  We formulate the Exact Renormalization Group on the string world sheet for
closed string backgrounds. The same techniques that were used for open strings
is used here. There are some subtleties. One is that holomorphic factorization
of the closed string vertex operators does not hold in the presence of a cutoff
on the Euclidean world sheet. This introduces extra terms in the Lagrangian at
the cutoff scale and they turn out to be crucial for implementing gauge
invariance. This naive generalization from open string to closed strings
requires a {\em massive} graviton and the gauge symmetry is Abelian, just as in
open string theory. Interestingly, it turns out that if one introduces a non
dynamical background metric (as in background field formalism) and combines a
gauge transformation on the field with a transformation on the coordinates and
background metric, the graviton can be massless. Some examples of background
coordinate covariant equations are worked out explicitly. A preliminary
discussion of massive modes, massive gauge transformations and the role of
world sheet regulator terms is given. Some of the gauge transformations can be
given a geometric meaning if space time is assumed to be complex at some level.
