  The theta dependence of the vacuum energy in large N Yang-Mills theory has
been studied some time ago by Witten using a duality of large N gauge theories
with string theory compactified on a certain space-time. We show that within
the field theory context vacuum fluctuations of the topological charge give
rise to the vacuum energy consistent with the string theory computation.
Furthermore, we calculate 1/N suppressed corrections to the string theory
result. The reconciliation of the string and field theory approaches is based
on the fact that the gauge theory instantons carry zerobrane charge in the
corresponding D-brane construction of Yang-Mills theory. Given the formula for
the vacuum energy we study certain aspects of stability of the false vacua of
the model for different realizations of the initial conditions. The vacuum
structure appears to be different depending on whether N is infinite or,
alternatively, large but finite.
