AdS2 Boundary Conditions and Fragmentation

AdS₂ is unusually sensitive to boundary data because finite energy changes the leading dilaton or electric field that supports the throat. A charged near-horizon region is therefore defined together with a fixed-charge or fixed-potential ensemble, a boundary trajectory, and Gauss-law constraints. Its two timelike boundaries cannot automatically be assigned two independent factorized Hilbert spaces.

Required background. Timelike Boundaries, Self-Adjoint Extensions, and AdS Boundary Conditions supplies the field-theory boundary problem. Gravitational Gauss Laws and Boundary Anchoring supplies the gravitational constraint.

Helpful background. Two-Sided Black Holes and Thermofield-Double States supplies the higher-dimensional comparison. Gauge Constraints, Centers, and Edge Data explains why constrained systems need not factorize across a cut.

Evidence cutoff: 25 July 2026.

Fields and ensembles at an AdS₂ boundary

In Poincaré coordinates,

$$ds^2=L_2^2\frac{dt^2-dz^2}{z^2}, \qquad z\to0,$$

a scalar has falloffs

$$\phi=z^{\Delta_-}\phi_{(s)}+z^{\Delta_+}\phi_{(v)}+\cdots, \qquad \Delta_\pm=\frac12\pm\sqrt{\frac14+m^2L_2^2}.$$

The usual source–response choice still requires a well-posed variational problem. The gravitational sector is more restrictive. A near-horizon charged throat carries an electric field

$$A_t=\frac{e_2}{z}+\mu_{\mathrm{IR}}+\cdots, \qquad \mathcal Q=-\frac{1}{g_2^2}\sqrt{-g}\,F^{zt},$$

whose leading potential diverges while the flux $\mathcal Q$ is finite. Fixing $\mathcal Q$ and fixing $\mu_{\mathrm{IR}}$ are different ensembles, related by a Maxwell boundary Legendre transform. The ensemble must be inherited from the higher-dimensional black hole rather than chosen after computing the on-shell action.

In a dilaton reduction,

$$\Phi=\Phi_0+\frac{\Phi_r}{z}+\cdots .$$

$\Phi_0$ controls the extremal entropy, while $\Phi_r$ sets the response to departures from extremality. Changing the finite energy changes the boundary trajectory relative to this growing dilaton. Strict AdS₂ with a constant dilaton consequently has no ordinary finite-energy excitation that leaves all leading boundary data untouched.

A charged throat as the first application

Fix the electric flux $\mathcal Q$ and the coefficient $\Phi_r$. Regulate the boundary by a curve $(t(u),z(u))$ whose induced metric is $du^2/\epsilon^2$. For a finite boundary reparametrization $t=f(u)$,

$$z(u)=\epsilon f'(u)+O(\epsilon^3).$$

The finite excitation is then encoded by the boundary trajectory and its Schwarzian energy,

$$E-E_0=-C\,\left\{\tan\frac{\pi f}{\beta},u\right\}_{\mathrm{ren}}, \qquad C\propto\frac{\Phi_r}{G_2},$$

rather than by an independent normalizable graviton in the AdS₂ interior. Matter inserted with energy $E_{\mathrm m}$ changes the constraint determining $f(u)$; keeping the original trajectory fixed while adding finite energy violates the gravitational equations. The nearly-AdS₂ reduction and this backreaction are developed by Maldacena, Stanford, and Yang 2016.

Two boundaries and fragmentation

Global AdS₂ has two timelike boundaries, but a spatial slice is connected. Gauss law relates the outward fluxes,

$$\mathcal Q_L+\mathcal Q_R+\mathcal Q_{\mathrm{bulk}}=0.$$

At fixed bulk charge, the two boundary fluxes cannot be varied independently. The gravitational Hamiltonian is also a boundary charge constrained by the bulk equations. These facts obstruct a naive tensor product $\mathcal H_L\otimes\mathcal H_R$ with independently specifiable energies and charges.

Multi-throat or fragmented configurations can arise in charged systems, but their existence depends on boundary conditions, topology, and the allowed charged sources. The original AdS₂ fragmentation solutions demonstrate multiple throats without proving a universal Hilbert-space factorization Maldacena, Michelson, and Strominger 1999.

Independent-excitation adversary

Try to increase the left energy while fixing the right energy, both fluxes, and the bulk state. The Hamiltonian and Gauss constraints become inconsistent unless an additional charged source or boundary degree of freedom is introduced. That added object changes the theory. Likewise, tracing a smooth two-boundary Penrose diagram does not supply a factorization map.

The strongest conclusion is conditional: a specified near-horizon reduction defines constrained boundary degrees of freedom and, in a nearly-AdS₂ regime, a reparametrization mode. It does not establish an ordinary two-copy dual, a unique microscopic Hilbert space, or a nonperturbative completion.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Maldacena, Juan, Jeremy Michelson, and Andrew Strominger. “Anti-de Sitter Fragmentation.” Journal of High Energy Physics 1999, 011 (1999). DOI.
• Maldacena, Juan, Douglas Stanford, and Zhenbin Yang. “Conformal Symmetry and Its Breaking in Two-Dimensional Nearly Anti-de Sitter Space.” Progress of Theoretical and Experimental Physics 2016, 12C104 (2016). DOI.