Baby Universe in a Coupled SYK Model

TL;DR

This paper explores three distinct saddle points in a coupled SYK model with a Maldacena-Qi interaction, revealing their dual bulk geometries and demonstrating how a baby universe can have a nontrivial Hilbert space.

Contribution

It develops explicit chord rules to analyze these saddle points and shows how a baby universe can support entanglement and a nontrivial Hilbert space.

Findings

Identifies three saddle points with different bulk geometries

Derives chord rules for probing these geometries

Shows entanglement between baby universe and external spacetime

Abstract

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.