Effective Chern-Simons theories of pfaffian and parafermionic quantum Hall states, and orbifold conformal field theories

We present a pure Chern-Simons formulation of families of interesting conformal field theories describing edge states of non-Abelian quantum Hall states. These theories contain two Abelian Chern-Simons fields describing the electromagnetically charged and neutral sectors of these models, respectively. The charged sector is the usual Abelian Chern-Simons theory that successfully describes Laughlin-type incompressible fluids. The neutral sector is a (2+1)-dimensional theory analogous to the (1+1)-dimensional orbifold conformal field theories. It is based on the gauge group O(2) which contains a Z₂ disconnected group manifold, which is the salient feature of this theory. At level q, the Abelian theory of the neutral sector gives rise to a Z_2q symmetry, which is further reduced by imposing the Z₂ symmetry of charge-conjugation invariance. The remaining Z_q symmetry of the neutral sector is the origin of the non-Abelian statistics of the (fermionic) q-pfaffian states.