Multiple radial SLE(0) and classical Calogero-Sutherland System

TL;DR

This work establishes a deterministic framework for multiple radial SLE(0) as the κ→0 limit of random multiple radial SLE(κ) processes in a simply connected domain, revealing that the traces are horizontal trajectories of residue-free quadratic differentials with prescribed zeros at the growth points. Central to the exposition are stationary relations and integrals of motion derived from a Coulomb gas/CFT viewpoint, which connect the trace geometry to the critical points of a master function for trigonometric KZ equations and to a topological radial link-pattern classification. The authors further show that, under a common capacity parametrization, the multiple radial SLE(0) dynamics coincide with a classical Calogero–Sutherland system on an appropriate submanifold defined by a Lax matrix, embedding the stochastic processes in an integrable-Hamiltonian setting. Extensions to spin are developed, yielding a spin-modified quadratic-differential framework and preserving the connection to CS dynamics; the results collectively bridge stochastic Loewner evolution, conformal field theory, algebraic geometry, and integrable systems with potential implications for the enumeration of KZ critical points and radial link patterns.

Abstract

We develop a theory of multiple radial SLE(0) -- a smooth system of curves in a simply connected domain $Ω$ with marked boundary points $z_1, \ldots, z_n \in \partial Ω$ and a marked interior point $q$ -- arising as the deterministic limit of random multiple radial SLE($κ$) systems. We construct multiple radial SLE(0) systems by starting from the stationary relations, which arise heuristically as the $κ\to 0$ limit of partition functions. By constructing the field integrals of motion for the Loewner dynamics, we show that the traces of multiple radial SLE(0) systems are the horizontal trajectories of an equivalence class of quadratic differentials. These trajectories have limiting ends at the boundary points $\{z_1, z_2, \ldots, z_n\}$. The stationary relations connect the classification of multiple radial SLE(0) systems to the enumeration of critical points of the master function of trigonometric Knizhnik--Zamolodchikov (KZ) equations. In the deterministic case $κ= 0$, we show that the Loewner dynamics with a common parametrization of capacity form a special class of classical Calogero--Sutherland systems, restricted to a submanifold of phase space defined by the Lax matrix.

Figures (26)

• Figure 2.1: Chordal SLE($\kappa$) $\boldsymbol{\sigma}^+= a \cdot \xi+ (2b-a) \cdot q$, $\boldsymbol{\sigma}^-= 0$
• Figure 2.2: Radial SLE($\kappa$) $\boldsymbol{\sigma}^+= a \cdot \xi+ (b-a) \cdot q$, $\boldsymbol{\sigma}^-= b \cdot q$
• Figure 4.1: $x_1=i,x_2=-i$
• Figure 4.2: $x_1=i,x_2=-i,\xi_1=-1$
• Figure 4.3: $x_1=1,x_2=e^{\frac{2\pi i}{3}},x_3=e^{\frac{-2\pi i}{3}}$

Theorems & Definitions (128)

• Definition 1.1
• Definition 1.2: Stationary relations
• Definition 1.3: Multiple radial SLE(0) Loewner chain
• Remark 1.4
• Definition 1.5: Quadratic differentials with prescribed zeros
• Theorem 1.6
• Theorem 1.7
• Definition 2.1: Conformal radius
• Definition 2.2: Capacity in $\mathbb{D}$
• Definition 2.3: Capacity in $\mathbb{H}$