  The loop variable technique (for open strings in flat space) is a gauge
invariant generalization of the renormalization group method for obtaining
equations of motion. Unlike the beta functions, which are only proportional to
the equations of motion, here it gives the full equation of motion. In an
earlier paper, a technique was described for adapting this method to open
strings in gravitational backgrounds. However unlike the flat space case, these
equations cannot be derived from an action and are therefore not complete. This
is because there are ambiguities in the method that involve curvature couplings
that cannot be fixed by appealing to gauge invariance alone but need a more
complete treatment of the closed string background. An indirect method to
resolve these ambiguities is to require symmetricity of the second derivatives
of the action. In general this will involve modifying the equations by terms
with arbitrarily high powers of curvature tensors. This is illustrated for the
massive spin 2 field. It is shown that in the special case of an AdS or dS
background, the exact action can easily be determined in this way.
