On the dimensional reduction of quadratic higher-derivative gravitational terms

  • M. D. PollockEmail author
Regular Article


Gravitational Lagrangian theories, that are formulated initially in \( D (\equiv 4+N)\) dimensions, produce scalar moduli \( \sigma_{A}\), \( \sigma_{B}\) upon reduction to four dimensions, via the metric decomposition \( \hat{g}_{AB} (X^{c}) = {\rm e}^{-\sqrt{2} \kappa_{0} \sigma_{A}} g_{ij}(x^{k})\)\( \otimes {\rm e}^{2 \kappa_{0} \sigma_{B}/\sqrt{N}} \bar{g}_{\mu\nu} (y^{\xi})\), where \( \kappa_{0}^{2}\) is the bare four-dimensional gravitational coupling, while \( g_{ij}(x^k)\) and \( \bar{g}_{\mu\nu} (y^{\xi})\) are the physical four- and internal-space N-metrics, respectively. After integration over the N-space, the four-Lagrangian resulting from the Einstein-Hilbert D-theory \( \hat{L} = -{\rm e}^{-2\phi} [\hat{R}\)\( +4(\hat{\nabla}\phi)^{2}]/2 \hat{\kappa}^{2}\) is \( L=-R/2\kappa^{2}\)\( +(\nabla \sigma_{A})^{2}/2 + (\nabla \sigma_{B})^{2}/2\), in which the kinetic-energy terms for \( \sigma_{A}\), \( \sigma_{B}\) have canonical coefficients 1/2. These coefficients are modified, however, if \( \hat{L}\) contains in addition quadratic higher-derivative terms \( \hat{\mathcal{R}}^{2} \equiv \hat{\alpha}_{1}\hat{R}^{2}\)\( + \hat{\alpha}_{2}\hat{R}_{AB} \hat{R}^{AB} + \hat{\alpha}_{3}\hat{R}_{ABCD}\hat{R}^{ABCD}\) , due to the rescaling under the conformal transformation \( \hat{g}_{AB}\rightarrow {\rm e}^{-\sqrt{2}\kappa_{0} \sigma_{A}} g_{ij}\), which is typically of the form \( \hat{\mathcal{R}}^{2} \rightarrow {\rm e}^{2\sqrt{2} \kappa_{0} \sigma_{A}} [{R}^{2}\)\(+\mathcal{R} (\nabla \sigma_{A,B})^2 + (\nabla \sigma_{A,B})^{4}]\) . Previously, we analyzed the effect of the terms \( \mathcal{R}(\nabla \sigma_{A,B})^{2}\) quadratic in \( \nabla \sigma_{A,B}\) , which in general lead to a mixing of \( \sigma_{A}\) and \( \sigma_{B}\), and consequently instability at high energies. Here, we consider the quartic terms \( (\nabla \sigma_{A,B})^{4}\) , that also give rise to instabilities, both for arbitrary \( \hat{\alpha}_n\) and in the specific case of the heterotic superstring theory, for which \( \hat{\alpha}_1= \hat{\alpha}_3 =-\hat{\alpha}_2/4=\kappa_{0}^{2}/{2}\), and become significant if \( \sigma_{A}\), \( \sigma_{B}\) behave as massless scalars.


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© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.V. A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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