Holographic Reconstruction from Broken Conformal Gauge Symmetry

Introduction

The AdS/CFT correspondence relates gravitational dynamics in an asymptotically anti-de Sitter spacetime to a conformal field theory defined on its boundary [1, 2, 3]. A central question in holography is how bulk gravitational information is encoded in boundary geometry and boundary observables.

In the gauge-theoretic formulation considered in our earlier work, four-dimensional gravity with a cosmological constant emerges after the breaking

for the anti-de Sitter case [4, 5]. The alternative breaking to $SO(4, 1)$, corresponding to the de Sitter case, is not considered here, since it leads to ghost degrees of freedom in the matter Lagrangian, as discussed in Ref. [4]. An essential feature of this gauge-theoretic construction is that the Euler term arises automatically from the gravitational gauge action, rather than being introduced as an independent topological term. In the broken phase, the relevant curvature-squared structure takes the form

Its expansion produces the Einstein–Hilbert term, the cosmological term, and the four-dimensional Euler density $\epsilon_{abcd}R^{ab} \wedge R^{cd}$ simultaneously. Thus, within this gauge-theoretic construction, the Euler sector is an intrinsic part of the gravitational action, with its coefficient fixed relative to the other terms by the original curvature-squared structure. This provides the starting point for our investigation of the bulk–boundary correspondence. The resulting gravitational action contains, in addition to the Einstein–Hilbert and cosmological terms, a quadratic-curvature Euler sector. It is important, however, not to identify this Euler term with the Pontryagin density relevant to the usual three-dimensional gravitational Chern–Simons functional. In differential-form notation,

Both are locally total derivatives, but they are different characteristic classes. The boundary form $Q_E$ associated with the Euler density depends, in an adapted radial decomposition, on both the intrinsic curvature and the extrinsic curvature of the boundary. By contrast, the intrinsic three-dimensional gravitational Chern–Simons functional, associated with the Pontryagin sector, has a metric variation proportional to the Cotton tensor. It should be distinguished from the full Pontryagin boundary form, which may also contain extrinsic-curvature contributions. This distinction changes the reconstruction problem. De Haro showed that in $\mathrm{AdS}_4/\mathrm{CFT}_3$ the Cotton tensor may be interpreted as magnetic gravity data and may be related, under an appropriate mixed boundary condition, to the normalizable Fefferman–Graham coefficient $g_{(3)ij}$ [6, 7, 8]. That construction belongs to the Pontryagin–Chern–Simons sector, not directly to the Euler sector of the gauge action.

The first purpose of the present paper is therefore to determine what bulk–boundary information is actually contained in the Euler sector. We explicitly expand its boundary form in Fefferman–Graham coordinates and ask whether the normalizable bulk mode $g_{(3)ij}$ survives in its finite boundary contribution. We shall find that it does not appear linearly in the finite Euler term. This suggests a useful distinction,

$E_4 = \mathrm{d}Q_E$	global/topological bulk–boundary structure,
$P_4 = \mathrm{d}Q_P$	local Pontryagin–Cotton structure.

Eq. (5) distinguishes the global/topological Euler structure inherited from the original gauge-theoretic action from the local Pontryagin–Cotton structure associated with gravitational Chern–Simons boundary dynamics. In particular, the latter does not follow automatically from the Euler sector.

Our aim is to determine which boundary data can be obtained from the original broken $\mathrm{SO}(4,2)$ gauge construction without adding an independent Pontryagin term or imposing self-duality. We examine the finite Euler boundary contribution and its variational properties using the Fefferman–Graham expansion, and distinguish the kinematical appearance of the Cotton tensor from a genuine boundary response. We then relate these results to holographic reconstruction, complementing the boundary–response approach of Refs. [9, 10].

The paper is organized as follows. § 2 identifies the Euler and Pontryagin sectors in broken conformal gauge gravity. § 3 evaluates the Euler boundary geometry and its holographic variation, including the Codazzi–Cotton contribution. § 4 compares the Pontryagin–Cotton construction with the Euler sector and examines the conditions for holographic reconstruction. § 5 discusses holographic imaging and explicit $\mathrm{AdS}_4$ examples. § 6 summarizes the implications and outstanding questions.

Euler and Pontryagin sectors in broken conformal gauge gravity

We first identify the topological term that follows from the original four-dimensional gauge formulation of gravity [4, 11]. Upon breaking the conformal gauge symmetry to the anti-de Sitter subgroup, the vierbein $e^a$ and the Lorentz connection $\omega^{ab}$ emerge as gravitational variables. In the torsion-free sector, the resulting action can be written from Eq. (2) as

Here the Euler term is inherited from the original SO(4,2) gauge-gravity construction rather than being introduced as an independent topological interaction [4]. Its coefficient is fixed relative to the Einstein–Hilbert and cosmological terms by the curvature-squared structure of the broken gauge action.

The two characteristic forms $E_4$ and $P_4$, defined in Eqs. (3) and (4), respectively, have distinct geometrical and topological meanings. The Euler density $E_4$ arises directly from the curvature-squared term of the broken gauge action, whereas the Pontryagin density $P_4$ is introduced here for comparison with the gravitational Chern–Simons construction. For a compact, oriented, closed four-manifold, the properly normalized integral of $E_4$ gives the Euler characteristic $\chi(M_4)$, while the normalized Pontryagin integral is related to the signature of the intersection form. The two characteristic classes are therefore not generally proportional. Their local Chern–Simons primitives $Q_E$ and $Q_P$ have already been introduced in Eqs. (3) and (4). We now examine their distinct boundary representations. With the curvature convention above, a local representative for the Euler primitive is

while the Pontryagin primitive can be expressed as

where the trace is taken in the Lorentz-vector representation, with its normalization and sign chosen consistently with Eq. (4). These are local formulas: the primitives need not be globally defined on a nontrivial bundle. In particular, $Q_E$ and $Q_P$ must not be identified merely because both are Chern–Simons three-forms.

For a manifold with boundary, the Chern–Gauss–Bonnet theorem also involves a boundary correction constructed from the intrinsic and extrinsic geometry. For the subsequent analysis of the Euler boundary contribution, it is useful to employ a representation adapted to a hypersurface with boundary. In the Chern–Gauss–Bonnet theorem, the Euler density is accompanied by a boundary correction that can be expressed as a transgression form between the bulk Lorentz connection and a reference connection adapted to the boundary. The difference between these connections is characterized by the extrinsic curvature. Consequently, the Euler boundary transgression form involves both the intrinsic and extrinsic geometry, rather than the intrinsic connection alone. Denoting the intrinsic curvature two-form by the calligraphic symbol $\mathcal{R}^{ab}$, to distinguish it from the four-dimensional Lorentz curvature $R^{ab}$, and the extrinsic-curvature one-form by $K^a$, we obtain, up to overall normalization, orientation, and curvature conventions,

This boundary transgression form is not identical, as a differential form, to the frame-dependent local primitive $Q_E$. Their relation requires a choice of reference connection and the corresponding transgression identity; the boundary orientation and normalization must be fixed consistently in the Gauss–Bonnet formula [12, 13]. We use $B_3^E$ below because it provides a covariant expression in terms of the geometry of a regulated radial hypersurface.

By contrast, the intrinsic three-dimensional gravitational Chern–Simons functional whose metric variation yields the Cotton tensor is associated with the Pontryagin-type invariant, not with the Euler boundary transgression. It can be discussed as a separate boundary structure, but it does not follow from the Euler term in Eq. (5) by simple identification of the two primitives. Our aim is therefore to analyze the boundary information contained in the Euler sector of the original gauge action without adding an independent Pontryagin term or imposing a self-duality condition. The Euler boundary term is evaluated in the next section; the distinct Pontryagin–Cotton construction is reviewed subsequently for comparison.

Euler boundary geometry and holographic response

We now investigate the Euler boundary term inherited from the original gauge action [4]. The boundary form $B_3^E$ in Eq. (8) is evaluated on a regulated radial hypersurface. Near the conformal boundary we use Fefferman–Graham coordinates [6, 7, 8],

with

Here the absence of a term linear in $z$ follows from the four-dimensional vacuum Einstein equations with a negative cosmological constant, rather than from the Fefferman–Graham gauge condition alone. The induced metric at fixed $z$ is

Choosing the unit normal $n = -(z/\ell)\partial_z$ (the opposite orientation reverses the overall sign),

The $g_{(2)ij}$ term cancels in this expression because its contribution to $h_{ij}$ is independent of $z$. Raising one index with $h^{ij}$, we obtain [14]

For four-dimensional Einstein–AdS gravity,

and

These relations follow from the radial constraints and the asymptotic expansion of the four-dimensional Einstein equation,

The coefficient $g_{(2)ij}$ is determined locally by the boundary metric, whereas $g_{(3)ij}$ is transverse and traceless and contains independent normalizable data proportional to the renormalized boundary stress tensor. In tensor notation the Euler boundary form (8) contains, up to normalization,

This formula allows us to test directly whether the Euler sector contains the normalizable coefficient $g_{(3)ij}$ in its finite boundary contribution.

Finite Euler boundary contribution and the normalizable mode

Consider first the cubic extrinsic-curvature combination in (17). In three dimensions,

Write

The term linear in $B$ in the determinant is proportional to $\operatorname{Tr}B$. (14) therefore gives

Thus the cubic $K$ sector does not retain the normalizable mode linearly in the finite boundary term. The remaining terms contain the intrinsic curvature,

Their asymptotic scaling is sufficient to determine the fate of a term linear in $g_{(3)}$. At fixed $z$,

Hence a curvature–extrinsic-curvature contribution linear in $g_{(3)}$ scales as

and vanishes as $z\to0$.

This estimate accounts for the contribution of the normalizable mode through the extrinsic curvature. The intrinsic curvature also depends on $g_{(3)ij}$. Since the corresponding variation of the mixed intrinsic Ricci tensor is of order $z^5$, its contribution to the curvature–extrinsic-curvature term is likewise of order $z^2$. Thus neither source of linear dependence on $g_{(3)ij}$ contributes to the finite $z^0$ term.

In particular, a structure such as

does not survive as a finite $z^0$ term.

We therefore obtain the useful negative result

for the standard Einstein–AdS Fefferman–Graham expansion. The Euler boundary form does not by itself provide a de Haro-type local condition that determines the normalizable mode from the boundary metric.

The result concerns the finite term linear in the normalizable mode, not the entire Euler contribution. The Euler class is a global invariant on a closed manifold; with a boundary, the bulk Euler integral and its properly normalized boundary correction must be considered together. In particular, the result does not assert that the Euler sector is dynamically trivial or that it supplies a local boundary equation. We next contrast this result with the separate Pontryagin–Cotton construction.

On-shell variation of the Euler-completed gauge action

We now examine the boundary variation of the full Euler-completed action, rather than the Euler boundary form in isolation. The distinction is important: the absence of a term linear in $g_{(3)}$ in the finite part of $B^{E}_{3}$ does not imply the vanishing of the variation of the full on-shell action. For a torsion-free AdS branch, write

where the last equality fixes the Einstein–Hilbert normalization in the stated curvature and orientation conventions. Varying the vierbein and independent Lorentz connection gives

In the torsion-free sector $DF^{ab}=0$¹ and, on the Einstein equations, the remaining bulk term vanishes. Thus

For an Einstein metric with $R_{\mu\nu}=-3\ell^{-2}g_{\mu\nu}$, the AdS curvature equals the Weyl curvature two-form, $F^{ab}=W^{ab}$. (28) therefore expresses the on-shell variation directly in terms of the Weyl tensor; it does not invoke Pontryagin descent or self-duality.

To identify its finite Dirichlet response, one must retain both the normal–tangential variation $\delta\omega^{a\perp}$ and the tangential variation $\delta\omega^{ab}$, and specify the boundary treatment of the first-order variables. In particular, discarding the tangential connection variation before fixing a boundary frame is not justified. The four-dimensional Euler regularization is known to reproduce the standard asymptotically locally AdS Dirichlet variational problem under the appropriate boundary prescription [14, 15]. With the convention in which the outward-normal Brown–York momentum is [16]

the Fefferman–Graham expansion gives, for the corresponding choice of orientation,

The terms at orders $z^{0}$ and $z^{2}$ cancel because $g_{(2)ij}=-P_{ij}[g_{(0)}]$, while the trace of $g_{(3)}$ vanishes. After conformal rescaling, and with the overall Euclidean/Lorentzian action sign chosen in the usual holographic convention, the renormalized response is

The sign in the intermediate momentum Eq. (30) depends on the outward normal and the definition of the gravitational action; the physical convention must be held fixed throughout the calculation. Equations (28) and (29) are obtained directly from the Euler-completed gauge action. By contrast, the passage from Eq. (28) to the specific Dirichlet momentum in Eq. (29) uses the established four-dimensional equivalence with holographic renormalization, rather than a separate complete reduction of all connection boundary terms in the original $SO(4,2)$ gauge system. Hence the stress-tensor agreement is a consistency result, not a claim of a new independent holographic prediction. A derivation including the boundary conditions of all fields of the unbroken conformal gauge theory would be needed to strengthen this conclusion.

Codazzi curvature, the boundary Cotton tensor, and its falloff

We now test whether the Cotton tensor can arise in the boundary variation of the Euler-completed action without adding a Pontryagin term or imposing (anti-)self-duality. This question is motivated by the appearance of the Schwarzian derivative in the asymptotic extrinsic curvature of $\mathrm{AdS}_{2}$ and by the role of the Cotton tensor in the conformal geometry of a three-dimensional boundary. The analogy is geometric; it does not by itself imply identical boundary dynamics.

Choose a Gaussian-normal orthonormal frame near a regulated hypersurface $z=\epsilon$. Let $e^{a}$ denote the induced coframe, $\widehat{\omega}^{ab}$ its torsion-free spin connection, and $K^{a}=K^{a}_{\ b}e^{b}$ the extrinsic-curvature one-form. Up to the orientation-dependent signs in the mixed connection, the pullbacks of the AdS curvature obey the Gauss–Codazzi decomposition

Consequently, the on-shell boundary variation in Eq. (28) contains both a normal–tangential connection variation paired with $F^{ab}$ and a tangential connection variation paired with $\widehat{D}K^{a}$. The latter is the possible Cotton-bearing contribution. Overall and relative signs depend on the convention for $\omega^{a\perp}$; the following falloff statement does not.

In Fefferman–Graham gauge, with outward unit normal $n=-(z/\ell)\partial_{z}$ and $h_{ij}=\ell^{2}z^{-2}g_{ij}$, we have

where $P_{ij}=\mathcal{R}_{ij}-\frac{1}{4}\mathcal{R}g_{(0)ij}$ is the three-dimensional Schouten tensor. Since the connection of $h$ on each constant-$z$ slice equals that of $g(z)$, its antisymmetric covariant derivative satisfies

Here $C^{i}{}_{jk}$ is the three-index Cotton tensor, with index placements and signs fixed by the definition in this equation. Thus the Codazzi curvature contains the Cotton tensor without a Pontryagin density or a self-duality condition.

It remains to check the weight of this term in the action variation. Writing $e^{a}=(\ell/z)\bar e^{a}+O(z)$, with $\bar e^{a}$ an orthonormal coframe of $g_{(0)}$, the mixed curvature has the coordinate-component asymptotics

The $O(z)$ estimate is for the differential two-form in a fixed boundary coordinate basis; it includes the $z^{-1}$ scaling of the orthonormal coframe. For variations of smooth boundary source metrics, the intrinsic connection variation is finite, $\delta\omega^{ab}=O(1)$. Therefore

This conclusion assumes a fixed compact integration domain (or sufficiently controlled falloff along a noncompact boundary), smooth Fefferman–Graham data, and standard Dirichlet variations [17]. It need not hold for singular boundary sources or for boundary conditions that scale the connection variation with the cutoff.

Equation (37) concerns only the tangential connection–Codazzi contribution, not the complete on-shell variation. The other contribution, involving the variation of extrinsic curvature and $F^{ab}$, must also be retained. Under the standard asymptotically AdS Dirichlet prescription, the finite response of the complete Euler-completed action is the usual holographic stress tensor in Eq. (31), determined by the transverse-traceless coefficient $g_{(3)ij}$ rather than by $C_{ij}[g_{(0)}]$ [15]. The identification with the Dirichlet response uses the established Euler regularization/counterterm equivalence, not an independent derivation of all boundary conditions of the original unbroken $SO(4,2)$ gauge system.

We therefore distinguish three statements: (i) the Cotton tensor occurs in the asymptotic Codazzi curvature; (ii) its contribution through the tangential-connection term vanishes in the standard Dirichlet limit; and (iii) neither fact implies a Cotton-type boundary equation of motion. A genuine Cotton boundary response would require a separately justified boundary variational prescription derived from the original gauge system. In particular, we do not add a Pontryagin term or impose self-duality to obtain it.

Pontryagin–Cotton structure and holographic reconstruction

Pontryagin descent and mixed boundary conditions

For a three-dimensional boundary metric $g_{(0)ij}$, the Cotton tensor is

It is symmetric, traceless, and covariantly conserved. The metric variation of the three-dimensional gravitational Chern–Simons functional is proportional to the Cotton tensor,

Consequently this Cotton structure belongs naturally to the descent of the Pontryagin density $P_{4}$, Eq. (4), rather than to the Euler density $E_{4}$ in Eq. (3) [18].

In the holographic description of de Haro [7], the renormalized stress tensor and Cotton tensor play roles analogous to the electric and magnetic components of the gravitational field,

With a gravitational Chern–Simons boundary term, the variational principle permits a mixed boundary condition of the schematic form

The proportionality coefficient depends on the normalization of the Chern–Simons term and on conventions. Equation (42) is not a consequence of the bulk Einstein equations alone; those equations constrain $g_{(3)ij}$ to be transverse and traceless but do not locally determine it from $g_{(0)ij}$. The Euler calculation in Sec. 3.1, summarized in Eq. (25), shows that the finite Euler boundary form contains no term linear in $g_3$. It therefore does not directly yield the mixed Cotton boundary condition of Eq. (41).

Two distinct bulk–boundary structures

The preceding calculation suggests that the Euler and Pontryagin sectors should be interpreted as two different bulk–boundary structures,

Euler sector	Pontryagin sector
$E_4=dQ_E$ (locally)	$P_4=dQ_P$ (locally)
$B^E_3=B^E_3(h,K)$	$Q_P=Q_P(\omega)$
intrinsic + extrinsic geometry	intrinsic gCS: $Q_{\mathrm{gCS}}(\widehat{\omega})$
global/topological information	Cotton tensor from intrinsic gCS
no direct linear fixing of $g_{(3)}$	$g_{(3)}\propto C$ under mixed BC

Here $Q_P$ denotes the full Pontryagin Chern–Simons form, which may also contain extrinsic-curvature terms, whereas $Q_{\mathrm{gCS}}$ is constructed solely from the intrinsic boundary connection $\widehat{\omega}$. This distinction provides the basis for examining whether the two sectors can be related by additional geometric constraints.

A possible unification arises in a self-dual sector. After symmetry breaking, define the dual on the four-dimensional Lorentz indices by

For a (anti-)self-dual Lorentz curvature, Euler and Pontryagin densities are no longer independent (with the precise sign and factors depending on Euclidean versus Lorentzian signature and on normalization). This raises the possibility that the Euler sector intrinsic to the broken gauge action and the Pontryagin–Cotton sector relevant to de Haro’s mixed boundary condition are two projections of a common chiral structure.

This observation also defines a sharper reconstruction problem. Rather than assuming the chain $E_4\to C_{ij}\to g_{(3)ij}$, which is not correct, one should investigate

and ask whether a self-duality, regularity, or gauge-theoretic principle relates the two rows and fixes their relative coefficient.

Implications for reconstruction

The Fefferman–Graham expansion separates the boundary source from normalizable gravitational data:

For four-dimensional Einstein gravity with negative cosmological constant, the radial constraints give

Thus $g_{(2)}$ is fixed locally by $g_{(0)}$, whereas the transverse-traceless tensor $g_{(3)}$ is not. In particular,

The holographic stress tensor supplies the missing state-dependent asymptotic information:

This relation is the renormalized Dirichlet response established in Eq. (31); it is not a consequence of the Cotton tensor alone.

A concrete reconstruction map can be exhibited in the linearized transverse-traceless sector about the Poincaré patch of $AdS_4$. For a Fourier mode with spacelike momentum $k^{2}=q^{2}>0$, the bulk Einstein equation and the condition of regularity at $z\to\infty$ select

so that

For a detailed derivation of Eqs. (47) and (48), see § 5.2.2. The regularity condition is essential: without it, the second independent radial solution allows $h_{(3)}$ to vary at fixed $h_{(0)}$. Eq. (48) therefore represents a particular bulk response map, not a universal local relation between the two Fefferman–Graham coefficients. The linearized boundary Cotton tensor for the same TT source is

with the orientation convention of Eq. (38). Unlike Eq. (48), this is a local, parity-odd third-derivative operation on the boundary metric. The two expressions are not identical, even though both scale as the third power of boundary momentum.

For example, if one additionally imposes a de Haro-type mixed boundary condition in the form

then regularity and the mixed condition must be compatible:

Here Eq. (51) is understood on the symmetric transverse-traceless subspace. It is a restriction on allowed momenta and polarizations, not an identity for arbitrary sources. Moreover, Eq. (50) is an additional boundary condition associated with the Pontryagin–Chern–Simons discussion of Eq. (41); it has not been derived from the Euler sector of the broken conformal gauge action.

The Euler boundary form $B^{E}_{3}(h,K)$ provides a distinct geometrical datum involving intrinsic and extrinsic boundary geometry. Its local finite term does not linearly fix the traceless normalizable coefficient $g_{(3)}$. The asymptotic quantities should therefore be distinguished as

The Schwarzschild–AdS$_4$ example in § 5 gives a particularly simple check: the conformal boundary metric and its Cotton tensor are independent of the black-hole mass, whereas $g_{(3)}$ and the boundary stress tensor retain that mass. These observations establish the limits of reconstruction from conformal boundary geometry alone. Reconstructing the complete bulk spacetime further requires the Einstein evolution equations and appropriate global, causal, and interior conditions; the asymptotic relations by themselves do not establish global uniqueness.

Holographic imaging and explicit AdS$_4$ examples

Relation to holographic imaging

The inverse problem considered by Hashimoto and collaborators [9, 10] provides a useful comparison with the boundary geometry of the broken conformal gauge theory. In particular, the holographic Einstein-ring construction of Ref. [10] uses the response of a scalar operator to a localized, time-dependent boundary source. It should not be identified with the metric stress-tensor response considered below.

For definiteness, consider a minimally coupled massless probe scalar $\Phi$ on a fixed asymptotically AdS$_4$ geometry,

In the standard quantization, $J$ is the boundary source and the normalizable coefficient $\Phi_{(3)}$ determines the renormalized one-point function, up to the conventional normalization and local contact terms:

The coefficient $\Phi_{(2)}$ is fixed locally by the source and the boundary metric, whereas $\Phi_{(3)}$ depends on the bulk solution and its interior boundary condition. At linear order one may therefore write the response as

where $G_R$ is the retarded boundary correlator when causal, ingoing conditions are imposed. In the black-hole example the bulk metric enters this kernel through the propagation of the probe field. The imaging procedure processes such a boundary response to extract information about the bulk geometry [10].

The gravitational reconstruction problem has a parallel, but distinct, source–response pair. For Dirichlet boundary conditions the source is $g_{(0)ij}$ and the state-dependent response is

in agreement with the renormalized Dirichlet relation derived in § 3.2. Thus the two schematic maps are

The second arrow in the metric line is the holographic dictionary; the first arrow is not determined by the near-boundary Einstein constraints alone. It requires a choice of bulk state and suitable regularity, causal, or interior conditions. The same distinction between source data and normalizable response appears in the probe problem.

The gauge-theoretic characteristic forms organize different boundary information:

Neither $B^E_3$ nor $C_{ij}[g_{(0)}]$ is, by itself, the scalar response kernel $G_R$. Furthermore, the Cotton tensor depends locally on the boundary metric and cannot generally replace the independent normalizable gravitational datum $g_{(3)ij}$. A Cotton–stress relation requires an additional mixed boundary condition, as discussed in § 4; it is not implied by the Euler boundary form.

We now examine two explicit examples illustrating the distinction between boundary conformal geometry and normalizable bulk data. The Schwarzschild–AdS$_4$ family shows that different bulk masses can share the same boundary Cotton tensor, whereas regular linearized perturbations of Poincaré AdS$_4$ provide an explicit momentum-dependent relation between the boundary source and the normalizable gravitational response.

Explicit asymptotically AdS geometries

Pure AdS$_4$ and Schwarzschild–AdS$_4$

Before considering a general reconstruction problem, it is useful to test the proposal for simple asymptotically AdS geometries.

For a conformally flat boundary,

Pure AdS$_4$ provides the reference solution with $C_{ij}=0$ and $g_{(3)ij}=0$. In contrast, Schwarzschild–AdS$_4$ has the same conformal boundary geometry but a nonvanishing normalizable mode $g_{(3)ij}$ when $M\ne0$. This example is trivial but important: it shows immediately that $C_{ij}=0$ by itself cannot distinguish all possible bulk states with conformally flat boundary geometry. Thus regularity and/or stress-tensor data must enter the complete reconstruction problem.

A concrete distinction between boundary conformal geometry and normalizable gravitational data is provided by Schwarzschild–AdS$_4$. This is also the black-hole background relevant to the imaging proposal of Ref. [10]. Consider

We introduce dimensionless quantities

and choose the dimensionless Fefferman–Graham coordinate $z$ so that $R\sim z^{-1}$ at the conformal boundary. The radial coordinate transformation obeys

Substituting $R=z^{-1}+az+bz^2+O(z^3)$ gives

The metric consequently takes the Fefferman–Graham form

with $g_{ij}=g_{(0)ij}+z^2g_{(2)ij}+z^3g_{(3)ij}+O(z^4)$. Writing $\gamma_{AB}$ for the metric of the unit two-sphere, we obtain

Thus

The mass appears first in $g_{(3)ij}$, which satisfies $g_{(0)}^{ij}g_{(3)ij}=0$ and $\nabla^{i}_{(0)}g_{(3)ij}=0$.

The boundary metric $\mathbb{R}\times S^2$ has Ricci tensor $\mathcal{R}_{\tau\tau}=0$, $\mathcal{R}_{AB}=\gamma_{AB}$, and scalar curvature $\mathcal{R}=2$. Its three-dimensional Schouten tensor is

Hence both the three-index Cotton tensor and its two-index dual vanish identically:

independently of $M$.

In contrast, the renormalized holographic stress tensor, (31), gives

This expression uses the dimensionless boundary representative $-d\tau^2+d\Omega_2^2$. In the physical conformal frame $-dt^2+\ell^2d\Omega_2^2$, the mixed components become

and the conserved energy is

Thus the same conformal boundary geometry, with $C_{ij}=0$, corresponds to different values of $g_{(3)ij}$ and different bulk masses. In particular,

This directly illustrates the non-implication discussed in § 4 without imposing a mixed boundary condition.

The Euler boundary sector contains additional extrinsic data. For the outward-normal convention of § 3, the Fefferman–Graham expansion of the mixed extrinsic curvature is

In this example it becomes

Although the mass-dependent normalizable mode occurs in $K^{i}{}_{j}$, its term linear in $g_{(3)}$ in the finite local Euler boundary density cancels by $\operatorname{tr}g_{(3)}=0$, as established in § 3. This local asymptotic cancellation must not be confused with mass independence of the integrated Euler invariant, which can involve global and horizon contributions.

Schwarzschild–$\mathrm{AdS}_{4}$ therefore exhibits explicitly the separation of the boundary Cotton tensor, the state-dependent holographic stress tensor, and the Euler intrinsic–extrinsic boundary geometry. Computing the dynamical imaging response itself would require solving a perturbation equation on this black-hole background, rather than inferring that response from the Cotton tensor alone.

Perturbed AdS4

We next consider a linearized gravitational perturbation of the Poincaré patch of $\mathrm{AdS}_{4}$. This example distinguishes the nonlocal response determined by the bulk equations and a regularity condition from the local Cotton tensor of the boundary source. We work in Fefferman–Graham gauge,

and restrict to transverse-traceless perturbations,

Writing $h_{ij}(z,x)=e^{ik\cdot x}H_{ij}(z,k)$, the linearized Einstein equations reduce to

For spacelike momentum $k^{2}=q^{2}>0$, the general solution is

Selecting the solution regular as $z\to\infty$ sets $B_{ij}=0. The near-boundary expansion is then

so that

The second relation is a consequence of the specified interior regularity condition; it is not a local Fefferman–Graham constraint valid for arbitrary states. Using the holographic stress tensor in Eq. (31) gives the corresponding linear response,

This expression is derived in the spacelike-momentum sector; retarded Lorentzian response requires the corresponding causal analytic continuation and horizon/interior prescription.

For comparison, define the boundary Cotton tensor by Eq. (38).

The overall sign depends on the orientation convention. At first order about the flat boundary metric, Eq. (78) implies

and hence

Thus the stress response and Cotton tensor are both third order in boundary momentum but act differently on the transverse-traceless polarizations: the former is proportional to $(k^2)^{3/2}$ times the source, whereas the latter includes the parity-odd operator $\epsilon_i{}^{kl}k_k$. In particular, the two tensors are not identical for a generic boundary perturbation.

We can test directly whether the Euler boundary three-form adds a finite contribution linear in the normalizable mode. Write the mixed extrinsic curvature, consistently with Eq. (13), as

The $z^3$ perturbation in Eq. (82) is

For the cubic extrinsic-curvature term in $B^E_3$, use the three-dimensional identity already established in Eq. (18). At first order in the normalizable perturbation,

where the would-be $z^3$ term vanishes because $\operatorname{Tr}h_{(3)}=0$. Since $\sqrt{|h|}=\ell^3z^{-3}\sqrt{|g_{(0)}|}[1+O(z^2)]$, the cubic sector has no finite $z^0$ density linear in $h_{(3)}$. The remaining intrinsic-curvature terms are of the form $\sqrt{|h|}(KR-2K^{ij}R_{ij})$. Their part linear in $h_{(3)}$ is at least

and likewise has no finite $z^0$ contribution. Thus,

This is a statement about the local asymptotic density, not about the full on-shell variation of the Euler-completed action. The latter also involves the connection variations discussed in § 3.2, and yields the standard renormalized stress tensor under the prescribed boundary conditions. The explicit comparison is therefore

No equality between $C_{ij}^{(1)}$ and $h_{(3)ij}$ follows from the Euler sector alone. A relation of that type requires an additional mixed boundary condition, a self-duality restriction, or some other independent dynamical input. The present calculation thus makes precise, at linear order, the distinction between the boundary response and the conformal boundary geometry.

Discussion and conclusions

The present analysis clarifies the distinct roles of the Euler and Pontryagin sectors in the holographic interpretation of gravity arising from broken $SO(4,2)$ gauge symmetry. Although both characteristic densities are locally total derivatives, their boundary structures are different:

The Euler boundary transgression involves both intrinsic and extrinsic geometry. The Pontryagin boundary form may also involve extrinsic curvature, while its intrinsic gravitational Chern–Simons counterpart has a Cotton metric variation.

Our explicit Fefferman–Graham calculation shows that the finite Euler boundary density contains no term linear in the normalizable coefficient $g_{(3)ij}$. The cubic extrinsic-curvature contribution vanishes because $\operatorname{Tr}g_{(3)}=0$, while the corresponding intrinsic-curvature contributions vanish in the conformal-boundary limit. This result does not imply the absence of a holographic stress-tensor response from the complete Euler-completed action. Rather, it shows that the Euler boundary form alone does not generate the local mixed condition $g_{(3)ij}\propto C_{ij}[g_{(0)}]$.

The Codazzi analysis provides a complementary result: the boundary Cotton tensor appears in the asymptotic mixed curvature, but its contribution through the tangential-connection variation vanishes under standard Dirichlet boundary conditions. Thus the kinematical appearance of the Cotton tensor must be distinguished from a dynamical Cotton boundary response.

These results suggest two complementary structures of bulk–boundary information:

In holographic reconstruction, state-dependent information is encoded in boundary response functions, while the gauge-theoretic characteristic sectors provide additional geometric and topological structure. A possible connection between the two sectors through self-duality or chirality of the Lorentz curvature remains an open question. Establishing such a connection, including its normalization and signature dependence, may clarify whether Cotton-type mixed boundary conditions can emerge from the broken gauge theory itself. A further connection may be found in the Ryu–Takayanagi prescription for holographic entanglement entropy [19]. In $AdS_{4}/CFT_{3}$, the entanglement entropy of a boundary region is related to the area of a bulk codimension-two extremal surface. The four-dimensional Euler term can also contribute a topological term associated with the Euler characteristic of this surface, provided the appropriate boundary contributions are included. This is distinct from the finite Euler boundary density and its Dirichlet variation examined in the present work. A direct derivation of the entanglement contribution from the Euler sector of the broken conformal gauge action, including its fixed normalization, remains an interesting problem for future investigation.