From the random geometry of conformally invariant systems to the K\"ahler geometry of universal Teichm\"uller space

TL;DR

This paper explores the deep connection between Loewner energy, a functional from conformally invariant systems, and the K"ahler geometry of universal Teichm"uller space, revealing new geometric insights.

Contribution

It elucidates the relationship between Loewner energy and the K"ahler geometry of universal Teichm"uller space, bridging conformal invariance and complex geometry.

Findings

Loewner energy is connected to the universal Liouville action.

The identity between Loewner energy and the Liouville action is established.

The paper provides a geometric interpretation of conformally invariant functionals.

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 K\"ahler geometry of universal Teichm\"uller space. We provide a background on Loewner energy, the universal Liouville action, and the intuition behind the proof of the identity between them.