A quasi-particle description of the M(3,p) models

Abstract

The M(3,p) minimal models are reconsidered from the point of view of the extended algebra whose generators are the energy-momentum tensor and the primary field ϕ_{2,1} of dimension $(p-2)/4$. Within this framework, we provide a quasi-particle description of these models, in which all states are expressed solely in terms of the ϕ_{2,1}-modes. More precisely, we show that all the states can be written in terms of ϕ_{2,1}-type highest-weight states and their phi_{2,1}-descendants. We further demonstrate that the conformal dimension of these highest-weight states can be calculated from the ϕ_{2,1} commutation relations, the highest-weight conditions and associativity. For the simplest models (p=5,7), the full spectrum is explicitly reconstructed along these lines. For $p$ odd, the commutation relations between the ϕ_{2,1} modes take the form of infinite sums, i.e., of generalized commutation relations akin to parafermionic models. In that case, an unexpected operator, generalizing the Witten index, is unravelled in the OPE of ϕ_{2,1} with itself. A quasi-particle basis formulated in terms of the sole ϕ_{1,2} modes is studied for all allowed values of p. We argue that it is governed by jagged-type partitions further subject a difference 2 condition at distance 2. We demonstrate the correctness of this basis by constructing its generating function, from which the proper fermionic expression of the combination of the Virasoro irreducible characters χ_{1,s} and χ_{1,p-s} (for 1\leq s\leq [p/3]+1) are recovered. As an aside, a practical technique for implementing associativity at the level of mode computations is presented, together with a general discussion of the relation between associativity and the Jacobi identities.