Multiple radial SLE($κ$) and quantum Calogero-Sutherland system

TL;DR

This work develops a comprehensive theory of multiple radial SLE$\kappa$ with $\kappa>0$ in a simply connected domain, incorporating boundary and interior marked points. By coupling SLE with Coulomb gas methods, it derives null vector equations and a rotation constraint, and constructs four families of screening solutions organized by link patterns, yielding a rich space of partition functions that correspond to eigenfunctions of the quantum Calogero–Sutherland Hamiltonian after a gauge transform. A key insight is the partition function’s classification into equivalence classes under interior-point dependence, with within-class representatives that exhibit conformal covariance; the connection to Calogero–Sutherland provides a unifying spectral perspective on the SLE system. The paper also develops rational and classical limits ($\kappa=0$, $\kappa>0$), clarifies transformation rules under conformal changes, and outlines a program toward pure partition functions and their relations to affine meander matrices, with implications for understanding the topology of link patterns in radial/chordal SLE$\kappa$ dynamics. Overall, the work advances a Coulomb gas–SLE/CFT framework that links stochastic growth processes to integrable quantum many-body systems, enabling precise spectral characterizations and topological classifications of multiple radial SLEs.

Abstract

We develop a theory for the multiple radial $\mathrm{SLE}(κ)$ systems with parameter $κ> 0$ -- a family of random multi-curve systems in a simply connected domain $Ω$, with marked boundary points $z_1, \ldots, z_n \in \partial Ω$ and a marked interior point $q$. As a consequence of the domain Markov property and conformal invariance, we show that such systems are characterized by equivalence classes of partition functions, which are not necessarily conformally covariant. Nevertheless, within each equivalence class, one can always choose a conformally covariant representative. When $Ω$ is taken to be the unit disk $\mathbb{D}$ and the marked interior point $q$ is set at the origin, we demonstrate that the partition function satisfies a system of second-order PDEs, known as the null vector equations, with a null vector constant $h$ and a rotation equation involving a constant $ω$. Motivated by the Coulomb gas formalism in conformal field theory, we construct four families of solutions to the null vector equations, which are naturally classified according to topological link patterns. For $κ> 0$, the partition functions of multiple radial $\mathrm{SLE}(κ)$ systems correspond to eigenstates of the quantum Calogero--Sutherland (CS) Hamiltonian beyond the states built upon the fermionic states.

Figures (8)

• Figure 1.1: Multiple radial SLE($\kappa$) systems in $\mathbb{D}$
• Figure 1.2: Multiple radial SLE($\kappa$) in $\mathbb{D}$
• Figure 1.3: Localization of multiple radial SLE($\kappa$)
• Figure 1.4: Integrate $\zeta_1,\zeta_2$ (yellow points) along two Pochhammer contour
• Figure 2.1: Chordal SLE($\kappa$) $\boldsymbol{\sigma}^+= a \cdot \xi+ (2b-a) \cdot q$, $\boldsymbol{\sigma}^-= 0$

Theorems & Definitions (88)

• Definition 1.1: Localization of Measures
• Definition 1.2: Local multiple radial SLE($\kappa$)
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Conformal radius
• Definition 2.2: Capacity in $\mathbb{D}$
• Definition 2.3: Capacity in $\mathbb{H}$
• Definition 2.4: Radial Loewner chain
• Definition 2.5: Radial SLE($\kappa$)