Conformal field theories and critical phenomena

In this article we present a brief review of the conformal symmetry and the two-dimensional conformal quantum field theories. As concrete applications of the conformal theories to the critical phenomena in statistical systems, we calculate the value of central charge and the anomalous scale dimensions of the Z ₂ symmetric quantum chain with boundary condition. The results are compatible with the prediction of the conformal field theories.     

Observations on the moduli space of superconformal field theories

Some aspects of the moduli space of superconformal field theories are discussed. It is helpful to consider the conformal field theory as a background for propagation of strings and to exploit the space-time interpretation. Using this point of view we show that the metric on the moduli space of N = 4 superconformal field theory with c = 6 is locally that of O(20,4)/O(20) × O(4). We also discover some properties of the moduli space of N = 2 superconformal field theories with c = 9. Particular examples of these conformal field theories are sigma models on four- and six-dimensional Calabi-Yau spaces. Therefore, we can use this technique to learn about the moduli space of these spaces. For c = 6 we recover the known moduli space of K₃. Our analysis of the c = 9 system leads to a new coupling in four dimensional supergravity. As a by-product, we prove that gauge couplings cannot depend on the moduli of N = 1 space-time supersymmetric compactifications.     

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.     

Exact g-function flow between conformal field theories

Exact equations are proposed to describe g-function flows in integrable boundary quantum field theories which interpolate between different conformal field theories in their ultraviolet and infrared limits, extending previous work where purely massive flows were treated. The approach is illustrated with flows between the tricritical and critical Ising models, but the method is not restricted to these cases and should be of use in unravelling general patterns of integrable boundary flows between pairs of conformal field theories.     

Chiral deformations of conformal field theories

We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W_1+∞ algebra, that is treated in detail.     

On bosonization of 2d conformal field theories

We show how bosonic (free field) representations for so-called degenerate conformal theories are built by singular vectors in Verma modules. Based on this construction, general expressions of conformal blocks are proposed. As an example, we describe new modules for the SL(2) Wess-Zumino-Witten model. They are, in fact, the simplest nontrivial modules in a full set of bosonized highest weight representations of the ŝl₂ algebra. The Verma and Wakimoto modules appear as boundary modules of this set. Our construction also yields a new kind of bosonization in 2d conformal field theories.     

Toward classification of conformal theories

By studying the representations of the mapping class groups which arise in 2D conformal theories we derive some restrictions on the value of the conformal dimension h_i of operators and the central charge c of the Virasoro algebra. As a simple application we show that when there are a finite number of operators in the conformal algebra, the h_i and c are all rational.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Conformal anomaly in 4D gravity-matter theories non-minimally coupled with a dilaton

The conformal anomaly for 4D gravity-matter theories, which are non-minimally coupled with the dilaton, is systematically studied. Special care is taken of rescaling of fields, treatment of total derivatives, hermiticity of the system operator and the choice of measure. Scalar, spinor and vector fields are taken as the matter quantum fields and their explicit conformal anomalies in the gravity-dilaton background are found. The cohomology analysis is carried out and some new conformal invariants and trivial terms, involving the dilaton, are obtained. The symmetry of the constant shift of the dilaton field plays an important role. The general structure of the conformal anomaly is examined. It is shown that the dilaton affects the conformal anomaly characteristically for each case: (1) [Scalar] The dilaton changes the conformal anomaly only by a new conformal invariant, I₄; (2) [Spinor] The dilaton does not change the conformal anomaly; (3) [Vector] The dilaton changes the conformal anomaly by three new (generalized) conformal invariants, I₄, I₂, I₁. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between earlier calculations and the result of Hawking and Bousso.     

Renormalization constants in asymptotically free field theories

Renormalization constants Z_i for asymptotically free field theories can be computed via renormalization group techniques from perturbation theory. We show that there exists a subclass of these theories in which, by virtue of a new eigenvalue condition on the gauge parameter, the Z_i are asymptotically gauge independent, and hence can vanish in all gauges.     

Thermal transport in chiral conformal theories and hierarchical quantum Hall states

Chiral conformal field theories are characterized by a ground-state current at finite temperature, that could be observed, e.g., in the edge excitations of the quantum Hall effect. We show that the corresponding thermal conductance is directly proportional to the gravitational anomaly of the conformal theory, upon extending the well-known relation between specific heat and conformal anomaly. The thermal current could signal the elusive neutral edge modes that are expected in the hierarchical Hall states. We then compute the thermal conductance for the Abelian multi-component theory and the W_1+∞ minimal model, two conformal theories that are good candidates for describing the hierarchical states. Their conductances agree to leading order but differ in the first, universal finite-size correction, that could be used as a selective experimental signature.     

Free fermion and parafermion theories in two dimensions

We show that the spectrum and the observables of a free massless O(N) parafermion theory of order Q in two dimensions is related to, but not identical with, a free fermion theory with O(N)×O(Q) symmetry. The Virasoro and the O(N) Kac-Moody algebras of these theories are given.     

Free energy and correlation lengths of quantum chains related to restricted solid-on-solid lattice models

An approach is presented for calculating the free energy as well as the correlation lengths of integrable quantum chains at arbitrary finite temperatures. The method is applied to critical Hamiltonians related to restricted solid-on-olid models comprising the hierarchy by Andrews, Baxter and Forrester, and generalizations hereof by the fusion procedure. The derived non-linear integral equations can be studied analytically in the low-temperature and high-temperature limits. The central charges and all primary conformal weights are obtained for the generalized minimal unitary series of conformal field theory and the Z_N parafermion theories. Thus an extension of the thermodynamic Bethe Ansatz is realized which recently has been speculated on in the literature.     

Dimensional regularization of massless theories in spherical space-time

The use of space-time curvature as an infra-red cut-off has been suggested for massless theories. In this paper we investigate the renormalization of massless theories in a spherical space-time (Euclidean version of de Sitter space) using dimensional regularization. Naive expectations are confirmed, namely that the coupling constant and wave-function renormalizations are independent of the curvature. Furthermore the curvature does not induce divergent mass terms or vacuum field values as would be possible on purely dimensional grounds. Although we have investigated only scalar field theories, φ⁴ theory in four dimensions and φ³ theory in six, these results are encouraging for an application of the method to gauge theories.Formally massless theories are conformally invariant so the formulation of the theory in a spherical space ought to be equivalent to its formulation in flat space. In fact the renormalization procedure breaks conformal invariance and removes this equivalence. We show that to achieve the flat space limit it is necessary to invoke the aid of the renormalization group. Thus the zero curvature limit can be achieved for infra-red stable theories (φ₄⁴) but not for infra-red unstable theories (φ₆³ as might be expected.     

Modular invariance and the spectrum of two dimensional conformal field theories

Modular invariance has recently emerged as a powerful tool in conformal field theory. In conjunction with the representation theory of infinite dimensional Lie algebras, the study of modular invariance gave the spectrum of several families of theories. These include the minimal conformal models (Cardy and others), WZW theories which describe string propagation on group manifolds (Gepner and Witten) and parafermionic field theories (Gepner and Qiu). The minimal conformal models models were shown to be a product of two SU(2) WZW theories (Gepner). These results represent a step towards a complete classification of conformal field theories, an important goal both for the study of critical phenomena and string theory.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

Unified field theories with U(N) internal symmetries: G

Gauge theories for extended SU(N) conformal supergravity are constructed which are invariant under local scale, chiral, proper conformal, supersymmetry and internal SU(N) transformations. The relation between intrinsic parity and symmetry properties of their generators of the internal vector mesons is established. These theories contain no cosmological constants, but technical problems inherent to higher derivative actions are pointed out.     

Unified approach to Thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories

We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero-temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a single nonlinear integral equation (NLIE) (a system of NLIE appears for nested BA). We derive the NLIE for: (a) the six-vertex model with twisted boundary conditions, (b) the XXZ chain in an external magnetic field h_z and (c) the massive Thirring sine-Gordon model (mT-sG) in a periodic box of size β ≡ T^-1 using the light-cone approach. This NLIE is solved by iteration in one regime (high T in the XXZ chain and low T in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present.     

The sigma model (dual) representation for a two-parameter family of integrable quantum field theories

We introduce and study two-parameter families of integrable field theories. The perturbative and non-perturbative methods are used to justify their factorized scattering theory in the form of the direct products of two S-matrices of the sine-Gordon model. The Bethe ansatz technique is applied for the calculation of the observables in the strong coupling region. The results are in the exact agreement with ones following from the sigma model action which is a two-parameter U(1) ⊗ (1) symmetrical deformation of the O(4) non-finear sigma model. The application of the sigma model representation to related perturbed conformal field theories is discussed.     

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.