Infinitely extended Kac table of solvable critical dense polymers

Abstract

Solvable critical dense polymers is a Yang–Baxter integrable model of polymers on the square lattice. It is the first member of the family of logarithmic minimal models . The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r, s = 1, 2, …. In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r, s) = (r, 1)⊗(1, s) and involve a boundary field ξ. Tuning the field ξ 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 for r, s = 1, 2, 3, … in the infinitely extended Kac table.