  Superbosonization is a new variant of the method of commuting and
anti-commuting variables as used in studying random matrix models of disordered
and chaotic quantum systems. We here give a concise mathematical exposition of
the key formulas of superbosonization. Conceived by analogy with the
bosonization technique for Dirac fermions, the new method differs from the
traditional one in that the superbosonization field is dual to the usual
Hubbard-Stratonovich field. The present paper addresses invariant random matrix
ensembles with symmetry group U(n), O(n), or USp(n), giving precise definitions
and conditions of validity in each case. The method is illustrated at the
example of Wegner's n-orbital model. Superbosonization promises to become a
powerful tool for investigating the universality of spectral correlation
functions for a broad class of random matrix ensembles of non-Gaussian and/or
non-invariant type.
