School of Mathematics and Statistics - Theses
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ItemCritical dense polymers on the strip and cylinder( 2013)This thesis is concerned with the study of solvable critical dense polymers using Yang-Baxter techniques in two distinct topologies, namely the strip and the cylinder. Critical dense polymers is the first member ${\cal LM}(1,2)$ of the Yang-Baxter integrable series of logarithmic minimal models ${\cal LM}(p,p')$. The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels $r,s=1,2,\ldots$. We begin by explicitly constructing the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion \mbox{labels} $(r,s)=(r,1)\otimes (1,s)$ and involve a boundary field $\xi$. Tuning the field $\xi$ appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by 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 which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized $q$-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge $c=-2$ and the Kac formula for the conformal weights in the infinitely extended Kac table $\Delta_{r,s}=\frac{(2r-s)^2-1}{8}$ for $r,s=1,2,3,\ldots$. 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 $\vec J$. The number of defects $\ell$ thus separates the theory into Ramond ($\ell/2$ even), Neveu-Schwarz ($\ell/2$ odd) and $\mathbb{Z}_4$ ($\ell$ odd) sectors. For the case of loop fugacity $\alpha=2$, the inversion identity is solved exactly sector by sector 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. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge $c=-2$ and conformal weights $\Delta,\bar\Delta=\Delta_t=(t^2-1)/8$. Here $t=\ell/2$ and $t=2r-s\in\mathbb{N}$ in the $\ell$ even sectors with Kac labels $r=1,2,3,\ldots; s=1,2$ while $t\in\mathbb{Z}-\half$ in the $\ell$ odd sectors. Strikingly, the $\ell/2$ odd sectors exhibit a ${\cal W}$-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.