  We address the finite temperature phase diagram of ultracold fermionic atoms
across a Feshbach resonance based on a functional integral for an atom-molecule
model. This allows to fully exploit the presence of the global symmetry of
phase rotations, U(1). Both the equation of state and the classification of the
thermodynamic phases are obtained from a symmetry consideration. We focus on
the universal aspects associated to narrow and broad resonances, in turn
connected by an additional form of crossover. The narrow resonance limit can be
solved exactly above the critical temperature. We also assess deviations from
the broad resonance universality which have recently been probed
experimentally.
