Gauge Fields, Currents, and the Graviton

The main idea

The scalar dictionary taught us that a bulk field $\phi$ near the AdS boundary encodes a scalar source and a scalar expectation value. Conserved currents and the stress tensor are more rigid. They are not just generic operators with spin. They are the operators that implement symmetries.

The holographic dictionary is:

$$\boxed{ A^a_M \quad \longleftrightarrow \quad J^\mu_a, \qquad G_{MN} \quad \longleftrightarrow \quad T^{\mu\nu}. }$$

Here $A^a_M$ is a bulk gauge field associated with a CFT global symmetry generator labeled by $a$, and $G_{MN}$ is the bulk metric. The boundary value of $A^a_M$ is a background gauge field sourcing the CFT current $J^\mu_a$. The boundary value of the bulk metric is the background metric sourcing the CFT stress tensor $T^{\mu\nu}$.

The deep slogan is:

$$\boxed{ \text{global symmetry in the CFT} \quad \longleftrightarrow \quad \text{gauge redundancy in the bulk}. }$$

This is not merely an analogy. Bulk gauge constraints become CFT Ward identities. Bulk radial canonical momenta become CFT one-point functions. Bulk diffeomorphism constraints become conservation and trace equations for the boundary stress tensor.

The boundary value $A^{a(0)}_\mu$ of a bulk gauge field sources the conserved current $J^\mu_a$, while the boundary metric $g^{(0)}_{\mu\nu}$ sources the stress tensor $T^{\mu\nu}$. Radial electric flux and extrinsic curvature are the bulk canonical momenta that determine the corresponding one-point functions after holographic renormalization.

This page develops the dictionary in a source-first way. The most important lesson is that conserved operators are protected because they are tied to redundancies: a conserved spin-one current has dimension $\Delta=d-1$, and the stress tensor has dimension $\Delta=d$.

Sources for currents and the stress tensor

Let the CFT live on a background metric $g^{(0)}_{\mu\nu}$ and let $A^{a(0)}_\mu$ be a nondynamical background gauge field for a global symmetry group $G_F$. The connected generating functional is

$$W[g^{(0)},A^{(0)},\cdots] = \log Z[g^{(0)},A^{(0)},\cdots].$$

Our convention for one-point functions is

$$\delta W = \int d^d x\sqrt{g_{(0)}} \left( \frac12 \langle T^{\mu\nu}\rangle\delta g^{(0)}_{\mu\nu} + \langle J^\mu_a\rangle\delta A^{a(0)}_\mu + \langle\mathcal O_I\rangle\delta\phi^I_{(0)} +\cdots \right).$$

Thus

$$\boxed{ \langle J^\mu_a(x)\rangle = \frac{1}{\sqrt{g_{(0)}}} \frac{\delta W}{\delta A^{a(0)}_\mu(x)}, \qquad \langle T^{\mu\nu}(x)\rangle = \frac{2}{\sqrt{g_{(0)}}} \frac{\delta W}{\delta g^{(0)}_{\mu\nu}(x)}. }$$

Some authors define the stress tensor by varying with respect to $g^{\mu\nu}_{(0)}$ rather than $g^{(0)}_{\mu\nu}$. Then a minus sign appears:

$$\langle T_{\mu\nu}\rangle = -\frac{2}{\sqrt{g_{(0)}}} \frac{\delta W}{\delta g_{(0)}^{\mu\nu}}.$$

This is only a convention. What matters is consistency between the source term, the variation of the renormalized action, and the definition of correlation functions.

For infinitesimal sources around flat space,

$$g^{(0)}_{\mu\nu}=\eta_{\mu\nu}+h^{(0)}_{\mu\nu},$$

one may write schematically

$$\delta S_{\mathrm{CFT}} = -\int d^d x\,A^{a(0)}_\mu J^\mu_a -\frac12\int d^d x\,h^{(0)}_{\mu\nu}T^{\mu\nu} +\cdots,$$

up to Euclidean versus Lorentzian sign conventions. The factor of $1/2$ in the metric coupling is not decorative; it comes from the symmetric variation of the metric.

Why the dimensions are fixed

A conserved current obeys

$$\partial_\mu J^\mu=0$$

in flat space without sources. In a unitary CFT, a conserved spin-one primary saturates the spin-one unitarity bound, and its scaling dimension is

$$\boxed{\Delta_J=d-1.}$$

This also follows from source dimensions. The source deformation is

$$\int d^d x\,A^{(0)}_\mu J^\mu.$$

The background gauge field has engineering dimension one, so $J^\mu$ must have dimension $d-1$.

Similarly, the metric source is dimensionless. Since

$$\frac12\int d^d x\,h^{(0)}_{\mu\nu}T^{\mu\nu}$$

must be dimensionless and $h^{(0)}_{\mu\nu}$ has dimension zero, the stress tensor has

$$\boxed{\Delta_T=d.}$$

These dimensions are protected. A generic vector operator need not have $\Delta=d-1$; it maps to a massive bulk vector field. A generic symmetric spin-two operator need not have $\Delta=d$; it maps to a massive spin-two field. The conserved current and stress tensor are special because they sit in shortened conformal multiplets.

For a massive vector field in AdS$_{d+1}$, the relation between mass and dimension is

$$m_A^2L^2=(\Delta-1)(\Delta-d+1).$$

A massless gauge field has $m_A^2=0$, giving the standard current branch $\Delta=d-1$ together with a second branch associated with boundary gauge-field behavior. In standard AdS/CFT quantization for $d>2$, the conserved-current interpretation uses $\Delta=d-1$.

Bulk gauge fields and boundary currents

Consider, for clarity, a bulk Maxwell field in an asymptotically AdS$_{d+1}$ background:

$$S_A = -\frac{1}{4g_F^2} \int d^{d+1}X\sqrt{-G}\,F_{MN}F^{MN}, \qquad F_{MN}=\partial_M A_N-\partial_N A_M.$$

For a non-Abelian symmetry, one replaces

$$F=dA+A\wedge A,$$

with the appropriate group indices and trace normalization. The coupling $g_F$ is a bulk gauge coupling. Its value determines the normalization of the current two-point function.

Near the boundary, use Fefferman-Graham/Poincaré coordinates

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+g_{\mu\nu}(z,x)dx^\mu dx^\nu\right), \qquad z\to0.$$

In radial gauge,

$$A_z=0,$$

the boundary expansion for $d>2$ takes the schematic form

$$\boxed{ A_\mu(z,x) = A^{(0)}_\mu(x) +\cdots +z^{d-2}A^{(d-2)}_\mu(x) +\cdots. }$$

The leading coefficient $A^{(0)}_\mu$ is the source. The coefficient $A^{(d-2)}_\mu$ is response data. In flat boundary metric and without logarithmic subtleties, the current one-point function is proportional to this response coefficient:

$$\boxed{ \langle J^\mu(x)\rangle = \frac{L^{d-3}}{g_F^2}(d-2)A_{(d-2)}^\mu(x) +\text{local counterterm contributions}. }$$

The precise local terms depend on the dimension, the background metric, and the counterterm scheme. The nonlocal part of the response is physical and determines separated-point correlators.

The on-shell variation makes the origin of the formula transparent. Varying the Maxwell action gives a boundary term

$$\delta S_A^{\mathrm{os}} = -\frac{1}{g_F^2} \int_{z=\epsilon} d^d x\sqrt{|\gamma|}\,n_MF^{M\mu}\delta A_\mu,$$

where $\gamma_{\mu\nu}$ is the induced metric on the cutoff surface and $n^M$ is its outward unit normal. The canonical momentum conjugate to $A_\mu$ is therefore radial electric flux:

$$\Pi^\mu = -\frac{1}{g_F^2}\sqrt{|\gamma|}\,n_MF^{M\mu}.$$

After adding counterterms and taking $\epsilon\to0$,

$$\boxed{ \langle J^\mu\rangle = \frac{1}{\sqrt{g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta A^{(0)}_\mu} \quad \text{is the renormalized boundary electric flux.} }$$

Depending on whether one writes $W_{\mathrm{CFT}}= -S^{E,\mathrm{ren}}_{\mathrm{bulk}}$ in Euclidean signature or $Z\sim e^{iS^{\mathrm{ren}}}$ in Lorentzian signature, the overall sign in this displayed formula may be adjusted. The flux interpretation is invariant.

Current conservation from the radial Maxwell constraint

The Maxwell equation is

$$\nabla_MF^{MN}=0.$$

The $N=z$ component is not an evolution equation for a propagating degree of freedom. It is a constraint:

$$\partial_\mu\left(\sqrt{-G}F^{\mu z}\right)=0.$$

Near the boundary, this becomes

$$\partial_\mu\langle J^\mu\rangle=0$$

in flat space with no charged sources. This is the simplest example of a general rule:

$$\boxed{ \text{radial bulk constraints} \quad \longrightarrow \quad \text{CFT Ward identities}. }$$

The same logic applies in a curved background and for non-Abelian symmetries. Gauge invariance of the generating functional under

$$\delta_\alpha A^{(0)}_\mu=D_\mu\alpha$$

gives

$$0 = \delta_\alpha W = \int d^d x\sqrt{g_{(0)}}\,\langle J^\mu_a\rangle D_\mu\alpha^a.$$

After integrating by parts, arbitrary $\alpha^a(x)$ implies

$$\boxed{ D_\mu\langle J^\mu_a\rangle=0 }$$

when there are no charged sources and no anomaly.

With charged scalar sources, the Ward identity is modified. Schematically,

$$D_\mu\langle J^\mu_a\rangle + (T_a\phi_{(0)})^I\langle\mathcal O_I\rangle =\mathcal A_a,$$

where $T_a$ is the symmetry generator acting on the source multiplet and $\mathcal A_a$ is a possible ‘t Hooft anomaly. Holographically, such anomalies are encoded by bulk Chern-Simons terms. For example, in an AdS$_5$ bulk, a term of the form

$$S_{\mathrm{CS}}\sim \int A\wedge F\wedge F$$

has a boundary gauge variation that reproduces a four-dimensional current anomaly.

This point is subtle but essential. The CFT symmetry may have an ‘t Hooft anomaly. That does not mean the bulk gauge theory is inconsistent. It means that gauge transformations which approach nonzero transformations at the boundary act on the CFT sources with the anomalous variation. Gauge transformations that vanish at the boundary remain true redundancies.

Boundary global symmetry versus bulk gauge redundancy

A boundary global symmetry transformation is a bulk gauge transformation whose parameter approaches a nonzero function at the conformal boundary:

$$\alpha(z,x)\longrightarrow \alpha_{(0)}(x).$$

If $\alpha_{(0)}(x)=0$, the transformation is pure redundancy. If $\alpha_{(0)}(x)$ is nonzero, it changes the boundary source by

$$A^{(0)}_\mu\to A^{(0)}_\mu+D_\mu\alpha_{(0)}.$$

For constant $\alpha_{(0)}$ in flat space with $A^{(0)}_\mu=0$, this is the ordinary global symmetry acting on CFT operators and states. The associated CFT charge is measured in the bulk by electric flux through a surface ending on the boundary.

This is the AdS/CFT version of Gauss’s law. Bulk charged objects cannot be hidden from the boundary: their total charge is visible as boundary flux. This is one reason why exact global symmetries are not expected in quantum gravity. What appears as a global symmetry in the boundary theory is represented in the gravitational bulk by a gauge field.

The metric and the stress tensor

The stress tensor is sourced by the background metric. On the bulk side, the corresponding field is the full dynamical metric $G_{MN}$. The gravitational action is

$$S_{\mathrm{grav}} = \frac{1}{16\pi G_{d+1}} \int_{\mathcal M} d^{d+1}X\sqrt{-G} \left(R+\frac{d(d-1)}{L^2}\right) +\frac{1}{8\pi G_{d+1}} \int_{\partial\mathcal M} d^d x\sqrt{|\gamma|}\,K +S_{\mathrm{ct}}.$$

Here $\gamma_{\mu\nu}$ is the induced metric on a cutoff surface, $K$ is the trace of its extrinsic curvature, and $S_{\mathrm{ct}}$ contains local boundary counterterms. The Gibbons-Hawking term is needed for a well-defined Dirichlet variational problem for the metric.

In Fefferman-Graham gauge,

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+g_{\mu\nu}(z,x)dx^\mu dx^\nu\right),$$

the near-boundary expansion is

$$\boxed{ \begin{aligned} g_{\mu\nu}(z,x) &=g_{(0)\mu\nu}(x) +z^2g_{(2)\mu\nu}(x) +\cdots \\ &\quad +z^d g_{(d)\mu\nu}(x) +z^d\log z^2\,h_{(d)\mu\nu}(x) +\cdots. \end{aligned} }$$

The logarithmic term appears in even boundary dimension and is tied to the Weyl anomaly. The leading coefficient $g_{(0)\mu\nu}$ is the CFT background metric. Lower coefficients such as $g_{(2)}$ are locally determined by $g_{(0)}$ and other sources through the bulk equations. The coefficient $g_{(d)}$ contains the state-dependent response data.

For flat $g_{(0)}$ and no other sources, the stress tensor takes the simple form

$$\boxed{ \langle T_{\mu\nu}\rangle = \frac{dL^{d-1}}{16\pi G_{d+1}}g_{(d)\mu\nu}. }$$

On a curved boundary or in the presence of sources, local curvature and source-dependent terms must be added. These are precisely the terms produced by holographic renormalization.

Brown-York stress tensor and counterterms

The gravitational analogue of radial electric flux is the Brown-York stress tensor. At a cutoff surface $z=\epsilon$, define

$$T^{\mathrm{BY}}_{\mu\nu} = \frac{1}{8\pi G_{d+1}} \left(K_{\mu\nu}-K\gamma_{\mu\nu}\right).$$

This expression diverges as $\epsilon\to0$ because the boundary lies at infinite proper distance and infinite redshift. Holographic renormalization adds local counterterms built from the induced metric $\gamma_{\mu\nu}$ and any boundary values of matter fields.

For pure Einstein gravity, the renormalized cutoff stress tensor has the schematic form

$$T^{\mathrm{ren}}_{\mu\nu} = \frac{1}{8\pi G_{d+1}} \left( K_{\mu\nu}-K\gamma_{\mu\nu} -\frac{d-1}{L}\gamma_{\mu\nu} +\frac{L}{d-2}\left(R_{\mu\nu}[\gamma]-\frac12R[\gamma]\gamma_{\mu\nu}\right) +\cdots \right),$$

where the curvature counterterm is present for $d>2$ and the ellipsis denotes additional terms needed in higher dimensions and matter-coupled systems. The CFT stress tensor is then obtained by the appropriate conformal rescaling and by taking $\epsilon\to0$.

Operationally,

$$\boxed{ \langle T^{\mu\nu}\rangle = \frac{2}{\sqrt{g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta g^{(0)}_{\mu\nu}}. }$$

Again, depending on Euclidean versus Lorentzian conventions, one may place a minus sign between $W_{\mathrm{CFT}}$ and $S_{\mathrm{ren}}$. The geometric object being varied is unambiguous: the renormalized on-shell gravitational action as a functional of the boundary metric.

Stress-tensor Ward identities from diffeomorphism invariance

Under a boundary diffeomorphism generated by $\xi^\mu$, the sources vary as

$$\delta_\xi g^{(0)}_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu, \qquad \delta_\xi A^{(0)}_\mu=\xi^\nu\nabla_\nu A^{(0)}_\mu+A^{(0)}_\nu\nabla_\mu\xi^\nu,$$

and scalar sources vary as

$$\delta_\xi\phi_{(0)}^I=\xi^\nu\nabla_\nu\phi_{(0)}^I.$$

Invariance of $W$ gives the diffeomorphism Ward identity. After using the current Ward identity, it can be written schematically as

$$\boxed{ \nabla_\mu\langle T^\mu{}_{\nu}\rangle = F^a_{\nu\mu}\langle J^\mu_a\rangle + \langle\mathcal O_I\rangle\nabla_\nu\phi^I_{(0)}. }$$

If all sources vanish except a fixed background metric, this reduces to

$$\nabla_\mu\langle T^\mu{}_{\nu}\rangle=0.$$

The right-hand side in the general identity has a simple physical interpretation. Background gauge fields exert a Lorentz force on charged matter, and spacetime-dependent couplings inject momentum into the system. The stress tensor of the CFT alone is conserved only when the background sources do not exchange momentum with it.

In the bulk, this boundary identity descends from the radial momentum constraint in Einstein’s equations. The radial Hamiltonian constraint gives the trace/Weyl identity.

Trace Ward identity and Weyl anomaly

Under a boundary Weyl transformation,

$$\delta_\sigma g^{(0)}_{\mu\nu}=2\sigma g^{(0)}_{\mu\nu},$$

an undeformed CFT in flat space has

$$\langle T^\mu{}_{\mu}\rangle=0.$$

With scalar sources, the trace identity contains source terms:

$$\boxed{ \langle T^\mu{}_{\mu}\rangle = \sum_I (d-\Delta_I)\phi^I_{(0)}\langle\mathcal O_I\rangle +\mathcal A[g_{(0)},A^{(0)},\phi_{(0)}]. }$$

The first term says that a source for a relevant or irrelevant operator explicitly breaks scale invariance. The second term $\mathcal A$ is the Weyl anomaly. It is present in even boundary dimension on curved backgrounds, even for an exact CFT.

Holographically, the Weyl anomaly is not an afterthought. It is the coefficient of the logarithmic divergence in the regulated on-shell action. In Fefferman-Graham language, it is tied to the $z^d\log z^2$ term in the metric expansion.

For a two-dimensional CFT dual to Einstein gravity in AdS$_3$, the anomaly is controlled by the Brown-Henneaux central charge

$$c=\frac{3L}{2G_3}.$$

For a four-dimensional CFT with an Einstein-gravity dual, the two central charges satisfy $a=c$ at leading large $N$, with

$$a=c=\frac{\pi L^3}{8G_5}$$

in standard AdS$_5$ normalization. Higher-derivative bulk interactions generally shift these relations and can make $a\ne c$.

Central charges and bulk couplings

The normalization of a CFT current two-point function is fixed, up to a coefficient $C_J$, by conformal symmetry:

$$\langle J_\mu(x)J_\nu(0)\rangle = C_J\frac{I_{\mu\nu}(x)}{|x|^{2(d-1)}}, \qquad I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2}.$$

Holographically,

$$\boxed{ C_J\propto \frac{L^{d-3}}{g_F^2}. }$$

The stress-tensor two-point function is similarly fixed up to a coefficient $C_T$:

$$\langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle =C_T\frac{\mathcal I_{\mu\nu,\rho\sigma}(x)}{|x|^{2d}},$$

where $\mathcal I_{\mu\nu,\rho\sigma}$ is the standard conformal tensor structure. Holographically,

$$\boxed{ C_T\propto \frac{L^{d-1}}{G_{d+1}}. }$$

Thus $L^{d-1}/G_{d+1}$ is the invariant measure of the number of degrees of freedom in the classical gravity limit. In matrix large-$N$ examples, it scales like $N^2$. This is the same parameter that suppresses bulk graviton loops.

The proportionality constants depend on conventions for $J$, $T_{\mu\nu}$, group generators, and the gravitational action. The scaling with $L$, $G_{d+1}$, and $g_F$ is the robust part.

Where bulk gauge fields come from

Bulk gauge fields can have several origins.

In the canonical AdS$_5\times S^5$ example, the isometry group of $S^5$ is

$$SO(6)\simeq SU(4)_R.$$

Kaluza-Klein reduction of the ten-dimensional metric on the sphere gives five-dimensional gauge fields. These are dual to the $SU(4)_R$ R-symmetry currents of $\mathcal N=4$ super Yang-Mills.

Flavor currents often come from gauge fields living on flavor branes. For example, adding probe D7-branes to a D3-brane system introduces fundamental matter, and the worldvolume gauge field on the D7-branes is dual to a flavor current. In the probe limit, $N_f\ll N$, the flavor sector contributes order $N_fN$ degrees of freedom and does not strongly backreact on the adjoint plasma.

Other generalized currents are sourced by higher-form gauge fields. A $p$-form global symmetry in the boundary theory is dual to a bulk $(p+1)$-form gauge field. The same philosophy applies: boundary global symmetry becomes bulk gauge redundancy, and charged objects are measured by generalized fluxes.

The graviton is universal

Every local relativistic QFT has a stress tensor if it can be coupled to a background metric. Therefore every ordinary AdS/CFT dual has a bulk graviton. This is not an optional extra field. It is the field sourced by the boundary metric.

However, two qualifications are important.

First, in standard AdS/CFT the boundary metric $g^{(0)}_{\mu\nu}$ is a nondynamical source. The CFT is placed on a chosen background geometry; it is not automatically coupled to dynamical $d$-dimensional gravity. Making the boundary metric dynamical requires changing the boundary conditions or adding a boundary gravitational action.

Second, the existence of a stress tensor does not guarantee a weakly curved Einstein-like bulk. Every CFT has $T_{\mu\nu}$, but only very special large-$N$ CFTs have a sparse enough single-trace spectrum for the bulk to be well approximated by local Einstein gravity plus a small number of light fields.

So the precise chain is:

$$\boxed{ T_{\mu\nu}\ \text{exists} \Rightarrow \text{a spin-two source exists}; \qquad \text{classical local graviton} \Rightarrow \text{large }C_T\text{ and sparse spectrum}. }$$

A preview: black-brane stress tensor

The stress-tensor dictionary becomes concrete for the planar AdS black brane,

$$ds^2 = \frac{L^2}{z^2} \left( - f(z)dt^2+d\vec x^{\,2}+\frac{dz^2}{f(z)} \right), \qquad f(z)=1-\left(\frac{z}{z_h}\right)^d.$$

After converting to Fefferman-Graham form and applying holographic renormalization, the dual CFT stress tensor has the form of a homogeneous thermal plasma:

$$\langle T^\mu{}_{\nu}\rangle = \mathrm{diag}(-\epsilon,p,p,\ldots,p),$$

with

$$\epsilon=(d-1)p, \qquad p\propto \frac{L^{d-1}}{G_{d+1}}\frac{1}{z_h^d}.$$

The relation $\epsilon=(d-1)p$ is just tracelessness of the stress tensor in a conformal plasma. The overall coefficient is fixed by the gravitational normalization. Later, in the finite-temperature module, this stress tensor will be matched to the horizon entropy and Hawking temperature.

Standard boundary conditions and alternatives

The standard dictionary uses Dirichlet boundary conditions:

$$A_\mu(z,x)\to A^{(0)}_\mu(x), \qquad g_{\mu\nu}(z,x)\to g^{(0)}_{\mu\nu}(x).$$

This means that the CFT global symmetry is not gauged; $A^{(0)}_\mu$ is a background source. It also means that the CFT does not include dynamical boundary gravity; $g^{(0)}_{\mu\nu}$ is a background metric.

Other boundary conditions are possible in some dimensions. For gauge fields in AdS$_4$, changing boundary conditions can correspond to gauging a three-dimensional global symmetry or performing operations related to particle-vortex duality. For gravity, Neumann or mixed metric boundary conditions can couple the boundary theory to dynamical gravity or modify the stress-tensor ensemble. These are important but not the default convention of this course.

Unless explicitly stated otherwise, this course uses standard Dirichlet sources for both $A^{(0)}_\mu$ and $g^{(0)}_{\mu\nu}$.

Common mistakes

Mistake 1: treating the CFT global symmetry as a bulk global symmetry

A CFT global symmetry is represented by a bulk gauge field, not by a bulk global symmetry. The bulk has gauge redundancy and Gauss-law constraints. Boundary global charges are measured by bulk flux.

Mistake 2: confusing a boundary source with a dynamical field

The source $A^{(0)}_\mu$ is a background gauge field unless one changes the boundary conditions. The source $g^{(0)}_{\mu\nu}$ is a background metric unless one couples the boundary theory to gravity. Standard AdS/CFT computes QFT observables as functionals of these nondynamical backgrounds.

Mistake 3: forgetting counterterms

The Brown-York tensor and Maxwell canonical momentum are divergent before renormalization. The physical current and stress tensor are obtained from the renormalized action, not from the raw cutoff variation.

Mistake 4: assuming conservation means zero response

Conservation says $\nabla_\mu J^\mu=0$ or $\nabla_\mu T^\mu{}_{\nu}=0$ under appropriate conditions. It does not say the current or stress tensor vanishes. Thermal states, charged states, rotating states, and states in background fields have nonzero conserved one-point functions.

Mistake 5: ignoring anomalies

Ward identities may contain anomaly terms. In holography, anomalies often arise from logarithmic divergences or Chern-Simons terms. These terms are physical and scheme-independent in their anomaly coefficients.

Exercises

Exercise 1: Current conservation from source gauge invariance

Let

$$\delta W[A]=\int d^d x\sqrt g\,\langle J^\mu\rangle\delta A_\mu.$$

Assume the generating functional is invariant under the Abelian background gauge transformation $\delta A_\mu=\nabla_\mu\alpha$ for arbitrary compactly supported $\alpha(x)$. Show that

$$\nabla_\mu\langle J^\mu\rangle=0.$$

Solution

Gauge invariance gives

$$0=\delta_\alpha W =\int d^d x\sqrt g\,\langle J^\mu\rangle\nabla_\mu\alpha.$$

Integrating by parts and assuming no boundary contribution,

$$0 =-\int d^d x\sqrt g\,\alpha\nabla_\mu\langle J^\mu\rangle.$$

Since $\alpha(x)$ is arbitrary,

$$\nabla_\mu\langle J^\mu\rangle=0.$$

Exercise 2: Maxwell falloffs in AdS

In radial gauge $A_z=0$, consider a zero-momentum transverse Maxwell perturbation $A_i(z)$ in Poincaré AdS$_{d+1}$ with $d>2$. Show that near the boundary the two independent behaviors are

$$A_i(z)=a_i+b_i z^{d-2}+\cdots.$$

Solution

For a transverse zero-momentum mode, the Maxwell equation reduces near the boundary to

$$\partial_z\left(\sqrt g\,g^{zz}g^{ii}\partial_z A_i\right)=0.$$

Using

$$\sqrt g=\frac{L^{d+1}}{z^{d+1}}, \qquad g^{zz}=g^{ii}=\frac{z^2}{L^2},$$

we get

$$\partial_z\left(L^{d-3}z^{3-d}\partial_z A_i\right)=0.$$

Thus

$$z^{3-d}\partial_z A_i=\text{constant},$$

so

$$\partial_z A_i\propto z^{d-3}.$$

Integrating gives

$$A_i(z)=a_i+b_i z^{d-2}+\cdots$$

for $d>2$. The coefficient $a_i$ is the source $A_i^{(0)}$, and $b_i$ is proportional to the current response.

Exercise 3: Dimensions of $J^\mu$ and $T^{\mu\nu}$ from sources

Use source dimensions to show that a conserved current has dimension $d-1$ and the stress tensor has dimension $d$.

Solution

The current source term is

$$\int d^d x\,A_\mu J^\mu.$$

Since $d^d x$ has dimension $-d$ and a background gauge field has dimension $1$, dimensional neutrality gives

$$1+\Delta_J-d=0.$$

Therefore

$$\Delta_J=d-1.$$

For the stress tensor, the source is the dimensionless metric perturbation $h_{\mu\nu}$:

$$\frac12\int d^d x\,h_{\mu\nu}T^{\mu\nu}.$$

Since $[h_{\mu\nu}]=0$,

$$\Delta_T-d=0,$$

so

$$\Delta_T=d.$$

Exercise 4: The Lorentz-force term in the stress-tensor Ward identity

Assume the generating functional depends on a background Abelian gauge field and a metric. Under an infinitesimal diffeomorphism generated by $\xi^\mu$,

$$\delta A_\mu=\xi^\nu F_{\nu\mu}+\nabla_\mu(\xi^\nu A_\nu),$$

where the second term is a gauge transformation. Using current conservation, show that diffeomorphism invariance implies

$$\nabla_\mu\langle T^\mu{}_{\nu}\rangle =F_{\nu\mu}\langle J^\mu\rangle.$$

Solution

The variation of $W$ is

$$\delta W =\int d^d x\sqrt g \left( \frac12\langle T^{\mu\nu}\rangle\delta g_{\mu\nu} + \langle J^\mu\rangle\delta A_\mu \right).$$

For a diffeomorphism,

$$\delta g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu.$$

The metric term becomes, after integrating by parts,

$$\int d^d x\sqrt g\,\langle T^{\mu\nu}\rangle\nabla_\mu\xi_\nu = -\int d^d x\sqrt g\,\xi_\nu\nabla_\mu\langle T^{\mu\nu}\rangle.$$

For the gauge-field term, use

$$\delta A_\mu=\xi^\nu F_{\nu\mu}+\nabla_\mu(\xi^\nu A_\nu).$$

The second term gives zero by current conservation, up to boundary terms. Thus

$$\delta W =\int d^d x\sqrt g\,\xi^\nu \left( -\nabla_\mu\langle T^\mu{}_{\nu}\rangle +F_{\nu\mu}\langle J^\mu\rangle \right).$$

Diffeomorphism invariance requires this to vanish for arbitrary $\xi^\nu$, so

$$\nabla_\mu\langle T^\mu{}_{\nu}\rangle =F_{\nu\mu}\langle J^\mu\rangle.$$

Exercise 5: Why counterterms are unavoidable

For pure AdS$_{d+1}$ with cutoff surface $z=\epsilon$, estimate the leading divergence of the induced volume element $\sqrt\gamma$. Explain why a local counterterm proportional to $\int d^d x\sqrt\gamma$ is needed in the gravitational action.

Solution

In Poincaré AdS,

$$ds^2=\frac{L^2}{z^2}(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu).$$

At $z=\epsilon$, the induced metric is

$$\gamma_{\mu\nu}=\frac{L^2}{\epsilon^2}\eta_{\mu\nu}.$$

Therefore

$$\sqrt\gamma=\left(\frac{L}{\epsilon}\right)^d.$$

The on-shell gravitational action contains an infinite volume divergence proportional to this factor. A local counterterm of the form

$$S_{\mathrm{ct}}\supset -\frac{d-1}{8\pi G_{d+1}L}\int d^d x\sqrt\gamma$$

cancels the leading divergence. Additional curvature counterterms cancel subleading divergences when the boundary metric is curved or when $d$ is sufficiently large.

Further reading

• E. Witten, “Anti De Sitter Space and Holography,” for the boundary-source formulation of AdS/CFT and the role of the boundary metric: arXiv:hep-th/9802150.
• D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Correlation functions in the CFT$_d$/AdS$_{d+1}$ correspondence,” for early explicit current and supergravity correlator calculations: arXiv:hep-th/9804058.
• M. Henningson and K. Skenderis, “The Holographic Weyl Anomaly,” for the holographic derivation of Weyl anomalies: arXiv:hep-th/9806087.
• V. Balasubramanian and P. Kraus, “A Stress Tensor for Anti-de Sitter Gravity,” for the counterterm stress tensor: arXiv:hep-th/9902121.
• S. de Haro, K. Skenderis, and S. N. Solodukhin, “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence,” for the Fefferman-Graham expansion and holographic one-point functions: arXiv:hep-th/0002230.
• K. Skenderis, “Lecture Notes on Holographic Renormalization,” for a systematic account of counterterms, Ward identities, and anomalies: arXiv:hep-th/0209067.