# Solvable Critical Dense Polymers on the Cylinder

Research paper by **Paul A. Pearce, Jorgen Rasmussen, Simon P. Villani**

Indexed on: **25 Feb '10**Published on: **25 Feb '10**Published in: **High Energy Physics - Theory**

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#### Abstract

A lattice model of critical dense polymers is solved exactly on a cylinder
with finite circumference. The model is the first member LM(1,2) of the
Yang-Baxter integrable series of logarithmic minimal models. The cylinder
topology allows for non-contractible loops with fugacity alpha that wind around
the cylinder or for an arbitrary number ell of defects that propagate along the
full length of the cylinder. Using an enlarged periodic Temperley-Lieb algebra,
we set up commuting transfer matrices acting on states whose links are
considered distinct with respect to connectivity around the front or back of
the cylinder. These transfer matrices satisfy a functional equation in the form
of an inversion identity. For even N, this involves a non-diagonalizable braid
operator J and an involution R=-(J^3-12J)/16=(-1)^{F} with eigenvalues
R=(-1)^{ell/2}. The number of defects ell separates the theory into sectors.
For the case of loop fugacity alpha=2, the inversion identity is solved exactly
for the eigenvalues in finite geometry. The eigenvalues are classified by the
physical combinatorics of the patterns of zeros in the complex
spectral-parameter plane yielding selection rules. The finite-size corrections
are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the
conformal partition functions and confirm the central charge c=-2 and conformal
weights Delta_t=(t^2-1)/8. Here t=ell/2 and t=2r-s in the ell even sectors with
Kac labels r=1,2,3,...; s=1,2 while t is half-integer in the ell odd sectors.
Strikingly, the ell/2 odd sectors exhibit a W-extended symmetry but the ell/2
even sectors do not. Moreover, the naive trace summing over all ell even
sectors does not yield a modular invariant.