Working in the dense loop representation, we use the planar Temperley-Lieb algebra to build integrable lattice models called logarithmic minimal models LM(p, p ′ ). Specifically, we construct Yang-Baxter integrable Temperley-Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley-Lieb algebra are inherently non-local and not (timereversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1 − , r, s = 1, 2, . . .. The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.

A lattice model of critical dense polymers is solved exactly for finite strips. The model is the first member of the principal series of the recently introduced logarithmic minimal models. The key to the solution is a functional equation in the form of an inversion identity satisfied by the commuting double-row transfer matrices. This is established directly in the planar Temperley-Lieb algebra and holds independently of the space of link states on which the transfer matrices act. Different sectors are obtained by acting on link states with s − 1 defects where s = 1, 2, 3, . . . is an extended Kac label. The bulk and boundary free energies and finite-size corrections are obtained from the Euler-Maclaurin formula. The eigenvalues of the transfer matrix are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields a selection rule for the physically relevant solutions to the inversion identity and explicit finitized characters for the associated quasi-rational representations. In particular, in the scaling limit, we confirm the central charge c = −2 and conformal weights ∆ s = (2−s) 2 −1 8 for s = 1, 2, 3, . . .. We also discuss a diagrammatic implementation of fusion and show with examples how indecomposable representations arise. We examine the structure of these representations and present a conjecture for the general fusion rules within our framework.

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 α 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 α = 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 ∆,∆ = ∆ t = (t 2 − 1)/8. Here t = ℓ/2 and t = 2r − s ∈ N in the ℓ even sectors with Kac labels r = 1, 2, 3, . . . ; s = 1, 2 while t ∈ 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.

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p, p ) considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an s (2) structure but require so-called Kac representations which are typically reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra p = 1 is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p = 1 and with Eberle and Flohr for (p, p ) = (2, 5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p, p ) with the augmented c p,p (minimal) models defined algebraically.

The bosonic βγ ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -due in part to its underlying su(2) −1/2 structure -of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this βγ system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2) −1/2 theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum.