Logarithmic M(2,p) minimal models, their logarithmic couplings, and duality

Abstract

A natural construction of the logarithmic extension of the M (2, p) (chiral) minimal models is presented, which generalises our previous model [R. Langlands, P. Pouliot, Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. 30 (1994) 1-61] of percolation (p = 3). Its key aspect is the replacement of the minimal model irreducible modules by reducible ones obtained by requiring that only one of the two principal singular vectors of each module vanish. The resulting theory is then constructed systematically by repeatedly fusing these building block representations. This generates indecomposable representations of the type which signify the presence of logarithmic partner fields in the theory. The basic data characterising these indecomposable modules, the logarithmic couplings, are computed for many special cases and given a new structural interpretation. Quite remarkably, a number of them are presented in closed analytic form (for general p). These are the prime examples of "gauge-invariant" data-quantities independent of the ambiguities present in defining the logarithmic partner fields. Finally, mere global conformal invariance is shown to enforce strong constraints on the allowed spectrum: It is not possible to include modules other than those generated by the fusion of the model's building blocks. This generalises the statement that there cannot exist two effective central charges in a c = 0 model. It also suggests the existence of a second "dual" logarithmic theory for each p. Such dual models are briefly discussed.