Maldacena-Milekhin-Popov Wormhole Construction

• The MMP construction is a framework in four-dimensional semiclassical gravity that uses negative Casimir energy from two-dimensional fermions and enhanced AdS2 symmetries to support traversable wormholes.
• It employs a perturbative solution to the Einstein-Maxwell equations around the near-horizon geometry of near-extremal magnetically charged black holes, ensuring smooth matching to asymptotically flat regions.
• The approach faces strict quantum energy inequality constraints, prompting extensions to holographic matrix models and raising critical questions about the semiclassical viability of such wormhole solutions.

The Maldacena-Milekhin-Popov (MMP) construction is a framework for realizing traversable wormholes in four-dimensional semiclassical gravity. It combines ingredients from quantum field theory, black hole physics, and negative Casimir energy to develop a perturbative solution of the Einstein-Maxwell equations around the near-horizon geometry of near-extremal magnetically charged black holes. The construction is characterized by leveraging the enhanced symmetries of the AdS$_2\times S^2$ throat and sourcing the geometry with negative vacuum (Casimir) energy from two-dimensional fermions, producing a long wormhole that connects two asymptotically flat regions. The MMP construction has served as a testbed for understanding wormhole physics, energy conditions, and holographic dualities, and has generated significant interest for its connections to quantum energy inequalities and holographic matrix models.

1. Geometric and Physical Background

At its core, the MMP construction operates within the near-horizon region of a near-extremal Reissner-Nordström black hole in four-dimensional Einstein-Maxwell theory. The action is

$$S=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R - \frac{1}{16\pi k}\int d^4x\,\sqrt{-g}\,F^2,$$

with nonzero magnetic charge $Q_m \neq 0$ and vanishing electric charge for the canonical MMP setup.

In the extremal limit, $GM^2 = k_m Q_m^2$, the near-horizon geometry approaches global AdS$_2\times S^2$: $$ds^2 = \bar \ell^2\left[ - (1+\rho^2)d\tau^2 + \frac{d\rho^2}{1+\rho^2} + d\Omega_2^2 \right],$$ where $\bar\ell = \sqrt{G k_m} Q_m$. The isometry group is enhanced to SO(2,1)$\times$SO(3), realized by the three AdS$_2$ Killing vectors

$\xi_1,\,\xi_2,\,\xi_3 = \partial_\tau$

(satisfying the $\mathfrak{so}(2,1)$ algebra), and the angular momentum operators $J_i$ on $S^2$. The unperturbed Maxwell field is

$$\bar F = - \frac{Q_m}{4\pi} \sin\theta\, d\theta \wedge d\phi,$$

preserving the same symmetry structure.

2. Role of Negative Casimir Energy

Traversability of the wormhole requires the introduction of localized negative energy violating the classical energy conditions. MMP employ the Casimir effect of massless two-dimensional fermions confined to the AdS$_2$ throat, wrapping the $S^2$ magnetic flux. The semiclassical stress-energy tensor for these fermions, consistent with all SO(2,1)$\times$SO(3) symmetries and energy-momentum conservation, is

$$\mathcal T_{\mu\nu} = \lambda\,\mathcal T^{(1)}_{\mu\nu} + O(\lambda^2), \qquad \mathcal T^{(1)}_{\mu\nu}dx^\mu dx^\nu = \mathcal E\left[(1+\rho^2)d\tau^2+\frac{d\rho^2}{1+\rho^2}\right],$$

where $\mathcal E < 0$ quantifies the Casimir energy density and $\lambda \ll 1$ is an expansion parameter. This negative energy is the unique source compatible with the throat's static geometry and symmetry (Kanai et al., 26 Nov 2025).

3. Perturbative Solution and Matching Conditions

The geometry and Maxwell field are perturbed around the AdS$_2\times S^2$ background: $$g_{\mu\nu}(\lambda) = \bar g_{\mu\nu} + \lambda h_{\mu\nu} + O(\lambda^2), \qquad F(\lambda) = \bar F + \lambda \delta F + O(\lambda^2),$$ with the ansatz (in SO(2,1)$\times$SO(3)-invariant gauge): $$g(\lambda) = \ell(\lambda)^2\left[ -\frac{1+\rho^2}{\gamma(\lambda;\rho)} d\tau^2 + \frac{\gamma(\lambda;\rho)}{1+\rho^2} d\rho^2 + A(\lambda;\rho)\,d\Omega^2 \right].$$ The throat is required to develop a minimal-area $S^2$ at $\rho=0$, and for large $|\rho|$, the metric must smoothly match to the near-extremal Reissner-Nordström solution, ensuring the wormhole cones into two asymptotic regions belonging to the same black hole family.

Solving the linearized Einstein equations, the crucial "flaring-out" equation reads

$$\partial_\rho^2 A^{(1)}(\rho) = -\frac{16\pi G\,\mathcal E}{(1+\rho^2)^2},$$

which, after imposing symmetry and matching conditions, yields explicit expressions for the metric perturbations. The wormhole length is determined as

$$\ell^{\mathrm{MMP}} = 4\pi G |q_m| \lambda |\mathcal E| + O(\lambda^2),$$

where $q_m$ is the magnetic charge quantum number. The throat length scales linearly with both the magnetic charge and the negative Casimir energy sourced by the fermion species (Kanai et al., 26 Nov 2025, Kontou, 9 May 2024).

4. Quantum Energy Inequality Constraints

Wormhole constructions require negative energy, but quantum field theory imposes restrictions via quantum energy inequalities (QEIs). For the MMP wormhole, the recently derived double-smeared null energy condition (DSNEC) places more stringent constraints than previously realized. The DSNEC, when applied to the MMP throat, leads to a lower bound on the number of fermion species $q$ required: $q \gtrsim \ell/r_e \gg 1,$ with $\ell$ the wormhole length and $r_e$ the characteristic magnetic length scale. This condition enforces $q$ to be orders of magnitude larger than originally intended, typically pushing $q$ into a regime where the semiclassical description is jeopardized. Moreover, the magnitude of the negative Casimir energy needed to satisfy the DSNEC would require densities much larger than those compatible with semiclassical reliability unless one invokes an unphysically large number of species. Thus, the MMP construction violates the double-smeared null energy bound under current semiclassical expectations (Kontou, 9 May 2024).

Energy Condition	Bound on $q$ or $\langle T_{tt}\rangle$	Semiclassical Feasibility
SNEC (single-smeared)	$B\,q = {\cal O}(1)$	Satisfied for $q \sim 10$
DSNEC (double-smeared)	$q \gtrsim \ell/r_e \gg 1$	Not feasible unless $q \gg 1$

5. Extensions and Nonperturbative Evidence in Matrix Models

The MMP construction generalizes to other settings, notably the D0-brane (BFSS) matrix model, where analogous ideas about ungauged sectors are realized. The ungauged matrix model allows for non-singlet (open-string) sectors, corresponding holographically to open strings with endpoints at the spacetime boundary. Maldacena and Milekhin conjectured that, at low temperature and strong coupling, these open-string contributions are exponentially suppressed: $$\Delta E = E_{U} - E_{G} \simeq d C N^{2} e^{-C/T},$$ with $d\in\mathbb{Z}^{+}$ and $C = \mathcal{O}(1)$. Nonperturbative Monte Carlo simulations confirm that the ungauged and gauged models become indistinguishable in the $T \to 0$, $N \to \infty$ limit, in harmony with the conjectured behavior and supporting the general philosophy inherent in the MMP construction (Berkowitz et al., 2018).

6. No-Go Theorems and Limitations

The MMP approach is highly constrained by the underlying symmetries of the near-horizon geometry. Effective field theory (EFT) methods that introduce higher-derivative corrections to the Einstein-Maxwell action cannot perturbatively generate traversable wormholes in this context. The enhanced SO(2,1)$\times$SO(3) symmetries of the AdS$_2\times S^2$ throat severely restrict the form of the effective energy-momentum tensor, precluding the required flare-out geometry. As a result, traversable wormholes of the MMP type cannot arise within the conventional EFT framework from standard black hole near-horizon regions; new matter sectors or a reduction of symmetry are required (Kanai et al., 26 Nov 2025).

7. Broader Implications and Open Questions

The MMP construction deepens the understanding of traversable wormholes, quantum inequalities, and their compatibility with semiclassical gravity. While the model demonstrates how negative Casimir energy can, in principle, support a long wormhole, the required violation of DSNEC imposes severe restrictions on its physical viability. A plausible implication is that consistent semiclassical wormholes of the MMP type may require either vastly more matter content than expected, new exotic quantum fields, or departure from standard effective field theory paradigms.

The extension to matrix models and holography suggests new avenues for realizing wormhole-like physics in lower dimensions and for numerical studies free of gauge constraints. However, the tension with quantum energy inequalities remains a central barrier, indicating the need for either further theoretical innovation or a re-evaluation of the semiclassical regime in which such solutions could exist.

Key references: (Kanai et al., 26 Nov 2025, Kontou, 9 May 2024, Berkowitz et al., 2018)