Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of its driving function. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of \(\hbox {SLE}_{8/3}\) ${\text{SLE}}_{8/3}$ loops touching the Jordan curve under quasiconformal deformations.