  The low-energy effective quantum field theory of the edge excitations of a
fully-gapped bulk topological phase corresponding to a local interaction
Hamiltonian must be local and unitary. Here it is shown that whenever all the
edge excitations propagate in the same direction with the same velocity, it is
a conformal field theory. In particular, this is the case in the quantum Hall
effect for model "special Hamiltonians", for which the ground state, quasihole,
and edge excitations can be found exactly as zero-energy eigenstates, provided
the spectrum in the interior of the system is fully gapped. In addition, other
conserved quantities in the bulk, such as particle number and spin, lead to
affine Lie algebra symmetries in the edge theory. Applying the arguments to
some trial wavefunctions related to non-unitary conformal field theories, it is
argued that the Gaffnian state and an infinite number of others cannot describe
a gapped topological phase because the numbers of edge excitations do not match
any unitary conformal field theory.
