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

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 ϕ2,1 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 ϕ2,1-modes. More precisely, we show that all the states can be written in terms of ϕ2,1-type highest-weight states and their ϕ2,1-descendants. We further demonstrate that the conformal dimension of these highest-weight states can be calculated from the ϕ2,1 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 ϕ2,1 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 unraveled in the OPE of ϕ2,1 with itself. A quasi-particle basis formulated in terms of the sole ϕ2,1 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 χ1,s and χ1,p−s (for 1⩽s⩽[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.