Polar-metric/axial-gauge perturbations of the Maldacena-Milekhin-Popov wormhole

Abstract

We study linear perturbations of the Maldacena-Milekhin-Popov (MMP) traversable wormhole. On a magnetically charged background, polar metric perturbations couple to axial perturbations of the gauge field. We treat this sector and reduce it to two coupled master equations of Zerilli-Moncrief type. The wormhole has two regions. In the mouths the geometry is a nearly extremal magnetic Reissner-Nordstrom black hole with no quantum source. In the throat the Casimir energy of the lowest Landau level of charged fermions holds the wormhole open. We show that the throat equations are consistent only if the fermions respond to the perturbation, computed exactly from the two-dimensional conformal anomaly with the full stress tensor including the trace part that MMP discard. This response must be accompanied by an induced Hall current of the fermions. With both effects included, the linearised Bianchi identities hold at first order in the fermionic backreaction $α$. Deriving the Hall current from the fermion effective action instead requires an Euler term with coefficient $c/48π$, which becomes a coupling between magnetic flux and curvature similar to a Wen-Zee term. At $α=0$ the throat is exact AdS$_2\times S^2$ and the modes decouple into two Poschl-Teller problems with masses $l(l-1)$ and $(l+1)(l+2)$, with levels spaced by $1/\ell$ in frequency. We give the $O(α)$ corrections in closed form. They mix the two modes and depend on frequency, and match the mouths in the overlap region. The symmetric part of the potential is positive throughout the wormhole, excluding purely exponential growth. The corrections lower the normal frequencies and split the levels shared by the two modes, with real shifts at first order in $α$. This sector has no growing mode for $l\ge2$ at this order.