  I present a pedagogical survey of a variety of quantum phases of the Hubbard
model. The honeycomb lattice model has a conformal field theory connecting the
semi-metal to the insulator with Neel order. States with fractionalized
excitations are linked to the deconfined phases of gauge theories. I also
consider the confining phases of such gauge theories, and show how Berry phases
of monopoles induce valence bond solid order. The triangular lattice model can
display a metal-insulator transition from a Fermi liquid to a deconfined spin
liquid, and I describe the theory of this transition. The bilayer triangular
lattice is used to illustrate another compressible metallic phase, the
`fractionalized Fermi liquid'. I make numerous connections of these phases and
critical points to the AdS/CFT correspondence. In particular, I argue that two
recent holographic constructions connect respectively to the Fermi liquid and
fractionalized Fermi liquid phases.
