  We compare the predictions of the Nambu-Goto effective string model with a
set of high precision Monte Carlo results for interfaces with periodic boundary
conditions in the 3D Ising model. We compute the free energy in the covariant
gauge exactly, up to the inclusion of the Liouville mode. The perturbative
expansion of this result agrees both with the result evaluated several years
ago by Dietz and Filk in the physical gauge and with a recent calculation with
the Polchinski-Strominger action. We also derive the effective string spectrum
which, because of the different boundary conditions, is very different from the
well known one of Arvis. Taking into proper account the effective string
corrections and exploiting some technical improvements in the simulations we
obtain precise estimate of the amplitude ratios T_c/\sqrt{sigma},
m_{0++}/\sqrt{\sigma} and sigma xi_{2nd}^2. We also discuss the behaviour of
the effective string free energy in the dimensional reduction limit (i.e., near
the deconfinement transition of the dual 3d gauge Ising model) and its
relationship with the 2d Ising model interfaces
