LA Probability Forum

I will discuss the joint work with Nikolai Makarov on multiple radial SLE(0) system which is the deterministic limit of the multiple radial SLE(\kappa) system.

By constructing the field integral of motion, we show that the traces of the multiple radial SLE(0) system are the horizontal trajectories of an equivalence class of quadratic differentials. The stationary relations establish a connection between the multiple radial SLE(0) systems and enumerative algebraic geometry. 

Our machinery can also be applied to various multiple SLE(0) systems such as multiple radial SLE(0) with spin and multiple chordal SLE(0) with arbitrary screening charges which extend the results in \cite{ABKM20}.

From a Hamiltonian perspective, we prove that the Loewner dynamics with equal growth weight in multiple radial SLE(0) systems are a special type of classical Calogero-Sutherland system. Furthermore, we interpret $n$ quadratic null vector equations as $n$ commutating Hamiltonian flows along the submanifolds defined as the intersection of level set of the Hamiltonian. 

The whole theory is classical but motivated by conformal field theory. Notably, the field integral motion can be seen heuristically as the classical limit of the screened martingale observable.