The study of Loewner energy lies at the interface of random conformal geometry, geometric function theory, and Teichmüller theory. Loewner energy of a loop is defined as Dirichlet energy of its Loewner driving function, motivated by the action functional of SLE.
In the first part of my talk, I will give an overview of the connections of Loewner energy to determinants of Laplacians and Weil-Petersson class of universal Teichmüller space.
In the second part, I will present some consequences of these connections. Including an identity of Loewner energy with a renormalized Brownian loop measure attached to the curve, and (ongoing project with F. Viklund) a deterministic proof of the fact that analogs of both SLE/GFF couplings, namely the quantum zipper and flow-line coupling, hold for finite energy curves.