Lorentz completion of effective string (and p-brane) action

The formation of a confining string (or a p-brane) in a Poincare' invariant theory breaks spontaneously this symmetry which is thereby realized non-linearly in the effective action of these extended objects. As a consequence the form of the action is strongly constrained. A new general method is described to obtain in a systematic way higher order Lorentz invariant contributions to this action. We find a simple recipe to promote a term invariant under the stability subgroup to an expression invariant under the whole Lorentz group. It is based on the following three steps: in the saturation of worldsheet (or worldvolume) indices replace the Minkowski metric with the inverse of the induced metric; in the saturation of indices of the transverse coordinates describing the position of the extended object replace the Euclidean metric with a certain new metric; finally replace the field derivatives of order higher than two with a certain covariant derivative. Lorentz invariance of the expression modified this way immediately follows. We find in particular that the leading bulk deviation of the Nambu Goto action in any space-time dimensions is proportional to the square of scalar curvature.