Comments on wormholes and factorization

TL;DR

This work analyzes how factorization can coexist with wormhole contributions in non-averaged holographic systems by studying α-states in topological gravity models (the MM model) and JT gravity. It introduces a disk-and-cylinder disk-and-cylinder (CGS) approximation that reduces complicated many-universe dynamics to a tractable description with a small number of effective boundaries Ψ, enabling factorization to emerge in α-states and aligning with ensemble intuitions from SYK. The authors develop a multi-species CGS framework, show explicit factorization identities, and then extend the approach to approximate α-states in MM and JT gravity, quantifying errors from non-Gaussian topologies and spectral discreteness. An effective description with random boundaries reconciles ensemble/time-averaged wormhole contributions with fixed α-states, and the JT gravity analysis connects to a coarse-grained density of states, suggesting broader applicability to non-averaged holography. Overall, the paper proposes a concrete, calculable mechanism by which factorization can persist in the presence of wormholes, via an exclusion-like structure encoded by random boundary data and a controlled disk-and-cylinder topological sector.

Abstract

In AdS/CFT partition functions of decoupled copies of the CFT factorize. In bulk computations of such quantities contributions from spacetime wormholes which link separate asymptotic boundaries threaten to spoil this property, leading to a "factorization puzzle." Certain simple models like JT gravity have wormholes, but bulk computations in them correspond to averages over an ensemble of boundary systems. These averages need not factorize. We can formulate a toy version of the factorization puzzle in such models by focusing on a specific member of the ensemble where partition functions will again factorize. As Coleman and Giddings-Strominger pointed out in the 1980s, fixed members of ensembles are described in the bulk by "$α$-states" in a many-universe Hilbert space. In this paper we analyze in detail the bulk mechanism for factorization in such $α$-states in the topological model introduced by Marolf and Maxfield (the "MM model") and in JT gravity. In these models geometric calculations in $α$ states are poorly controlled. We circumvent this complication by working in $\textit{approximate}$ $α$ states where bulk calculations just involve the simplest topologies: disks and cylinders. One of our main results is an effective description of the factorization mechanism. In this effective description the many-universe contributions from the full $α$ state are replaced by a small number of effective boundaries. Our motivation in constructing this effective description, and more generally in studying these simple ensemble models, is that the lessons learned might have wider applicability. In fact the effective description lines up with a recent discussion of the SYK model with fixed couplings arXiv:2103.16754. We conclude with some discussion about the possible applicability of this effective model in more general contexts.

Figures (38)

• Figure 1: Here we picture a spacetime wormhole which contributes to $Z_L Z_R$.
• Figure 3: Here we show an example contribution to $Z_n$ (for $n\geq 4)$. Each spacetime contributing to $Z_n$ has $n$ circular boundaries, and may have any number of connected components. The lower set of dots represent the $n-4$ boundaries not explicitly drawn.
• Figure 4: Here we illustrate some examples of contributions to $Z_0$, labeled by the number of connected components for each genus, $m_g$.
• Figure 5: Here we have pictured the leading contributions to $\tilde{Z}_1$, which is given by a sum over spacetimes with one connected component and a single circular boundary. We have labeled the contributions by their Euler character.
• Figure 6: Here we have pictured the leading contributions to $\tilde{Z}_2$, which is given by a sum over spacetimes with two circular boundaries, including only spacetimes where each connected component has at least one of these two boundaries. The cylinder contribution, with $\chi=0$, is an example of a spacetime wormhole, connecting two separate boundaries.