Uniqueness of Malliavin-Kontsevich-Suhov measures

About 20 years ago, Kontsevich & Suhov conjectured the existence and uniqueness of a family of measures on the set of Jordan curves, characterised by conformal invariance and another property called "conformal restriction". This conjecture was motivated by (seemingly unrelated) works of Schramm, Lawler & Werner on stochastic Loewner evolutions (SLE), and Malliavin, Airault & Thalmaier on "unitarising measures". The existence of this family was settled by works of Werner-Kemppainen and Zhan, using a loop version of SLE. The uniqueness was recently obtained in a joint work with Jego. I will start by reviewing the different notions involved before giving some ideas of our proof of uniqueness: in a nutshell, we construct a family of "orthogonal polynomials" which completely characterise the measure. In the remaining time, I will discuss the broader context in which our construction fits, namely the conformal field theory associated with SLE.