Spin-2 spectrum of defect theories

Abstract

We study spin-2 excitations in the background of the recently-discovered type-IIB solutions of D'Hoker et al. These are holographically-dual to defect conformal field theories, and they are also of interest in the context of the Karch-Randall proposal for a string-theory embedding of localized gravity. We first generalize an argument by Csaki et al to show that for any solution with four-dimensional anti-de Sitter, Poincaré 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 differential equation known as Heun's equation. We solve this equation numerically, and exhibit the spectrum as a function of the dilaton-jump parameter ∆φ. In the limit of large ∆φ a nearly-flat linear-dilaton dimension grows large, and the Janus geometry becomes effectively five-dimensional. We also discuss the difficulties 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.

AdS-sliced flavor branes and adding flavor to the Janus solution

“…The sign reversal is for convenience, so that we plot mostly positive masses, as an overall sign can be introduced in the dual gauge theory by a chiral rotation. For reference, we list the fit parameters for the example solutions shown in Figure 1 To obtain plots for studying the phase structure of this system, we numerical solve and fit (18) successively from µ b = 1.4 down to µ b = 0.1 in steps of 0.01, then adjust the step size successively to 10 −5 , 10 −7 , and 10 −12 for reasons that will become apparent. Recall that in [4] with arcsine solutions the mass was given by the inverse of the position where the brane ended.…”

On the spin-2 Kaluza-Klein spectrum of AdS 4 × S 2 ℬ 4 $$ {\mathrm{AdS}}_4\times {S}^2\left({\mathrm{\mathcal{B}}}_4\right) $$

“…Equation (3.4) of the present paper then reduces to (2.4) of [6] provided we specialize to an AdS space of unit radius, L = 1; this corresponds to setting k = −1 in [6]. Note also that (5.3) of [2] reduces to (3.3) of the present paper upon setting M = 0 in that reference and identifying λn there with…”

“…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., AdS d , R 1,d−1 or dS d ) 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 M 6 .…”

“…It then follows from the analysis of [6] that the metric (3.1) obeys the ten-dimensional linearized Einstein equations, independently of the form of the energy-momentum tensor for the matter fields (i.e., all fields of the theory other than the metric). Normalizable spin-2 excitations correspond to eigenmodes g n of the modified Laplacian (3.3) for which…”

“…3 Equation (3.3) is equivalent to (2.20) of [6] where m and ψ of that reference are identified respectively with LMn and gn here. Equation (3.4) of the present paper then reduces to (2.4) of [6] provided we specialize to an AdS space of unit radius, L = 1; this corresponds to setting k = −1 in [6].…”