Spin-2 spectrum of defect theories

Constantin Bachas 1 John Estes 0 Open Access 0 Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven , Celestijnenlaan 200D B-3001 Leuven, Belgium 1 Laboratoire de Physique Tehorique de l'Ecole Normale Sueprieure , 24 rue Lhomond, 75231 Paris cedex, France We study spin-2 excitations in the background of the recently-discovered typeIIB solutions of D'Hoker et al. These are holographically-dual to defect conformal eld theories, and they are also of interest in the context of the Karch-Randall proposal for a string-theory embedding of localized gravity. We rst generalize an argument by Csaki et al to show that for any solution with four-dimensional anti-de Sitter, Poincaer or de Sitter invariance the spin-2 excitations obey the massless scalar wave equation in ten dimensions. For the interface solutions at hand this reduces to a Laplace-Beltrami equation on a Riemann surface with disk topology, and in the simplest case of the supersymmetric Janus solution it further reduces to an ordinary dierential equation known as Heun's equation. We solve this equation numerically, and exhibit the spectrum as a function of the dilaton-jump parameter . In the limit of large a nearly-at linear-dilaton dimension grows large, and the Janus geometry becomes eectively ve-dimensional. We also discuss the diculties of localizing four-dimensional gravity in the more general backgrounds with NS5-brane or D5-brane charge, which will be analyzed in detail in a companion paper. Contents 1 Introduction 2 Universal equation for spin-2 excitations 2.1 Ansatz for the Kaluza-Klein spin-2 modes 2.2 Reduction of the Einstein equations 2.3 Normalizability and the zero mode 3 Interface solutions and their charges 3.1 Local solutions: general form 3.2 AdS5 S5 and supersymmetric Janus 3.3 Non-vanishing ve-brane charge 4 Localized gravity: what to look for? 4.1 The Karch-Randall model 4.2 Warp factors and a limiting geometry 5 Spectral problem on the strip 5.1 Reduction of the eigenmode equation 5.2 The case of AdS5 S5 6 Spectrum of supersymmetric Janus 6.1 Heuns equation 6.2 Numerical solution 7 Discussion A Summary of type IIB supergravity 1 3 1 Introduction Whether gravity can be localized [1{3] is a question of great interest for cosmology. It is closely related to the question of whether the graviton can have a mass [4{8], or more generally whether Einsteins theory | with or without a cosmological constant term | can be consistently modied at cosmic-distance scales. Despite many interesting results (see e.g. [9{16] and references therein), there is still no denitive answer to these questions. The diculties with low-energy descriptions of localized and/or massive gravity typically arise when one considers classical non-linearities, and quantum corrections. Problems include the non-decoupling of heavy mass scales (the Planck scale, the bulk curvature or the brane thickness) and/or the appearance of ghosts. Clearly, an embedding in an ultraviolet-complete theory like string theory could shed important light on these issues. A promising proposal for such an embedding was made some time ago by Karch and Randall [17, 18]. These authors considered an AdS4 S2 brane in an AdS5 S5 bulk, and argued that a nearly-massless graviton can be obtained by tuning the brane tension. In their calculation Karch and Randall had to approximate the background geometry by two AdS5 slices glued together along a \thin" AdS4 brane. The exact type-IIB supergravity solutions, describing the full back-reaction of branes on geometry, were discovered only recently by DHoker et al [19, 20] (see also [21, 22] for related work). A central motivation for our work here was to check whether the conclusions of Karch and Randall survive in these exact string-theory backgrounds. The supergravity solutions of DHoker et al are holographically-dual to defect conformal eld theories [ 23{29], i.e. to two or more conformal theories interacting along a scale-invariant domain wall.1 The \locally-localized graviton" has been discussed from the viewpoint of the defect CFTs in [32] (for earlier perspectives see [33{36], and for a recent one [37] ). Other than their potential relevance to the localization of gravity, our results are thus also of interest in this dual context. In this paper we will rst set up the spectral problem for spin-2 excitations in the interface geometries of refs. [19, 20]. These have the structure of an AdS4 S2 S2 spacetime, bered over a Riemann surface ( ; g^) with disk topology. The excitations which are constant on the spheres, and which have spin 2 in AdS4, obey Neumann boundary conditions on the surface . Their shifted mass-squared operator, m2 + 2, is, as we will see, the Laplace-Beltrami operator on the disk with metric given by g = e 2Ag^, where A is the AdS4 warping factor. In deriving this result, we will generalize slightly an argument of Csaki et al [38] to show that the spin-2 excitations obey the massless scalar-wave equation in ten dimensions. The argument, rst made in the context of at thick branes and scalar elds, 2 generalizes easily to all supergravity solutions with 4D anti-de Sitter, Poincaer or de Sitter symmetry, and with arbitrary p-form ux backgrounds. The universality of the spin-2 wave equation should be contrasted with what happens for spins 1, whose wave equations depend a priori on non-metric features of the backgrounds. Solving the spectral problem on the surface ( ; g) is, in general, a dicult task. It simplies considerably for the supersymmetric Janus solution [which describes a supersymmetric dilaton domain wall], for which the relevant eigenmode equation is separable. The reduced ordinary dierential equation has four regular singular points, and is known in the mathematics literature as Heuns equation [40]. We will solve this equation numerically, and exhibit the spectrum as a function of the dilaton-jump parameter . As we will see, Janus cannot localize four-dimensional gravity and its only interesting limit, ! 1, is a limit in which a at fth dimension decompacties. This behavior can be attributed to the vanishing superpotential of the dilaton eld, whose domain walls have no intrinsic tension or thickness. Solutions with NS5-brane and/or D5-brane charge are in this respect 1To be distinguished from conformal theories interacting via multi-trace marginal operators in the bulk, and which have been argued to be the duals of multi-gravity models [13, 30, 31]. Their embedding in string theory is more problematic, as it most likely involves quantum-gravity eects. 2For branes of co-dimension one, the observation was actually made earlier in ref. [39]. more promising, in line with the original proposal of Karch and Randall [18]. Generating a large scale hierarchy without decompactifying extra dimensions looks, however, a priori a dicult task. The detailed analysis of the backgrounds with ve-brane charge, for which the eigenmode equation on the strip does not fac (...truncated)