Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality

TL;DR

This paper constructs a logarithmic extension of M(2,p) minimal models, systematically analyzing indecomposable modules, computing logarithmic couplings, and exploring duality constraints within conformal field theory.

Contribution

It introduces a new construction of logarithmic M(2,p) models using reducible modules and fusion, providing explicit calculations of logarithmic couplings and insights into duality.

Findings

Logarithmic couplings computed for many cases.

Closed-form expressions for gauge-invariant data.

Global conformal invariance constrains the spectrum and suggests dual models.

Abstract

A natural construction of the logarithmic extension of the M(2,p) minimal models is presented, which generalises our previous model [0708.0802] 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.