Concomitants theory and renormalization in even dimensions

International Journal of Modern Physics D

Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under these transformations the following applications are given respectively: (1) Natural time gauges for multidimensional geometry, (2) conformally equivalent Lagrangian models for geometry coupled to a spatially homogeneous scalar field, and (3) the conformal Laplace operator on the n-dimensional manifold ℳ of minisuperspace for multidimensional geometry and the Wheeler-de Witt equation. The conformal coupling constant ξc is critically distinguished among arbitrary couplings ξ, for both, the equivalence of Lagrangian models with D-dimensional geometry and the conformal geometry on n-dimensional minisuperspace. For dimension D=3, 4, 6 or 10...

Classical and Quantum Gravity, 1990

We derive field equations for sixth-order gravity with the Lagrangian containing the terms R D R and R 2 and show its conformal equivalence to the Einstein theory with two interacting scalar fields. Their masses are calculated in the' weak-field limit. This equivalence is generalised to higher order gravity in the linearised case. Inflationary regimes are investigated.

Physics Letters B, 1997

The second alternative conformal limit of the recently proposed general higher derivative dilaton quantum theory in curved spacetime is explored. In this version of the theory the dilaton is transformed, along with the metric, to provide the conformal invariance of the classical action. We find the corresponding quantum theory to be renormalizable at one loop, and the renormalization constants for the dimensionless parameters are explicitly shown to be universal for an arbitrary parametrization of the quantum field. The renormalization group equations indicate an asymptotic freedom in the IR limit. In this respect the theory is similar to the well-known model based on the anomaly-induced effective action.

Annals of Physics, 1980

The regularization and renormalization of an interacting scalar field + in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, "vacuum" counter terms must also be introduced. These involve zero, first, and second powers of the Riemann curvature tensor R,B*. Moreover, the renormalizability of the theory requires that the Lagrange function couple @ to the curvature scalar R with a coupling constant 7. The coupling 7 must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n-4. The coefficient functions of the counter terms determine the construction of 4" and 4" in terms of renormalized composite operators 1, [+2], and [I$"]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [+"I and [V] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R2 interaction which is of fifth order in the coupling constant h.

Classical and Quantum Gravity, 1995

The most general version of a renormalizable d = 4 theory corresponding to a dimensionless higher-derivative scalar field model in curved spacetime is explored. The classical action of the theory contains 12 independent functions, which are the generalized coupling constants of the theory. We calculate the one-loop beta functions and then consider the conditions for finiteness. The set of exact solutions of power type is proven to consist of precisely three conformal and three nonconformal solutions, given by remarkably simple (albeit nontrivial) functions that we obtain explicitly. The finiteness of the conformal theory indicates the absence of a conformal anomaly in the finite sector. The stability of the finite solutions is investigated and the possibility of renormalization group flows is discussed as well as several physical applications.

2008

This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.

Trends in Mathematics, 2014

Presented in honour of Daniel Sternheimer, on the occasion of his 75th birthday.

Nuclear Physics B, 1994

We study quantum gravity in 2 + dimensions in such a way to preserve the volume preserving diffeomorphism invariance. In such a formulation, we prove the following trinity: the general covariance, the conformal invariance and the renormalization group flow to Einstein theory at long distance. We emphasize that the consistent and macroscopic universes like our own can only exist for matter central charge 0 < c < 25. We show that the spacetime singularity at the big bang is resolved by the renormalization effect and universes are found to bounce back from the big crunch. Our formulation may be viewed as a Ginzburg-Landau theory which can describe both the broken and the unbroken phase of quantum gravity and the phase transition between them.

Journal of High Energy Physics

We extend Maldacena’s argument, namely, obtaining Einstein gravity from Conformal Gravity, to six dimensional manifolds. The proof relies on a particular combination of conformal (and topological) invariants, which makes manifest the fact that 6D Conformal Gravity admits an Einstein sector. Then, by taking generalized Neumann boundary conditions, the Conformal Gravity action reduces to the renormalized Einstein-AdS action. These restrictions are implied by the vanishing of the traceless Ricci tensor, which is the defining property of any Einstein spacetime. The equivalence between Conformal and Einstein gravity renders trivial the Einstein solutions of 6D Critical Gravity at the bicritical point.

arXiv (Cornell University), 2022

We study a conformally coupled scalar-tensor theory with a quartic potential possessing local conformal symmetry up to a boundary term. We show that requiring the restoration of the full local conformal symmetry fixes the counterterms that render the on-shell action finite. The building block of the resulting action is a conformally covariant tensor which is constructed out of the metric and the scalar field and it has the same conformal weight as the Weyl tensor. This allows us to obtain the counterterms for the scalar-tensor sector in a closed form. The finiteness of the conformally complete version of the action is suggestive on the validity of the Conformal Renormalization prescription. We extend this theory by adding the Conformal Gravity action and also the Einstein-AdS action written in McDowell-Mansouri form. Even though the latter breaks the conformal symmetry, we find that the action is still renormalized provided a suitable falloff of the scalar field when considering asymptotically locally anti-de Sitter solutions. Black hole solutions in these theories are studied, for which the Hawking temperature and the partition function to first order in the saddle-point approximation are calculated, providing a concrete example of this renormalization scheme.