Gravitational Effective Action Research Articles

Ambiguity in Determining the Effective Action for String-Corrected Einstein Gravity

We attempt to clarify some issues dealing with the determination of some of the coefficients in the low-energy gravitational effective action for closed strings. An explicit calculation is generalized demonstrating that the coefficients of the R2µν and R2 terms of completely general quadratic curvature corrections to the Einstein-Hillbert action cannot be determined by matching the on-shell S-matrix of the effective theory with those of the string theories. Although ghost poles will appear in the graviton propagator for general quadratic curvature corrections, the effective theory remains unitary, the pole dependence cancelling completely from the scattering amplitude. The manifest absence of ghosts is achieved by choosing the Gauss-Bonnet combination of quadratic curvatures, but the requirement of unitarity does not uniquely specify this combination.

Superstring effective actions and the central charge of the virasoro algebra on a Kähler manifold

We study higher-order corrections to the gravitational string effective action within a supersymmetric σ-model approach. We show that on a Kähler manifold the corrections to the Einstein-Hilbert action vanish to all orders in α'. Moreover on such spaces the central charge of the Virasoro algebra is not pertubatively renormalized, once the gravitational equations of motion are satisfied.

Curvature cubed terms in string theory effective actions

Starting from the three- and four-point amplitudes we determine the R 3-terms in the gravitational effective actions in the Bose, heterotic and type II superstring theories. We prove the absence of unambiguous R 3-terms (including the Gauss-Bonnet R 3-invariant) in the heterotic and superstring cases. As a by-product, we check the presence of the square of the Lorentz Chern-Simons term in the heterotic string effective action and obtain the three-loop contribution to the β-function of the bosonic σ-model. We also discuss the structure of the field redefinition ambiguity in a gravitational effective action.

Gravity from Strings

We obtain the Einstein action plus quadratic curvature corrections generated by closed bosonic, heterotic and supersymmetric strings by matching the four-graviton amplitude (to first order in the slope parameter and fourth power of momenta) with an effective local gravitational action. The resulting corrections are first shown to be of the Gauss-Bonnet form. It is then noted that, by the very nature of the slope expansion, the field-redefinition theorem applies. Consequently, only the curvature-squared term is determined, while squares of its contractions are explicitly seen not to contribute. This latter property has a generalization to all orders which implies that the effective gravitational action is unavoidably ghost-free.The properties of solutions to these corrected theories are then examined. First neglecting dilatons, we find the explicit "Schwarzschild" metrics. Both asymptotically flat and de Sitter solutions are present. The latter are however shown to be unstable. The former have horizons and singularities which are respectively smaller and less violent than in Einstein gravity; the correct sign of the slope parameter also ensures absence of naked singularities. When dilatons are included, the cosmological vacua are gratifyingly excluded.

Classical stability for spontaneous compactification in higher derivative gravity

Quantum theories of gravity (e.g. string theories) lead to higher derivative terms in the effective gravitational action. We present a general discussion of classical stability for compactification solutions of higher derivative gravity (also applicable for other bosonic fields and in four dimensions). In general there is no need for higher derivative terms to appear as generalized Euler forms. As a specific example we discuss solutions M× S D for the most general four-derivative approximation of pure higher dimensional gravity. For a certain range of parameters, instabilities for low momentum fluctuations may be absent. For the allowed parameter range, however, higher poles of propagators appear at momenta only slightly larger than the compactification scale, indicating insufficiency of the four-derivative approximation at this scale.

Vector field effective action in the open superstring theory

Using a path integral technique we find a closed expression (to all orders in α′) for the abelian, constant field strenght limit of the (tree) effective action for the massless vector field in the open superstring theory. The result is a modification of the Born-Infeld action found in the Bose string theory case. One-loop correction to the effective action is computed and shown to be finite if the gauge group is SO(32). It is demonstrated how the on-shell superstring scattering amplitudes can be calculated in the path-integral approach. We determine the leading (O( α') and O( α′ 2)) terms in the full non-abelian effective action starting from the known results for the 3-point and 4-point amplitudes. We find that because of the equivalence theorem the coefficients of some of the invariant structures in the effective lagrangian cannot be fixed from the S-matrix. In the path integral approach this ambiguity manifests itself as a 2d renormalization scheme (and Weyl gauge choice) ambiguity. We also discuss the leading terms in the gravitational effective actions in the closed (super) string theories and point out that whether or not the R 2 terms form the “Gauss-Bonnet” combination depends on choice of a renormalization (massless exchange subtraction) scheme.

Ambiguity in the effective action in string theories

We point out that there is an ambiguity in some of the coefficients in the “low-energy” effective action (EA) for the massless string modes (in particular, the coefficients of the R μν 2- and R 2-terms in the gravitational EA's are ambiguous). In the S-matrix approach this ambiguity is a consequence of an off-shell ambiguity in the subtraction of the massless exchanges (manifesting itself as a freedom of field redefinitions which leave the S-matrix invariant because of the equivalence theorem). In the world-sheet path integral approach the ambiguity is due to the freedom in the choice of a 2D renormalization scheme and a Weyl gauge. We also comment on the relation with the second-quantized string approach.

Massless fermions and (2+1)-dimensional gravitational effective action.

The three-dimensional gravitational effective action due to a massless fermion coupled to a weak gravitational field is calculated. Radiative corrections induce the parity-violating gravitational Chern-Simons term.

Parity violation and the effective gravitational action in three dimensions

A diagrammatic computation of the parity-breaking gravitational effective action induced by massless fermions in three dimensions is presented. The lowest-order approximation to the Chern-Simons action is obtained, in agreement with a general index-theoretic result.

Beta-functions and superstring compactifications

It is shown that quartic curvature modifications to the gravitational effective action derived from superstrings agree on Ricci-flat Kähler backgrounds with a recently-proposed four-loop counterterm in N = 2 Ricci-flat σ-models. These results seem to suggest that Calabi-Yau manifolds may not provide consistent compactifications of the superstring.

Solutions to the effective field equations for a proposed gravitational effective action

A proposal by DeWitt for the full effective action for pure quantum gravity is discussed with the metric restricted to Robertson-Walker form, and with gravity coupled to a classical perfect fluid of massless particles. A class of nonsingular solutions to the effective field equations are obtained and discussed. These are simple generalizations of the classical solution. Some restrictions are established on any other solutions which may exist.

On the consistency of self-consistent dimensional reduction

Several aspects of self-consistent dimensional reduction in Kaluza–Klein theory are addressed. First, the validity of the one-loop approximation in quantum gravity with a cosmological constant is discussed. Second, a distinction is made between mathematical self-consistency and physical self-consistency. Finally, the possible gauge dependence of the physical predictions for the radius and gauge coupling constant is analyzed within the context of recent theorems concerning the gauge invariance of the one-loop gravitational effective action.

Superstring modifications of Einstein's equations

The modifications of the classical equations of motion of the gravitational field in type II string theory are derived by studying tree-level gravitational scattering amplitudes. The effective gravitational action is determined through quartic order in the Riemann tensor. It is shown that generic Ricci-flat manifolds do not solve the modified equations, unless in addition the manifolds are Kähler (2 N-dimensional manifolds of SU( N) holonomy). Translated into sigma model language, this calculation would indicate that the N = 1 supersymmetric sigma model in 2 dimensions, with a Ricci-flat target space, is not conformally invariant, but has a nonzero beta function at four-loop order.

Quantized antisymmetric tensor fields and self-consistent dimensional reduction in higher-dimensional spacetimes

The problem of quantizing an antisymmetric tensor field on a higher-dimensional gravitational background of the form M 4 × S N (where M 4 is an arbitrary 4-dimensional spacetime) is examined. Terms appearing in the effective gravitational action are found for such a theory. This effective gravitational action is then used to find the necessary conditions for the existence of solutions to the higher-dimensional dield equations of the form R 4 × S N . The cases where the background antisymmetric tensor field vanishes or else is of the Freund-Rubin form are both considered. It is mentioned how this analysis is modified when gauge fields are included in a Kaluza-Klein metric on the higher-dimensional space. Two appendices contain details of our conventions concerning differential forms, and results for the eigenvalues (and their multiplicities) of the Hodge de Rham operator acting on exact and coexact forms on a sphere. In a third appendix it is shown how the vacuum energy for photons in the Einstein static universe may be obtained using the methods of this paper.

Gauge dependence and the gravitational effective action

The effective action in a gauge theory is in general gauge dependent. In this paper sufficient conditions are derived under which the effective action is the same in a wide class of gauges at the one-loop order.

Einstein gravity as a symmetry-breaking effect in quantum field theory

This article gives a pedagogical review of recent work in which the Einstein-Hilbert gravitational action is obtained as a symmetry-breaking effect in quantum field theory. Particular emphasis is placed on the case of renormalizable field theories with dynamical scale-invariance breaking, in which the induced gravitational effective action is finite and calculable. A functional integral formulation is used throughout, and a detailed analysis is given of the role of dimensional regularization in extracting finite answers from formally quadratically divergent integrals. Expressions are derived for the induced gravitational constant, for the induced cosmological constant, and for quantized matter theories on a background manifold, and a strategy is outlined for computing the induced constants in the case of an SU(n) gauge theory. By use of the background field method, the formalism is extended to the case in which the metric is also quantized, yielding a derivation of the semiclassical Einstein equations as an approximation to quantum gravity, as well as general formulas for the induced (or renormalized) gravitational and cosmological constants.