On thermodynamic approaches to conformal field theory

We present the thermodynamic Bethe ansatz as a way to factorize the partition function of a 2d field theory, in particular, a conformal field theory and we compare it with another approach to factorization due to Schoutens which consists of diagonalizing matrix recursion relations between the partition functions at consecutive levels. We prove that both are equivalent, taking as examples the SU(2) spinons and the 3-state Potts model. In the latter case we see that there are two different thermodynamic Bethe ansatz equation systems with the same physical content, of which the second is new and corresponds to a one-quasiparticle representation, as opposed to the usual two-quasiparticle representation. This new thermodynamic Bethe ansatz system leads to a new dilogarithmic formula for the central charge of that model.     

On conformal invariance in quantum field theory

We study the transformation law of interacting fields under the universal convering group of the conformal group. It is defined with respect to the representations of the discrete series. These representations are field representations in the sense that they act on finite component fields defined over Minkowski space. The conflict with Einstein causality is avoided as in the case of free fields with canonical dimension. Furthermore, we determine the conformal invariant two-point function of arbitrary spin. Our result coincides with that obtained by Rühl. In particular, we investigate the two-point function of symmetric and traceless tensor fields and give the explicit form of the trace terms.     

On the application of conformal symmetry to quantum field theory

To leading order in α_s(Q²) conformal symmetry specifies the eigensolutions of the evolution equation for meson distribution amplitudes, the wavefunctions which control large-momentum-transfer exclusive mesonic processes in QCD. We find that at next to leading order, the eigensolutions in various field theories depend on the regularization scheme, even for zero β-function. This is contrary to the expectations of conformal symmetry.     

Adelic conformal field theory

We stress the use of modular forms in obtaining adelic formulations of field theoretical problems. Supersymmetry then appears in the real section with thep-adic parts as arithmetic completions. We first show how the Casimir effect is naturally interpreted adelically and the coefficient arises from dimensional analysis. We then suggest looking at the zero slope limit of adelic string amplitudes. Finally, we interpret the rationality of the critical exponents for conformal field theories in terms of p-adic analyticity of correlation functions.     

On the structure of unitary conformal field theory. I. Existence of conformal blocks

We study the general mathematical structure of unitary rational conformal field theories in two dimensions, starting from the Euclidean Green functions of the scaling fields. We show that, under certain assumptions, the scaling fields of such theories can be written as sums of products of chiral fields. The chiral fields satisfy an algebra whose structure constants are the matrix elements of Yang-Baxter- or braid-matrices whose properties we analyze. The upshot of our analysis is that two-dimensional conformal field theories of the type considered in this paper appear to be constructible from the representation theory of a pair of chiral algebras.     

On the large N limit of conformal field theory

Following recent advances in large N matrix mechanics, I discuss here the free (Cuntz) algebraic formulation of the large N limit of two-dimensional conformal field theories of chiral adjoint fermions and bosons. One of the central results is a new affine free algebra which describes a large N limit of affine Lie algebra. Other results include the associated free-algebraic partition functions and characters, a free-algebraic coset construction, free-algebraic construction of , free-algebraic vertex operator constructions in the large N Bose systems, and a provocative new free-algebraic factorization of the ordinary Koba-Nielsen factor.     

Classical and quantum conformal field theory

We define chiral vertex operators and duality matrices and review the fundamental identities they satisfy. In order to understand the meaning of these equations, and therefore of conformal field theory, we define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish. The classical limit of the equations for the duality matrices in rational field theory together with some results of category theory, suggest that (quantum) conformal field theory should be regarded as a generalization of group theory.On leave of absence from the Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel     

Operator algebras and conformal field theory

We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III₁ factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a background-independent formulation of conformal field theories.     

Logarithmic conformal field theory through nilpotent conformal dimensions

We study logarithmic conformal field theories (LCFTs) through the introduction of nilpotent conformal weights. Using this device, we derive the properties of LCFTs such as the transformation laws, singular vectors and the structure of correlation functions. We discuss the emergence of an extra energy momentum tensor, which is the logarithmic partner of the energy momentum tensor.     

Rationality in conformal field theory

We show that if the one-loop partition function of a modular invariant conformal field theory can be expressed as a finite sum of holomorphically factorized terms thenc and all values ofh are rational.     

Introduction to string theory and conformal field theory

A concise survey of noncritical string theory and two-dimensional conformal field theory is presented. A detailed derivation of a conformal anomaly and the definition and general properties of conformal field theory are given. Minimal string theory, which is a special version of the theory, is considered. Expressions for the string susceptibility and gravitational dimensions are derived.     

A conformal holomorphic field theory

A formulation of a field theory on the complex Minkowski space in terms of complex differential geometry is proposed. It is also shown that our model of field theory differs from the standard model on the real Minkowski space only in the limit of high energy.     

Two-dimensional conformal quantum field theory

On leave of absence from the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria.     

Fay's trisecant identity and conformal field theory

We study the correlation functions of a system of free chiral fermions on a compact Riemann surface using techniques of algebraic geometry. Fay's trisecant identity arises as a consequence of the proof of the uniqueness of correlation functions.     

Four-dimensional heterotic strings and conformal field theory

The techniques of (super) conformal field theory are applied to 4-dimensional heterotic string theories. We discuss certain aspects of 4-dimensional strings in the framework of the bosonic lattice approach such as the realization of superconformal symmetry, character valued partition functions, construction of vertex operators and ghost picture changing. As an application we compute all possible 3- and 4-point tree amplitudes of the massless fields and derive from them the low energy effective action of the massless modes. Some effects for the massless spectrum due to one-loop string effects are also mentioned.     

Temperature corrections to conformal field theory

We consider finite temperature dynamical correlation functions in the interacting delta-function Bose gas. In the low-temperature limit the asymptotic behaviour of correlation functions can be determined from conformal field theory. In the present work we determine the deviations from conformal behaviour at low temperatures. Received: 14 January 1998 / Accepted: 17 March 1998