Abstract
We perform a numerical study of the four-dimensional spin-2 Kaluza-Klein spectrum of supersymmetric AdS$_4\times S^2(\mathcal{B}_4)$ vacua and show that they do not exhibit scale separation. Our methods are generally applicable to similar problems where the compactification geometry is not known analytically, hence an analytic treatment of the spectrum of Kaluza-Klein masses is not available.
Highlights
JHEP12(2014)144 ratio becomes necessarily of order one in a neighborhood of the equator
In order to settle the question of scale separation we examine directly the KK spectrum of fourdimensional spin-2 excitations
We have performed a numerical study of the four-dimensional spin-2 KK spectrum in the supersymmetric AdS4 vacua of [1]
Summary
For the analysis of the KK spectrum of four-dimensional spin-2 fields (massive “gravitons”) we will draw upon the results of [6] where it was shown (generalizing earlier work of [7]) that the spin-2 excitations of any ten-dimensional background containing a d-dimensional factor with maximal symmetry (i.e., AdSd, R1,d−1 or dSd) obey the massless scalar tendimensional wave equation. In particular for supergravity backgrounds of the form (2.1) this result correlates the KK mass of four-dimensional gravitons to the eigenvalues of a modified Laplacian of M6. With gpq the metric of M6 and g6 its determinant; h(μnν)(x) in the expansion (3.2) is assumed to be transverse and traceless and to obey the Pauli-Fierz equations for a massive spin-2 particle of mass Mn in an AdS4 space of radius of curvature L (see, e.g., [8]):. Where ∇ ̄ is the covariant derivative with respect to the Christoffel connection of gμν It follows from the analysis of [6] that the metric (3.1) obeys the ten-dimensional linearized. Normalizable spin-2 excitations correspond to eigenmodes gn of the modified Laplacian (3.3) for which d6y√g6e2A|gn|2 < ∞.
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