AdS Black Holes Are Short-Lived inside the Spectral Form Factor

We analyze the contribution of AdS black hole saddles to the spectral form factor of $d$-dimensional CFTs with a gravity dual. At high temperatures, the analytically continued gravitational action of the large AdS black hole is expected to give a leading large-$N$ approximation to the 'slope' of the spectral form factor. When the CFT is put on a $(d-1)$-dimensional spatial sphere of radius $\ell$, we show that such an evaluation implies unphysical behavior of $Z(\beta +it)$ at large $t\gg \ell$ for $d\equiv5 \,({\rm mod}\; 4)$. A Picard--Lefschetz analysis of a minisuperspace approximation to $Z(\beta+it)$ does reveal the required Stokes phenomena that restore consistency. It is found that for $d>3$, the large AdS black hole saddle loses dominance and subsequently disconnects from the relevant integration cycle at $t=O(\beta)$. For $d=3$, dominance is lost at a time of order $\beta$, while the Stokes disconnection takes place at $t=O(\ell)$. After the large black hole saddle disconnects from the path-integral contour, the slope of the spectral form factor becomes dominated by the low-energy end of the spectrum.