2 research outputs found
A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra
We study finite loop models on a lattice wrapped around a cylinder. A section
of the cylinder has N sites. We use a family of link modules over the periodic
Temperley-Lieb algebra EPTL_N(\beta, \alpha) introduced by Martin and Saleur,
and Graham and Lehrer. These are labeled by the numbers of sites N and of
defects d, and extend the standard modules of the original Temperley-Lieb
algebra. Beside the defining parameters \beta=u^2+u^{-2} with u=e^{i\lambda/2}
(weight of contractible loops) and \alpha (weight of non-contractible loops),
this family also depends on a twist parameter v that keeps track of how the
defects wind around the cylinder. The transfer matrix T_N(\lambda, \nu) depends
on the anisotropy \nu and the spectral parameter \lambda that fixes the model.
(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 family of periodic XXZ Hamiltonians is extended to depend on this new
parameter v and the relationship between this family and the loop models is
established. The Gram determinant for the natural bilinear form on these link
modules is shown to factorize in terms of an intertwiner i_N^d between these
link representations and the eigenspaces of S^z of the XXZ models. This map is
shown to be an isomorphism for generic values of u and v and the critical
curves in the plane of these parameters for which i_N^d fails to be an
isomorphism are given.Comment: Replacement of "The Gram matrix as a connection between periodic loop
models and XXZ Hamiltonians", 31 page
Fusion rules and boundary conditions in the c=0 triplet model
The logarithmic triplet model W_2,3 at c=0 is studied. In particular, we
determine the fusion rules of the irreducible representations from first
principles, and show that there exists a finite set of representations,
including all irreducible representations, that closes under fusion. With the
help of these results we then investigate the possible boundary conditions of
the W_2,3 theory. Unlike the familiar Cardy case where there is a consistent
boundary condition for every representation of the chiral algebra, we find that
for W_2,3 only a subset of representations gives rise to consistent boundary
conditions. These then have boundary spectra with non-degenerate two-point
correlators.Comment: 50 pages; v2: changed formulation in section 1.2.1 and corrected
typos, version to appear in J. Phys.