Three-dimensional Einstein gravity and Chern-Simons-Higgs matter both admit Chern-Simons descriptions, but their holonomies encode different physics. We prove that charge conjugation reverses the vortex winding $n\to-n$ while leaving the metric and the gravitational connections pointwise unchanged, so every closed gravitational holonomy character, the BTZ charges, the Bañados data and the leading geometric mixed-state hierarchy are exactly even in $n$. On a fixed marked Euclidean replica filling the same change of variables gives $Z_q[n;\mathcal J_+]=Z_q[-n;\mathcal J_+]$, so the uncharged Rényi hierarchy, the Faulkner-Lewkowycz-Maldacena one-loop correction, the renormalized generalized entropy and the quantum extremal surfaces coincide for the two orientations. The complementary sector is fixed exactly: with an Aharonov-Bohm angle $μ$ conjugate to the entangling-region charge, $Z_q[n;μ]=Z_q[-n;-μ]$, so the symmetry-resolved spectrum obeys $\mathcal Z_q(\mathcal Q;n)=\mathcal Z_q(-\mathcal Q;-n)$ and every odd charged cumulant reverses with the winding. The orientation is therefore stored entirely in the charge asymmetry of the entanglement spectrum, and is read at topological order by a linked charged matter line. The pair is a bulk logical qubit on which the neutral gravitational-replica algebra acts as a multiple of the identity. Exact BTZ cross sections show that the purification plateau and the entanglement wedge disconnection transition are orientation blind, the latter closing with the cutoff-independent jump $(c/3)\log(1+\sqrt2)$. Numerical Bogomolny solutions verify odd flux and even energy and spin; used as linearized charge-matching data they give the even asymptotic shifts $δ\widehat M=16πGv^2|n|$ and $|δ\widehat j|=16πGv^2n^2/(mL)$ and, conditionally, a finite continuous winding envelope.
Three-dimensional Einstein gravity and Chern-Simons-Higgs matter both admit Chern-Simons descriptions, but their holonomies encode different physics. We prove that charge conjugation reverses the vortex winding n→−n while leaving the metric and the gravitational connections pointwise unchanged, so every closed gravitational holonomy character, the BTZ charges, the Bañados data and the leading geometric mixed-state hierarchy are exactly even in n. On a fixed marked Euclidean replica filling the same change of variables gives Zq[n;J+]=Zq[−n;J+], so the uncharged Rényi hierarchy, the Faulkner-Lewkowycz-Maldacena one-loop correction, the renormalized generalized entropy and the quantum extremal surfaces coincide for the two orientations. The complementary sector is fixed exactly: with an Aharonov-Bohm angle μ conjugate to the entangling-region charge, Zq[n;μ]=Zq[−n;−μ], so the symmetry-resolved spectrum obeys Zq(Q;n)=Zq(−Q;−n) and every odd charged cumulant reverses with the winding. The orientation is therefore stored entirely in the charge asymmetry of the entanglement spectrum, and is read at topological order by a linked charged matter line. The pair is a bulk logical qubit on which the neutral gravitational-replica algebra acts as a multiple of the identity. Exact BTZ cross sections show that the purification plateau and the entanglement wedge disconnection transition are orientation blind, the latter closing with the cutoff-independent jump (c/3)log(1+2). Numerical Bogomolny solutions verify odd flux and even energy and spin; used as linearized charge-matching data they give the even asymptotic shifts δM=16πGv2∣n∣ and ∣δj∣=16πGv2n2/(mL) and, conditionally, a finite continuous winding envelope.
Black-hole hair can affect boundary mixed-state entanglement through local stress-energy and through topological information invisible to the classical metric. We separate these channels for a Nielsen--Olesen vortex on a nonrotating BTZ black hole. A superselection theorem shows that Shannon sector entropy cancels from the Markov gap, while standard U(1) symmetry-resolved reflected entropy in a thermal U(1)k CFT is equipartitioned at leading order, excluding a universal log∣n∣ term in the imbalance-resolved gap. We therefore define a Wilson-threaded reflected moment in a probe U(1)k Chern--Simons completion coupled to the compact vortex-flux class. In a reflected-replica sector with linking number ν, its normalised phase is 2πκpnν/k, where κ∈Z is the mixed topological coupling; for gcd(κν,k)=1, a discrete Fourier transform reconstructs nmodk. The RT connectivity transition switches the specified linked contour on or off, whereas the independent scale ℓ⋆r+/L2=1.128378… determines the direction of the vortex-induced shift of that transition. The charged-moment modulus and the ordinary Markov gap remain geometric observables and are computed from a horizon-anchored Einstein-Abelian-Higgs solution with a complete first-variation kernel including the motion of the entanglement-wedge cross-section endpoints. This phase and modulus separation distinguishes topological hair from gravitational dressing without assigning an unsupported winding-dependent entropy.
A bstract We consider single-particle excitations of the massive Chern-Simons field of mass M in AdS 3 and evaluate their contribution at the first sub-leading order in G N to the entanglement entropy across the Ryu-Takayanagi surface. Quantizing the Chern-Simons field in AdS 3 , we evaluate the corrections to the holographic entanglement entropy using the Faulkner-Lewkowycz-Maldacena formula. The massive Chern-Simons field also obeys the equations of motion of a massive vector in AdS 3 . The lowest-energy single-particle excitation of this field is dual to the primary operator of conformal dimension M + 1 with spin one in the dual CFT; all other single-particle excitations are dual to its global descendants. We compare the entanglement entropy result from the FLM formula to the single-interval entanglement entropy in large-charge holographic CFT obtained using the replica trick for the primary and its tower of holomorphic descendants. The two results agree precisely in the leading and sub-leading terms of the short interval expansion. We evaluate the contribution of the edge mode to the vacuum-subtracted entanglement and show that it vanishes, which is crucial for the FLM formula to agree with the CFT result. On taking the massless limit, the result coincides with the contribution of a U (1) current to the single interval entanglement entropy. This is surprising since an earlier calculation in the literature reproduced the CFT result entirely from the edge U (1) degrees of freedom on the RT surface.
Abstract We consider symmetry-resolved entanglement entropy in AdS3/CFT2coupled to U(1) Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS3, namely the Ryu-Takayanagi geodesic minimally coupled to the U(1) Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged U(1) vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the U(1) Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincaré patch and global AdS3, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons levelk, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of u^(1)k û1k Kac-Moody type. Employing the u^(1)k û1k Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, K, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3 we find that the K-dependent holographic entanglement entropy is governed by cm=2GN3ℓ. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=2GN3ℓ2Kℓ−K2ℓ2−4, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉ like operator - and find SEE=3ceffln(ε2Rsinϕ0). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceff while clarifying that this state is not the state dual to global AdS.
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
We propose that spacetime is five-dimensional — (x, y, z, t, e) — where the fifth dimension e is circular, periodic, and corresponds to the U(1) fiber in a principal bundle over four-dimensional spacetime. Our four-dimensional experience is a projection of this five-dimensional reality, and every quantum phenomenon is a geometric consequence of the projection. Within this framework, superposition is extension in the e-dimension, entanglement is conservation of the e-coordinate (e₁ + e₂ = C), momentum is helical pitch (∂e/∂x = p/ℏ), spin is helical chirality, and measurement is e-space sampling. The wave function is not an abstract probability amplitude — it is the literal shape of the particle in five-dimensional spacetime. We present three original results. First, the Aharonov-Bohm effect is derived as the holonomy of the e-bundle around a topological defect, establishing the physical reality of the vector potential from geometry. Second, the spin-statistics theorem is derived from a single geometric fact: the winding number of a particle's helix through the circular e-dimension determines both its spin (via the Noether theorem) and its exchange statistics (via parallel transport). Integer winding gives bosons; half-integer gives fermions; fractional winding in two dimensions gives anyons — confirmed experimentally in the fractional quantum Hall effect. Third, the Born rule P(i) = |αᵢ|² is reproduced within the five-dimensional density interpretation and the projection postulate, and the framework reproduces the quantum correlations that violate the CHSH Bell inequality (|S| = 2√2), with the non-locality sourced by the e-conservation constraint through the fifth dimension. The same compact e-circle also resolves the central obstacle to quantum gravity. The five-dimensional Einstein-Hilbert action on the e-bundle reduces via Kaluza-Klein decomposition to four-dimensional gravity coupled to electromagnetism and a scalar field in a single metric. The compactness of the e-circle converts the continuous UV momentum integrals of 4D quantum gravity into discrete Kaluza-Klein mode sums that admit zeta regularization. Under zeta regularization of the KK mode sums, the leading UV divergence vanishes identically at every loop order: S₀^{(L)} = [1 + 2ζ(0)]^L = 0. The subleading terms are Epstein zeta functions evaluated at non-positive integers. For the two-loop R³ counterterm of linearized 5D gravity on M⁴ × S¹ — corresponding to the Goroff-Sagnotti divergence that proved 4D Einstein gravity non-renormalizable in 1986 — the result under zeta regularization is stronger than expected: the R³ coefficient is identically zero at every order in the mass expansion, not just at leading order. The sunset topology produces the Epstein zeta of the Eisenstein lattice, E₂(s; n²+nm+m²) = 6ζ(s)L(s,χ₋₃), which vanishes at every negative integer through complementary trivial zeros of ζ(s) and L(s,χ₋₃). The figure-eight and vertex correction topologies vanish independently through the trivial zeros of the Riemann zeta at even negative integers (forced by the perfect-square structure of KK masses). Ghost contributions, carrying the same Eisenstein quadratic form through KK number conservation, vanish by the same mechanisms. Under zeta regularization of the KK mode sums, linearized 5D gravity on M⁴ × S¹ is perturbatively predictive — established through two loops by explicit computation, and conjectured to all orders by the Epstein-Terras structure — with counterterm coefficients determined from two parameters: G₄ and the e-circle circumference L. The spin structure of the e-circle (bosons periodic, fermions anti-periodic) implies a natural Z₂ orbifold with two fixed-point branes. The visible brane at φ = 0 supports Standard Model matter; the hidden brane at φ = π gravitates normally but couples to no SM force — geometric dark matter. The Casimir energy of bulk fields on the orbifold, with three right-handed neutrinos in the bulk to ensure positive energy, matches the observed dark energy density for a brane separation R ≈ 12 μm and predicts Yukawa-type gravitational deviations at that scale. The same bulk neutrinos give neutrino masses at the meV scale through the bulk seesaw mechanism. Kinetic mixing between the two branes' U(1) gauge fields is predicted at ε ~ 5 × 10⁻⁴ (from the KK tower mediation: α_EM × π²/6 × exp(−π)), testable by LDMX and LHCb Run 3. A conjectured Z₃ orbifold structure gives three SM generations, with conjectured exponential mass hierarchies from a warp factor k ≈ 2. A speculative identification of the Planck-scale electromagnetic coupling with the inverse configuration torus area, 1/α(M_P) = 4π², combined with three generations of SM running, gives 1/α(0) ≈ 137. The e-circle, Z₆-orbifolded and warped, provides a geometric account of twenty-two phenomena at three levels of rigor. Eight are derived from the basic e-circle (M⁴ × S¹): quantum mechanics, electromagnetism, gravity, the spin-statistics theorem, perturbative UV finiteness, the hydrogen spectrum, black hole entropy, and the CPT theorem. Eight more follow from the Z₂ orbifold extension: dark energy (Casimir energy), dark matter (hidden-brane matter), neutrino masses (bulk seesaw), kinetic mixing with a dark photon (ε ~ 5 × 10⁻⁴), the strong CP problem (topological resolution: π₄(SU(3)) = 0 in 5D), the Hubble tension (H₀ ≈ 68.7–69.5 km/s/Mpc from hidden-brane dark radiation; in 3–4σ tension with ACT DR6 N_eff = 2.86 ± 0.13), the Casimir effect, and baryon asymmetry (derived in Paper 2 via bulk leptogenesis). Six are conjectured: the dark energy equation of state (thawing dilaton, derived quantitatively in Paper 2), the fine structure constant α ≈ 1/137 (from the configuration torus area), α stability, three fermion generations (Z₃ orbifold), the fermion mass hierarchy (warp factor k ≈ 2), and normal neutrino mass ordering (from Z₃ geometry, testable by JUNO). Seven testable predictions follow: two are already consistent with current data (observed α stability, and the dark energy equation of state w ≈ −1 — though DESI DR2 shows 4.2σ tension with static w = −1, resolvable via thawing dilaton quintessence). The naive dilaton prediction (ΔN_eff ≈ 0.57) appeared in tension with combined ACT+SPT+Planck measurements (N_eff = 2.81 ± 0.18), but this tension is resolved by intra-tower KK neutrino decays (Gonzalo et al. 2024), reducing the effective ΔN_eff to ~0.05 and giving N_eff ≈ 3.09, consistent with all CMB data. The remaining four — short-range gravitational deviations at ~12 μm, the dark photon signal at ε ~ 5 × 10⁻⁴, neutrino masses and normal mass ordering (testable by JUNO within 6 years), and mirror dark matter — are within reach of experiments in the next 3–10 years. The orbifold extensions are speculative but quantitative, with specific falsifiable predictions. What is established is a geometric framework in which the same compact circle that makes quantum mechanics necessary also makes perturbative quantum gravity finite. A companion computation (Paper 2) applies the CAMB Boltzmann solver to the framework's cosmological sector (to appear separately) with parameters derived entirely from the e-circle geometry — fitting zero parameters to CMB data. The central result is a scaling law from bulk leptogenesis on the Z₂ orbifold: the three bulk right-handed neutrinos deposit baryon asymmetries on both branes, but the hidden brane is colder (ξ = T_hidden/T_visible). The entropy asymmetry (1/ξ³) and washout suppression (1/ξ²) combine to give: Ω_DM / Ω_b = 1/ξ² Inverting against the observed ratio 5.36 determines ξ = 0.432 at leading order; the washout correction shifts this to ξ ≈ 0.49, converging with the independently θ*-matched value ξ = 0.47. Three scenarios bracket the prediction. The CAMB-computed results: t₀ = 13.47–13.60 Gyr, H₀ = 68.7–69.5 km/s/Mpc, S8 = 0.753–0.785 (resolving the 4σ weak lensing tension without tuning), r_d = 145.8–148.0 Mpc, and the CMB angular scale θ* matched to within 0.5–6.6 arcseconds of Planck's measurement. The thawing dilaton (w₀ = −0.85, w_a = −0.23) drives the expansion history H(z) to peak 4–6% above ΛCDM at z ~ 0.3–0.7 — a specific fingerprint testable by DESI DR3 at 8σ. The cosmic coincidence Ω_DM/Ω_b ≈ 5 is not a coincidence: it is 1/ξ², where ξ is fixed by geometry and baryogenesis. All three scenarios predict N_eff = 3.31–3.39, in 3–4σ tension with ACT DR6 (N_eff = 2.86 ± 0.13) — the framework's primary open issue, definitively resolved by CMB-S4 (σ(N_eff) ≈ 0.03).