  We model the underdoped cuprates using fermions moving in a background with
local antiferromagnetic order. The antiferromagnetic order fluctuates in
orientation, but not in magnitude, so that there is no long-range
antiferromagnetism, but a 'topological' order survives. The normal state is
described as a fractionalized Fermi liquid (FL*), with electron-like
quasiparticles coupled to the fractionalized excitations of the fluctuating
antiferromagnet. The electronic quasiparticles reside near pocket Fermi
surfaces enclosing total area x (the dopant density), centered away from the
magnetic Brillouin zone boundary. The violation of the conventional Luttinger
theorem is linked to a 'species doubling' of these quasiparticles. We describe
phenomenological theories of the pairing of these quasiparticles, and show that
a large class of mean-field theories generically displays a nodal-anti-nodal
`dichotomy': the interplay of local antiferromagnetism and pairing leads to a
small gap near the nodes of the d-wave pairing along the Brillouin zone
diagonals, and a large gap in the anti-nodal region.
