$_κ$ with spiral of constant spiraling rate or a chordal SLE$_κ$ weighted by a power of the conformal radius of its complement. Namely, for fixed $κ$ and starting points, we have exactly two one-parameter continuous families of locally commuting 2-radial SLE. Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. We also discuss the semiclassical limit of the commutation relation as $κ\to 0$. In particular, we show that the limit for the second family with an appropriately chosen power of conformal radius is a chord that minimizes a modified chordal Loewner energy, which is unique only when the endpoints are not antipodal.

Figures (6)

• Figure 1: Domain Markov property of radial SLE.
• Figure 2: Simulation by Minjae Park. Two-sided radial SLE around the origin with $\kappa=0.01$ and $\mu=5$ in the left panel. Two-sided radial SLE around the origin with $\kappa=2$ and $\mu=5$ in the right panel.
• Figure 3: The curves generated by $\mathcal{U}^\lambda$ from Proposition \ref{['prop::semiclassical_pf']}. They correspond to the energy minimizers from Proposition \ref{['prop:confRadMinimizer']}. On the left, $\theta_2-\theta_1\approx \pi$, and $\lambda=0.1,\ 0.2,\ 0.5,\ 1,\ 2,\ 5,\ 10,\ 20,\ 50,\ 100,\ 200,\ 500$ ($\lambda=0.1$ corresponds to the innermost green curve and $\lambda=500$ corresponds to the outermost orange curve). On the right $\lambda=10$ while $\theta_2-\theta_1$ varies.
• Figure 5: We have $g_{\mathbf{t}}=g_{\mathbf{t}, 1}\circ g_{t_1}^{(1)}=g_{\mathbf{t}, 2}\circ g_{t_2}^{(2)}$.
• Figure 6: Plot of $\theta\mapsto \mathcal{Z}_{\alpha}(\theta)$ with $\kappa=4$ and $\theta=\theta_2-\theta_1$ for different $\alpha$'s. When $\alpha>1/2$, it is not always positive.

Theorems & Definitions (69)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3
• Proposition 1.4
• Remark 2.1
• Proposition 2.2
• proof : Proof of Proposition \ref{['prop:radial_comm']}
• Proposition 2.3: Radial BPZ equations
• proof
• Theorem 2.4