"Filtering" CFTs at large N: Euclidean Wormholes, Closed Universes, and Black Hole Interiors

Despite its successes, the large-$N$ holographic dictionary remains incomplete. Key features of gravitational path integrals–most notably Euclidean wormholes and the associated failure of factorization–lack a clear interpretation in the standard large-$N$ framework. A related challenge is the possibility of erratic $N$-dependence in CFT observables, behavior with no evident semiclassical gravitational counterpart. We argue that these puzzles point to a missing ingredient in the dictionary: a large-$N$ filter. This filter projects out the erratic $N$-dependence of CFT quantities when mapping them to semiclassical bulk physics, providing an intrinsic boundary definition of gravitational “averages.” It also offers a boundary explanation of wormhole contributions and a boundary prediction of their amplitudes, thereby giving a natural resolution of the factorization puzzle. In addition, we derive an infinite tower of inequalities constraining wormhole amplitudes and argue that internal wormholes do not induce random couplings in the low-energy effective theory. Beyond resolving factorization, the large-$N$ filter supplies a generalized framework from which richer Lorentzian spacetime structures can emerge, including closed universes and black hole interiors. We argue that, as a consequence of erratic large-$N$ behavior, both closed universes and black hole interiors are quantum volatile, and that an AdS spacetime entangled with a baby universe is likewise quantum volatile. This volatility may allow an observer in AdS to infer the existence of the baby universe, whereas for an infalling observer, the ability to make measurements near a black hole horizon may become fundamentally limited–even if they may not live long enough to notice.

💡 Research Summary

The paper addresses a critical gap in the large-$N$ holographic dictionary, specifically focusing on the tension between Euclidean wormholes and the principle of factorization in AdS/CFT. In the standard framework, the presence of wormholes leads to a failure of factorization, where the expectation value of a product of operators does not equal the product of their individual expectation values ($\langle Z^2 \rangle \neq \langle Z \rangle^2$). This phenomenon lacks a clear interpretation within the conventional large-$N$ limit of Conformal Field Theories (CFTs).

To resolve this, the authors propose the concept of a “large-$N$ filter.” They identify that CFT observables can exhibit “erratic $뮬$ dependence”—fluctuations that vary unpredictably with the number of degrees of freedom $N$. The proposed filter acts as a mathematical projection that smooths out these erratic fluctuations when mapping boundary CFT quantities to semiclassical bulk physics. By doing so, the filter provides a boundary-side definition of the “averages” that correspond to the semiclassical gravitational path integral. This mechanism naturally explains how wormhole contributions are integrated into the boundary theory and provides a predictive framework for wormhole amplitudes through a series of infinite inequalities.

Furthermore, the paper extends this logic to more complex spacetime geometries, such as closed universes and black hole interiors. The authors introduce the concept of “quantum volatility,” arguing that these structures are inherently unstable due to the erratic $N$-dependence of the underlying CFT. This volatility has profound implications for observers: while an observer in the AdS bulk might be able to infer the existence of a baby universe through these fluctuations, an infalling observer approaching a black hole horizon may face fundamental limitations in performing measurements. This suggests that the very nature of spacetime geometry near a horizon is tied to the mathematical stability of the large-$N$ limit, offering a new perspective on the information paradox and the fundamental limits of observability in quantum gravity.