Clustering properties,Jack polynomials and unitary conformal field theories

Recently, Jack polynomials have been proposed as natural generalizations of Zk${\mathbb{Z}}_k$ Read–Rezayi states describing non-Abelian fractional quantum Hall systems. These polynomials are conjectured to be related to correlation functions of a class of W-conformal field theories based on the Lie algebra A_k−1$A_{k-1}$. These theories can be considered as non-unitary solutions of a more general series of CFTs with Zk${\mathbb{Z}}_k$ symmetry, the parafermionic theories. Starting from the observation that some parafermionic theories admit unitary solutions as well, we show, by computing the corresponding correlation functions, that these theories provide trial wavefunctions which satisfy the same clustering properties as the non-unitary ones. We show explicitly that, although the wavefunctions constructed by unitary CFTs cannot be expressed as a single Jack polynomial, they still show a fine structure where the mathematical properties of the Jack polynomials play a major role.