During the last 20 years, random conformal geometry that arises from 2D critical statistical mechanics has attracted many attentions. For instance, random curves (Schramm-Loewner evolution, SLE) and random surfaces (Liouville Quantum gravity, LQG) appear naturally as universality classes of scaling limits of discrete models.  

On the SLE side, its action functional called Loewner energy, encrypts the skeleton of their distribution. I will outline how this energy is related to Teichmueller theory, more precisely, the Weil-Petersson class of universal Teichmueller space.