Asymptotic Symmetries and Charges in AdS

AdS asymptotic symmetries are the transformations preserving a chosen timelike-boundary phase space, modulo proper gauge. Their charges can be computed by Hamiltonian, covariant phase-space, electric-Weyl, or counterterm methods. These methods agree only after boundary conditions, conformal representative, reference normalization, action counterterms, and flux assumptions have been aligned.

Helpful background. Lorentzian Metrics and Causal Cones supplies boundary causal geometry; Geodesics and Jacobi Fields supplies asymptotic limits; Causality and Global Structure supplies conformal completion; Exact Solutions of Einstein’s Equations supplies AdS backgrounds; Formulations of General Relativity supplies charge methods; Gravitational Actions and Boundary Terms supplies boundary variations.

Boundary conditions select the symmetry group

Put an asymptotically locally AdS$_{d+1}$ metric in Fefferman–Graham form,

$$d\widetilde s^2 =\frac{\ell^2}{z^2} \left(dz^2+g_{ij}(z,x)dx^i dx^j\right).$$

Under Dirichlet-type gravitational boundary conditions, the boundary conformal class $[g_{(0)}]$ is fixed. An allowed asymptotic diffeomorphism induces a boundary vector field $\zeta^i$ satisfying

$$\mathcal L_\zeta g_{(0)ij}=2\sigma(x)g_{(0)ij},$$

so it is a conformal Killing field of the chosen representative. Transformations with trivial action and zero charge are quotiented. For boundary dimension $d\gt 2$, the Einstein-cylinder conformal Killing algebra is $\mathfrak{so}(d,2)$, matching the exact AdS isometry algebra. Globally, the symmetry group is the appropriate cover or quotient of $SO_0(d,2)$ determined by whether AdS time is unwrapped and by any bulk or boundary identifications.

If one fixes a specific boundary metric rather than its conformal class, only its Killing fields may preserve the source. If mixed or Neumann boundary conditions allow $g_{(0)}$ to vary, boundary diffeomorphisms and Weyl transformations can become gauge, physical, or anomalous depending on the symplectic structure. The phrase “the AdS asymptotic group is the conformal group” therefore assumes the standard Dirichlet conformal phase space.

In bulk dimension three, the asymptotic symmetry group can be infinite dimensional: Brown–Henneaux boundary conditions yield two Virasoro algebras with a central charge. That exceptional enhancement is dimension- and falloff-dependent; it is not the generic higher-dimensional result.

Covariant and Hamiltonian charges

For an allowed generator $\xi$ extending $\zeta$ into the bulk, covariant phase space defines

$$\delta H_\xi=\int_S k_\xi[g;\delta g]$$

on a boundary cut $S$. Finiteness can require counterterms or a renormalized symplectic current. Integrability requires vanishing phase-space curl after accounting for field-dependent generators. If symplectic flux through the timelike boundary is nonzero, the cut charge changes according to a balance law or the phase space must include boundary degrees of freedom.

In a canonical slicing, the same generator arises as a Hamiltonian boundary term added to the bulk constraints. Under matching falloffs and normalization, the Hamiltonian and covariant charges agree. General derivations and ambiguities are developed in Covariant Phase Space, Boundaries, and Charges and Compère and Fiorucci 2019, §§1.2–1.5, pp. 10–32.

For exact AdS normalized as the reference, every charge vanishes in a background-subtraction convention. Other renormalization schemes can assign a nonzero Casimir energy to global AdS in odd bulk dimension/even boundary dimension. The difference is a local boundary term and must be kept fixed when comparing states.

Counterterm stress-tensor charges

Add the Gibbons–Hawking–York term and local intrinsic counterterms to the regulated action. Varying with respect to the induced boundary metric gives a finite renormalized stress tensor $T^{\rm ren}_{ij}$. A boundary conformal Killing field $\zeta^i$ defines

$$Q_\zeta[S] =\int_S d^{d-1}x\sqrt{\sigma}\, u^iT^{\rm ren}_{ij}\zeta^j,$$

where $u^i$ is the future unit normal to the cut inside the timelike boundary and $\sigma$ its induced metric determinant. To decide whether this cut integral is conserved, use both Ward identities. With no matter-source contribution,

$$D^iT^{\rm ren}_{ij}=0.$$

For $j^i=T^{{\rm ren}\,i}{}_j\zeta^j$ and $D_{(i}\zeta_{j)}=\alpha g_{(0)ij}$, however,

$$D_i j^i =T^{{\rm ren}\,ij}D_{(i}\zeta_{j)} =\alpha\,T^{{\rm ren}\,i}{}_i.$$

Thus a Killing field ($\alpha=0$) gives a conserved charge on compact cuts with no additional flux. A proper conformal Killing field requires the trace term to vanish or to be included in the balance law. In even boundary dimension, a nonzero trace anomaly can obstruct conservation of that current even though $D^iT^{\rm ren}_{ij}=0$. The divergence and trace are separate inputs; conformal symmetry of a representative alone does not settle the charge de Haro, Skenderis, and Solodukhin 2001, §3.

Finite local counterterms shift scheme-dependent stress-tensor pieces and vacuum energy. Charges associated with genuine isometries and differences between states with identical sources are often more robust than the absolute reference value.

The Fefferman–Graham coefficient $g_{(d)}$ supplies the state-dependent part of $T^{\rm ren}_{ij}$, plus local curvature functionals of $g_{(0)}$. This connects charge computation to the asymptotic expansion without making a holographic duality assumption.

Electric-Weyl charges

For sufficiently regular asymptotically AdS Einstein metrics, the appropriately rescaled electric part of the Weyl tensor has a finite boundary limit. Contracting it with a boundary symmetry and cut normal gives an Ashtekar–Magnon–Das-type charge,

$$Q_\zeta^{\rm E}[S] \propto\frac{\ell}{G} \int_S\mathcal E_{ij}u^i\zeta^j\,dS,$$

with dimension-dependent normalization. Because exact AdS has zero Weyl tensor, this prescription naturally sets its charge to zero.

Under standard Dirichlet falloffs, the electric-Weyl charge agrees with the state-dependent part of the counterterm or Hamiltonian charge. In dimensions where the counterterm stress tensor includes a vacuum Casimir term, the absolute values differ by that local reference contribution. Comparing formulas without stating this offset creates a false disagreement.

The Weyl method also assumes adequate conformal smoothness. Logarithmic terms, matter falloffs, or weakened boundary conditions can require modifications. It is not automatically finite for every asymptotically locally AdS phase space.

Flux, matter, and varying sources

If the boundary metric or other sources depend on time, the corresponding charge can exchange energy or momentum with them. The Ward identity becomes

$$D^iT^{\rm ren}_{ij} =\sum_A\mathcal O_A D_jJ^A$$

schematically, so a boundary generator that does not preserve the sources has a work term. Transparent conditions likewise allow bulk flux through the boundary. A nonconserved charge can be correct; the balance law must include its source.

Mixed gravitational boundary conditions may promote the boundary metric to a dynamical variable. Then its stress, symplectic form, and gauge transformations belong to the combined phase space. A Dirichlet counterterm charge cannot be imported without modification.

Matter can contribute directly to Hamiltonian and stress-tensor charges and can alter admissible falloffs. Scalar fields near the BF window are a standard example: finite counterterms and mixed conditions shift the energy. Positivity depends on both bulk potential and boundary functional.

Comparison and checks

To compare two AdS charge formulas, hold fixed:

• bulk and boundary dimension;
• Newton’s constant and AdS radius;
• boundary conformal representative and cut orientation;
• field content and falloffs;
• Dirichlet, Neumann, mixed, or transparent conditions;
• counterterm scheme and reference solution; and
• whether vacuum/Casimir terms are included.

Then check exact AdS, a Schwarzschild–AdS solution, and algebra representation. The mass should be cut independent under stationary reflective conditions, vanish or equal the declared vacuum value on the reference background, and generate the correct time translation. Proper gauge transformations must have zero charge.

Exercise

Suppose two valid prescriptions give $Q_1[\mathrm{AdS}]=0$ and $Q_2[\mathrm{AdS}]=E_0$, while for every state with the same boundary metric $Q_2-Q_1=E_0$. Do they disagree on physical energy differences?

Solution

No. For two states $A$ and $B$ in the same phase space, using the same generator and sources,

$$Q_2[A]-Q_2[B] =Q_1[A]+E_0-Q_1[B]-E_0 =Q_1[A]-Q_1[B].$$

The prescriptions choose different additive vacuum normalizations, often through a finite local counterterm or Casimir convention. They agree on variations and state differences. A state-dependent offset would change those differences. Dependence on the fixed sources or on which generator is used instead means that the cancellation must be checked anew when comparing different sources or different charges; it is not automatically a physical inconsistency.

AdS Causality, Trapping, and Stability Interfaces next explains why conserved charges and reflecting boundaries constrain, but do not settle, long-time nonlinear evolution.

References

• Ashtekar, Abhay, and Saurya Das, “Asymptotically Anti-de Sitter Space-Times: Conserved Quantities”, Classical and Quantum Gravity 17, §§II–IV, pp. L17–L30, 2000.
• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§1.2–1.5, pp. 10–32, Springer, 2019.
• de Haro, Sebastián, Kostas Skenderis, and Sergey N. Solodukhin, “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence”, Communications in Mathematical Physics 217, §§2–4, pp. 595–622, 2001.