A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

Abstract

In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007), and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley--Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L_0 Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and sl(2|1) supersymmetric spin chains with extra spins at the boundary, together with the "mirror" spin chain introduced by Martin and Woodcock (2004).

65 pages, 19 figures. New appendix

References in corpus (18)

• Logarithmic extensions of minimal models: characters and modular transformations
• Associative-algebraic approach to logarithmic conformal field theories
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
• On the SU(2|1) WZW model and its statistical mechanics applications
• Conformal boundary loop models
• From boundary to bulk in logarithmic CFT
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Logarithmic observables in critical percolation
• Fusion Algebras of Logarithmic Minimal Models
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Free fermion resolution of supergroup WZNW models
• Instantons beyond topological theory II
• W-Extended Logarithmic Minimal Models
• Combinatorial aspects of boundary loop models
• SLE local martingales in logarithmic representations
• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Commuting Families in Temperley-Lieb Algebras

Cited by in corpus (24)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Logarithmic Conformal Field Theory: a Lattice Approach
• Holographic applications of logarithmic conformal field theories
• Correlation functions in loop models
• Boundary algebras and Kac modules for logarithmic minimal models
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Local logarithmic correlators as limits of Coulomb gas integrals
• Lattice W-algebras and logarithmic CFTs
• Jordan cells of periodic loop models
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• 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
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• Blob algebra approach to modular representation theory
• Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Integrable Floquet systems related to logarithmic conformal field theory
• On graded decomposition numbers of Cherednik algebras
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
• The Nichols algebra $\mathfrak{B}(V_{abe})$ and a class of combinatorial numbers
• Staggered and affine Kac modules over $A_1^{(1)}$