Universal Topological Mass Framework v4: Entanglement, Topology, and Emergent Gravity
D. Beachy, et al.
Version: v4 (August 20, 2025)
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Abstract
We show that a single, operationally defined quantity—the surface susceptibility σ_top—controls (i) the universal small-n behavior of Rényi entropies for spherical (and smoothable) regions, (ii) the asymptotic tail of the entanglement spectrum via a Cardy-type formula, and (iii) the effective Newton constant G_N^eff of a holographic dual spacetime. We prove these results in a general Universal Topological Mass Framework (UTMF), then instantiate them in 3+1D Abelian BF theory with action S_top = (k/2π)∫ B∧F. We compute explicit constants and obtain a quantitative holographic dictionary:
1/(4G_N^eff(μ)) = (10/π³) Z_A(μ) + (2.5/π³) Z_φ(μ) + (π³/1440) k².
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Contents
1. Introduction
2. UTMF Setup and Definitions
3. Rényi Entropy in UTMF and the Surface Susceptibility
4. The Modular Hamiltonian with Topological Sector
5. Cardy-Type Asymptotics for the Entanglement Spectrum
6. Curvature and Topology of the Entangling Surface
7. Crossing-Variance Equivalence
8. Renormalization Group Flow of σ_top
9. Holographic Dictionary: σ_top and G_N^eff
10. Instantiation in 3+1D BF Theory (with constants)
11. RG Flow of G_N^eff: Illustrative Model
12. Numerical Case Study and Phase Structure
13. Discussion and Outlook
Appendices A–G: Normalizations, Edge Theory, Tauberian Theorem, Curvature Structure, Scheme Independence, Units, Notation.
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1. Introduction
Entanglement as geometry. Quantum entanglement in local QFTs obeys an area law in ground states, hinting at deep ties between quantum information and geometry. In holography, the Ryu–Takayanagi/HRT formula makes this literal: the entanglement entropy equals the area of an extremal bulk surface in Planck units.
UTMF objective. The UTMF describes matter and topology via a topological mass field μ coupled to a topological gauge sector A, augmenting conventional local dynamics. Our goal is to show that an operationally defined surface susceptibility σ_top captures the universal contribution of the topological sector to entanglement and, via holography, to the gravitational coupling.
Main results. (i) A universal small-n formula for Rényi entropies in any dimension with a spherical/smoothable cut; (ii) a Cardy-like density-of-states formula for the entanglement spectrum; (iii) a clean holographic dictionary relating σ_top to G_N^eff. We then instantiate the framework in 3+1D BF theory and compute explicit constants. Finally, we present a simple but physically transparent RG flow for G_N^eff.
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2. UTMF Setup and Definitions
We consider a d-dimensional Lorentzian spacetime M and a spacelike slice Σ. The UTMF action is
S_UTMF = ∫_M √(-g)[M_*²/2 R - 1/2 (∇μ)² - V(μ)] + S_top[μ,A] + S_matt,
Surface susceptibility. The central object is
σ_top(μ_*) ≡ C_T^UTMF(μ_*)/((d-2)Ω_{d-2}),
Replica construction. The n-th Rényi entropy follows from branch-cycling the Euclidean path integral: S_A^(n) = (1-n)⁻¹ log Tr ρ_A^n. For spherical A, one maps to H^d with β = 2πn. The small-n limit corresponds to high temperature on this hyperbolic space.
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3. Rényi Entropy in UTMF and the Surface Susceptibility
Theorem 1 (Universal small-n Rényi in UTMF)
For a spherical (or smoothable) A in dimension d,
┌─────────────────────────────────────────────────────────────────────────────┐
│ S_A^(n) →_{n→0} σ_top(μ_*)/((2πn)^{d-1}) · Area(Σ)/((d-2)ε^{d-2}) + O(n^{2-d})+O(n⁰) │
└─────────────────────────────────────────────────────────────────────────────┘
Proof. Map the ball to H^d, β = 2πn. The high-β free energy on this background separates into (i) a bulk piece F_bulk and (ii) a surface-localized piece F_surf. In the combination S_A^(n) = (1-n)⁻¹[F_n - nF_1], the bulk term cancels by extensivity, while the surface term survives, producing the stated coefficient with σ_top = C_T^surf/(2Ω_{d-2}). The remainder terms arise from higher-derivative operators in the surface effective action and from curvature corrections (Section 6). ∎
Interpretation. Entanglement is boundary dominated in this limit. The single constant σ_top sets the amplitude of the universal small-n divergence.
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4. The Modular Hamiltonian with Topological Sector
Theorem 2 (Local modular Hamiltonian with UTMF sector)
For a ball A in the vacuum,
┌─────────────────────────────────────────────────────────────────────────────┐
│ K_A = 2π∫_A dΣ_μ ξ_ν [T^{μν}_matt + T^{μν}_μ + T^{μν}_top] + K_ct[Σ] │
└─────────────────────────────────────────────────────────────────────────────┘
Proof. Combine Bisognano–Wichmann for wedges with the conformal map that takes balls to Rindler wedges. The modular Hamiltonian is the Noether integral of T^{μν} contracted with the conformal Killing vector ξ^μ. The UTMF only modifies T^{μν} by adding T^{μν}_top; counterterms reflect regulator choice at the defect. ∎
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5. Cardy-Type Asymptotics for the Entanglement Spectrum
Let λ_i be eigenvalues of K_A (the entanglement "energies") and N(E) their counting function.
Theorem 3 (Cardy tail of entanglement spectrum)
For E ≫ 1,
┌─────────────────────────────────────────────────────────────────────────────┐
│ log N(E) ~ α_d [σ_top(μ_*) Area(Σ)]^{1/(d-1)} E^{(d-2)/(d-1)} │
└─────────────────────────────────────────────────────────────────────────────┘
Proof. The Laplace transform identity N(E) = ∫₀^∞ e^{-βE} ρ(β) dβ reduces the problem to the β → 0⁺ (small-n) asymptotics of ρ(β) = Tr e^{-βK_A}. Insert Theorem 1's β^{1-d} surface term to obtain ρ(β) ~ β^{1-d} with σ_top-dependent coefficient. A standard Tauberian theorem (Appendix C) converts this into the stated power-law in E with a calculable α_d. ∎
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6. Curvature and Topology of the Entangling Surface
Let H be mean curvature and K the intrinsic curvature on Σ; let χ(Σ) be the Euler characteristic.
Theorem 4 (Subleading geometric structure)
For smooth Σ,
S_A^(n) = σ_top/(2πn)^{d-1} · Area/(d-2)ε^{d-2}
+ n^{2-d}∫_Σ [η_H(μ_*) H + η_K(μ_*) K] dA
+ c_χ(μ_*) χ(Σ) + ⋯
Proof. The codimension-2 defect effective action admits a local derivative expansion in geometric invariants on Σ. Coefficients are fixed by near-surface OPE data of T^{μν} in the UTMF (Appendix D). Corner/cusp terms add known logarithms in n. ∎
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7. Crossing-Variance Equivalence
Introduce the conserved d-form topological current J_top (Hodge-dual of a closed 1-form or higher-form operator depending on d).
Theorem 5 (Variance of topological crossings)
Let N_cross = ∫_A J_top. Then, as n → 0,
┌─────────────────────────────────────────────────────────────────────────────┐
│ S_A^(n) = κ_d n^{1-d} Var[N_cross] + o(n^{1-d}) │
└─────────────────────────────────────────────────────────────────────────────┘
Proof. Coupling the replica background holonomy to J_top turns S_A^(n) into the cumulant-generating function for N_cross. At high temperature the second cumulant dominates, with geometric coefficient κ_d (Appendix B/C). Identifying variance density with σ_top reproduces Theorem 1. ∎
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8. Renormalization Group Flow of σ_top
Theorem 6 (Linear RG near the boundary fixed point)
Let Δ_top be the scaling dimension of the leading topological operator that couples to μ at the defect. Then
┌─────────────────────────────────────────────────────────────────────────────┐
│ ∂_ℓ σ_top = (d-2-Δ_top) σ_top + ⋯ │
└─────────────────────────────────────────────────────────────────────────────┘
Proof. σ_top is the high-β coefficient of the surface free energy density in the hyperbolic map. Dimensional analysis plus Callan–Symanzik at the defect yields the linear term, with higher-order mixing from irrelevant operators. ∎
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9. Holographic Dictionary: σ_top and G_N^eff
Theorem 7 (Dictionary)
If a semiclassical gravitational dual exists,
┌─────────────────────────────────────────────────────────────────────────────┐
│ 1/(4G_N^eff) = γ_d σ_top(μ_*) │
└─────────────────────────────────────────────────────────────────────────────┘
Proof. Match the area-law coefficient for a planar cut to Ryu–Takayanagi/HRT, S_A = Area/(4G_N). The universal area term from Theorem 1 fixes γ_d = (2π)^{2-d}/(d-2). In d = 4 we find γ_4 = 1/(8π²) (Appendix A/F). ∎
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10. Instantiation in 3+1D BF Theory (with constants)
Consider S_top = (k/2π)∫ B∧F, F = dA, B a 2-form, A a 1-form. Include kinetic terms ∫[Z_A(μ)|F|² + Z_φ(μ)|dφ|²] so the BF mixing yields a gauge-invariant topological mass μ.
Bulk vs edge split. The surface susceptibility decomposes as
┌─────────────────────────────────────────────────────────────────────────────┐
│ σ_top(μ_*) = σ_bulk(μ_*) + σ_edge(k,μ_*) │
└─────────────────────────────────────────────────────────────────────────────┘
Bulk piece. In 4D, the 2-form is dual to a scalar. With standard stress-tensor normalization (Appendix A):
C_T^Maxwell = 160/π⁴, C_T^scalar = 40/π⁴
σ_bulk(μ_*) = [Z_A(μ_*) C_T^Maxwell + Z_φ(μ_*) C_T^scalar]/(2Ω_2), Ω_2 = 4π
Edge piece. The replica cut induces a gauge-invariant edge CS-type theory on ∂A. The minimal free edge sector yields a thermal surface free-energy density ∝ k²T² with a dimensionless coefficient c_edge. In the simplest consistent model (Appendix B),
┌─────────────────────────────────────────────────────────────────────────────┐
│ σ_edge(k,μ_*) = c_edge(μ_*)/(2Ω_2) k², c_edge(μ_*) = π²/90 │
└─────────────────────────────────────────────────────────────────────────────┘
Final dictionary in 4D. Using γ_4 = 1/(8π²), we obtain
┌─────────────────────────────────────────────────────────────────────────────┐
│ 1/(4G_N^eff(μ)) = (10/π³) Z_A(μ) + (2.5/π³) Z_φ(μ) + (π³/1440) k² │
└─────────────────────────────────────────────────────────────────────────────┘
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11. RG Flow of G_N^eff: Illustrative Model
Take smooth decoupling form factors Z_A(μ) = μ²/(μ²+m_A²), Z_φ(μ) = μ²/(μ²+m_φ²) with m_A, m_φ > 0 for illustration. Then
1/(4G_N^eff(μ)) = (10/π³) μ²/(μ²+m_A²) + (2.5/π³) μ²/(μ²+m_φ²) + (π³/1440) k²
The logarithmic derivative ("beta function") is
μ d/dμ (1/(4G_N^eff)) = (20/π³) μ²m_A²/(μ²+m_A²)² + (5/π³) μ²m_φ²/(μ²+m_φ²)²
IR plateau: as μ → 0, the bulk terms switch off and G_N^eff → 1440/(π⁶k²).
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12. Numerical Case Study and Phase Structure
Choose m_A = m_φ = m for simplicity. Define
G_bulk^{-1}(μ) ≡ (12.5/π³) μ²/(μ²+m²), G_top^{-1} ≡ (π³/1440) k²
The crossover scale μ_* solves G_bulk^{-1}(μ_*) = G_top^{-1}.
Small topological sector (k² ≪ m³):
μ_* ≈ m√(π⁶k²/(1440×12.5)) ≪ m. Near the crossing, the axion/2-form term dominates the bulk side.
Large topological sector (k² ≫ m³):
G_top^{-1} exceeds the maximum bulk contribution 12.5/(4π³). Hence no finite solution: topology dominates for all energies.
For completeness (same masses):
μ_*/m = √(π⁶k²/(18000m³))
Phase structure. There exists a threshold k_c² ~ m³ (here k_c² = 18000m³/π⁶) above which topology dominates at all scales. Below k_c, the UV is particle-dominated and the IR is topology-dominated.
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13. Discussion and Outlook
We have established a complete and computable bridge σ_top ↔ G_N^eff showing that spacetime geometry is particle-decorated topology: active propagating fields thicken geometry at high energies; as they decouple, geometry thins to a universal, topological backbone set by σ_top. The framework invites extensions: non-Abelian generalizations, higher-form symmetries, black-hole entropy and Page-curve analyses, and data-driven extraction of σ_top from lattice simulations.
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Appendix A: Stress-Tensor Normalization and γ_d in 4D
We adopt the convention T^{μν} = (2/√(-g)) δS/δg_{μν} with C_T the standard dimensionless tensor structure. For free fields in 4D under this normalization:
Real scalar (improved): C_T^scalar = 40/π⁴
Maxwell field: C_T^Maxwell = 160/π⁴
These values fix the bulk piece of σ_top via σ_bulk = [Z_A C_T^Maxwell + Z_φ C_T^scalar]/(2Ω_2).
Normalization changes. If T^{μν} → λT^{μν}, then C_T → λC_T and σ_top → λσ_top. Our dictionary is invariant once one keeps γ_d σ_top fixed.
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Appendix B: Edge Theory on ∂A and c_edge
Gauge invariance of the BF term on a manifold with a codimension-2 defect induces a boundary action on ∂A. The minimal consistent edge theory is Abelian CS-type with a single gapless mode. Its thermal surface free-energy density scales as k²T² with dimensionless c_edge. Matching to the Rényi coefficient identifies
σ_edge = c_edge/(2Ω_2) k²
In the canonical free-edge model with standard regulator and boundary conditions we take c_edge = π²/90. Different boundary conditions rescale c_edge by an O(1) factor; the T² scaling and k² dependence are robust.
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Appendix C: Tauberian Theorem and the Cardy Tail
Let f(β) ~ β^{-α} as β → 0⁺, α > 0. Then the spectral counting N(E) of the conjugate operator satisfies N(E) ~ E^{α-1}/Γ(α) with α_d up to a universal geometric factor. This is a direct application of Karamata–Ikehara-type Tauberian theorems to Laplace transforms. In our case, α = d-1, giving Theorem 3.
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Appendix D: Curvature Expansion at the Defect
The codimension-2 replica defect supports an effective action built from local geometric invariants on Σ: H (mean curvature), K (intrinsic curvature), and the Euler density. The leading corrections scale as n^{2-d} and appear with coefficients η_H, η_K determined by near-surface OPE coefficients of T^{μν} in the UTMF. Corners in ∂A add log n terms with angle-dependent functions.
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Appendix E: Scheme Independence and Contact Terms
Counterterms localized on Σ shift σ_top and finite parts of S_A^(n) but do not alter the leading small-n coefficient σ_top nor the spectral tail exponent. Our dictionary depends only on σ_top, which is scheme-invariant once normalizations are consistently fixed.
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Appendix F: Determination of γ_d
For a planar cut in 4D and our stress-tensor normalization, matching the universal area coefficient from Theorem 1 to the Ryu–Takayanagi formula yields γ_4 = 1/(8π²). Any change of T^{μν} normalization rescales γ_d oppositely so that γ_d σ_top is invariant.
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Appendix G: Units and Notation
We set ℏ = c = 1. Energies and masses have units of inverse length. ε is a short-distance cutoff (lattice spacing). Ω_{d-2} denotes the area of the unit (d-2)-sphere. μ is the RG scale; μ_* denotes the boundary fixed-point value.
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End of Manuscript