On Staggered Indecomposable Virasoro Modules

Abstract

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L_0 is assumed to be non-diagonalisable, possessing Jordan blocks of rank two. Moreover, the module is further assumed to have a highest weight submodule, the "left module", and that the quotient by this submodule yields another highest weight module, the "right module". Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centres on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. The text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit.

54 pages, 6 figures, 16 examples. v2: small changes including new historical footnote after Eq. (3.6). Not the same as the published version (we gave up correcting the errors)

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• Lattice fusion rules and logarithmic operator product expansions
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• A BMS-invariant free scalar model
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• Bosonic Ghosts at $c=2$ as a Logarithmic CFT
• sl(2)_{-1/2} and the Triplet Model
• A modular invariant bulk theory for the c=0 triplet model
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• The puzzle of bulk conformal field theories at central charge c=0
• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• Boundary algebras and Kac modules for logarithmic minimal models
• Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• The Verlinde formula in logarithmic CFT
• Takiff superalgebras and Conformal Field Theory
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• A BMS-invariant free fermion model
• Logarithmic Superconformal Minimal Models
• Logarithmic conformal invariance in the Abelian sandpile model
• Local logarithmic correlators as limits of Coulomb gas integrals
• The Jordan Structure of Two Dimensional Loop Models
• Jordan cells of periodic loop models
• Unitary and non-unitary $N=2$ minimal models
• From Jack polynomials to minimal model spectra
• On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Some Properties of Sandpile Models as Prototype of Self-Organized Critical Systems
• Logarithmic Exotic Conformal Galilean Algebras
• Finite-size corrections for logarithmic representations in critical dense polymers
• Physical ageing and new representations of some Lie algebras of local scale-invariance
• Sandpile models in the large
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• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Two-point boundary correlation functions of dense loop models
• Logarithmic Correlations in Quantum Hall Plateau Transitions
• Free Field Realisations of Staggered Modules in 2D Logarithmic CFTs
• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
• Staggered modules of $N=2$ superconformal minimal models
• Multipoint correlators in the Abelian sandpile model
• Planar UST Branches and $c=-2$ Degenerate Boundary Correlations
• Discrete symplectic fermions on double dimers and their Virasoro representation