Conformal invariance of σ-models and classical string physics

We show that the identification of the conformal anomaly of the general bosonic two-dimensional non-linear σ-model as the generating functional for on-shell string scattering amplitudes is correct up toO(α′) terms. The absence, in the loop corrections to the spacetime effective action, of contributions from the explicit coupling to the dilaton field is suggested as a general feature for σ-models describing tree-level string physics.     

Conformal supergravity in ten dimensions

We present the complete off-shell structure of conformal supergravity in ten dimensions. It is based on 128 + 128 degrees of freedom and its formulation requires differential constraints. We study how these constraints are resolved in four and five dimensions. Covariant conditions are given that restrict conformal supergravity to its on-shell Poincaré counterpart. In ten dimensions the relationship between the two theories has new and unusual aspects, which we explore in a variety of ways. We rewrite on-shell Poincaré supergravity in a superconformally invariant form, from which we deduce that its off-shell version must contain at least a scalar (chiral) multiplet. We analyze some aspects of the non-linear structure of the field representation based on the conformal fields combined with one scalar multiplet.     

Tractors,mass, and Weyl invariance

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus—a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. Breitenlohner–Freedman stability bounds for Anti-de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories—which rely on the interplay between mass and gauge invariance—are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s?2$s?2$ are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s?2$s?2$ we give tractor equations of motion unifying massive, massless, and partially massless theories.     

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.     

Subquark pregeometry with spontaneously broken conformal invariance

We present a model of subquark pregeometry in which the Higgs scalar as well as the space-time metric appears as a composite of subquark-antisubquark pairs. The effective langrangian becomes the einstein one with no cosmological term when conformal invariance is broken spontaneously.     

Equations of motion for N = 1 massless super-Poincaré representations in D = 3,4 space-time

We derive the complete set of equations of motion for an arbitrary N = 1 massless super-Poincaré representation in D = 3 and D = 4 space-time. These equations are derived from the constraints imposed by the on-shell conformal invariance of a massless super-Poincaré representation.     

Constraints for anomalous dimensions of local light-cone operators in [ϕ 3]6 theory

Using the MS scheme, we derive in [? ³]₆ theory the collinear conformal Ward identity for the Green's functions of local light-cone operators of leading twist. The Ward identity for special collinear conformal transformations and renormalization group invariance give constraints for the off-diagonal part of the anomalous dimension matrix for the general case of β#0. We compute the anomaly of special conformal transformation in lowest loop order and obtain from the constraints the off-diagonal part of the anomalous dimension in 2-loop order.     

Scalar tilt from broken conformal invariance

Within recently proposed scenario which explains flatness of the spectrum of scalar cosmological perturbations by a combination of conformal and global symmetries, we discuss the effect of weak breaking of conformal invariance. We find that the scalar power spectrum obtains a small tilt which depends on both the strength of conformal symmetry breaking and the law of evolution of the scale factor.     

Killing spinors for the bosonic string and Kaluza–Klein theory with scalar potentials

The paper consists mainly of two parts. In the first part, we obtain well-defined Killing spinor equations for the low-energy effective action of the bosonic string with the conformal anomaly term. We show that the conformal anomaly term is the only scalar potential that one can add into the action that is consistent with the Killing spinor equations. In the second part, we demonstrate that Kaluza–Klein theory can be gauged so that the Killing spinors are charged under the Kaluza–Klein vector. This gauging process generates a scalar potential with a maximum that gives rise to an AdS spacetime. We also construct solutions of these theories.     

Towards conformal cosmology

Approximate de Sitter symmetry of inflating Universe is responsible for the approximate flatness of the power spectrum of scalar perturbations. However, this is not the only option. Another symmetry that can explain nearly scale-invariant power spectrum is conformal invariance. We give a short review of models based on conformal symmetry that lead to the scale-invariant spectrum of the scalar perturbations. We discuss also potentially observable features of these models.     

Gauge fixing problem in the conformal QED

The gauge fixing problem in the conformal (spinor and scalar) QED is examined. For the analysis, we generalize Dirac's manifestly conformal-covariant formalism. It is shown that the (vector and matter) fields must obey a certain mixed (conformal and gauge) type of transformation law in order to fix the local gauge symmetry preserving the conformal invariance in the Lagrangian.     

Construction of exponential Liouville field operators for closed string models

Operators for arbitrary exponentials exp(λφ) of a periodic Liouville field φ(τ,σ) are represented iteratively by an infinite power series in terms of a periodic scalar free field. Necessary quantum corrections of the Liouville operators with respect to their classical expressions are fixed by conformal covariance and locality. Canonical commutation relations for the Liouville field quantities are valid when the canonical quantization of the scalar free field is imposed. A quantum correction of the energy momentum tensor can be avoided thus preserving the conformal invariance of the Liouville theory.     

Renormalization of Lorentz non-invariant actions and manifest T-duality

We study general two-dimensional σ-models which do not possess manifest Lorentz invariance. We show how demanding that Lorentz invariance is recovered as an emergent on-shell symmetry constrains these σ-models. The resulting actions have an underlying group-theoretic structure and resemble Poisson–Lie T-duality invariant actions. We consider the one-loop renormalization of these models and show that the quantum Lorentz anomaly is absent. We calculate the running of the couplings in general and show, with certain non-trivial examples, that this agrees with that of the T-dual models obtained classically from the duality invariant action. Hence, in these cases solving constraints before and after quantization are commuting operations.     

General relativity as a conformally invariant scalar gauge field theory

The global symmetry implied by the fact that one can multiply all masses with a common constant is made into a local, gauge symmetry. The matter action then becomes Conformally invariant and it seems natural to choose for the corresponding scalar gauge field the action for a conformally invariant (massless) scalar field. The resulting conformally invariant theory turns out to be equivalent to general relativity. Since this means that the usual Einstein-Hilbert action is not, in fact, a true gauge action for the space-time geometry, the full theory ought to be supplied with such a term. Gauge-theoretic arguments and conformal invariance requirements dictate its form.     

Exact solutions of Einstein-conformal scalar equations

The massless scalar field which satisfies a conformally invariant equation is in some respects more interesting than the ordinary one. Unfortunately, few, if any, exact solutions of Einstein's equations for a conformal scalar stress-energy have appeared previously. Here we present a theorem by means of which one can generate two Einstein-conformal scalar solutions from a single Einstein-ordinary scalar solution (of which many are known). As an example we show how to obtain Weyl-like solutions with a conformal scalar field. We obtain and analyze in some detail two families of spherically symmetric static Einstein-conformal scalar solutions. We also exhibit a family of static spherically symmetric Einstein-Maxwell-conformal scalar solutions (parametrized by both electric and scalar charge), which have black-hole geometries but are not genuine black holes. Finally, we present all the Robertson-Walker cosmological models which contain both incoherent radiation and a homogeneous conformal scalar field. One class of these represents open universes which bounce and never pass through a singular state; they circumvent the “singularity theorems” by violating the energy condition.     

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.     

Conformally coupled scalar field solutions and the cosmological constant

Solutions are presented for a scalar field coupled conformally to Einstein gravity with a nonvanishing cosmological constant, in the case that the spacetime metric is spatially homogeneous and isotropic. Since the cosmological constant destroys the conformal invariance of the action, these solutions cannot be obtained by solving the flat space wave equation for the scalar field. It turns out that the metric is determined entirely by the cosmological constant, while the scalar field acquires an apparent mass squared which is proportional to the cosmological constant. It is conjectured that the cosmological constant in the universe at present may thus be disguised as the mass of some scalar field.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Equivalence of the Hawking temperature in conformal frames

The conformal invariance of the Hawking temperature, conjectured for the asymptotically flat and stationary black holes by Jacobson and Kang, is semiclassically evaluated for a simple particular case of symmetrical spherically and non-asymptotically flat black hole. By using the Bogoliubov coefficients, the metric euclideanization, the reflection coefficient and the gravitational anomaly, as methods of calculating the Hawking temperature, we find that it is invariant under a specific conformal transformation of the metric. We briefly discuss the results for each method.