The double cone geometry is stable to brane nucleation

TL;DR

The paper investigates whether the bulk double cone geometry, which captures the ramp feature of the spectral form factor in gauge/gravity duality, remains stable to brane nucleation in UV-complete theories. By analyzing probe-brane actions on complex double-cone saddles across AdS$_{d+1}$ spaces (including AdS$_5\times S^5$, AdS$_4$, and AdS$_7$ contexts) and enforcing microcanonical boundary conditions, the authors show that the relevant brane saddles either do not lie on a deformable global descent contour or yield suppressed contributions, signaling stability of the double cone at the probe level. Extensions to charged and rotating black holes (Cveti\v{c}-Lu-Pope) corroborate the stability, with phase-transition behavior of dominant saddles and consistent uplift to higher dimensions. The work also clarifies how positive boundary curvature contrasts with negative curvature, which can trigger brane instabilities, and discusses back-reaction and ensemble interpretations. Overall, the double cone appears robust as the semiclassical bulk saddle in the ramp regime, supporting its role in modeling spectral statistics and informing AdS/CFT factorization considerations, while leaving open questions about back-reaction, negative modes, and flat-boundary cases.

Abstract

In gauge/gravity duality, the bulk double cone geometry has been argued to account for a key feature of the spectral form factor known as the ramp. This feature is deeply associated with quantum chaos in the dual field theory. The connection with the ramp has been demonstrated in detail for two-dimensional theories of bulk gravity, but it appears natural in higher dimensions as well. In a general bulk theory the double cone might thus be expected to dominate the semiclassical bulk path integral for the boundary spectral form factor in the ramp regime. While other known spacetime wormholes have been shown to be unstable to brane nucleation when they dominate over known disconnected (factorizing) solutions, we argue that the double cone is stable to semiclassical brane nucleation at the probe-brane level in a variety of string- and M-theory settings. Possible implications for the AdS/CFT factorization problem are briefly discussed.

Figures (7)

• Figure 1: An example showing failure of factorization due to spacetime wormholes. The top line represents a path integral $\langle Z\rangle$. Although we have drawn the configuration as connected, it may include contributions from disconnected spacetimes as well. In any case, the natural path integral $\langle Z^2 \rangle$ associated with a pair of boundaries yields all terms generated by squaring $\langle Z\rangle$, but also contains additional contributions connecting the two boundaries as indicated by the second term in the bottom line.
• Figure 2: Schematic illustration showing the identifications in the Kruskal spacetime that lead to the double cone geometry.
• Figure 3: A keyhole-type contour in the complex $r$ plane that defines the double cone.
• Figure 4: A density plot of $\mathrm{Re}\left(\mathrm{i} I_{D3}/L^3T\right)$ in AdS$_5$ showing saddles and relevant contours for $r_+=L$. The red disks are singuarlities, with $\rho=0$ being the orbifold singularity of the double cone at the black hole bifurcation surface. The magenta dashed line is slightly below the real axis, and we take it to be the defining contour for our path integral. The blue square corresponds to $\rho^{3\;j}_\star$, and the blue line through it is the steepest descent contour. The defining contour can be deformed to this, so we find a contribution to the integral from this saddle. But \ref{['eq:rho3']} has negative real part, so this contribution is suppressed; it does not induce a brane nucleation instability. The black triangle corresponds to $\rho^{2\;j}_\star$, and the purple diamonds correspond to $\rho^{1\;j}_\star$. The steepest descent path through the purple saddle is also shown, but any deformation to the defining contour is obstructed both by regions of large brane amplitude (shaded in red) and the black hole singularity (associated with the red disk below the real axis). The constant phase contour (not shown) that descends from the black triangle in fact also passes through the blue dot, near which it is the contour of steepest ascent; the amplitude then grows without bound as one continues further along this contour.
• Figure 5: A density plot of $\mathrm{Re}\left(\mathrm{i} I_{\mathrm{M}2}\right)$ in AdS$_4$ showing saddles and relevant contours in the complex $r$ plane. We take $r_+ = L$. The red dots are branch points for the action, which now include all singularities of the double cone spacetime (though some branch points also occur at smooth points of the double cone spacetime). The dashed magenta curve is the defining contour for our integral. The blue dots are saddles, and one of these has a global descent contour (solid blue line) that can be deformed to the dashed magenta curve. The relevant saddle has $\mathrm{Re}\left(\mathrm{i} I_{\mathrm{M}2}\right) <0$. As described in section \ref{['sec:saddlesanddescents']}, no other saddle can be more dominant (and we have again checked explicitly that the other saddles fail to lie on useful descent contours), so the system is stable to brane nucleation.