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.
Highlights
A connectivity operator Ok,x (r) in the loop model is described by its number of defects 2k ∈ and its twist parameter x ∈ ×
We propose an alternative construction of the Graham-Lehrer homomorphisms between standard modules for generic values of q
We formulated a new prescription for the fusion of standard modules of the enlarged periodic Temperley-Lieb algebra EPTLN (−q − q−1)
Summary
In the study of critical two-dimensional statistical models, the conceptual framework and computational tools of Conformal Field Theory (CFT) have proven very efficient [1]. A connectivity operator Ok,x (r) in the loop model is described by its number of defects 2k ∈ and its twist parameter x ∈ × It is naturally associated, through the state-operator correspondence, with a standard module Wk,x (N ) over EPTLN (β), spanned by link states with 2k defects attached to a marked point. From the resulting module decompositions, it is readily observed that the two proposals [37, 38] and [39] are inequivalent It is presently unclear whether these two prescriptions for fusion are physically useful to compute the operator product expansion of the bulk connectivity operators. The properties of the Jones-Wenzl projectors are reviewed in Appendix A, and certain more technical computations of Section 4 are relegated to Appendix B
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