Modified BFV quantization of membranes and BRST invariant boundary conditions on the ghost fields

The covariant BRST off-shell effective action for the open and closed membranes is obtained. Our result generalizes Fujikawa's on-shell effective action for membranes. We found boundary conditions on the ghost fields, which are necessary in order to obtain the off-shell nilpotency of the BRST charge for the open membrane. We also proved that these boundary conditions on the ghost fields are invariant under BRST transformations and are the generalization to the Kato - Ogawa boundary conditions for the open bosonic string.