Asymptotically Locally AdS Fields and Fefferman–Graham Expansions

The Fefferman–Graham expansion separates near-boundary bulk data into three kinds: coefficients fixed locally by the boundary sources, logarithmic coefficients fixed by anomalies or resonances, and normalizable coefficients that contain state-dependent response. It is an asymptotic expansion, not a globally convergent coordinate system, and the first undetermined coefficient is precisely where interior regularity or a state choice must enter.

Required background. Asymptotically locally AdS boundary data fixes conformal boundary conditions and defining functions. Helpful background. Curvature counterterms and operator mixing explains why local curvature terms and logarithms are compatible with renormalization.

First application. Solve the scalar radial equation through the first subleading orders and identify where a logarithm appears when the two asymptotic exponents differ by an integer.

Radial expansions near an AdS boundary

Work first in Euclidean signature with bulk dimension $d+1$, AdS radius $L$, and boundary at $z=0$. Fefferman–Graham gauge is

$$\mathrm ds^2 =\frac{L^2\,\mathrm dz^2}{z^2} +\frac{1}{z^2}g_{ij}(z,x)\,\mathrm dx^i\mathrm dx^j.$$

A change of defining function $z\mapsto e^{\sigma(x)}z+O(z^3)$ induces a Weyl transformation of the leading boundary metric. Consequently $g_{(0)ij}$ is a representative of a conformal class, not an invariant metric selected by the bulk alone.

For an Einstein metric, the formal expansion has the structure

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

Odd powers can occur when matter, boundary conditions, or the chosen defining function permit them; the displayed parity pattern is the standard pure-gravity one. The radial Einstein equations determine $g_{(2)},\ldots,g_{(d-2)}$ locally from $g_{(0)}$. In even boundary dimension, $h_{(d)}$ is the local obstruction associated with the Weyl anomaly. The trace and divergence of $g_{(d)}$ are constrained, while its transverse-traceless part is normalizable data related to the boundary stress tensor de Haro, Solodukhin, and Skenderis 2001, §§2–4.

The coordinate gauge may fail at caustics or before reaching an interior horizon. Nothing in the expansion proves that a smooth global filling exists.

Scalar recursion and the first resonance

On Poincaré AdS with flat boundary metric, a free scalar obeys

$$\left[z^2\partial_z^2-(d-1)z\partial_z +z^2\Box_{(0)}-m^2L^2\right]\phi=0, \qquad m^2L^2=\Delta(\Delta-d).$$

For the standard branch, write

$$\phi=z^{d-\Delta} \left(\phi_{(0)}+z^2\phi_{(2)}+\cdots\right) +z^\Delta\left(\phi_{(2\Delta-d)}+\cdots\right).$$

Inserting this ansatz gives, away from resonance,

$$\phi_{(2)} =\frac{\Box_{(0)}\phi_{(0)}}{2(2\Delta-d-2)}.$$

Higher coefficients are local differential operators acting on $\phi_{(0)}$ until the recursion denominator vanishes. If $\nu=\Delta-d/2$ is a nonnegative integer, the two radial series resonate and a term $z^\Delta\log z^2\,\psi_{(2\nu)}$ is required. Its coefficient is local in the source and produces the scale dependence of the renormalized generating functional. By contrast, $\phi_{(2\Delta-d)}$ is not fixed by the near-boundary recursion; regularity, an ingoing condition, or another state prescription fixes it.

This distinction survives interactions, although powers can mix and nonlinear resonances can generate additional logarithms. A coefficient being normalizable is not by itself a choice of state: one must also impose the interior or Lorentzian condition appropriate to the observable.

Source, response, and a controlled application

For standard quantization, $\phi_{(0)}$ is the source for an operator of dimension $\Delta$, and the renormalized one-point function has the form

$$\langle\mathcal O\rangle_{\phi_{(0)}} =(2\Delta-d)\phi_{(2\Delta-d)} +\mathcal F_{\mathrm{local}}[\phi_{(0)},g_{(0)}],$$

where $\mathcal F_{\mathrm{local}}$ records scheme-dependent contact terms and anomaly contributions. The response coefficient is therefore meaningful only after the action normalization and finite counterterm scheme are fixed.

As a diagnostic, take a massless scalar in AdS$_{d+1}$, so $\Delta=d$. The first correction is

$$\phi_{(2)}=\frac{\Box_{(0)}\phi_{(0)}}{2(d-2)} \qquad(d\ne2).$$

At $d=2$ the denominator vanishes and the correct solution contains $z^2\log z^2\,\Box_{(0)}\phi_{(0)}$. Attempting to keep a pure power series fails directly in the field equation. This is the simplest adversarial fixture for a recursion routine: it must switch to the logarithmic branch rather than divide by zero.

Validity checks and breakdown

• Indicial check: both exponents solve $\alpha(\alpha-d)=m^2L^2$.
• Dimensional check: each $z^{2k}\phi_{(2k)}$ has the same boundary scaling as $\phi_{(0)}$.
• Pure-AdS check: flat $g_{(0)}$ with no sources gives $g_{(2k)}=0$ and vanishing normalizable data in the vacuum patch.
• Constraint check: the trace and divergence of the metric response reproduce the Weyl and diffeomorphism Ward identities after renormalization.
• Domain check: large boundary momentum with $z\lvert p\rvert\not\ll1$, caustics, singular fillings, or nonlinear backreaction can invalidate a truncated expansion.

The expansion organizes boundary data and divergences. It does not select a unique bulk state, prove a smooth filling, or make the normalizable coefficient local in the source.

Exercises

For a massless scalar in AdS$_5$, derive the coefficient of $z^2$ in terms of the four-dimensional boundary Laplacian.

Solution

Here $d=4$ and $\Delta=4$. Substitution in the recursion gives

$$\phi_{(2)}=\frac{\Box_{(0)}\phi_{(0)}}{2(8-4-2)} =\frac14\Box_{(0)}\phi_{(0)}.$$

The result also follows by inserting $\phi=\phi_{(0)}+z^2\phi_{(2)}+\cdots$ directly into the radial equation.

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, S., Solodukhin, S. N., and Skenderis, K. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. DOI. arXiv.
• Fefferman, C., and Graham, C. R. “Conformal Invariants.” In Élie Cartan et les Mathématiques d’Aujourd’hui, 95–116. Astérisque, 1985. Numdam.
• Skenderis, K. “Lecture Notes on Holographic Renormalization.” Classical and Quantum Gravity 19 (2002): 5849–5876. DOI. arXiv.