From the random geometry of conformally invariant systems to the Kähler geometry of universal Teichmüller space

Abstract

The goal of this expository article is to explain how a fundamental functional on the space of Jordan curves arising from SLE - Loewner energy - is connected to a seemingly far apart subject: the Kähler geometry of universal Teichmüller space. We provide a background on Loewner energy, the universal Liouville action, and the intuition behind the proof of the identity between them.

arXiv comment14 pages, 2 figures, invited article for Notices of the American Mathematical Society. This version is revised according to the referees' comments and added further discussions