  The path integral computation of field strength correlation functions for two
dimensional Yang-Mills theories over Riemann surfaces is studied. The
calculation is carried out by abelianization, which leads to correlators that
are topological. They are nontrivial as a result of the topological
obstructions to the abelianization. It is shown in the large N limit on the
sphere that the correlators undergo second order phase transitions at the
critical point. Our results are applied to a computation of contractible Wilson
loops.
