  The one--loop determinant computed around the kink solution in the 3D
$\phi^4$ theory, in cylindrical geometry, allows one to obtain the partition
function of the interface separating coexisting phases. The quantum
fluctuations of the interface around its equilibrium position are described by
a $c=1$ two--dimensional conformal field theory, namely a 2D free massless
scalar field living on the interface. In this way the capillary wave model
conjecture for the interface free energy in its gaussian approximation is
proved.
