Conformal boundary loop models

Abstract

We study a model of densely packed self-avoiding loops on the annulus, related to the Temperley–Lieb algebra with an extra idempotent boundary generator. Four different weights are given to the loops, depending on their homotopy class and whether they touch the outer rim of the annulus. When the weight of a contractible bulk loop x ≡ q + q −1 ∈ ( − 2 , 2 ] , this model is conformally invariant for any real weight of the remaining three parameters. We classify the conformal boundary conditions and give exact expressions for the corresponding boundary scaling dimensions. The amplitudes with which the sectors with any prescribed number and types of non-contractible loops appear in the full partition function Z are computed rigorously. Based on this, we write a number of identities involving Z which hold true for any finite size. When the weight of a contractible boundary loop y takes certain discrete values, y r ≡ [ r + 1 ] q [ r ] q with r integer, other identities involving the standard characters K r , s of the Virasoro algebra are established. The connection with Dirichlet and Neumann boundary conditions in the O ( n ) model is discussed in detail, and new scaling dimensions are derived. When q is a root of unity and y = y r , exact connections with the A m type RSOS model are made. These involve precise relations between the spectra of the loop and RSOS model transfer matrices, valid in finite size. Finally, the results where y = y r are related to the theory of Temperley–Lieb cabling.