Baby Universes in AdS$_3$

TL;DR

The paper investigates Euclidean geometries in AdS$_3$ whose Lorentzian slices yield baby universes of genus $g\ge 2$ within a holographic CFT on a higher-genus boundary. It shows that in the standard CFT-path-integral construction these baby-universe geometries exist but are subdominant saddles, so they do not provide a semiclassical description of the state. By applying a microscopic prescription that contracts the density matrix with OPE data, the authors render a state whose leading bulk dual includes a baby universe, at the cost of the CFT state becoming mixed; crucially, the fluctuations remain small and the baby universe is semiclassical. The Virasoro TQFT interpretation clarifies how the baby universe is encoded in an auxiliary conformal-block Hilbert space and discusses canonical purification, observers, and purification structure. The work offers a coherent framework to analyze baby universes in AdS$_3$/CFT$_2$, outlining extensions to include matter and higher-genus geometries and addressing longstanding puzzles about semiclassical descriptions and purifications in quantum gravity.

Abstract

We discuss Euclidean geometries in AdS$_3$ whose Lorentzian slicing gives rise to closed baby universes with a spatial geometry given by genus $g\geq 2$ surfaces. Our setup only involves a two-dimensional holographic CFT defined on a higher genus Riemann surface and thus provides a well-posed alternative to shell states whose microscopic duals are less well understood. We find that geometries giving rise to baby universes are always subdominant. It follows that the baby universe does not provide a semi-classical description of the state since it is encoded in an exponentially suppressed part of the wave function. We then apply a prescription developed in \cite{Belin:2025wju} to make the baby universe geometry the leading saddle. In the process, the CFT state becomes mixed, in agreement with the qualitative gravitational picture. We show that the fluctuations in the baby universe are small, even at fixed central charge, making the geometry reliable in the semi-classical limit. Finally, we discuss the interpretation of this mixed state in pure gravity from the perspective of the Virasoro TQFT.

Figures (5)

• Figure 1: The difference between the AS${}^2$ setup (left) and our setup (right). These geometries correspond to the overlap of the state with itself. In the AS${}^2$ setup, the state is prepared by a path integral on a cylinder with a shell inserted. The shell propagates inside the Euclidean geometry, and the $t=0$ slice corresponds to two disconnected AdS regions, entangled with a baby universe on which the shell lives. In our setup, the shell is replaced by topology. Unlike shells, the geometry shown here is not the only geometry with these boundary conditions, there are others, and these other saddles dominate over the one that is represented here.
• Figure 2: The preparation of the state $\ket{\frac{1}{2} [g=5]}$ using the CFT path integral. The path integral is performed over half of a genus-5 surface, and this preparation yields an entangled state in the tensor-product of two copies of the CFT Hilbert space.
• Figure 3: The geometry of the handlebody at genus-5. The boundary of this bulk three-dimensional geometry is the genus-5 surface that corresponds to the overlap (\ref{['overlap']}). The geometry is obtained by filling in a choice of cycles, as shown here. On the $t=0$$\mathbb{Z}_2$-symmetric slice where we cut the overlap to get the state, we simply have two disconnected disks. The quantum fields on these disks are entangled, just like in thermal AdS, but the entanglement pattern is more complicated than the familiar thermal one.
• Figure 4: The geometry of the non-handlebody at genus-5. The boundary of this bulk three-dimensional geometry is the genus-5 surface that corresponds to the overlap (\ref{['overlap']}). Unlike a handlebody, the cycles of the bottom and top genus-2 surfaces are not contractible. Instead, the bottom and top genus-2 surfaces are connected by a Maoz-Maldacena wormhole. On the $\mathbb{Z}_2$-symmetric slice drawn as the dashed line, we can slice the geometry and continue it to Lorentzian signature. This will lead to two asymptotically AdS disks, and a closed universe made by a genus-2 surface.
• Figure 5: A picture of the 3-geometry which prepares the canonical purification of the baby universe state. We have a Maoz-Maldacena wormhole, with each asymptotic genus-2 boundary having two cylinders attached to it. The topology is such that the four cylinders are filled, and the time-slice drawn as a blue cut has four disks. The left two disks are those of the original baby universe state, while the two other disks are those coming from the purification. The time-cut of the original baby-universe state, before being purified, corresponds to the union of the left blue segment until it reaches the red cut, and then it extends along the red cut. The genus-2 surface on the red cut is a sort of bottleneck for entanglement between the left and right sides.