  We propose a theory for the underdoped hole-doped cuprates, focusing on the
"nodal-anti-nodal dichotomy" observed in recent experiments. Our theory begins
with an ordered antiferromagnetic Fermi liquid with electron and hole pockets.
We argue that it is useful to consider a quantum transition at which the loss
of antiferromagnetic order leads to a hypothetical metallic "algebraic charge
liquid" (ACL) with pockets of charge -e and +e fermions, and an emergent U(1)
gauge field; the instabilities of the ACL lead to the low temperature phases of
the underdoped cuprates. The pairing instability leads to a superconductor with
the strongest pairing within the -e Fermi pockets, a d-wave pairing signature
for electrons, and very weak nodal-point pairing of the +e fermions near the
Brillouin zone diagonals. The influence of an applied magnetic field is
discussed using a proposed phase diagram as a function of field strength and
doping. We describe the influence of gauge field and pairing fluctuations on
the quantum Shubnikov-de Haas oscillations in the normal states induced by the
field. For the finite temperature pseudogap region, our theory has some
similarities to the phenomenological two-fluid model of -2e bosons and +e
fermions proposed by Geshkenbein, Ioffe, and Larkin [cond-mat/9609209], which
describes anomalous aspects of transverse transport in a magnetic field.
