Domain walls, fusion rules and conformal field theory in the quantum Hall regime

We provide a simple way to obtain the fusion rules associated with elementary quasi-holes over quantum Hall wave functions, in terms of domain walls. The knowledge of the fusion rules is helpful in the identification of the underlying conformal field theory describing the wave functions. We obtain the fusion rules, and explicitly give a conformal field theory description, for a two-parameter family (k,r) of wave functions. These include the Laughlin, Moore-Read and Read-Rezayi states when r=2. The `gaffnian' wave function is the prototypical example for r>2, in which case the conformal field theory is non-unitary.