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

Abstract

We develop a theory for the multiple radial $\mathrm{SLE}(\kappa)$ systems with parameter $\kappa > 0$ -- a family of random multi-curve systems in a simply connected domain $\Omega$, with marked boundary points $z_1, \ldots, z_n \in \partial \Omega$ 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 $\Omega$ 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 $\omega$. 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 $\kappa > 0$, the partition functions of multiple radial $\mathrm{SLE}(\kappa)$ systems correspond to eigenstates of the quantum Calogero--Sutherland (CS) Hamiltonian beyond the states built upon the fermionic states.