  We compare different versions of a bosonic description for systems of
interacting fermions, with particular emphasis on the free energy functional.
The bosonic effective action makes the issue of symmetries particularly
transparent and we present for the Hubbard model an exact mapping between
repulsive and attractive interactions. A systematic expansion for the bosonic
effective action starts with a solution to the lowest order Schwinger-Dyson or
gap equation. We propose a two particle irreducible formulation of an exact
functional renormalization group equation for computations beyond leading
order. On this basis we suggest a renormalized gap equation. This approach is
compared with functional renormalization in a partially bosonized setting.
