Asymptotically Locally AdS Boundary Data

An asymptotically locally AdS (AlAdS) solution is defined by its controlled approach to an AdS conformal boundary, not by having a pure-AdS interior. The leading metric coefficient fixes a boundary conformal class, leading field coefficients supply sources, and selected subleading coefficients contain state-dependent responses subject to radial constraints. Interior regularity, topology, and global state data remain separate inputs.

Required background. Timelike-boundary causality and boundary-value problems supplies the admissibility and flux conditions. Helpful background. Boundaries, surface counterterms, and boundary stress explains why finite observables require more than an asymptotic series.

Fefferman–Graham data near an AlAdS boundary

Choose a defining function $z$ and Lorentzian Fefferman–Graham gauge,

$$\mathrm ds^2=\frac{L^2}{z^2} \left(-\mathrm dz^2+g_{ij}(z,x)\,\mathrm dx^i\mathrm dx^j\right),$$

where $g_{(0)ij}=g_{ij}(0,x)$ has signature $(+,-,\ldots,-)$. For Einstein gravity the formal expansion has the structure

$$g(z,x)=g_{(0)}+z^2g_{(2)}+\cdots +z^d\bigl(g_{(d)}+\log z^2\,h_{(d)}\bigr)+\cdots.$$

The logarithmic coefficient occurs in the appropriate even boundary dimensions and encodes the conformal anomaly. Lower coefficients are locally determined by $g_{(0)}$ and the equations of motion. The transverse and trace parts of $g_{(d)}$ are constrained, while its remaining data determine the renormalized stress-tensor expectation value after counterterms and scheme are fixed. This source/response split is derived systematically by de Haro, Solodukhin, and Skenderis 2001, §§2–4, while the geometric expansion is developed in Fefferman and Graham 2012, ch. 3.

A change of defining function $z\mapsto e^{\sigma(x)}z+\cdots$ induces a boundary Weyl transformation. Hence the invariant datum is the conformal class $[g_{(0)}]$, together with anomaly and source information; a particular representative is needed for dimensionful energies and correlators.

First application: scalar source and response

For a scalar with $\nu=\sqrt{d^2/4+m^2L^2}$ and nonresonant $2\nu$, the near-boundary equation gives

$$\phi(z,x)=z^{\Delta_-} \left(\alpha(x)+z^2\alpha_{(2)}(x)+\cdots\right) +z^{\Delta_+} \left(\beta(x)+\cdots\right),$$

where $\Delta_\pm=d/2\pm\nu$. In standard quantization, $\alpha$ is fixed source data. The coefficients before $\beta$ are local functionals of $\alpha$ and $g_{(0)}$ whenever the recursion denominators are nonzero. The renormalized one-point function has the schematic form

$$\langle\mathcal O\rangle_{\alpha} =(2\nu)\mathcal N_\phi\,\beta +\text{local source terms},$$

where $\mathcal N_\phi$ is the complete bulk kinetic normalization in the chosen dimensionful coordinates. Alternate quantization, when admissible, exchanges the source/response roles through a Legendre transform. Neither the bare coefficient $\beta$ nor the proportionality above is scheme independent at coincident points; the counterterm analysis is given by Skenderis 2002, §§3–4.

The two radial falloffs therefore classify fixed versus dynamical boundary data without any assumption that the interior is exactly AdS. To obtain a unique bulk solution one must still impose interior regularity, horizon conditions, or Lorentzian initial-state data.

Metric constraints and what remains free

For gravity, the radial Hamiltonian and momentum constraints imply boundary Ward identities. Schematically,

$$\nabla^i\langle T_{ij}\rangle =\sum_A\langle\mathcal O_A\rangle\,\nabla_j J_A +\text{gauge-source terms},$$

and the trace is fixed by explicit source breaking plus the anomaly. Thus $g_{(d)}$ cannot be specified as an arbitrary tensor. Conversely, $g_{(0)}$ alone does not determine $g_{(d)}$: different normalizable data or states can share the same boundary geometry.

The same distinction applies to gauge fields. A leading boundary potential is a source modulo boundary gauge transformations; the radial electric coefficient carries current data, subject to Gauss constraints. Global bundle and holonomy sectors are not visible in a purely local power series.

Adversarial check: resonances and slower falloffs

If the gap $2\nu$ coincides with an order generated by the source recursion—for the free even-power scalar recursion, when $\nu$ is a positive integer—the two branches resonate and can generate terms such as

$$z^{\Delta_+}\log z\,\widetilde\beta(x).$$

Interactions can produce additional logarithms or slower falloffs, and fields saturating a bound require separate treatment. Applying the nonresonant formula would misidentify local anomaly data as an independent response. The correct response is obtained only after the logarithmic counterterms and variational problem are included.

An even stronger failure occurs if a proposed falloff changes the asymptotic symmetry group or gives divergent renormalized flux. Then the solution belongs to a different boundary problem, not to the original AlAdS phase space. The strongest surviving claim is its formal asymptotic behavior; the standard source/operator assignment is no longer licensed.

Controlled limits and handoff

This classification fixes asymptotic data before solving the interior and before assigning finite renormalized one-point functions. It applies within the declared AlAdS phase space and must be modified for logarithmic branches or relaxed falloffs. The GKPW Generating-Functional Dictionary uses the scalar source/response split; the next chapter derives the full Fefferman–Graham recursion and counterterms.

Exercise

Why can two AlAdS states with identical $g_{(0)}$ have different stress tensors?

Solution

$g_{(0)}$ fixes the boundary conformal geometry and determines the local lower-order coefficients. The unconstrained part of $g_{(d)}$ is normalizable state data. After counterterms, it contributes to $\langle T_{ij}\rangle$. Different choices of this data—such as vacuum and thermal states—can therefore share $g_{(0)}$ while having different energy and stress profiles.

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

• de Haro, Sebastian, Sergey N. Solodukhin, and Kostas Skenderis. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. arXiv. DOI.
• Fefferman, Charles, and C. Robin Graham. The Ambient Metric. Annals of Mathematics Studies 178. Princeton: Princeton University Press, 2012. arXiv. DOI.
• Skenderis, Kostas. “Lecture Notes on Holographic Renormalization.” Classical and Quantum Gravity 19 (2002): 5849–5876. arXiv. DOI.