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

TL;DR

This paper develops a quasi-particle framework for the M(3,p) minimal models using an extended algebra, enabling explicit spectrum reconstruction and revealing new operator structures, with implications for fermionic character formulas.

Contribution

It introduces a novel quasi-particle description based solely on _{2,1}-modes, providing explicit spectrum calculations and uncovering generalized commutation relations and operators.

Findings

Explicit spectrum reconstructed for p=5,7 models

Generalized commutation relations akin to parafermions

Fermionic expressions of Virasoro characters derived

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 \phi_{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 \phi_{2,1}-modes. More precisely, we show that all the states can be written in terms of \phi_{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 \phi_{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 \phi_{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 \phi_{2,1} with itself. A quasi-particle basis formulated in terms of the sole \phi_{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 \chi_{1,s} and \chi_{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.