Spin-2 spectrum of defect theories

TL;DR

This work establishes a universal spin-2 equation for defect-theory holographic backgrounds, showing that TT spin-2 fluctuations obey the massless scalar wave equation in the full ten-dimensional geometry and reduce to a Laplace–Beltrami problem on the Riemann surface base Σ. In the tractable supersymmetric Janus case, the internal eigenproblem becomes Heun’s equation, which the authors solve numerically to map the spin-2 spectrum as a function of the dilaton jump Δφ, and they analyze the large-Δφ limit where the geometry effectively develops a nearly-flat fifth dimension. The analysis reveals that the Janus geometry cannot localize 4D gravity, while backgrounds with NS5/D5 charge are more favorable for localization, though a fully controlled realization in string theory remains subtle. Overall, the paper provides a precise spectral framework for spin-2 modes in holographic interface geometries and connects the gravity localization question to the detailed geometry of the internal space and brane charges, with implications for defect CFT duals.

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, Poincare 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 $Δφ$. In the limit of large $Δφ$ 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.

Figures (6)

• Figure 1: The AdS$_4$ warp factor ($f_4$) of the Karch-Randall geometry with $L=1$ for $l = 1, 2, 4$ and $8$ (blue, purple, yellow, and green, respectively).
• Figure 2: The AdS$_4$ warp factor ($f_4$) and the sphere radii ($f_1, f_2$) of the Janus geometry for dilaton jumps $\Delta\phi = 0, 1, 4$ and $10$ (blue, purple, brown and green curves, respectively). The horizontal axis gives the invariant distance from the strip center, along the line $y= \pi/4$. The asymptotic AdS$_5$ radii have been set to one, and the dilaton in the left asymptotic region has been set to zero. Note the absence of a local maximum of the warp factor.
• Figure 3: The warp factor along the $y = \pi/4$ line for an NS5-brane stack (see section \ref{['5chargesection']}), with charge $Q_{NS5} = 16 \pi^2 \gamma$ where $\gamma = 0,1/2,1,5$ (blue, purple, yellow and green curves, respectively). The horizontal axis gives the invariant distance from the strip center. The asymptotic AdS$_5$ radius has been set to one, and the asymptotic dilaton has been set to zero. Note, as can be seen in the lower left figure, that the local maximum of $f_4$ does not exceed the asymptotic value of the AdS$_5$ radius.
• Figure 4: Warp factor along the $y = \pi/4$ line for intersecting D5-brane and NS5-brane stacks (see section \ref{['5chargesection']}), with brane charges $Q_{D5} = Q_{NS5} = 16 \pi^2 \gamma$ where $\gamma = 0,1/2,1,5$ (blue, purple, yellow and green curves, respectively). The conventions are the same as in figure \ref{['fig:warp-5brane']}. Note that on the line $y = \pi /4$ the $S^2$ warp factors are identical ($f_1 = f_2$).
• Figure 5: Numerical results for the AdS$_4$ masses of the first four normalizable modes for $n=0$. The dashed lines at $\xi\simeq \infty$ correspond to the $AdS_5$ values $m= 2, \sqrt{10}, \sqrt{18}, \sqrt{28}$.