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

TL;DR

This paper introduces new algebraic representations in the periodic Temperley-Lieb algebra to better understand the fusion and operator product expansion of connectivity operators in two-dimensional loop models, revealing their scaling properties.

Contribution

It presents a novel family of algebraic representations with marked points, providing insights into the fusion of standard modules and the structure of connectivity operators in loop models.

Findings

Decomposition of new representations on standard modules for generic parameters.

Structure of the operator product expansion of connectivity operators.

Enhanced understanding of scaling properties in dense O(n) 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.