Topology sums, sectorwise holography, and horizon normalcy

Abstract: The ``holography of information'' (HoI) principle argues that gravity can encode information redundantly in asymptotic observables. Although HoI is ultimately a nonperturbative claim, its standard motivation uses semiclassical gravitational constraints, the boundary nature of the Hamiltonian, and vacuum-sector cyclicity. We ask what happens when the same semiclassical path-integral reasoning allows topology sums that generate baby-universe or $α$-sector data. Our analysis is conditional: such sectors need not survive in every unitary completion, and the Baby Universe Hypothesis of McNamara and Vafa instead suggests $\dim\mathcal H_{\rm BU}=1$ in consistent $d>3$ quantum gravity. If $\mathcal H_{\rm BU}$ is nontrivial, as in the Marolf--Maxfield formulation and in ensemble-like examples such as JT gravity, then HoI is naturally refined to an $α$-sectorwise statement, $\overline{\mathcal A_\infty^{(α)}|0_α\rangle}=\mathcal H_α$, rather than completeness on the full topology-summed Hilbert space. In a fixed $α$-sector, HoI may obstruct AMPS factorization and allow a smooth horizon; in an unconditioned topology-summed state, the sector-independent obstruction is not automatic. A Bell-pair diagnostic shows that a sector-independent smooth interior requires aligned interior reconstructions, or access to the sector label. Thus the HoI-based absence of firewalls becomes conditional on global sector data, in tension with the generally covariant expectation emphasized by Bousso that horizon normalcy should be determined by local semiclassical geometry. If the exact theory collapses $\mathcal H_{\rm BU}$ to one dimension, the obstruction discussed here is absent.

• The paper formulates sectorwise holography by showing how topology sums induce baby-universe sectors that obstruct global holographic information completeness.
• It demonstrates that non-factorizing amplitudes from topology sums prevent a universal Hilbert space factorization, making horizon normalcy conditional on sector alignment.
• Using JT gravity as a diagnostic example, the study highlights implications for firewall resolutions and the structural decomposition of gravitational Hilbert spaces.

Topology Sums, Sectorwise Holography, and Horizon Normalcy: A Formal Analysis

Holography of Information: Representation-Wise Completeness

The holography of information (HoI) principle claims that gravitational theories manifest a distinct Hilbert-space structure compared to local quantum field theory, enabling asymptotic observables to redundantly encode bulk information. This principle aims to undermine the factorization premise in the AMPS paradox by asserting that the asymptotic boundary algebra is complete within the physical Hilbert space associated with a given vacuum sector. Formally, the completeness statement is cyclicity: $H_{\mathrm{vac}} = \mathcal{A}_{x}|0\rangle$, where $\mathcal{A}_{x}$ is the asymptotic algebra. This framework leverages the boundary nature of the gravitational Hamiltonian and semiclassical constraints.

The argument is inherently representation-wise, with assumptions on the identity and accessibility of vacua near null infinity, and the spectral properties of the Hamiltonian. Standard AdS/CFT and asymptotically flat settings reinforce this sectorwise completeness, referencing the Reeh-Schlieder theorem as adapted to gravitational systems.

Topology-Summed Hilbert Spaces and Sector Obstruction

The paper scrutinizes the scenario in which the gravitational path integral includes topology sums, potentially generating baby-universe or $\alpha$-sector superselection structure. The central question is whether HoI completeness automatically extends from the vacuum sector to the full topology-summed Hilbert space. The analysis distinguishes two sector notions: ordinary superselection tied to boundary conditions, and those stemming from topology sums (e.g., baby universes).

A trichotomy of nonperturbative interpretations is delineated:

1. Nontrivial Baby-Universe Hilbert Space: Topology sums induce genuine superselection sectors, with the asymptotic algebra acting within but not across $\alpha$-sectors. The Marolf-Maxfield construction exemplifies this framework, where boundary-insertion operators commute, are diagonalized by $\alpha$-states, and amplitudes factorize only in fixed $\alpha$-states [5].
2. Collapse to a Single Sector: The exact theory, via the Baby Universe Hypothesis (McNamara-Vafa), restricts $\dim H_{\mathrm{BU}} = 1$ for $d > 3$, eliminating baby-universe sectors in unitary quantum gravity [6].
3. Effective/Open-System Description: The topology-summed path integral is not the Hilbert space of a closed unitary theory, as in JT gravity or AdS coupled to a bath.

If nontrivial $\alpha$-sector structure survives, HoI completeness is refined to a sectorwise statement: $\mathcal{A}_\alpha |0_\alpha\rangle = H_\alpha$ for each sector, rather than a global statement over $\mathcal{A}_{x}$0. The sector-preserving asymptotic algebra cannot connect distinct sectors, as shown through operator commutator analysis. Consequently, global HoI completeness is obstructed unless sector alignment, projection, or access to sector labels is achieved.

Diagnostic Example: JT Gravity and Sector Structure

JT gravity demonstrates the canonical sectorwise Hilbert-space structure arising from topology sums. Matrix integral averages, Euclidean wormholes, and the resulting non-factorizing amplitudes reveal the embedding of baby-universe Hilbert space into the gravitational description [11, 12]. In a fixed $\mathcal{A}_{x}$1-sector, amplitudes factorize, but the Hartle-Hawking "no-boundary" state induces an ensemble decomposition over sectors. The asymptotic algebra acts only within these sectors, confirming sectorwise completeness and algebraic obstruction to global HoI.

Sectorwise HoI and Conditional Horizon Normalcy

The primary physical consequence pertains to black hole interiors. The HoI argument seeks to evade the firewall dilemma in AMPS by denying that interior and exterior are independent tensor factors. Within each fixed $\mathcal{A}_{x}$2-sector, this logic holds; HoI completeness restricts factorization, enabling a smooth horizon.

However, in the full topology-summed Hilbert space, completeness is not automatic. The sectorwise construction, relying on baby-universe superselection, necessitates conditioning on $\mathcal{A}_{x}$3, sector alignment, or sector resolution for sector-independent horizon smoothness. Without such mechanisms, horizon normalcy—expected as a local, generally covariant property—is rendered representation-dependent, contravening Bousso’s assertion that normalcy should be state-independent and governed solely by local semiclassical geometry [8].

The paper formalizes this conditionality through a Bell-pair fidelity diagnostic. In the presence of sector-dependent interior reconstructions, the fidelity with sector-independent smooth states can be strictly less than unity, and may vanish if sector reconstructions form a unitary one-design. Thus, sectorwise smoothness does not guarantee the existence of a sector-independent smooth horizon, quantitatively undermining the local validity of semiclassical geometry for horizon-crossing events.

Implications and Theoretical Consequences

The analysis reveals that semiclassical saddle expansions do not by themselves exclude non-factorizing wormholes and associated baby-universe sectors; global input from the nonperturbative definition (exact CFT, contour restrictions, null-state quotients) is required. In standard AdS/CFT, such constraints may exclude baby universes, reconciling HoI completeness in the relevant representation.

If the exact theory collapses the baby-universe Hilbert space to one dimension, as conjectured by McNamara and Vafa, the sectorwise obstruction disappears and HoI remains globally valid. Otherwise, a sectorwise HoI statement is insufficient for a universally covariant, state-independent resolution of the firewall paradox. The validity of horizon normalcy (and the logical status of firewalls) becomes conditional upon a nonperturbative sector structure, the accessibility of sector labels, or the alignment of sector-dependent interior reconstructions.

Speculation on Future Developments

Advancement in understanding whether topology-summed path integrals in higher-dimensional gravity induce genuine baby-universe sectors or collapse to a single sector will clarify the fate of sectorwise HoI and the universality of horizon normalcy. Greater integration between semiclassical analysis, top-down holographic constraints, and operator algebra frameworks is required to address the interplay of sector structure and information holography. Practical implications include refined criteria for quantum gravity models, potential constraints on swampland theories, and deeper insight into Hilbert-space decomposition in the presence of spatial topology fluctuations.

Conclusion

This work rigorously examines the intersection of the holography of information principle and sector structure arising from topology sums in quantum gravity. Sectorwise HoI completeness, induced by baby-universe superselection, makes the criterion for horizon normalcy conditional on global sector data, undermining generally covariant, state-independent local geometry. Sector-independent avoidance of firewalls is only achievable if sector labels are accessible, projected out, or interior alignments are universally maintained. The outcome hinges on the nonperturbative Hilbert-space structure dictated by the exact quantum gravity theory; in closed unitary theories lacking nontrivial sectors, these obstructions vanish. Otherwise, additional input is essential to restore universally covariant horizon normalcy and a robust global completeness principle in gravitational theories.