Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue –a₂⁽¹⁾ and a₂⁽²⁾. By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory (with or without RSOS restriction) indeed has some fundamental problems, but that these problems are of two different sorts. For a₂⁽¹⁾, the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a₂⁽²⁾ however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found earlier [Nucl. Phys. B 489 [FS] (1997) 557]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.     

On differential equation on four-point correlation function in the conformal Toda field theory

The properties of completely degenerate fields in the conformal Toda field theory are studied. It is shown that a generic four-point correlation function that contains only one such field does not satisfy an ordinary differential equation, in contrast to the Liouville field theory. Some additional assumptions for other fields are required. Under these assumptions, we write such a differential equation and solve it explicitly. We use the fusion properties of the operator algebra to derive a special set of three-point correlation functions. The result agrees with the semiclassical calculations.     

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.     

Aspects of affine Toda field theory

Several of the recently discovered classical and quantum features of affine Toda field theory are briefly reviewed, with particular emphasis on the Lie algebraic structure of masses, conserved quantities and S-matrices.     

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.     

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.     

The significance of conformal inversion in quantum field theory

The 2-point functions of Euclidean conformal invariant quantum field theory are looked at as intertwining kernels of the conformal group. In this analysis a fundamental role is played by a two-element groupW, whose non-identity element =R·I consists of the conformal inversionR multiplied by a space-time reflectionI. The propagators of conformal invariant quantum field theory are determined by the requirement of -covariance. The importance of the -inversion in the theory of Zeta-functions is mentioned.     

Logarithmic operators in conformal field theory

Conformal field theories with correlation functions which have logarithmic singularities are considered. It is shown that those singularities imply the existence of additional operators in the theory which together with ordinary primary operators form the basis of the Jordan cell for the operator L₀. An example of the field theory possessing such correlation functions is given.     

Yangian superalgebras in conformal field theory

Quantum Yangian symmetry in several sigma models with supergroup or supercoset as target is established. Starting with a two-dimensional conformal field theory that has current symmetry of a Lie superalgebra with vanishing Killing form we construct non-local charges and compute their properties. Yangian axioms are satisfied, except that the Serre relations only hold for a subsector of the space of fields. Yangian symmetry implies that correlation functions of fields in this sector satisfy Ward identities. We then show that this symmetry is preserved by certain perturbations of the conformal field theory.     

The ultraviolet behaviour of integrable quantum field theories,affine Toda field theory

We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against the numerical solutions. We address the question of existence and uniqueness of the solutions of the TBA equations, derive precise error estimates and determine the rate of convergence for the applied numerical procedure. A general expression for the Fourier transformed kernels of the TBA equations allows one to derive the related Y-systems and a reformulation of the equations into a universal form.     

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.     

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.     

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.     

Quantum group structure of quantum Toda conformal field theories (I)

We consider the quantization of non-affine Toda field theories in the light-cone and lattice formalisms. The vertex operators are constructed and their braiding is found to be a consequence of the fundamental commutation relations satisfied by the monodromy matrix. For certain values of the coupling, which correspond to the minimal models, the truncation of the operator algebra is closely tied to the quantum group structure.     

The Toda lattice field theory hierarchies and zero-curvature conditions in Kac-Moody algebras

The two-dimensional Toda lattice field theories possess an infinite number of local conserved quantities in involution. These can be used as hamiltonians to define a consistent simultaneous evolution in the infinite number of associated times. Our previous explicit construction of the corresponding zero-curvature gauge potentials is used to extend the zero curvature to the complete infinite-dimensional space defined by these times by means of the Yang-Baxter equations. This result is elevated to the full Kac-Moody algebra with central extension thereby providing a link with the work of the Kyoto school.