Conformal invariance of chiral edge theories

The low-energy effective quantum-field theory of the edge excitations of a fully gapped bulk topological phase corresponding to a local 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 wave functions related to nonunitary 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.