  It has recently been pointed out that the existence of massless chiral edge
excitations has important strong coupling consequences for the topological
concept of an instanton vacuum. In the first part of this paper we elaborate on
the effective action for ``edge excitations'' in the Grassmannian $U(m+n)/U(m)
\times U(n)$ non-linear sigma model in the presence of the $\theta$ term. This
effective action contains complete information on the low energy dynamics of
the system and defines the renormalization of the theory in an unambiguous
manner. In the second part of this paper we revisit the instanton methodology
and embark on the non-perturbative aspects of the renormalization group
including the anomalous dimension of mass terms. The non-perturbative
corrections to both the $\beta$ and $\gamma$ functions are obtained while
avoiding the technical difficulties associated with the idea of {\em
constrained} instantons. In the final part of this paper we present the
detailed consequences of our computations for the quantum critical behavior at
$\theta = \pi$. In the range $0 \leq m,n \lesssim 1$ we find quantum critical
behavior with exponents that vary continuously with varying values of $m$ and
$n$. Our results display a smooth interpolation between the physically very
different theories with $m=n=0$ (disordered electron gas, quantum Hall effect)
and $m=n=1$ (O(3) non-linear sigma model, quantum spin chains) respectively, in
which cases the critical indices are known from other sources. We conclude that
instantons provide not only a {\em qualitative} assessment of the singularity
structure of the theory as a whole, but also remarkably accurate {\em
numerical} estimates of the quantum critical details (critical indices) at
$\theta = \pi$ for varying values of $m$ and $n$.
