Loewner energy, determinants of Laplacians, and Brownian loop measure

The Loewner energy of a Jordan curve is the Dirichlet energy of its driving function. In this talk, I will show the relation between Loewner energy, Schramm-Loewner evolution, zeta-regularized determinants of Laplacians, and Brownian loop measure. It is in the spirit of interpreting probabilistically analytic objects. In particular, we derive an identity between the Loewner energy and a renormalized total mass of the Brownian loops attached to the curve.