How do fermions perturb the Maldacena-Milekhin-Popov wormhole?

A crosswise parity pairing completes the perturbation theory

On a magnetically charged background, gravitational and electromagnetic perturbations pair crosswise: a polar metric perturbation couples to an axial gauge perturbation, and an axial metric perturbation couples to a polar gauge perturbation. A companion paper treated the first pairing. This paper treats the second, which carries the axial gravitational mode together with the electric field along the magnetic field lines. Together the two papers exhaust the l ≥ 2 perturbations of the Maldacena-Milekhin-Popov (MMP) wormhole.

The MMP construction joins two magnetically charged near-extremal Reissner-Nordström mouths of opposite charge by a long AdS₂ × S² throat. Massless charged fermions in the monopole background fill a lowest Landau level (LLL) of q modes, one per unit of flux, and move along the field lines as two-dimensional massless fermions. Their Casimir energy on the closed field lines holds the throat open. The expansion is controlled by a perturbation amplitude ε and by α = g_c²/4π²q, which measures the fermionic backreaction.

The fermion response here differs from the companion sector. In the companion sector the perturbation deforms the two-dimensional metric seen by the fermions on each field line, and the conformal anomaly fixes the response. Here that two-dimensional metric is untouched: the axial metric perturbation has no (τ, ρ) block and does not change the sphere radius. A two-dimensional stress tensor has no linear response to a gauge field by charge conjugation, so the fermions must respond through their charge instead.

The Schwinger current gives the electric field a screening mass

The gauge components a_τ and a_ρ, proportional to the spherical harmonic Y(θ), pull back to a two-dimensional gauge field on the worldsheet of every field line. The exact linear response of a massless two-dimensional charged fermion is the Schwinger current, written in bosonised form as j^a = (1/π) ε^{ab} ∂_b χ with □₂χ = E, where E is the worldsheet electric field.

Summing over field lines and lifting to four dimensions gives the electric field a mass m_S² = g_c² q / 4π² = αq². The factors are explicit: q/4π is the line density, 1/π normalises the bosonised current, and g_c² is the gauge coupling. The same mass follows from dimensional reduction, where the Maxwell term integrated over a sphere of radius R gives a two-dimensional coupling e₂² = g_c²/4πR², and the Schwinger mass of q unit-charge flavours is q e₂²/π.

This mass is not small and it survives in the mouth, so it is not a throat-only artefact. The Schwinger current has no components across the field lines, because LLL fermions move only along them. The currents that cross field lines later in the throat have a different origin.

A useful structural consequence: the Schwinger current is the curl of a longitudinal polarisation, so the Maxwell equations see the electric flux only through a displacement flux F_D = F − m_S² χ. Gravity couples to that displacement flux rather than to the electric flux, which is why the master variable is built from F_D in all regions.

Tilted worldsheets force a current across field lines

In the throat the perturbation tilts the worldsheets, and the Casimir stress tensor acquires mixed components. Force balance then requires a current across the field lines, and charge conservation fixes an accompanying polarisation current — the counterpart of the Hall current in the polar sector. These currents close the Bianchi identities at first order in α.

The paper is careful about what this logic does and does not establish. Consistency alone does not select the fermionic constitutive relation. The authors run control experiments in which parts of the response are removed: dropping the cross-line current and the polarisation current together, or dropping the tilted stress tensor, produces equal and opposite residuals in the τφ Einstein equation, so the two must be fixed as a pair. But a response-free covariant closure also passes the same test, so consistency cannot decide whether the pair is present at all.

Locality, covariance and the index theorem make that choice. The tilted worldsheet is identified as the unique timelike invariant plane of F, the drift lift is the local Lorentz transformation of the background stress tensor, and every other covariant closure is a magnetisation that the index theorem forbids. The entire imprint of the guiding-centre drift on the master dynamics at this order is a specific matrix that grows as αρ², subleading to the ρ³ background deformation in the overlap region.

Three coupled master equations match the fermion-free mouth

With these sources the system reduces to three coupled master equations, given in closed form, with a vanishing first-derivative matrix at α = 0 and frequency-independent potentials. In the mouth the decoupled potentials are Chandrasekhar's V₁,₂^(±), derived here on the magnetic background, so electric-magnetic duality emerges as an output rather than an assumption.

The throat solution is matched to the fermion-free mouth at a fiducial junction. The check is quantitative: the relative mismatch divided by (αρ̃)² stays at 8.33, 8.88 and 6.17 in the gg, DD and ΞΞ entries as the junction is moved through δ_J = 10⁻⁹ to 10⁻², until αρ̃ ~ 10⁻³ where the next order sets in. The full coefficient matrices agree with the mouth to (αρ̃)² ~ 10⁻¹¹ in all three channels. The authors note that this junction test validates the corrected background, the reduction and the Schwinger sector, but cannot discriminate between covariant closures of the drift sector.

At α = 0 the throat splits into three Pöschl-Teller channels

Setting α = 0 while keeping m_S² collapses the system onto the exact AdS₂ × S² throat. Unlike the companion polar-metric/axial-gauge sector, this throat is not pure Einstein-Maxwell: the polar gauge field is a worldsheet gauge field, and the LLL charge still responds through the Schwinger current. The system diagonalises globally in the eigenbasis of a constant matrix, and each eigenchannel is a Pöschl-Teller problem.

At m_S² = 0 the masses are the integers l(l−1) and (l+1)(l+2), matching the polar sector. At finite m_S² one mode is screened, one tends to the non-integer mass l² + l + 2, and one neutral charge-density wave stays exactly massless — the gravitationally dressed Wess-Zumino-Witten sector of the MMP appendices. For l ≥ 2 the two nonzero eigenvalues are positive for every m_S² ≥ 0, giving bounded oscillatory evolution with normal frequencies Ω² = (n + 1/2 + ½√(1+4λ))² under standard AdS₂ boundary conditions. No exponentially growing mode exists in the α = 0 throat for any value of the Schwinger mass.

Fermionic backreaction shifts massive frequencies downward

The O(α) shifts of the massive normal frequencies are computed by first-order perturbation theory, with a remap term that accounts for the difference between the throat coordinate x and arctan ρ. Every diagonal shift is a rational number plus 2Ω₀² log 2, and every one is negative. For l = 2, the Z₋ channel with Ω₀ = 2, 3, 4 gives δ = dΩ²/dα equal to −226/15 + 8 log 2, −433/10 + 18 log 2 and −1316/15 + 32 log 2, numerically −9.521, −30.82 and −65.55.

The shifts grow with n because higher modes sample larger ρ, where the corrected background lowers the potential. Levels shared by Z₊ and Z₋ split through a real 2×2 problem; for l = 2 and Ω₀ = 4 the eigenvalues are −27.16 and −65.62. The off-diagonal product is positive on every shared level examined, up to Ω₀ = 8 for l = 2 and on the lowest three shared levels for l = 3 to 6, which makes the eigenvalues real whatever the diagonal entries. A direct numerical solution at α = 10⁻³, 3×10⁻⁴ and 10⁻⁴ reproduces every value to O(α) and finds no complex eigenvalue.

The massless channel has no finite first-order shift in this idealisation. Its O(α) potential grows as ρ³ while its free eigenfunctions fall as cos²x, so the perturbative integral diverges logarithmically at the throat ends. That mode is controlled by the mouths, not by the throat interior.

The two parity sectors are isospectral only without fermions

In the fermion-free Einstein-Maxwell limit the two parity sectors are exactly isospectral: in the mouth through Chandrasekhar's Darboux pair, and in the throat through identical Pöschl-Teller potentials. Fermions break this at leading order in α in three distinct ways. The Schwinger sector of the axial throat adds a massless channel and moves the masses to {0, λ±} while the polar masses stay at {l(l−1), (l+1)(l+2)}. The mouth acquires a neutral channel with no polar counterpart and a region of negative potential. And the common Einstein-Maxwell-Casimir block is split at O(α) by rational differences.

The paper's reading is that duality survives only as a statement about Einstein-Maxwell theory alone. Fermions couple to A_μ and not to its dual; the polar gauge field is a worldsheet gauge field for them and the axial one is not, so the two parities see different fermionic operators. The authors connect this to the same conclusion reached for one-loop corrections to charged black holes, here with a backreaction that is O(1) in the throat.

The neutral channel is attractive but does not bind

The neutral channel is the place where a naive reading would predict instability. In the throat its O(α) potential has λ₁ₐ(0) = −m_S²(l̂+6)/2H < 0 at the centre, so the wormhole weakly focuses its own neutral charge-density waves near the throat centre. Continued into the mouth, that well is only the inner tail of an O(1) attractive region between r_e and 2r_e; the throat interior contributes O(α/ℓ) ~ 10⁻⁷⁰ in mouth units, so a frequency shift computed inside the throat alone has no meaning.

The global check is numerical. Counting negative eigenvalues of the stiffness matrix by Sylvester's law of inertia gives zero growing modes for l = 2, 3, 4 on two different grids, and for the full three-channel system at m_S² = 1, 10² and 10⁴. A single-channel estimate would have failed: taking the smallest eigenvalue of the symmetrised light potential as a single-channel potential produces a spurious bound state at ω²r_e² = −9.4×10⁻⁵, but the eigenvector rotates between the Ψ_g and Ξ directions, and the geometric term |∂_{r*}n|² integrates to 0.026/r_e, exceeding the well depth 0.019/r_e in magnitude. With that term included, the single channel also has no bound state.

For the massive channels, positivity of the symmetric parts of V₀ + αV₁ₐ and 1 + αV₁_b excludes a real negative Ω² for every p at once. A numerical scan over l = 2, 3, 4, α ∈ {10⁻⁶, 10⁻³, 10⁻²}, |p| ∈ {0, 1, 3} and the validity window αρ ≤ 0.1 gives min eig V_s ≥ 0.978 × l(l−1); positivity first fails at αρ ≈ 0.26, 0.28 and 0.29 for l = 2, 3, 4, inside the mouth-matching region where the positive mouth potentials take over.

What the axial sector now makes computable

The paper's own priority is not a stability verdict but a set of quantities that are now calculable in this sector. The finite-m_S² deformation of the throat spectrum, the loss of reflectionlessness, the mass l² + l + 2, the O(α) shifts and the massless charge-density channel threading the wormhole will all imprint on the echo structure of the MMP wormhole, and each is now available in closed or tabulated form. The neutral channel is free along the whole throat and sees only the mouth structure, so its transmission through a mouth is a clean scattering problem.

The stated scope is narrow and worth keeping in view when using these numbers. Only the lowest Landau level responds; higher levels are gapped by ~√q/R, and the Schwinger mass lies below that gap by the factor g_c/2π, which holds for q ≫ 1 and l ≪ √q. The background t_ab is the Casimir tensor on the circle L_c = π, transported by the anomaly, with corrections of order d/ℓ from the length d of the ambient part of each line dropped. The LLL stress tensor is taken to be a local functional of (g, F) up to gradient corrections of order l/√q. The calculation is to first order in ε and in α with m_S² unexpanded, for l ≥ 2 and m = 0, and the neutral-channel stability is established numerically at extremality with the two mouths treated as decoupled across the free throat.

Within those conditions, the sharpest comparison the paper offers is between the two parity sectors: isospectrality holds exactly at α = 0 and fails at leading order in α, with the axial throat gaining a massless channel the polar throat does not have. The observable that would test the drift closure is the matrix of Eq. (7.19), the entire imprint of the guiding-centre drift at this order — the quantity a first-principles Landau-level computation of ⟨T_{aφ}⟩ in crossed fields would have to reproduce.

Sources used in this answer