Baby Universes in AdS$_3$

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.

• The paper demonstrates that baby universes exist in AdS3 geometries but are exponentially suppressed as subleading saddles in the gravitational path integral.
• It employs holographic CFT formulations on higher genus Riemann surfaces to identify gravitational saddle contributions and resolve factorization puzzles.
• The study clarifies bulk-boundary duality by showing that only engineered CFT manipulations can promote baby universes to dominant, yet mixed, semiclassical states.

Euclidean AdS$_3$ Geometries with Baby Universes: Dominance, Mixedness, and Semi-Classicality

Introduction and Motivation

The paper "Baby Universes in AdS$_3$" (2512.02098) investigates the role and realization of baby universes in three-dimensional gravity within the AdS/CFT framework. The study is embedded in the context of topologically nontrivial Euclidean geometries, where Lorentzian slicing can yield closed baby universes characterized by genus $g \geq 2$ spatial geometry. The authors provide an alternative to previous constructions involving shell states by formulating setups purely from holographic CFTs defined on higher genus Riemann surfaces, thereby ensuring a transparent and microscopically well-posed duality perspective.

A central puzzle relates to the dimensionality of the Hilbert space associated with closed universes and the extent to which these baby universes represent semi-classical bulk states. This is particularly relevant in the aftermath of wormhole studies and the ensuing factorization and AR (Antonini-Rath) puzzles, which question the purity/mixedness of bulk states as prepared by the boundary CFT.

Geometric Saddle Dominance and Wavefunction Structure

The paper demonstrates that for states prepared by the CFT path integral on higher genus surfaces, baby universe geometries generically exist but are never the dominant saddle in gravitational path integrals. The dominant contributions are handlebody geometries tied to the identity Virasoro blocks, while non-handlebody solutions admitting baby universes are exponentially suppressed (by factors such as $e^{-3S(E)}$, where $S(E)$ is the entropy at the relevant moduli). Notably, the presence of a baby universe in the bulk does not reflect a semi-classical description for the CFT-prepared state, as its occurrence is restricted to an exponentially small segment of the wavefunction.

Crucially, the leading saddle in the low-energy (AS$^2$-regime) is a geometry lacking any baby universe, resulting in a pure (but possibly partially entangled) state for two copies of the CFT. The exponentially subdominant saddles with baby universes correspond to mixed states and do not contradict the expected holographic behavior.

Microscopic Prescriptions and Mixedness

By applying a microscopic prescription akin to that proposed in [Belin:2025wju], the paper exhibits how one can engineer a CFT state whose leading gravitational geometry contains a baby universe. This construction relies on performing OPE contractions directly at the level of the density matrix, distinguishing it from traditional Euclidean path integral preparations. This operation yields a mixed state in the CFT, aligning with the qualitative gravitational expectation and resolving the AR puzzle for these specially engineered states. The mixed nature reflects the entanglement between the asymptotic AdS regions and the degrees of freedom residing inside the baby universe.

Additionally, the paper finds that the fluctuations in the baby universe wavefunction coefficients are small even at fixed central charge. Unlike shell state setups where large fluctuations pervade and complicate semi-classical interpretations, the engineered mixed state with dominant baby universe geometry displays reliable semi-classicality, without requiring any averaging over the central charge.

Virasoro TQFT Interpretation and Purification

For pure 3D gravity (i.e., no matter), where local degrees of freedom are absent and the theory is topological, the Hilbert space associated with the baby universe is interpreted as an auxiliary Hilbert space akin to that found in Virasoro TQFT. This auxiliary space is labeled by Virasoro representations and serves as a factor with which the AdS boundaries are entangled. The canonical purification (GNS construction) of the mixed state is discussed, with the full gravitational path integral realized as a wormhole geometry connecting four AdS regions, and the entanglement bottleneck interpreted via genus-2 topological states. Thus, the baby universe does not correspond to local QFT excitations but to a set of Virasoro conformal blocks that encode topological entanglement structure.

Implications and Future Directions

The results have strong implications for our understanding of bulk-boundary duality in theories with nontrivial topologies. The exponential suppression of baby universe contributions in standard CFT path integral states clarifies the absence of semi-classical baby universes, resolving apparent contradictions in the literature concerning the purity of bulk states. Only by direct microscopic manipulation—departing from path integral state preparation—can one achieve dominance and semi-classical reliability of baby universe geometries, necessarily accompanied by mixedness in the CFT state.

The study sets a template for exploring the interplay of topology, saddle dominance, and Hilbert space structure in AdS$_3$ gravity. It points toward several open avenues:

• Analysis of matter-coupled holographic CFTs to track how bulk matter populates the baby universe Hilbert space and influences saddle structure.
• Generalization to higher genus and multi-baby universe configurations to elucidate combinatorial and entropic emergent phenomena at large central charge.
• Systematic classification of non-handlebody solutions and their statistical representation in holographic OPE data, potentially elucidating sum-over-topologies approaches.
• Investigation of over-completeness and black hole entropy counting via explicit geometric construction in the CFT path integral framework.

Conclusion

This work rigorously establishes that in AdS$_3$/CFT$_2$ contexts, baby universes generically exist as subleading geometries and do not contribute semi-classically to states prepared via Euclidean path integrals. Only through explicit manipulation of the microscopic state can baby universe geometries be made dominant, at the cost of generating mixed states. The auxiliary Hilbert space structure revealed in pure gravity is interpreted through Virasoro TQFT, suggesting non-local entanglement sources for the asymptotic regions. The findings clarify fundamental aspects of holographic duality, reinforce predictions from semiclassical gravity, and lay the groundwork for deeper exploration of topology in quantum gravity.