  Recent quantum oscillation experiments on SmB$_6$ pose a paradox, for while
the angular dependence of the oscillation frequencies suggest a 3D bulk Fermi
surface, SmB$_6$ remains robustly insulating to very high magnetic fields.
Moreover, a sudden low temperature upturn in the amplitude of the oscillations
raises the possibility of quantum criticality. Here we discuss recently
proposed mechanisms for this effect, contrasting bulk and surface scenarios. We
argue that topological surface states permit us to reconcile the various data
with bulk transport and spectroscopy measurements, interpreting the low
temperature upturn in the quantum oscillation amplitudes as a result of surface
Kondo breakdown and the high frequency oscillations as large topologically
protected orbits around the X point. We discuss various predictions that can be
used to test this theory.
