  The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner
driving term. It is finite if and only if the curve is a Weil-Petersson
quasicircle. In this paper, we describe cutting and welding operations on
finite Dirichlet energy functions defined in the plane, allowing expression of
the Loewner energy in terms of Dirichlet energy dissipation. We show that the
Loewner energy of a unit vector field flow-line is equal to the Dirichlet
energy of the harmonically extended winding. We also give an identity involving
a complex-valued function of finite Dirichlet energy that expresses the welding
and flow-line identities simultaneously. As applications, we prove that
arclength isometric welding of two domains is sub-additive in the energy, and
that the energy of equipotentials in a simply connected domain is monotone. Our
main identities can be viewed as action functional analogs of both the welding
and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian
free field.

Complex Variables;Probability