Nonlocal operator basis for the path representation of the unitary minimal models

We propose a new operator interpretation for the lattice paths of the unitary minimal models M(k + 1, k + 2). These operators are akin to the modes of a complex fermion, but at the same time, they capture some characteristics of Z k parafermions in that there is a bound on the number of allowed successive operators (which is k − 1). The operators act non-locally on the path: Their action modifies the path from its insertion point to its right extremity. In particular, they modify the path boundary condition. The states in a given module (i.e., with specified boundary conditions) are described by chargeless strings of operators. This description in terms of operators is captured by simple rules that can be viewed as defining a basis. Even when no reference is made to the paths, this operator formulation keeps a trace of non-locality in that the charge of the complete string of operators describing a particular state must remain within definite limits. The combinatorics underlying this operator representation leads to the known fermionic character formulae. These results generalize directly to the close relatives of the unitary models (in terms of path description), the M(k + 1, 2k + 3) models.