Associative-algebraic approach to logarithmic conformal field theories

We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non-semisimple associative algebras appearing in their lattice regularizations (as discussed in a companion paper [N. Read, H. Saleur, Enlarged symmetry algebras of spin chains, loop models, and S-matrices, cond-mat/0701259]). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras gl(n|n)$\mathrm{gl}(n|n)$ and gl(n+1|n)$\mathrm{gl}(n+1|n)$, respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits are boundary conformal field theories (CFTs) with central charge c=−2$c=-2$ and c=0$c=0$ respectively, and in the loop interpretation they describe dense polymers and the boundaries of critical percolation clusters, respectively. We also discuss the case of dilute (critical) polymers as another boundary CFT with c=0$c=0$. Within the supersymmetric formulations, these boundary CFTs describe the fixed points of certain nonlinear sigma models that have a supercoset space as the target manifold, and of Landau–Ginzburg field theories. The submodule structures of indecomposable representations of the Virasoro algebra appearing in the boundary CFT, representing local fields, are derived from the lattice. A central result is the derivation of the fusion rules for these fields.     

Singular vectors in logarithmic conformal field theories

Null vectors are generalized to the case of indecomposable representations which are one of the main features of logarithmic conformal field theories. This is done by developing a compact formalism with the particular advantage that the stress energy tensor acting on Jordan cells of primary fields and their logarithmic partners can still be represented in form of linear differential operators. Since the existence of singular vectors is subject to much stronger constraints than in regular conformal field theory, they also provide a powerful tool for the classification of logarithmic conformal field theories.     

Disordered loop model and coupled conformal field theories

A family of models for fluctuating loops in a two-dimensional random background is analyzed. The models are formulated as O(n) spin models with quenched inhomogeneous interactions. Using the replica method, the models are mapped to the M→0 limits of M-layered O(n) models coupled each other via _1,3 primary fields. The renormalization group flow is calculated in the vicinity of the decoupled critical point, by an epsilon expansion around the Ising point (n=1), varying n as a continuous parameter. The one-loop beta function suggests the existence of a strongly coupled phase (0<n<n_*) near the self-avoiding walk point (n=0) and a line of infrared fixed points (n_*<n<1) near the Ising point. For the fixed points, the effective central charges are calculated. The scaling dimensions of the energy operator and the spin operator are obtained up to two-loop order. The relation to the random-bond q-state Potts model is briefly discussed.     

Origin of logarithmic operators in conformal field theories

We study logarithmic operators in Coulomb gas models, and show that they occur when the “puncture” operator of the Liouville theory is included in the model. We also consider WZNW models for SL(2,R), and for SU(2) at level 0, in which we find logarithmic operators which form Jordan blocks for the current as well as the Virasoro algebra.     

Towards the construction of local logarithmic conformal field theories

Although logarithmic conformal field theories (LCFTs) are known not to factorise many previous findings have only been formulated on their chiral halves. Making only mild and rather general assumptions on the structure of an chiral LCFT we deduce statements about its local non-chiral equivalent. Two methods are presented how to construct local representations as subrepresentations of the tensor product of chiral and anti-chiral Jordan cells. Furthermore we explore the assembly of generic non-chiral correlation functions from generic chiral and anti-chiral correlators. The constraint of locality is studied and the generality of our method is discussed.     

Two-dimensional fractional supersymmetric conformal and logarithmic conformal field theories and two point functions

A general two-dimensional fractional supersymmetric conformal field theory is investigated. The structure of the symmetries of the theory is studied. Applying the generators of the closed subalgebra generated by and , the two point functions of the component fields of supermultiplets are calculated. Then the logarithmic superconformal field theories are investigated and the chiral and full two point functions are obtained. Received: 15 December 2000 / Revised version: 19 March 2001 / Published online: 18 May 2001     

Disordered Dirac fermions: the marriage of three different approaches

We compare the critical multipoint correlation functions for two-dimensional (massless) Dirac fermions in the presence of a random su(N) (non-Abelian) gauge potential, obtained by three different methods. We critically reexamine previous results obtained using the replica approach and in the limit of infinite disorder strength and compare them to new results (presented here) obtained using the supersymmetric approach to the N=2 case. We demonstrate that this ménage à trois of different approaches leads to identical results. Remarkable relations between apparently different conformal field theories (CFTs) are thereby obtained. We further establish a connection between the random Dirac fermion problem and the c=−2 theory of dense polymers. The presence of the c=−2 theory may be seen in all three different treatments of the disorder.     

A local logarithmic conformal field theory

The local logarithmic conformal field theory corresponding to the triplet algebra at c = -2 is constructed. The constraints of locality and duality are explored in detail, and a consistent set of amplitudes is found. The spectrum of the corresponding theory is determined, and it is found to be modular invariant. This provides the first construction of a non-chiral rational logarithmic conformal field theory, establishing that such models can indeed define bona fide conformal field theories.     

De Sitter and conformal field theories

After a brief review of conformal gravity and conformal anomalies in field theories, this paper deals with elementary particles and quantum field theories in 3 + 2 de Sitter space and in conformal space. The importance of working in realistic space-time rather than Euclidean or spherical models, is demonstrated. The parton-like representations Di and Rac give rise to gauge theories of scalar and spinor fields, and a theory of interacting massless particles with all spins. This theory (in 4-dimensional de Sitter space) is constructed on the basis cf a conformally invariant field theory in 3-dimensional space-time. Conformally invariant field theories in 3 and 4 dimensional are reviewed and examined, and new proposals are made for the interpretation of massless field theories in general.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Sewing conformal field theories II

We show that any Riemann surface M with punctures can be constructed by sewing three-punctured speres. Correspondingly, any correlation function on M can be obtained by sewing three-point functions on a sphere. There is no unique way of sewing three-punctured spheres to construct M, and the resulting correlation functions may depend on the precise way of sewing. We show that this dependence is absent, if we assume that four-point correlation function on a sphere and one-point functions on a torus are determined unambiguously.     

Chiral deformations of conformal field theories

We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W_1+∞ algebra, that is treated in detail.     

Meromorphicc=24 conformal field theories

Modular invariant conformal field theories with just one primary field and central chargec=24 are considered. It has been shown previously that if the chiral algebra of such a theory contains spin-1 currents, it is either the Leech lattice CFT, or it contains a Kac-Moody sub-algebra with total central charge 24. In this paper all meromorphic modular invariant combinations of the allowed Kac-Moody combinations are obtained. The result suggests the existence of 71 meromorphicc=24 theories, including the 41 that were already known.     

Integrability of coupled conformal field theories

The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin ladders along the c < 1 discrete series.     

Indecomposability parameters in chiral logarithmic conformal field theory

Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc.LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the ‘b-number’ introduced initially by Gurarie for the c=0 case. The determination of these indecomposability parameters (or logarithmic couplings) has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to gl(n+m|m) and osp(n+2m|2m)-invariant superspin chains and to non-linear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas.We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarie?s, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models LM(1,p) and LM(p,p+1).