Factorizing Holographic Theory

• The paper demonstrates that strict factorization in holographic theories necessitates a one-dimensional closed-universe Hilbert space, reconciling wormhole-induced non-factorization with a single dual quantum system.
• It examines how ensemble averaging, random matrix theory, and topological quantum field theory provide mechanisms to account for non-factorizing wormhole corrections and restore consistency in bulk-boundary dynamics.
• The work employs SYK and matrix models to show that half-wormhole saddle contributions cancel non-factorizing terms, offering an algorithmic and semiclassical framework for the emergence of spacetime locality.

Factorizing holographic theory concerns the structural requirement that observables and partition functions in quantum gravitational systems with AdS/CFT duals decompose into products associated with each boundary component, and the implications of this requirement for the Hilbert space of states, the algebra of observables, and the physical meaning of wormhole contributions, ensemble averaging, and global symmetries. Recent advances have linked the factorization structure to rigorous features of black hole microstates and the emergence of spacetime locality, using tools from random matrix theory, topological quantum field theory, and semiclassical gravity.

1. The Factorization Puzzle in Holography

In AdS/CFT, the quantum gravity path integral on manifolds with multiple asymptotic boundaries formally sums over connectivity, including wormhole geometries. The boundary CFT partition function computed from such a bulk path integral would appear to violate factorization, since wormhole saddles produce non-factorizing, connected contributions: $$Z_{\mathrm{grav}}[\partial M_1 \sqcup \partial M_2] \neq Z_{\mathrm{grav}}[\partial M_1] Z_{\mathrm{grav}}[\partial M_2]$$ This is in tension with the basic tenet of holography: if the dual is truly a single, fixed quantum mechanical system (not an ensemble), then path-integral-computable partition functions and correlators must strictly factorize for disconnected boundaries. Resolving this "factorization puzzle" is essential for consistent quantum gravity, unitarity, and the statistical interpretation of black hole entropy (Usatyuk et al., 2024, Banerjee et al., 2024, Benini et al., 2022).

2. Structure of Closed-Universe States and Factorization Constraints

In a fixed (non-ensemble) holographic theory, the requirement of bulk-to-boundary factorization imposes drastic constraints on allowable closed-universe states. For a compact spatial slice $\Sigma$ with no asymptotic boundary (a "closed universe"), the path-integral-prepared Hartle-Hawking wave function $\Psi_{\mathrm{HH}}[\Sigma]$ is unique up to normalization. Any boundary preparation leading to a state $\Psi_\alpha[\Sigma]$ must satisfy

$$\Psi_\alpha[\Sigma] = C_\alpha\,\Psi_{\mathrm{HH}}[\Sigma]$$

for some $C_\alpha$ depending only on the boundary data. The inner product for closed-universe states thus factorizes,

$$\langle\Sigma_1|\Sigma_2\rangle = \Psi_{\mathrm{HH}}[\Sigma_1]\Psi_{\mathrm{HH}}[\Sigma_2]$$

with the Gram matrix of closed-universe states having rank one. All geometrically distinct $\Sigma$ thus correspond to the same quantum state, rendering the closed-universe Hilbert space one-dimensional (Usatyuk et al., 2024). This rigidity is a direct consequence of the strict factorization of boundary observables in a single, fixed dual.

3. Resolution via Ensemble Averaging and Wormhole Contributions

For theories admitting ensemble duals (e.g., JT gravity with a random matrix ensemble), ensemble averaging reintroduces non-trivial closed-universe structure. The averaged Hartle-Hawking wave function

$$\langle \Psi_{\mathrm{HH}}(b) \rangle = \sum_{g=1}^\infty e^{S_0\chi}\,V_g(b)$$

(where $V_g(b)$ are Weil–Petersson volumes) smooths out the erratic, fine-grained fluctuations from chaotic spectra and supports an infinite-dimensional Hilbert space of closed-universe states distinguishable by different boundary preparations. In this context, wormhole geometries correspond to ensemble correlations or superselection sectors (e.g., baby universes), and physically distinct states emerge only in the ensemble-averaged theory—a violation of strict factorization (Usatyuk et al., 2024, Torres et al., 7 Oct 2025).

The ensemble-averaged scenario breaks bulk factorization but provides a semiclassical, many-state closed-universe Hilbert space, resolving the factorization puzzle via topological quantum field theory structure or statistical mechanisms (Benini et al., 2022).

4. Factorization and the Hilbert Space of Black Holes

Factorization has direct consequences for the Hilbert space structure in the presence of black holes. In the two-sided (eternal) AdS black hole geometry, the boundary Hilbert space must be a tensor product, $$\mathcal H_{\mathrm{CFT}} = \mathcal H_L\otimes \mathcal H_R$$. However, naive semiclassical bulk quantization produces a non-factorizing, overcomplete Hilbert space with gauge constraints correlating left and right excitations. When non-perturbative wormhole corrections are accounted for (e.g., via the resolvent in random-matrix language), overlaps between ostensibly orthogonal microstates become nonzero at order $e^{-S_{\mathrm{BH}}}$, and the effective Hilbert space dimension is truncated to $e^{S_{\mathrm{BH}}}$. This leads to a transition from non-factorizing (Type III) to factorizing (Type I) von Neumann algebras for the bulk observables (Banerjee et al., 2024, Boruch et al., 2024, Balasubramanian et al., 2024).

The locus of "factorization" in the trace of two-sided observables,

$$\operatorname{Tr}_{LR}[f(H_L)g(H_R)] = \operatorname{Tr}_L[f(H_L)]\,\operatorname{Tr}_R[g(H_R)]$$

implies tensor-product structure and the full decoupling of left and right sectors at the microscopic level, matching the Bekenstein–Hawking entropy bound for black holes (Balasubramanian et al., 2024, Boruch et al., 2024).

5. Symmetry, Non-Factorization, and Topological Structures

Gauging higher-form global symmetries (especially non-invertible ones) in the bulk eliminates wormhole contributions and enforces exact factorization of boundary partition functions. In 3d Chern–Simons models, gauging the one-form symmetry trivializes the bulk Hilbert space and projects onto a unique boundary RCFT; failure to do so manifests as ensemble averaging or non-factorizing answers (Benini et al., 2022). More generally, "SymTFT entanglement" arises when multiple CFTs share a common symmetry subcategory, leading to entangled topological boundary conditions that explicitly violate holographic factorization. Such S-entanglement is realized by gauging a shared symmetry, leading to bulk global symmetries (e.g., unsplittable Wilson lines) and superselection of the baby-universe type. These non-factorizing setups naturally reproduce ensemble-averaged behavior, $\alpha$-state structure, and refined dualities for eternal black holes (Torres et al., 7 Oct 2025).

The following table summarizes the effects of specific mechanisms on factorization, wormhole contributions, and the Hilbert space structure:

Mechanism	Factorization Status	Closed Universe Hilbert Space
Single fixed theory	Strictly factorizing	One-dimensional; unique Hartle–Hawking state
Ensemble averaging (worms)	Violated (due to wormholes/ensemble correlations)	Infinite-dimensional; smooth wave functions
Gauging one-form symmetry	Factorization restored, wormholes eliminated	Single-state; modular-invariant boundary CFT
S-entanglement (SymTFT)	Non-factorizing, superselection over sectors	Dimension equals order of symmetry averaging

6. Matrix and SYK Models: Factorization at Large N

Recent toy model analyses (SYK, quantum matrix models) support the universality of these mechanisms. In fixed-coupling SYK or matrix models, correct factorization of partition functions requires the inclusion of an erratic "half-wormhole" saddle along with the usual wormhole contribution: $$Z_L(j)Z_R(j) \approx \langle Z_L Z_R \rangle_J + \Phi(j)$$ where $\Phi(j)$ fluctuates with the couplings, vanishes under ensemble averaging, and precisely cancels the non-factorizing contributions for typical couplings (Mukhametzhanov, 2021). The structure is robust across deterministic and random ensembles under large $N$ or large matrix size limits, demonstrating that factorization requires careful accounting of both connectivity in spacetime and couplings in the dual boundary description.

7. Emergent Factorization, Quantum Graphs, and Algorithmic Structure

Emergent factorization phenomena have also been formalized in the collective dynamics of large $N$ matrix models, where the low-energy Hilbert space splits into distinct Fock spaces ("boxes" and "anti-boxes"), providing an operational model for black hole complementarity, thermofield doubles, and the formation of islands in the Page curve scenario (Jevicki et al., 2024).

On the algorithmic side, the "holographic transformation" for classical and quantum factor graphs unifies the language of partition function factorization, recasting intractable sums as tractable linear algebraic objects. The quantum Holant theorem provides a mechanism for factorization in polynomial time for special cases and clarifies when message-passing schemes or Gaussian integral techniques yield exact results (Mori, 2014).

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In summary, factorizing holographic theory elucidates the algebraic and physical conditions under which the quantum gravitational path integral respects or violates strict factorization, tying together ensemble averaging, wormhole corrections, random matrix theory, global symmetries, and the microstate structure of black holes. Resolution of the factorization puzzle is central to the consistency of AdS/CFT, the reality of black hole microstates, and the emergence of semiclassical spacetime locality.