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.     

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.     

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.     

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.     

Constraints on representations of conformal field theories

We consider conformal field theories with generalized blocks which satisfy Ising-type algebras. By requiring the correct analytic properties of the correlators, we find constraints on the central charge and conformal dimensions of the fields.     

K-matrices for 2D conformal field theories

In this paper we examine fermionic type characters (Universal Chiral Partition Functions) for general 2D conformal field theories with a bilinear form given by a matrix of the form . We provide various techniques for determining these K-matrices, and apply these to a variety of examples including (higher level) WZW and coset conformal field theories. Applications of our results to fractional quantum Hall systems and (level restricted) Kostka polynomials are discussed.     

Gravity duals for nonrelativistic conformal field theories

We attempt to generalize the anti-de Sitter/conformal field theory correspondence to nonrelativistic conformal field theories which are invariant under Galilean transformations. Such systems govern ultracold atoms at unitarity, nucleon scattering in some channels, and, more generally, a family of universality classes of quantum critical behavior. We construct a family of metrics which realize these symmetries as isometries. They are solutions of gravity with a negative cosmological constant coupled to pressureless dust. We discuss realizations of the dust, which include a bulk superconductor. We develop the holographic dictionary and find two-point correlators of the correct form. A strange aspect of the correspondence is that the bulk geometry has two extra noncompact dimensions.     

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.     

Structure constants for rational conformal field theories

Exploiting the associativity of the operator product algebra, we obtain the structure constants for the affine SU(2) complementary and exceptional series, uniquely determined in terms of the diagonal ones. The result extends to the non-diagonal unitary minimal series with c<1. We investigate the more general cases of class I (extended diagonal) and class II (non-diagonal) algebras and find a relation between them based on the symmetries of their fusion rules.     

Boundary conditions in rational conformal field theories

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalisation of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk–boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.