Baby Universe in a Coupled SYK Model

We analyze three saddle points of the path integral computing the partition function of the SYK model with a Maldacena-Qi coupling in the double scaling limit. The three saddle points are holographically dual to three topologically different spacetimes: a pair of Euclidean black holes (two thermal disks), a thermal AdS$_2$ (a cylinder), and a thermal AdS$_2$ with a baby universe (a cylinder with a handle). We develop explicit chord rules that span and probe these three bulk geometries. We derive the rules by expanding the effective $G,\Sigma$ action in powers of the coupling $\mathcal{J}$ and writing the partition function as a weighted sum of chord diagrams. By slicing the diagrams open, we generate a Hilbert space description on a spatial slice for each saddle point. The Hartle-Hawking chord state for the third saddle point has genuine entanglement between the baby universe and the external spacetimes, providing evidence that a closed universe can support a nontrivial Hilbert space.