  Loewner's equation provides a way to encode a simply connected domain or
equivalently its uniformizing conformal map via a real-valued driving function
of its boundary. The first main result of the present paper is that the
Dirichlet energy of this driving function (also known as the Loewner energy) is
equal to the Dirichlet energy of the log-derivative of the (appropriately
defined) uniformizing conformal map.
  This description of the Loewner energy then enables to tie direct links with
regularized determinants and Teichm\"uller theory: We show that for smooth
simple loops, the Loewner energy can be expressed in terms of the
zeta-regularized determinants of a certain Neumann jump operator. We also show
that the family of finite Loewner energy loops coincides with the
Weil-Petersson class of quasicircles, and that the Loewner energy equals to a
multiple of the universal Liouville action introduced by Takhtajan and Teo,
which is a K\"ahler potential for the Weil-Petersson metric on the
Weil-Petersson Teichm\"uller space.
