  Recent experiments on a $\2d$ Fermi gas find an undamped breathing mode at
twice the trap frequency over a wide range of parameters. To understand this
seemingly scale-invariant behavior in a system with a scale, we derive two
exact results valid across the entire BCS-BEC crossover at all temperatures.
First, we relate the shift of the mode frequency from its scale-invariant value
to $\gamma_d \equiv (1+2/d)P-\rho(\partial P/\partial\rho)_s$ in $d$
dimensions. Next, we relate $\gamma_d$ to dissipation via a new low-energy bulk
viscosity sum rule. We argue that $\2d$ is special, with its logarithmic
dependence of the interaction on density, and thus $\gamma_2$ is small in both
the BCS and BEC regimes, even though $P - 2\varepsilon/d$, sensitive to the
dimer binding energy that breaks scale invariance, is not.

Quantum Gases;High Energy Physics - Phenomenology;Nuclear Theory