  We study the two-dimensional attractive Hubbard model using the mapping onto
the half-filled repulsive Hubbard model in a uniform magnetic field coupled to
the fermion spins. The low-energy effective action for charge and pairing
fluctuations is obtained in the hydrodynamic regime. We recover the action of a
Bose superfluid where half the fermion density is identified as the conjugate
variable of the phase of the superconducting order parameter. By integrating
out charge fluctuations, we obtain a phase-only action. In the zero-temperature
superconducting state, this action describes a collective phase mode smoothly
evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov
mode of a Bose superfluid at strong coupling. At finite temperature, the
phase-only action can be used to extract an effective XY model and thus obtain
the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also
identify a renormalized classical regime of superconducting fluctuations above
the BKT phase transition, and a regime of incoherent pairs at higher
temperature. Special care is devoted to the nearly half-filled case where the
symmetry of the order parameter is enlarged to SO(3) due to strong ${\bf
q}=(\pi,\pi)$ charge fluctuations. The low-energy effective action is then an
SO(3) non-linear sigma model with a (symmetry breaking) magnetic field
proportional to the doping. In the strong-coupling limit, the attractive
Hubbard model can be mapped onto the Heisenberg model, from which we recover
the Gross-Pitaevskii equation in the low-density limit.
