  Topological insulators in three spatial dimensions are known to possess a
precise bulk/boundary correspondence, in that there is a one-to-one
correspondence between the 5 classes characterized by bulk topological
invariants and Dirac hamiltonians on the boundary with symmetry protected zero
modes. This holographic characterization of topological insulators is studied
in two dimensions. Dirac hamiltonians on the one dimensional edge are
classified according to the discrete symmetries of time-reversal,
particle-hole, and chirality, extending a previous classification in two
dimensions. We find 17 inequivalent classes, of which 11 have protected zero
modes. Although bulk topological invariants are thus far known for only 5 of
these classes, we conjecture that the additional 6 describe edge states of new
classes of topological insulators. The effects of interactions in two
dimensions are also studied. We show that all interactions that preserve the
symmetries are exactly marginal, i.e. preserve the gaplessness. This leads to a
description of the distinct variations of Luttinger liquids that can be
realized on the edge.
