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

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.

25 pages, 1 figure; v2 has several changes including more references and a new appendix; v3 has only minor changes (NPB published version)

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Cited by in corpus (34)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• Logarithmic Conformal Field Theory: a Lattice Approach
• Lattice fusion rules and logarithmic operator product expansions
• On Staggered Indecomposable Virasoro Modules
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Phenomenology of ageing in the Kardar-Parisi-Zhang equation
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• sl(2)_{-1/2} and the Triplet Model
• Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters
• Boundary algebras and Kac modules for logarithmic minimal models
• Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model
• On the Percolation BCFT and the Crossing Probability of Watts
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• On logarithmic extensions of local scale-invariance
• Factorization of correlations in two-dimensional percolation on the plane and torus
• Dynamical symmetries and causality in non-equilibrium phase transitions
• SLE local martingales in logarithmic representations
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
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• Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus
• Modularity of Bershadsky-Polyakov minimal models
• Logarithmic Exotic Conformal Galilean Algebras
• Physical ageing and new representations of some Lie algebras of local scale-invariance
• Logarithmic correlators or responses in non-relativistic analogues of conformal invariance
• Topological effects and conformal invariance in long-range correlated random surfaces
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Logarithmic Correlations in Quantum Hall Plateau Transitions
• Staggered modules of $N=2$ superconformal minimal models
• Free Field Realisations of Staggered Modules in 2D Logarithmic CFTs
• Staggered and affine Kac modules over $A_1^{(1)}$