Self-dual tensors and gravitational anomalies in 4n + 2 dimensions

Starting from a manifestly Lorentz- and diffeomorphism-invariant classical action we perform a perturbative derivation of the gravitational anomalies for chiral bosons in 4n + 2 dimensions. The manifest classical invariance is achieved using a newly developed method based on a scalar auxiliary field and two new bosonic local symmetries. The resulting anomalies coincide with the ones predicted by the index theorem. In the two-dimensional case, moreover, we perform an exact covariant computation of the effective action for a chiral boson (a scalar) which is seen to coincide with the effective action for a two-dimensional complex Weyl fermion. All these results support the quantum reliability of the new, at the classical level manifestly invariant, method.     

Drinfeld-Sokolov Construction of Heterotic Toda Model

A free field representation for the left-right asymmetric conformal Toda theory based on simplf-laced even-rank Lie algebras is given. It is shown that the classical chiral exchange algebra for such theories can be reconstructed from free chiral bosons via Drinfeld-Sokolov linear systems, and is a bit more complicated than that of the standard Toda due to some additional δ-function terms and extra degrees of freedom.     

On new conformal field theories with affine fusion rules

Some time ago, conformal data with affine fusion rules were found. Our purpose here is to realize some of these conformal data, using systems of free bosons and parafermions. The so constructed theories have extended W algebras which are close analogues of affine algebras. Exact character formulae are given, and the realizations are shown to be full-fledged unitary conformal field theories.     

Gravitational Frames and Scalar Field Dynamics

Scalar fields describe interesting phenomena such as Higgs bosons, dark matter and dark energy, and are found to be quite common in physical theories. These fields are susceptible to gravitational forces so that being massless is not enough to remain conformal invariant. They should also be connected directly to the scalar curvature. Because of this characteristics, we investigated the structure and interactions of scalar fields under the conformal transformations. We show how to reduce the quadratic quantum contributions in the single scalar field theory. In the multi-scalar field theories, we analyzed interactions in certain limits. We suggest a new method for stabilizing Higgs bosons.

     

Introduction to Conformal Field Theory

An elementary introduction to conformal field theory is given. Topics include free bosons and fermions, orbifolds, affine Lie algebras, coset conformal field theories, superconformal theories, correlation functions on the sphere, partition functions and modular invariance.     

The operator algebra of orbifold models

We analyze the chiral properties of (orbifold) conformal field theories which are obtained from a given conformal field theory by modding out by a finite symmetry group. For a class of orbifolds, we derive the fusion rules by studying the modular transformation properties of the one-loop characters. The results are illustrated with explicit calculations of toroidal andc=1 models.     

On the algebraic structure ofN=2 string theory

InN=2 string theory the chiral algebra expresses the generations and anti-generations of the theory and the Yukawa couplings among them and is thus crucial to the phenomenological properties of the theory. Also the connection with complex geometry is largely through the algebras. These algebras are systematically investigated in this paper. A solution for the algebras is found in the context of rational conformal field theory based on Lie algebras. A statistical mechanics interpretation for the chiral algebra is given for a large family of theories and is used to derive a rich structure of equivalences among the theories (dihedralities). The Poincaré polynomials are shown to obey a resolution series which cast these in a form which is a sum of complete intersection Poincaré polynomials. It is suggested that the resolution series is the proper tool for studying allN=2 string theories and, in particular, exposing their geometrical nature.     

On classification ofN=2 supersymmetric theories

We find a relation between the spectrum of solitons of massiveN=2 quantum field theories ind=2 and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricN=2 conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A-D-E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory.     

Realizations of pseudo bosonic theories with non-diagonal automorphisms

Pseudo conformal field theories are theories with the same fusion rules, but different modular matrix as some conventional field theory. One of the authors defined these and conjectured that, for bosonic systems, they can all be realized by some actual RCFT, which is that of free bosons. We complete the proof here by treating the non-diagonal automorphism case. It is shown that for characteristic p≠2, they are equivalent to a diagonal case, fully classified in our previous publication. For p=2ⁿ we realize the non-diagonal cases, establishing this theorem.     

Nonabelions in the fractional quantum hall effect

Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.     

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.     

Purely transmitting defect field theories

We define an infinite class of integrable theories with a defect which are formulated as chiral defect perturbations of a conformal field theory. Such theories are massless in the bulk and are purely transmitting through the defect. The integrability of these theories requires the introduction of defect degrees of freedom. Such degrees of freedom lead to a novel set of Yang-Baxter equations. The defect degrees of freedom are identified through folding the chiral defect theories onto massless boundary field theories. The examples of the sine-Gordon theory and Ising model are worked out in some detail.     

On modular invariant partition functions for tensor products of conformal field theories

We give two results concerning the construction of modular invariant partition functions for conformal field theories constructed by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type SU(2)_KASU(2)_KB, finding all consistent theories for K_A, K_B odd. Second we show how known diagonal modular invariants can be used to construct some inherently asymmetric ones where the holomorphic and antiholomorphic theories do not share the same chiral algebra. Some explicit examples are given.     

parafermionic current algebras of torus/orbifold models

We study the quantum field theory of bosons on the torus and the orbifold. When the torus is in special moduli, the representations of the theory are equivalent to those of some rational conformal field theories. We show that there are parafermonic current algebras in Z_N orbifold models.     

On representations of conformal field theories and the construction of orbifolds

We consider representations of meromorphic bosonic chiral conformal field theories and demonstrate that such a representation is completely specified by a state within the theory. The necessary and sufficient conditions upon this state are derived and, because of their form, we show that we may extend the representation to a representation of a suitable larger conformal field theory. In particular, we apply this procedure to the (untwisted) lattice conformal field theories (i.e. corresponding to the propagation of a bosonic string on a torus), and deduce that Dong's proof of the uniqueness of the twisted representation for the reflection-twisted projection of the Leech lattice conformal field theory generalises to an arbitrary even (self-dual) lattice. As a consequence, we see that the reflection-twisted lattice theories of Dolan, Goddard and Montague are truly self-dual, extending the analogies with the theories of lattices and codes which were being pursued. Some comments are also made on the general concept of the definition of an orbifold of a conformal field theory in relation to this point of view.     

Asymptotic Completeness for Infraparticles in Two-Dimensional Conformal Field Theory

We formulate a new concept of asymptotic completeness for two-dimensional massless quantum field theories in the spirit of the theory of particle weights. We show that this concept is more general than the standard particle interpretation based on Buchholz’ scattering theory of waves. In particular, it holds in any chiral conformal field theory in an irreducible product representation and in any completely rational conformal field theory. This class contains theories of infraparticles to which the scattering theory of waves does not apply.     

Derivation of Effective Chiral Lagrangian Involving PSGB and Vector Bosons from QCD

The effective chiral Lagrangian for a matter field content consisting of pseudo-scalar Goldstone bosons and vector bosons (with hidden symmetry) is derived from the underlying QCD theory. No approximations are made. All the free parameters of the effective chiral Lagrangian are expressed in terms of QCD-based Green's functions. These may be regarded as the QCD definitions of these Lagrangian coefficients.     

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.     

Characterizing invariants for local extensions of current algebras

ParisA of local quantum field theories are studied, whereA is a chiral conformal quantum field theory and is a local extension, either chiral or two-dimensional. The local correlation functions of fields from have an expansion with respect toA into conformal blocks, which are non-local in general. Two methods of computing characteristic invariant ratios of structure constants in these expansions are compared: (a) by constructing the monodromy representation of the braid group in the space of solutions of the Knizhnik-Zamolodchikov differential equation, and (b) by an analysis of the local subfactors associated with the extension with methods from operator algebra (Jones theory) and algebraic quantum field theory. Both approaches apply also to the reverse problem: the characterization and (in principle) classification of local extensions of a given theory.     

Fusion rings and geometry

The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising inN=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the typeSU(N) _k /SU(n–1) are theN=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism.Supported in part by NSF grant PHY 89-04035 supplemented by funds from NASA