Asymptotically Locally AdS Boundary Data

An asymptotically locally AdS (AlAdS) solution approaches AdS in a controlled neighborhood of conformal infinity, but its interior need not be pure AdS—or even share pure AdS’s topology or isometries. Its near-boundary coefficients have four different jobs: some are prescribed sources, some are local consequences of those sources, some are fixed only through constraint equations, and the remaining response data require a state or interior condition. Keeping those roles separate is the central result of this page.

Required background. Timelike-boundary causality and boundary-value problems supplies the admissibility and flux conditions that turn an asymptotic falloff into a boundary problem. Helpful background. Boundaries, surface counterterms, and boundary stress explains why finite observables require a complete variational problem, not merely a formal series.

Conformal infinity without a pure-AdS interior

Let $M$ be the interior of a manifold with boundary $\overline M$, and let $G_{MN}$ be a Lorentzian bulk metric of signature $(+,-,\ldots,-)$. A defining function is a function $z$ such that

$$z>0\quad\text{in }M, \qquad z=0\quad\text{on }\partial M, \qquad \mathrm dz\neq0\quad\text{on }\partial M.$$

The metric is conformally compact when, for some such $z$,

$$\overline G_{MN}=\frac{z^2}{L^2}G_{MN}$$

extends nondegenerately to the boundary. For an AlAdS Einstein metric, one can choose $z$ to be geodesic near the boundary, so that

$$\overline G^{MN}\partial_Mz\,\partial_Nz=-1$$

there. The normal is spacelike in the site’s mostly-minus convention, so the conformal boundary is timelike. These conditions are local near $z=0$: they neither prescribe the boundary topology nor assert that Fefferman–Graham coordinates reach a horizon, center, or second asymptotic region. The conformal-compactness definition and local normal form are developed in Skenderis 2002, §3.1, eqs. (3.6)–(3.11), PDF, while the geometric normal-form theorem appears in Fefferman and Graham 2012, ch. 4, especially eqs. (4.3), (4.6), and theorem 4.8, open manuscript. Skenderis’s positive-radial convention becomes the minus sign above after the site’s signature translation.

In a neighborhood without radial caustics, Fefferman–Graham gauge is

$$\mathrm ds^2 =\frac{L^2}{z^2} \left( -\mathrm dz^2+g_{ij}(z,x)\,\mathrm dx^i\mathrm dx^j \right), \qquad g_{(0)ij}(x)=g_{ij}(0,x).$$

The word locally matters. The representative $g_{(0)}$ can be curved and need not be the Minkowski metric or the Einstein-cylinder metric. The bulk curvature approaches the AdS value near each regular boundary point, while normalizable excitations, black holes, topology, and other global information remain possible in the interior.

The defining function chooses a conformal frame

The defining function is not unique. Replacing it by

$$z'=e^{\sigma(x)}z+O(z^3)$$

and making the compensating tangential shift $x'^i=x^i+O(z^2)$ needed to preserve $G_{z'i'}=0$ gives

$$g'_{(0)ij}=e^{2\sigma(x)}g_{(0)ij}.$$

This residual Penrose–Brown–Henneaux diffeomorphism is why the conformal completion canonically supplies a class $[g_{(0)}]$, not one preferred metric. The radial rescaling alone is insufficient when $\sigma$ varies: without the tangential shift it generally leaves Fefferman–Graham gauge. The explicit gauge-preserving transformation is given in Imbimbo, Schwimmer, Theisen, and Yankielowicz 2000, §2, eqs. (2.1)–(2.5), PDF.

A holographic calculation nevertheless chooses a representative $g_{(0)}$, because proper lengths, temperatures, energies, and the normalization of dimensionful sources are frame dependent. In even boundary dimension, a Weyl transformation can also change the renormalized generating functional by the local anomaly Imbimbo, Schwimmer, Theisen, and Yankielowicz 2000, §§3–4, PDF. Thus “only the conformal class is intrinsic to the completion” does not mean that every representative gives numerically identical renormalized observables.

Metric coefficients: source, recursion, constraint, response

For vacuum Einstein gravity near the boundary, the metric has a schematic polyhomogeneous expansion: it is built from powers of $z$ and, when required, powers multiplied by logarithms.

$$\begin{aligned} g_{ij}(z,x) ={}&g_{(0)ij}(x) +\sum_{1\leq k<d/2}z^{2k}g_{(2k)ij}(x)\\ &+z^d\left[ g_{(d)ij}(x) +\log\!\left(z^2\mu^2\right)h_{(d)ij}(x) \right]+\cdots . \end{aligned}$$

Here $\mu$ is an inverse reference length. For the standard pure-gravity expansion, $h_{(d)}=0$ when the boundary dimension $d$ is odd. In even $d$ a logarithmic coefficient may appear; when nonzero, it is the local obstruction proportional to the metric variation of the integrated Weyl anomaly.

The pure-gravity obstruction vanishes in $d=2$ even though the two-dimensional trace anomaly need not vanish. The sum is empty when no positive integer satisfies $k<d/2$, so the formula does not double-count $g_{(d)}$ in that case. Matter sources, nonlinear resonances, or relaxed boundary conditions can introduce additional powers and logarithms.

The radial Einstein equations do not treat every coefficient alike:

• $g_{(0)}$ is prescribed boundary geometry, modulo boundary diffeomorphisms and the choice of conformal frame.
• Coefficients $g_{(2k)}$ with $2k<d$ are local curvature functionals of the sources in the standard Einstein expansion. They are not independent state data.
• The logarithmic coefficient $h_{(d)}$, when present, is also source-local. Changing $\mu$ repartitions the displayed $g_{(d)}$ and $h_{(d)}\log(z^2\mu^2)$ terms without changing the bulk metric.
• At order $z^d$, the radial constraints fix the divergence and trace of $g_{(d)}$. They do not fix all of $g_{(d)}$. With no matter sources, the undetermined part may be represented by transverse-traceless data; with sources, the constraint equations are the sourced Ward identities.

This hierarchy follows from the Einstein equations in Fefferman–Graham gauge de Haro, Skenderis, and Solodukhin 2001, §2, especially eqs. (2.3) and (2.5)–(2.8), PDF. The phrase “subleading coefficient” therefore does not identify a physical role: one must ask at which order it appears and which radial equation determines it.

First application: the two scalar falloffs

Take a real scalar with action

$$S_\phi =\frac{\mathcal N_\phi}{2} \int_M\mathrm d^{d+1}x\sqrt{\lvert G\rvert} \left( G^{MN}\partial_M\phi\,\partial_N\phi-m^2\phi^2 \right),$$

so that $(\Box_G+m^2)\phi=0$. Only the leading AlAdS metric is needed to find the indicial roots. Near $z=0$ the equation begins as

$$\left[ -z^2\partial_z^2 +(d-1)z\partial_z +z^2\Box_{(0)} +m^2L^2 +\text{subleading AlAdS terms} \right]\phi=0.$$

This first pass treats $\phi$ on a fixed AlAdS background. If its stress tensor backreacts, the source can generate additional metric powers—often beginning with $z^{2\Delta_-}$—and logarithms, so the vacuum metric series above cannot simply be reused unchanged de Haro, Skenderis, and Solodukhin 2001, §5.2, especially eqs. (5.17)–(5.18), PDF.

Substituting $\phi=z^\lambda f(x)+\cdots$ and keeping the leading power gives

$$\lambda(\lambda-d)=m^2L^2.$$

Hence

$$\Delta_\pm=\frac d2\pm\nu, \qquad \nu=\sqrt{\frac{d^2}{4}+m^2L^2}.$$

The Breitenlohner–Freedman bound $m^2L^2\geq-d^2/4$ makes $\nu$ real Klebanov and Witten 1999, §2.1, eqs. (2.1)–(2.3), PDF. Assume first that $\nu>0$ and that the actual radial recursion is nonresonant; for a free scalar in the standard even-power expansion, this means $\nu\notin\mathbb Z_{>0}$. Then the slow-branch recursion never lands on the fast exponent, and

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

In standard quantization, $\alpha$ is fixed source data for an operator of dimension $\Delta_+$; $\beta$ is the first coefficient not fixed by the local source recursion. Interior regularity can determine $\beta$ as a nonlocal functional of $\alpha$ in a specified Euclidean saddle. In Lorentzian signature, initial-state data and any horizon prescription also enter de Haro, Skenderis, and Solodukhin 2001, §1, printed p. 3, PDF. Thus “undetermined by the asymptotic recursion” does not mean “arbitrary after the full boundary and state problem has been posed.”

The defining-function change introduced above also fixes the scalar Weyl weights. Invariance of the bulk field gives, up to the coordinate pullback,

$$\alpha'=e^{-\Delta_-\sigma}\alpha, \qquad \beta'=e^{-\Delta_+\sigma}\beta.$$

Logarithmic cases acquire additional source-local mixing, exactly as an anomalous Weyl transformation should Skenderis 2002, §5.6, eqs. (5.25)–(5.31), PDF.

The falloff classification is signature independent, but response signs must be attached to a declared variational convention. Choose here the Euclidean source convention used by the GKPW generating-functional page, $W=-S_{E,\mathrm{ren}}$; the black-brane comparison below returns to Lorentzian signature. Away from resonance, define $\beta$ so that the renormalized variation gives

$$\langle\mathcal O\rangle_\alpha =2\nu\,\mathcal N_\phi L^{d-1}\beta +\mathcal F_{\mathrm{local}}[\alpha,g_{(0)}].$$

The coefficient of the nonlocal response is fixed once the bulk normalization and source convention are fixed. Finite local counterterms shift $\mathcal F_{\mathrm{local}}$—and hence contact terms—but do not change the bulk coefficient $\beta$ in a fixed asymptotic coordinate expansion. At resonance, the reference scale used to split a logarithm from the power coefficient must also be declared. The scalar expansion and one-point relation are derived in de Haro, Skenderis, and Solodukhin 2001, §5.1, especially eqs. (5.2)–(5.11), PDF.

A concrete nonresonant scalar

For $d=3$ and $m^2L^2=-2$,

$$\nu=\frac12, \qquad \Delta_-=1, \qquad \Delta_+=2,$$

so

$$\phi(z,x)=z\,\alpha(x)+z^2\beta(x)+\cdots .$$

Standard quantization fixes $\alpha$ and interprets $\beta$ as response data. In particular, the source-free condition $\alpha=0$ does not force $\beta=0$: Lorentzian normalizable excitations can have nonzero $\beta$. This mass also lies in the open alternate-quantization window, but exchanging source and response is a different boundary problem, developed on Boundary Conditions, Alternate Quantization, and Deformations and in Klebanov and Witten 1999, §2.1, especially eqs. (2.1)–(2.3), PDF.

Metric response and radial constraints

After renormalization, choose the source variation

$$\delta W =\int\mathrm d^d x\sqrt{\lvert g_{(0)}\rvert} \left( \frac12\langle T^{ij}\rangle\delta g_{(0)ij} +\langle\mathcal O\rangle\delta\alpha +\langle J^i\rangle\delta A_{(0)i} \right).$$

The local part of the radial equations determines the lower coefficients and counterterm structures. The radial momentum and Hamiltonian constraints instead become boundary Ward identities. In the displayed source convention, a neutral scalar and a background gauge source obey schematically

$$\nabla_i\langle T^i{}_j\rangle =\langle\mathcal O\rangle\partial_j\alpha +F^{(0)}_{ji}\langle J^i\rangle+\cdots,$$

and

$$\langle T^i{}_i\rangle =(d-\Delta_+)\alpha\langle\mathcal O\rangle +\mathcal A[g_{(0)},\alpha,A_{(0)}].$$

Here $A_{(0)i}$ is the boundary gauge potential, $F^{(0)}=\mathrm dA_{(0)}$, and $\mathcal A$ is the Weyl-anomaly density in a stated scheme. Charged sources add their gauge variations. These identities fix the divergence and trace of the response; they do not manufacture its remaining state-dependent components. Their source signs follow directly by varying the displayed $W$ and provide a convention check against the radial constraints de Haro, Skenderis, and Solodukhin 2001, §5.2, especially eqs. (5.20)–(5.23), PDF and Bianchi, Freedman, and Skenderis 2002, §4.5, especially eqs. (4.18)–(4.21), PDF.

For Einstein gravity the renormalized stress tensor has the structure

$$\langle T_{ij}\rangle =\frac{dL^{d-1}}{2\kappa^2}\,g_{(d)ij} +X_{ij}[g_{(0)},\text{other sources}],$$

for the two-derivative Einstein action normalized by $2\kappa^2=16\pi G_{d+1}$ and the site’s Fefferman–Graham convention de Haro, Skenderis, and Solodukhin 2001, eq. (1.3) and §3, PDF. The tensor $X_{ij}$ is local and scheme sensitive where finite counterterms are allowed. Matter couplings and alternate normalizations modify this formula. Its full renormalized derivation belongs to Metric Counterterms and the Boundary Stress Tensor; the data classification here already shows why $g_{(0)}$ cannot determine the stress tensor.

A concrete comparison makes this visible. Poincaré AdS and the planar AdS black brane both have the flat boundary representative $g_{(0)}=\eta$. In the site’s signature the latter may be written

$$\mathrm ds^2 =\frac{r^2}{L^2} \left[ f(r)\,\mathrm dt^2-\mathrm d\mathbf x^2 \right] -\frac{L^2}{r^2f(r)}\,\mathrm dr^2, \qquad f(r)=1-\left(\frac{r_h}{r}\right)^d.$$

With $z\sim L^2/r$, the first difference from the Poincaré vacuum occurs at order $z^d$ and is proportional to $r_h^d$. The two geometries therefore share the same leading source metric but have different allowed response data and different stress tensors. Conservation and the trace condition constrain the black-brane response; they do not set it equal to the vacuum response. The thermal interpretation and normalization are derived on AdS Black Branes and Holographic Thermodynamics.

Adversarial check: when a logarithm replaces division

The simplest way to see why “leading coefficient equals source, subleading coefficient equals response” needs qualifications is to expose a failed recursion. On a flat boundary, the free scalar equation above gives, for $\nu\neq1$,

$$\alpha_{(2)} =\frac{\Box_{(0)}\alpha}{4(1-\nu)}.$$

At $\nu=1$, the denominator vanishes because $\Delta_-+2=\Delta_+$. A pure power series cannot solve the equation for a generic spacetime-dependent source. The correct ansatz is

$$\phi =z^{d/2-1} \left\{ \alpha +z^2\left[ \beta +\psi_{(2)}\log\!\left(z^2\mu^2\right) \right] +\cdots \right\},$$

and direct substitution gives, in the site’s Lorentzian convention,

$$\psi_{(2)}=\frac14\Box_{(0)}\alpha.$$

The logarithmic coefficient is fixed locally by the source; the coefficient $\beta$ at the same order remains the independent response datum. Calling the entire $z^{\Delta_+}$ order “the vev” would therefore mix anomaly data with state data. The detailed free-scalar recursion and its higher resonances are given in Skenderis 2002, §5.1, eqs. (5.4)–(5.13), PDF.

At the BF point $\nu=0$, the roots themselves coalesce and the expansion instead begins

$$\phi =z^{d/2} \left[ \alpha_{\log}\log\!\left(\frac z{\ell_\mu}\right) +\beta_{\mathrm{BF}}+\cdots \right].$$

The standard source is then the logarithmic datum in the displayed convention Klebanov and Witten 1999, §2.2, discussion preceding eq. (2.21), PDF, and the finite response must be extracted from the renormalized radial momentum Skenderis 2002, §4.5, eqs. (4.7)–(4.9), PDF. One must not obtain the endpoint by setting $\nu=0$ in the nonresonant one-point formula. The distinct BF-saturating branch and its backreaction are analyzed in Henneaux, Martínez, Troncoso, and Zanelli 2004, §2, especially eqs. (2.1)–(2.7), PDF.

Slower falloffs can change the boundary problem

Logarithms are not the only failure mode. Three checks prevent an asymptotic expression from being overinterpreted:

1. Backreaction. For an irrelevant operator, $\Delta_+>d$ and therefore $\Delta_-<0$. A finite source grows toward the boundary and can alter the leading metric rather than remain a harmless perturbation. In dynamical gravity such sources are commonly treated perturbatively; the unmodified AlAdS expansion is not automatically valid de Haro, Skenderis, and Solodukhin 2001, §5.2, PDF.
2. Flux and asymptotic symmetry. A relaxed falloff must give a finite renormalized symplectic structure and finite conserved charges and must preserve the declared asymptotic symmetry group. When those tests fail, the falloff defines a different phase space or is inadmissible; a logarithmic falloff is not rejected merely because it is logarithmic. A BF-saturating logarithmic branch that retains AdS asymptotic symmetries and finite total charges is exhibited in Henneaux, Martínez, Troncoso, and Zanelli 2004, §2, eqs. (2.1)–(2.7), PDF, while the renormalized symplectic test is developed in Compère and Marolf 2008, §§2–3, especially eqs. (2.3)–(2.9), PDF.
3. Nonlinear resonance. Interactions can make products of slow modes land on another characteristic exponent, producing powers or logarithms absent from the free recursion. The boundary conditions, counterterms, and response variable must then be recomputed together Henneaux, Martínez, Troncoso, and Zanelli 2007, §6, especially eqs. (6.1)–(6.4), PDF.

When one of these tests fails, the formal leading behavior may still be a useful local ansatz. What no longer survives automatically is the original source–operator assignment, finite variational problem, or claim that the solution belongs to the same AlAdS phase space.

What the asymptotic data do—and do not—supply

The AlAdS data determine a formal near-boundary solution to the order supported by the field equations. They identify sources, source-local descendants, logarithmic obstruction data, constraint equations, and the slots in which state-dependent responses enter.

They do not by themselves prove that the series converges, that a smooth global filling exists, that the filling is unique, or that a particular topology, horizon condition, Lorentzian state, or Euclidean saddle has been selected. Even in Euclidean signature, one boundary conformal class can admit more than one filling; in Lorentzian signature, normalizable modes make boundary sources alone insufficient. This local-versus-global distinction is emphasized in de Haro, Skenderis, and Solodukhin 2001, §1, printed p. 3, PDF.

Across metric and scalar fields, the decisive question is which equation fixes a coefficient—not merely how far down the displayed series it appears. The comparison below collects the four roles after all of its symbols have been defined. On a narrow screen, scroll horizontally without shrinking the text.

Near-boundary data classified by how they are fixed and what they determine

Field	Near-boundary datum	What fixes it	Boundary meaning	What it does not determine
Metric	g(0) and [g(0)]	Boundary problem and conformal-frame choice	Source geometry for the stress tensor	State, filling, or topology
	g(2k) with 2k < d	Local radial recursion	Source-local curvature data	An independent stress response
	h(d), when nonzero	Local obstruction equation	Obstruction associated with the anomaly and scale dependence	A freely chosen state
	Divergence and trace of g(d)	Radial constraints	Diffeomorphism and trace Ward data	All components of g(d)
	Remaining allowed part of g(d)	State or interior condition	Stress-tensor response data	A global solution by itself
Scalar	Source α in standard quantization	Prescribed boundary condition	Source for the operator O	The response β
	Source descendant α(2k) before resonance	Local scalar recursion	Derivative and curvature contact data	New state data
	Log coefficient ψ(2ν)	Local resonance equation	Obstruction and scale-dependent local data	The independent response
	Response coefficient β	State or interior condition	Contributes to πren, the momentum conjugate to α, and hence to the one-point response	Topology, existence of a filling, or local scheme terms

Common pitfalls

Treating Fefferman–Graham gauge as a global chart. It is a near-boundary normal form. Radial geodesics can form caustics, and the chart need not cross a horizon.

Calling every subleading term a response. Coefficients below the first undetermined order, and logarithmic obstruction coefficients, are local functionals of the sources. Only the unconstrained response data require state or interior information.

Calling a response coefficient an observable before renormalization. The physical object is the renormalized, scheme-declared response obtained from the variational problem. Finite counterterms can shift local one-point terms even though they cannot change separated-point nonlocal data.

Assuming the conformal class fixes a state. A boundary geometry specifies where the QFT lives, not which state it occupies. Pure AdS and an AdS black brane provide the basic counterexample.

Exercises

Transfer the scalar classification to AdS₆

For $d=5$ and $m^2L^2=-6$, derive $\Delta_\pm$, identify the standard source and response, and decide whether a source-free solution must have zero response.

Solution: scalar source and response

The indicial equation gives

$$\nu=\sqrt{\frac{25}{4}-6}=\frac12, \qquad \Delta_-=2, \qquad \Delta_+=3.$$

Thus $\phi=z^2\alpha+z^3\beta+\cdots$. Standard quantization fixes $\alpha$ as the source of a dimension-three operator. The local radial equation does not fix $\beta$; a state or interior condition does, and the renormalized one-point function is proportional to $\beta$ plus local source terms. Therefore $\alpha=0$ does not force $\beta=0$: a normalizable Lorentzian excitation can be source free.

Separate boundary geometry from state

Why can two AlAdS solutions with the same $g_{(0)}$ and the same non-normalizable matter sources have different stress tensors?

Solution: constrained but state-dependent metric data

The fixed sources determine the lower local coefficients. At order $z^d$, the radial constraints fix only the divergence and trace of $g_{(d)}$. Its remaining allowed part is state-dependent response data. Pure Poincaré AdS and a planar black brane can therefore share flat $g_{(0)}$, with all other non-normalizable sources absent, while carrying different $g_{(d)}$ and different renormalized stress tensors.

Change the defining function

Let $z'=e^\sigma z$ with constant $\sigma$. Derive the Weyl weights of $g_{(0)}$, $\alpha$, and $\beta$ in the nonresonant scalar expansion.

Solution: conformal-frame weights

Because $\overline G'=(z'^2/L^2)G=e^{2\sigma}\overline G$, the boundary representative transforms as $g'_{(0)}=e^{2\sigma}g_{(0)}$. Scalar invariance requires

$$z^{\Delta_-}\alpha =z'^{\Delta_-}e^{-\Delta_-\sigma}\alpha, \qquad z^{\Delta_+}\beta =z'^{\Delta_+}e^{-\Delta_+\sigma}\beta.$$

Hence $\alpha'=e^{-\Delta_-\sigma}\alpha$ and $\beta'=e^{-\Delta_+\sigma}\beta$. For nonconstant $\sigma$, the compensating tangential shift also generates derivative terms in subleading coefficients; logarithmic cases add local mixing.

Diagnose the first resonance

Why does $\alpha_{(2)}=\Box_{(0)}\alpha/[4(1-\nu)]$ fail at $\nu=1$, and which datum remains free after the logarithm is added?

Solution: the logarithmic branch

At $\nu=1$, the exponent $\Delta_-+2$ equals $\Delta_+$. The slow-branch recursion therefore collides with the second homogeneous solution, so its algebraic denominator vanishes. Substitution of

$$z^{\Delta_+} \left[ \beta+\psi_{(2)}\log(z^2\mu^2) \right]$$

gives $\psi_{(2)}=\Box_{(0)}\alpha/4$. This logarithmic coefficient is source-local; $\beta$ is still the coefficient not fixed by the near-boundary recursion, although the renormalized response also contains local scale- and scheme-dependent terms.

Controlled handoffs

Bulk Fields and Boundary Operators develops the mass–dimension and representation map. The GKPW Generating-Functional Dictionary turns fixed sources into connected correlators, while Boundary Conditions, Alternate Quantization, and Deformations changes the source assignment within its allowed window.

Asymptotically Locally AdS Fields and Fefferman–Graham Expansions derives the radial recursion in detail; the rest of that chapter constructs counterterms, renormalized momenta, and Ward identities. Rigorous asymptotic existence and uniqueness questions belong to Mathematical QFT.

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

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• Fefferman, Charles, and C. Robin Graham. The Ambient Metric. Annals of Mathematics Studies 178. Princeton: Princeton University Press, 2012. DOI. Open manuscript, ch. 4 and theorem 4.8.
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• Skenderis, Kostas. “Lecture Notes on Holographic Renormalization.” Classical and Quantum Gravity 19(22) (2002): 5849–5876. DOI. Open PDF.