Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models

Abstract

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.

v2: 46 pages (including appendices)

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• Spaces of states of the two-dimensional O(n) and Potts models
• Critical loop models are exactly solvable
• From combinatorial maps to correlation functions in loop models
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Fusion of irreducible modules in the periodic Temperley--Lieb algebra
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors