Alice in Warpland: KK modes, Warped Compactifications and the Swampland

Abstract

We investigate the asymptotic behavior of Kaluza-Klein (KK) towers in warped compactifications to Minkowski space. Focusing on the overall decompactification limit, we derive the scaling of KK masses at large KK momentum for scalar fluctuations in lower-dimensional Planck units. In codimension-one warped backgrounds sourced by a higher-dimensional exponential potential, we solve explicitly for the internal profiles and obtain a closed expression for the exponential mass decay rate $λ_{\rm KK}$ of the tower in terms of the moduli space distance. We find that warping reduces $λ_{\rm KK}$ relative to the unwarped case, in such a way that sufficiently strong warping could in principle violate the Sharpened Distance Conjecture bound. Remarkably, this sharpened bound is still satisfied precisely when the higher-dimensional potential obeys the condition forbidding asymptotic accelerated expansion, establishing a direct link between the Sharpened Distance Conjecture and the Strong de Sitter condition in one higher dimension. We also argue that for higher-codimension warped backgrounds the asymptotic KK scaling remains unmodified.

Figures (6)

• Figure 1: Brane arrangement in the massive type IIA orientifold with gauge groups $SO(32)$. The Romans mass is non-vanishing, and backreacts on the metric and 10d dilaton $\Phi$. In the limiting case of maximal warping, the dilaton diverges at the location of the opposite O8$^{-}$ plane, giving the enhancement $SO(32)\to SO(32)\times SU(2)$, see Polchinski:1995dfAharony:2007duEtheredge:2023odp.
• Figure 2: Comparison of exponential rates for the KK modes of the warped one-dimensional compact dimension, the tower of KK$'$ modes in $d+1$ dimensions (assuming that the higher-dimensional scalar field is a radion measuring the volume of $n$ additional compact unwarped dimensions), and the Planck mass $M_{{\rm Pl},d+1}$. We illustrate the results in two cases, with $(d,n)=(8,1)$ and $(d,n)=(4,4)$. The exponential rates correspond to the highly warped limits with fixed impact parameter. The unwarped values, $\sqrt{\frac{d-1}{d-2}}$ for $\lambda_{\rm KK}$ and $\frac{1}{\sqrt{(d-1)(d-2)}}$ for $\lambda_{{\rm Pl},d+1}$, are in dashed lines and are recovered as $\gamma\to\infty$.
• Figure 3: Comparison of exponential rates for the KK modes, the string oscillation modes, and the winding modes from the fundamental string (assuming that the higher-dimensional scalar field is the higher-dimensional dilaton $\Phi_{d+1}$) and the Planck mass $M_{{\rm Pl},d+1}$ for warped codimension-one compactifications with $d=4$ and $d=9$. The rates correspond to the highly warped limits with fixed impact parameter. The unwarped values, $\sqrt{\frac{d-1}{d-2}}$, $\frac{1}{\sqrt{(d-1)(d-2)}}$, and $\frac{3-d}{\sqrt{(d-1)(d-2)}}$ are in dashed lines, and are recovered as $\gamma\to\infty$. The dashed vertical gray line corresponds to $\gamma=\frac{2}{\sqrt{d-1}}$.
• Figure 4: $\zeta$-vectors for the KK modes (in blue) and the string oscillator modes (in red) in the maximum warping regimes with vanishing impact parameter. Since in this regime the tangent vector $\hat{T}\propto \partial_{\hat{B}}$ is proportional to $\vec{\zeta}_{\rm KK}$, we have that $\lambda_{\rm KK}=|\vec{\zeta}_{\rm KK}|$ and that $\lambda_{\rm osc}=\hat{T}\cdot\vec{\zeta}_{\rm osc}$ is the projection of $\vec{\zeta}_{\rm osc}$ over the $\vec{\zeta}_{\rm KK}$ direction. In lighter blue we depict for reference the scaling vectors of the KK modes in the unwarped case, $\vec{\zeta}_{\rm KK}^{\rm (unwarp.)}$, as well as the sliding direction as we move in the highly warped direction.
• Figure 5: Type IIB string theory on the pillowcase manifold $\mathbb{S}^2(2222)\simeq \mathbb{T}^2/\mathbb{Z}_2$ with 16 D7-branes on one of the O7$^{-}$ singularities. This is T-dual to the massive type IIA configuration of Figure \ref{['fig.typeIprime']} compactified on an additional $\mathbb{S}^1$.