Alexi Morin-Duchesne, Yvan Saint-Aubin
Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, $T_N$, is an element of the periodic Temperley-Lieb algebra $\eptl(\beta, \alpha)$, where $N$ is the number of sites on a section of the cylinder, and $\beta=-q-q^{-1} = 2 \cos \lambda$ and $\alpha$ the weights of contractible and non-contractible loops. The thermodynamic limit of $T_N$ is believed to describe a conformal field theory of central charge $c=1-6\lambda^2/(\pi(\lambda-\pi))$. The abstract element $T_N$ acts naturally on (a sum of) spaces $\tilde V_N^d$, similar to those upon which the standard modules of the (classical) Temperley-Lieb algebra act. These spaces known as {\em sectors} are labeled by the numbers of defects $d$ and depend on a {\em twist parameter} $v$ that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of $T_N$ both between sectors with distinct defect numbers and within a given sector.
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http://arxiv.org/abs/1302.5483
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