Irreducible automorphic functions and then-point functions of conformal field theory

Irreducible automorphic functions for a compact Riemann surface of arbitrary genus are used to expand two- and three-point functions of conformal quantum field theory. The divergence of this expansion at coinciding arguments is studied in particular.     

Correlation functions of the chiral Potts chain from conformal field theory and finite-size corrections

We study the incommensurate phase of the N = 3 superintegrable chiral Potts chain by computing the leading finite-size corrections to the low-lying energy levels. These results are used to study the asymptotic behavior of the correlation functions. In particular we analyze the difference between the regions above and below the self-dual point.     

Constraining conformal field theory with higher spin symmetry in four dimensions

We analyze the constraints on the general form and the singularity structure of the correlation functions of the symmetric, traceless and conserved stress-energy tensor implied by conformal invariance and higher spin symmetry in four dimensions. In particular, we show that all these correlation functions will have at most double pole singularities. We then compute the 4-, 5- and 6-point functions of the stress-energy tensor and find that they are linear combinations of the three free field expressions (scalar, fermion and Maxwell field). This is a strong indication that all such theories are essentially free.     

Globally conformal invariant gauge field theory with rational correlation functions

Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields V_κ(x₁,x₂) of dimension (κ,κ). For a globally conformal invariant (GCI) theory we write down the OPE of V_κ into a series of twist (dimension minus rank) 2κ symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field.

We argue that the theory of a GCI hermitian scalar field of dimension 4 in D=4 Minkowski space such that the 3-point functions of a pair of 's and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density .     

Symmetries induced by conserved vector currents in the theory of a scalar field

Let us consider a quantum theory of one scalar, real, local, Poincaré covariant fieldA(x) with the restricted spectrum condition (massive one particle states and a unique vacuum). The asymptotic fieldsA _{in out} (x) are assumed to be irreducible. Our conjecture is that under some technical assumptions the charge of every real, hermitean, locally conserved, Poincaré covariant quantum (pseudo) vector fieldj (x) relatively local toA(x), appearing in this theory-vanishes. This means that in a theory of one scalar, real field with a massive particle one can not expect to get symmetry groups induced by conserved (pseudo) vector currents, only by global, selfadjoint, Poincaré invariant generators.Our arguments can be easily extended to a theory of one complex scalar field, in this case the only symmetry transformation induced by a current can be the gauge transformation.We prove also that under very weak assumptions two fields related to each other by a unitary (or similarity) transformation are equal barring some patological cases.     

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.     

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.     

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.     

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.     

The conserved currents for the Maxwellian field

We classify the conserved currents for the Maxwellian field. There are four families: (1) the classical currents derived using Noether's theorem from conformal invariance (2) certain Noetherian currents based on translations in field space, (3,4) two more kinds not equivalent to any Noetherian form.     

Analytical properties of conformal invariant four point functions

On the basis of the known group theoretical structure of the conformal invariant four point functions in the case of identical scalar fields ?(x) of scale dimension d,

$\text{<0|?(x}{}_1\text{)?(x}{}_2\text{)?(x}{}_3\text{)?(x}{}_4\text{)|0〉 = [(x}{}_1\text{?x}{}_4\text{)}{}^2\text{(x}{}_2\text{?x}{}_3\text{)}{}^2\text{]}{}^{\mathrm{-d}}\text{g(A,B),}$

the analytical properties of g(A, B) as a function of the harmonic ratios A and B are investigated. By imposing the conditions of spectrality and locality, and using invariance under complex dilatations, it is shown that the function g(A, B) must be homomorphic in the whole complex A-plane and B-plane with exception of the values A=0 and A=∞, B arbitrary, and B=0 and B=∞, A arbitrary.