  We invent a method that exploits the geometry in the space of couplings and
the known all-loop effective action, in order to calculate the exact in the
couplings anomalous dimensions of composite operators for a wide class of
integrable $\sigma$-models. These involve both self and mutually interacting
current algebra theories. We work out the details for important classes of such
operators. In particular, we consider the operators built solely from an
arbitrary number of currents of the same chirality, the composite operators
which factorize into a chiral and an anti-chiral part, as well as those made up
of three currents of mixed chirality. Remarkably enough, the anomalous
dimensions of the former two sets of operators turn out to vanish. In our
approach, loop computations are completely avoided.
