Solvable critical dense polymers on the cylinder

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 a that wind around the cylinder or for an arbitrary number l 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)(l/2). This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects l thus separates the theory into Ramond (l/2 even), Neveu-Schwarz (l/2 odd) and Z(4) (l odd) sectors. For the case of loop fugacity a = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler-Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = -2 and conformal weights Delta(Delta) over bar = Delta(t) = (t(2) -1)/8. Here t = l/2 and t = 2r -s is an element of N in the l even sectors with Kac labels r = 1, 2, 3,...; s = 1, 2 while t is an element of Z -1/2 in the l odd sectors. Strikingly, the l/2 odd sectors exhibit a W-extended symmetry but the l/2 even sectors do not. Moreover, the naive trace summing over all l even sectors does not yield a modular invariant.