  A generalized Hubbard-Stratonovitch transformation relating an integral over
random unitary N times N matrices to an integral over Efetov's unitary sigma
model manifold, is introduced. This transformation adapts the supersymmetry
method to disordered and chaotic systems that are modeled not by a Hamiltonian
but by their scattering matrix or time-evolution operator. In contrast to the
standard method, no saddle-point approximation is made, and no massive modes
have to be eliminated. This first paper on the subject applies the generalized
Hubbard-Stratonovitch transformation to Dyson's Circular Unitary Ensemble. It
is shown how to use a supersymmetric variant of the
Harish-Chandra-Itzykson-Zuber formula to compute, in the large-N limit, the
n-level correlation function for any n. Nontrivial applications to random
network models, quantum chaotic maps, and lattice gauge theory, are expected.
