  In this note we revisit the subject of anomaly cancelation in string theory
and M-theory on manifolds with String structure and give three observations.
First, that on String manifolds there is no E8 x E8 global anomaly in heterotic
string theory. Second, that the description of the anomaly in the phase of the
M-theory partition function of Diaconescu-Moore-Witten extends from the Spin
case to the String case. Third, that the cubic refinement law of
Diaconescu-Freed-Moore for the phase of the M-theory partition function extends
to String manifolds. The analysis relies on extending from invariants which
depend on the Spin structure to invariants which instead depend on the String
structure. Along the way, the one-loop term is refined via the Witten genus.
