The GKPW Prescription

The main idea

The previous pages built the ingredients of the local operator dictionary:

$$\text{CFT source } J(x) \quad \longleftrightarrow \quad \text{leading boundary value } \phi_{(0)}(x) \text{ of a bulk field } \phi.$$

The GKPW prescription turns this dictionary into a calculational rule. In its most compact form,

$$\boxed{ Z_{\mathrm{CFT}}[J] = Z_{\mathrm{string}}[\phi \to J]. }$$

The left-hand side is the CFT generating functional with sources for local operators. The right-hand side is the full string-theory path integral over bulk fields whose asymptotic boundary values are fixed by those sources. In the regime where the bulk is well described by classical supergravity, the string path integral is approximated by a saddle point:

$$\boxed{ Z_{\mathrm{CFT}}[J] \simeq \exp\left(-S^E_{\mathrm{ren}}[\phi_{\mathrm{cl}};J]\right) }$$

in Euclidean signature. Equivalently, if

$$W_{\mathrm{CFT}}[J]=\log Z_{\mathrm{CFT}}[J],$$

then

$$\boxed{ W_{\mathrm{CFT}}[J] = -S^E_{\mathrm{ren}}[\phi_{\mathrm{cl}};J] + \text{bulk loop and string corrections}. }$$

This is the practical heart of the dictionary. To compute CFT correlators at large $N$ and strong coupling, solve a classical boundary-value problem in AdS, evaluate the renormalized on-shell action, and differentiate with respect to boundary sources.

The GKPW prescription identifies the CFT generating functional $Z_{\mathrm{CFT}}[J]$ with the string partition function with asymptotic boundary condition $\phi_{(0)}=J$. In the classical supergravity limit, the bulk path integral is dominated by a solution of the bulk equations of motion, and $W_{\mathrm{CFT}}[J]$ is computed by the renormalized on-shell action.

Three warnings should be kept visible from the beginning.

First, the exact statement is an equality of partition functions, not an equality between a classical field and a quantum operator. A classical bulk field appears only after taking a semiclassical limit.

Second, the object that appears in the saddle formula is not the naive on-shell action. It is the renormalized on-shell action. Near the AdS boundary, the bulk volume diverges, and the on-shell action contains divergences. These are removed by local covariant counterterms on cutoff surfaces.

Third, boundary conditions contain physics. The same asymptotic source $J$ can lead to different CFT states depending on the bulk interior condition: Euclidean regularity, Lorentzian normalizability, black-hole horizon infalling behavior, or a more general Schwinger-Keldysh contour.

Source convention used on this page

Sign conventions in this subject are famously slippery. The safest approach is to state the source convention explicitly.

On this page, for a scalar operator $\mathcal O$, define the Euclidean CFT generating functional by

$$Z_{\mathrm{CFT}}[J] = \left\langle \exp\left(\int d^d x\sqrt{g_{(0)}}\,J(x)\mathcal O(x)\right) \right\rangle,$$

so that

$$W[J]=\log Z[J], \qquad \langle\mathcal O(x)\rangle_J = \frac{1}{\sqrt{g_{(0)}(x)}} \frac{\delta W[J]}{\delta J(x)}.$$

With this convention, the Euclidean classical GKPW relation is

$$W[J]= -S^E_{\mathrm{ren}}[J].$$

Therefore

$$\boxed{ \langle\mathcal O(x)\rangle_J = -\frac{1}{\sqrt{g_{(0)}(x)}} \frac{\delta S^E_{\mathrm{ren}}[J]}{\delta J(x)}. }$$

Some papers instead define the source by deforming the Euclidean action as

$$S_E \longrightarrow S_E + \int d^d x\sqrt{g_{(0)}}\,J\mathcal O.$$

Then $Z[J]=\langle e^{-\int J\mathcal O}\rangle$, and the sign of the variational formula changes. This is not physics. It is bookkeeping. The invariant rule is: differentiate the generating functional in the convention in which you defined the source.

In Lorentzian signature, the analogous saddle relation is schematically

$$Z_{\mathrm{CFT}}[J] \simeq \exp\left(iS^L_{\mathrm{ren}}[\phi_{\mathrm{cl}};J]\right),$$

but the word “schematically” is important: real-time correlators require a choice of contour and state. Retarded correlators, for example, are obtained by imposing infalling boundary conditions at a black-hole horizon. We postpone the systematic real-time story to the correlator and finite-temperature chapters.

The exact statement and the classical limit

Let $\mathcal O_i$ be a set of single-trace CFT operators, and let $J^i(x)$ be sources. The CFT generating functional is

$$Z_{\mathrm{CFT}}[J] = \left\langle \exp\left(\sum_i\int d^d x\sqrt{g_{(0)}}\,J^i(x)\mathcal O_i(x)\right) \right\rangle.$$

The dual bulk theory contains fields $\phi^i$ whose asymptotic boundary data are determined by $J^i$. The exact GKPW statement is

$$Z_{\mathrm{CFT}}[J] = \int_{\phi^i\to J^i}\mathcal D\Phi\; e^{-S^E_{\mathrm{string}}[\Phi]},$$

where $\Phi$ denotes all bulk fields, not only the fields explicitly sourced. The condition $\phi^i\to J^i$ means that the leading non-normalizable part of $\phi^i$ is fixed by the source. For a scalar operator of dimension $\Delta_i$ in standard quantization,

$$\phi^i(z,x) = z^{d-\Delta_i}\left(J^i(x)+\cdots\right) + z^{\Delta_i}\left(A^i(x)+\cdots\right), \qquad z\to0.$$

Here $z$ is a Fefferman-Graham radial coordinate, with the conformal boundary at $z=0$. The coefficient $J^i$ is specified externally. The coefficient $A^i$ is determined dynamically by the bulk equations and by the interior condition. After holographic renormalization, $A^i$ gives the nonlocal part of $\langle\mathcal O_i\rangle_J$.

The classical supergravity approximation consists of two simplifications:

$$\int_{\phi\to J}\mathcal D\Phi\;e^{-S^E_{\mathrm{string}}[\Phi]} \quad\longrightarrow\quad \exp\left(-S^E_{\mathrm{sugra}}[\Phi_{\mathrm{cl}}]\right),$$

and then

$$S^E_{\mathrm{sugra}}[\Phi_{\mathrm{cl}}] \quad\longrightarrow\quad S^E_{\mathrm{ren}}[\Phi_{\mathrm{cl}}].$$

The first step uses a saddle-point approximation. The second step is holographic renormalization.

In the canonical AdS$_5$/CFT$_4$ example, this limit is controlled by

$$N\gg1, \qquad \lambda=g_{\mathrm{YM}}^2N\gg1.$$

Large $N$ suppresses bulk loops, while large $\lambda$ suppresses stringy $\alpha'$ corrections. The leading classical action scales as

$$S_{\mathrm{bulk}}\sim \frac{L^{d-1}}{G_{d+1}}\sim N^2$$

for matrix large-$N$ theories with Einstein-like duals. Thus the saddle approximation is also the statement that the CFT generating functional is large:

$$W_{\mathrm{CFT}}[J]\sim N^2.$$

The regulated bulk problem

The boundary of AdS lies at infinite proper distance, so the on-shell action is usually divergent. The regulated problem has three steps.

Step 1: introduce a cutoff surface

In Fefferman-Graham coordinates,

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

introduce a cutoff surface

$$\Sigma_\epsilon: z=\epsilon.$$

The induced metric on $\Sigma_\epsilon$ is

$$\gamma_{\mu\nu}(\epsilon,x) = \frac{L^2}{\epsilon^2}g_{\mu\nu}(\epsilon,x).$$

For a scalar field dual to an operator of dimension $\Delta$, the Dirichlet data on the cutoff surface are chosen as

$$\phi(\epsilon,x) = \epsilon^{d-\Delta}J(x)+\cdots,$$

where the omitted terms include local derivative corrections required by the equations of motion.

Step 2: solve the bulk equations

Solve

$$\frac{\delta S_{\mathrm{bulk}}}{\delta\Phi}=0$$

with the specified boundary behavior. In Euclidean vacuum calculations, one usually demands smoothness or regularity in the interior. In global AdS, regularity at the center selects the vacuum response. In Euclidean black-hole geometries, smoothness at the contractible thermal circle selects the thermal state.

In Lorentzian signature, specifying boundary data near $z=0$ is not enough. One must also specify a state. For retarded finite-temperature correlators, the correct black-hole prescription is infalling behavior at the future horizon. For time-ordered correlators, one should use an appropriate real-time contour. The Euclidean formula is cleanest, but it is not a substitute for the Lorentzian state problem.

Step 3: evaluate and renormalize the on-shell action

The regulated on-shell action is

$$S_{\mathrm{reg}}[\epsilon,J] = S_{\mathrm{bulk}}[\Phi_{\mathrm{cl}}] +S_{\mathrm{boundary}}[\Phi_{\mathrm{cl}}] \quad \text{with the radial integral cut off at } z=\epsilon.$$

For gravity, $S_{\mathrm{boundary}}$ includes the Gibbons-Hawking-York term. For matter fields, boundary terms may also be needed for a well-posed variational principle, especially for gauge fields, fermions, higher-derivative actions, alternate quantization, or mixed boundary conditions.

The regulated action has an asymptotic expansion of the form

$$S_{\mathrm{reg}}[\epsilon,J] = \sum_k \epsilon^{-a_k}S_{(a_k)}[J] +S_{\log}[J]\log\epsilon +S_{\mathrm{finite}}[J] +o(1).$$

The divergent terms are local functionals of the induced fields at the cutoff surface. One cancels them with local counterterms:

$$S_{\mathrm{ct}}[\epsilon] = -\sum_k \epsilon^{-a_k}S_{(a_k)}[J] -S_{\log}[J]\log\epsilon +S_{\mathrm{finite,local}}[J].$$

The renormalized action is

$$\boxed{ S_{\mathrm{ren}}[J] = \lim_{\epsilon\to0} \left(S_{\mathrm{reg}}[\epsilon,J]+S_{\mathrm{ct}}[\epsilon,J]\right). }$$

The finite local part $S_{\mathrm{finite,local}}$ is a renormalization-scheme choice. It changes contact terms in correlators, not separated-point nonlocal physics.

Scalar example: the on-shell action is a boundary term

Take a scalar field in Euclidean AdS$_{d+1}$,

$$S_\phi = \frac12\int d^{d+1}x\sqrt g\, \left(g^{ab}\partial_a\phi\partial_b\phi+m^2\phi^2\right).$$

Integrating by parts gives

$$S_\phi = \frac12\int d^{d+1}x\sqrt g\,\phi(-\nabla^2+m^2)\phi +\frac12\int_{\partial M}d^d x\sqrt\gamma\,\phi n^a\partial_a\phi,$$

where $n^a$ is the outward-pointing unit normal to the boundary of the regulated region. On a classical solution,

$$(-\nabla^2+m^2)\phi_{\mathrm{cl}}=0,$$

so the regulated on-shell action reduces to

$$S_{\mathrm{reg}}[\phi_{\mathrm{cl}}] = \frac12\int_{z=\epsilon}d^d x\sqrt\gamma\,\phi_{\mathrm{cl}}n^a\partial_a\phi_{\mathrm{cl}}.$$

This is why the boundary behavior of fields is so powerful. Once the classical solution is known as a functional of the source, the entire quadratic generating functional is encoded in its near-boundary canonical momentum.

Define the regulated radial momentum by

$$\pi_\epsilon(x) = \sqrt\gamma\,n^a\partial_a\phi(\epsilon,x).$$

Then the variation of the regulated on-shell action is

$$\delta S_{\mathrm{reg}} = \int_{z=\epsilon}d^d x\,\pi_\epsilon(x)\delta\phi(\epsilon,x),$$

up to possible variations of explicit boundary terms. The renormalized momentum is obtained by adding the counterterm contribution and taking the limit:

$$\Pi_{\mathrm{ren}}(x) = \lim_{\epsilon\to0} \left[\text{appropriate power of }\epsilon\right] \left( \pi_\epsilon(x)+\frac{\delta S_{\mathrm{ct}}}{\delta\phi(\epsilon,x)} \right).$$

The phrase “appropriate power of $\epsilon$” is not a nuisance; it reflects the fact that the source is not the cutoff value $\phi(\epsilon,x)$ itself but the rescaled coefficient $J(x)$ in the asymptotic expansion.

For a scalar with noninteger

$$\nu=\Delta-\frac d2>0,$$

the near-boundary solution in standard quantization has the schematic form

$$\phi(z,x) = z^{d-\Delta}\left(J(x)+z^2J_{(2)}(x)+\cdots\right) + z^\Delta\left(A(x)+z^2A_{(2)}(x)+\cdots\right).$$

The coefficients $J_{(2)},J_{(4)},\ldots$ are local functionals of $J$ fixed by the near-boundary equations. The coefficient $A(x)$ is nonlocal in $J$ and depends on the interior condition. In the common normalization where the source convention is adjusted so that the one-point function is positive, one writes

$$\boxed{ \langle\mathcal O(x)\rangle_J = (2\Delta-d)A(x)+\text{local terms}. }$$

With the explicit $W=-S^E_{\mathrm{ren}}$ convention of this page, this means that the nonlocal variation of the renormalized action is

$$\delta S^E_{\mathrm{ren}} = -\int d^d x\sqrt{g_{(0)}}\,(2\Delta-d)A(x)\delta J(x) +\text{local variations}.$$

The local terms are scheme-dependent contact terms. The nonlocal dependence of $A$ on $J$ is the physical content of the correlator.

Two-point functions from the quadratic action

For a free scalar in Euclidean Poincare AdS,

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

the scalar wave equation in momentum space is solved by

$$\phi(z,p) = J(p)\,\mathcal K_\Delta(z,p),$$

where regularity in the interior selects the bulk-to-boundary mode

$$\mathcal K_\Delta(z,p) = \frac{2^{1-\nu}}{\Gamma(\nu)} \,p^\nu z^{d/2}K_\nu(pz), \qquad \nu=\Delta-\frac d2.$$

The small-$z$ expansion of $K_\nu$ contains both the source falloff $z^{d-\Delta}$ and the response falloff $z^\Delta$. After renormalization, the nonlocal part of the quadratic action is of the form

$$S^E_{\mathrm{ren,quad}} = -\frac12\int\frac{d^d p}{(2\pi)^d}\, J(p)\,\mathcal G_E(p)\,J(-p) +\text{local terms},$$

with

$$\mathcal G_E(p) \propto (2\nu)\frac{\Gamma(-\nu)}{\Gamma(\nu)} \left(\frac p2\right)^{2\nu}$$

for noninteger $\nu$, up to the normalization of the bulk kinetic term and the normalization chosen for $\mathcal O$.

Since $W=-S^E_{\mathrm{ren}}$, the connected two-point function is

$$\boxed{ \langle\mathcal O(p)\mathcal O(-p)\rangle_c = \frac{\delta^2 W}{\delta J(p)\delta J(-p)} = \mathcal G_E(p) +\text{contact terms}. }$$

In position space, conformal invariance fixes the separated-point answer to be

$$\boxed{ \langle\mathcal O(x)\mathcal O(0)\rangle = \frac{C_\mathcal O}{|x|^{2\Delta}}, \qquad x\ne0. }$$

The holographic calculation determines the normalization $C_\mathcal O$ in terms of the bulk kinetic coefficient. Different normalizations of $\mathcal O$ correspond to rescaling the bulk field and its action.

Higher-point functions and Witten diagrams

The GKPW prescription also explains why Witten diagrams compute CFT correlators.

Suppose the bulk scalar has interactions,

$$S_{\mathrm{bulk}} = \frac12\int\phi\mathcal D\phi +\frac{g_3}{3!}\int d^{d+1}x\sqrt g\,\phi^3 +\frac{g_4}{4!}\int d^{d+1}x\sqrt g\,\phi^4 +\cdots.$$

The classical solution can be expanded perturbatively in the source:

$$\phi_{\mathrm{cl}} = \phi^{(1)}[J] +\phi^{(2)}[J^2] +\phi^{(3)}[J^3] +\cdots.$$

Substituting this solution into the on-shell action produces an expansion

$$S_{\mathrm{ren}}[J] = S^{(2)}[J^2] +S^{(3)}[J^3] +S^{(4)}[J^4] +\cdots.$$

Functional differentiation gives connected correlators. Diagrammatically:

bulk contribution	CFT object	large-$N$ order
quadratic on-shell action	two-point function	leading planar order
cubic contact vertex	three-point OPE coefficient	tree level in bulk
exchange diagram	four-point function with single-trace exchange	tree level in bulk
bulk loop diagram	$1/N$ correction	quantum gravity correction
stringy higher-derivative term	finite-$\lambda$ correction	$\alpha'$ correction

At tree level, the external legs of Witten diagrams are bulk-to-boundary propagators, while internal lines are bulk-to-bulk propagators. The statement “Witten diagrams compute CFT correlators” is simply the perturbative expansion of the classical and quantum bulk path integral with fixed asymptotic boundary conditions.

One-point functions and states

A source is not the same thing as a state. This distinction becomes crucial after the first few computations.

For a scalar in standard quantization,

$$\phi(z,x) = z^{d-\Delta}J(x)+z^\Delta A(x)+\cdots.$$

The source $J$ deforms the CFT Lagrangian or background. The response $A$ encodes the expectation value in the state selected by the bulk interior condition. In the vacuum with no source, one normally has

$$J=0, \qquad A=0, \qquad \langle\mathcal O\rangle=0,$$

for an operator not forced to have a vev by symmetry breaking or anomalies. But a normalizable bulk mode can have

$$J=0, \qquad A\ne0,$$

which corresponds to a state with a nonzero expectation value, not to a deformation of the Hamiltonian.

For the metric, the analogous statement is even more familiar. A black brane can have a flat boundary metric $g_{(0)\mu\nu}=\eta_{\mu\nu}$ but a nonzero stress tensor:

$$\langle T_{\mu\nu}\rangle \ne0.$$

The boundary metric is the source; the subleading normalizable coefficient contains the energy density, pressure, and other state data.

The gravitational part of the prescription

For correlators involving the stress tensor, the bulk action must include the gravitational boundary terms needed for a Dirichlet variational principle. For Einstein gravity,

$$S_E = -\frac{1}{16\pi G_{d+1}} \int_M d^{d+1}x\sqrt g\,(R-2\Lambda) - \frac{1}{8\pi G_{d+1}} \int_{\partial M} d^d x\sqrt\gamma\,K + S_{\mathrm{ct}}.$$

The signs depend on Euclidean conventions, but the structural point is fixed: the Gibbons-Hawking-York term makes the variational problem well posed when the induced metric is held fixed.

The boundary metric $g_{(0)\mu\nu}$ is the source for the CFT stress tensor:

$$\delta W = \frac12\int d^d x\sqrt{g_{(0)}}\, \langle T^{\mu\nu}\rangle\delta g_{(0)\mu\nu} +\cdots.$$

Therefore

$$\boxed{ \langle T^{\mu\nu}\rangle = \frac{2}{\sqrt{g_{(0)}}} \frac{\delta W}{\delta g_{(0)\mu\nu}} = -\frac{2}{\sqrt{g_{(0)}}} \frac{\delta S^E_{\mathrm{ren}}}{\delta g_{(0)\mu\nu}} }$$

in the Euclidean source convention used here.

For gauge fields, the analogous formula is

$$\boxed{ \langle J^\mu_A\rangle = \frac{1}{\sqrt{g_{(0)}}} \frac{\delta W}{\delta A^{(0)A}_\mu} = -\frac{1}{\sqrt{g_{(0)}}} \frac{\delta S^E_{\mathrm{ren}}}{\delta A^{(0)A}_\mu}. }$$

The radial Hamiltonian constraints of the bulk theory become the CFT Ward identities. For example, if scalar sources are present, the diffeomorphism Ward identity has the schematic form

$$\nabla_\mu\langle T^\mu{}_{\nu}\rangle = \sum_i\langle\mathcal O_i\rangle\nabla_\nu J^i + F^{(0)A}_{\nu\mu}\langle J^\mu_A\rangle +\text{anomaly terms}.$$

The trace Ward identity has the schematic form

$$\langle T^\mu{}_{\mu}\rangle = \sum_i(d-\Delta_i)J^i\langle\mathcal O_i\rangle +\mathcal A[g_{(0)},J,A^{(0)}],$$

where $\mathcal A$ is the conformal anomaly in even boundary dimension. These identities are not optional checks; they are consequences of the bulk constraints plus correct counterterms.

What is fixed, what is solved for?

A common source of confusion is whether the boundary value of a bulk field is fixed or dynamical. The answer depends on which coefficient one is discussing.

For a scalar in standard quantization:

quantity	bulk meaning	CFT meaning	status in GKPW
$J(x)$	leading non-normalizable coefficient	source for $\mathcal O$	fixed boundary data
$A(x)$	normalizable response coefficient	determines $\langle\mathcal O\rangle_J$	solved for
$S_{\mathrm{ren}}[J]$	renormalized on-shell action	$-W[J]$ in this convention	generating functional
$\delta^n W/\delta J^n$	response to boundary data	connected $n$-point function	observable

For gauge fields:

quantity	bulk meaning	CFT meaning
$A^{(0)}_\mu$	boundary value of bulk gauge field	background gauge field source
radial electric flux	canonical momentum	current expectation value
gauge transformation nonzero at boundary	background symmetry transformation	Ward identity

For the metric:

quantity	bulk meaning	CFT meaning
$g_{(0)\mu\nu}$	conformal boundary metric	source for $T^{\mu\nu}$
$g_{(d)\mu\nu}$ plus local terms	normalizable metric data	stress-tensor expectation value
radial Einstein constraints	bulk constraint equations	stress-tensor Ward identities

The key lesson is simple: fix sources, solve for responses, differentiate the renormalized action.

Alternate quantization and Legendre transforms

The previous statements assumed standard quantization. For scalars with mass in the Breitenlohner-Freedman window

$$-\frac{d^2}{4}<m^2L^2<-\frac{d^2}{4}+1,$$

both falloffs are normalizable. The two possible dimensions are

$$\Delta_+=\frac d2+\nu, \qquad \Delta_-=\frac d2-\nu, \qquad 0<\nu<1.$$

In standard quantization, $J$ is the coefficient of the $z^{\Delta_-}$ falloff and the operator has dimension $\Delta_+$. In alternate quantization, the roles of source and response are exchanged: the operator has dimension $\Delta_-$, and the generating functional is related to the standard one by a Legendre transform.

This is not a small technicality. It is the simplest example showing that the phrase “the boundary value of the field is the source” must be interpreted through the variational principle. The source is the coefficient held fixed in the chosen quantization.

Mixed boundary conditions describe multi-trace deformations. For a double-trace deformation

$$\delta S_{\mathrm{CFT}} = \frac f2\int d^d x\sqrt{g_{(0)}}\,\mathcal O^2,$$

the bulk boundary condition relates the two asymptotic coefficients schematically as

$$A(x)=fJ(x)+\text{external source terms}$$

or the inverse relation, depending on the quantization. The correct statement is again variational: add the boundary functional corresponding to the multi-trace deformation, then impose that the total variation vanish under the allowed boundary variations.

Contact terms and scheme dependence

Holographic counterterms are local functionals of boundary sources. Therefore finite counterterms can change local terms in correlation functions.

For example, adding a finite scalar counterterm

$$\Delta S_{\mathrm{ren}} = \frac c2\int d^d x\sqrt{g_{(0)}}\,J(x)\Box J(x)$$

changes the two-point function in momentum space by a polynomial:

$$G_E(p)\longrightarrow G_E(p)+c p^2$$

up to signs determined by the source convention. In position space, this is a contact term proportional to derivatives of $\delta^{(d)}(x)$.

This is why separated-point correlators and nonanalytic momentum dependence are more invariant than the full distributional answer. In even dimensions, logarithmic counterterms encode conformal anomalies, and the coefficient of the logarithm is universal. But finite local terms remain scheme-dependent.

A good practical rule is:

$$\boxed{ \text{power divergences and finite local terms are scheme-sensitive;} \quad \text{nonlocal terms and anomaly coefficients are physical.} }$$

Large-$N$ counting in the prescription

The GKPW prescription packages the large-$N$ expansion geometrically. Suppose the bulk action has the form

$$S_{\mathrm{bulk}}=N^2\,s_{\mathrm{bulk}}$$

after choosing fields of order one. Then

$$W[J]\sim N^2.$$

If single-trace operators are normalized so that their two-point functions scale as $N^0$, one usually includes corresponding powers of $N$ in the definition of the source or operator. Different communities use different normalizations. The invariant content is the hierarchy:

$$\text{classical bulk tree diagrams} \quad\longleftrightarrow\quad \text{leading large-}N\text{ connected correlators},$$ $$\text{bulk loops} \quad\longleftrightarrow\quad 1/N\text{ corrections},$$ $$\text{higher-derivative string terms} \quad\longleftrightarrow\quad 1/\lambda\text{ or }\alpha'\text{ corrections}.$$

For canonically normalized single-trace operators in a matrix large-$N$ theory, connected correlators often scale as

$$\langle\mathcal O_1\cdots\mathcal O_n\rangle_c \sim N^{2-n}.$$

If instead one normalizes $\mathcal O$ so that $\langle\mathcal O\mathcal O\rangle\sim N^2$, then all formulas shift by powers of $N$. The GKPW prescription does not remove the need to state operator normalization. It makes the normalization calculable once the bulk kinetic terms are fixed.

Euclidean, Lorentzian, and thermal prescriptions

The Euclidean prescription is the cleanest starting point:

1. choose Euclidean boundary geometry and sources;
2. find a smooth Euclidean bulk saddle with those boundary conditions;
3. evaluate $S^E_{\mathrm{ren}}$;
4. differentiate.

This computes Euclidean correlators in the state or ensemble prepared by the Euclidean path integral.

For Lorentzian correlators, the prescription is richer. One must specify both boundary sources and a state. At zero temperature in the vacuum, the $i\epsilon$ prescription selects the desired Green function. At finite temperature, the bulk geometry often contains a horizon. Then:

CFT correlator	bulk prescription
Euclidean thermal correlator	smooth Euclidean black-hole saddle
retarded Green function	infalling boundary condition at future horizon
advanced Green function	outgoing boundary condition
time-ordered correlator	appropriate Lorentzian contour / analytic continuation
Schwinger-Keldysh correlators	doubled boundary contour and matching conditions

The infalling rule for retarded functions is one of the most useful technical tools in holographic transport. But it should not be confused with the Euclidean GKPW formula. They are related by analytic continuation when the continuation is well defined, but the Lorentzian prescription knows about causality and state preparation.

A practical computation checklist

A holographic correlator computation should pass the following checklist.

1. Identify the operator and source

Specify the CFT operator $\mathcal O_i$, its source $J^i$, and the source convention. For the stress tensor and currents, include the factors of $1/2$, $\sqrt{g_{(0)}}$, and possible gauge-group generators.

2. Identify the dual bulk field and normalization

Write the relevant part of the bulk action, including the kinetic coefficient. This fixes the normalization of the CFT correlator.

3. Choose the background and state

Vacuum AdS, global AdS, thermal AdS, black brane, charged black hole, domain wall, time-dependent geometry, and horizon prescription all compute different physics.

4. Solve the linear or nonlinear boundary-value problem

For two-point functions, solve linearized equations. For higher-point functions, either solve perturbatively in sources or use Witten diagrams.

5. Evaluate the on-shell action

Do not use the bulk Lagrangian naively. Use the on-shell boundary term plus any necessary gravitational, gauge-field, or fermionic boundary terms.

6. Add counterterms

Cancel divergences with local covariant counterterms at $z=\epsilon$. Keep track of finite counterterm choices.

7. Differentiate the renormalized action

Use

$$\langle\mathcal O_i\rangle_J = \frac{1}{\sqrt{g_{(0)}}} \frac{\delta W}{\delta J^i}, \qquad G_{ij}(x,y) = \frac{1}{\sqrt{g_{(0)}(x)}\sqrt{g_{(0)}(y)}} \frac{\delta^2 W}{\delta J^i(x)\delta J^j(y)}.$$

In the Euclidean convention of this page, $W=-S^E_{\mathrm{ren}}$.

8. Interpret the answer

Separate universal nonlocal terms from contact terms. State the regime of validity: large $N$, large coupling, probe limit, hydrodynamic limit, low frequency, near-boundary expansion, or whatever approximation was used.

Common mistakes

Mistake 1: differentiating the unrenormalized on-shell action

The regulated on-shell action contains divergences. Differentiating it before renormalization gives divergent or cutoff-dependent correlators. The correct object is $S_{\mathrm{ren}}$.

Mistake 2: fixing both source and vev

For a second-order bulk equation, the leading and subleading asymptotic coefficients are not both arbitrary after imposing an interior condition. In standard GKPW, the source is fixed and the response is solved for.

Mistake 3: treating all boundary terms as optional

Boundary terms determine the variational principle. For gravity, omitting the Gibbons-Hawking-York term gives the wrong stress-tensor variational problem. For alternate quantization or mixed boundary conditions, additional boundary functionals are essential.

Mistake 4: confusing contact terms with physical disagreement

Finite counterterms shift contact terms. Two calculations that differ by local polynomials in momentum space may agree on all separated-point observables.

Mistake 5: using Euclidean regularity for a retarded correlator

Retarded correlators are causal Lorentzian objects. At black-hole horizons, they require infalling boundary conditions. Euclidean smoothness computes Euclidean thermal correlators; analytic continuation must be handled carefully.

Mistake 6: forgetting operator normalization

The overall coefficient of a two-point function depends on the normalization of the bulk kinetic term and on the normalization of the CFT operator. Universal ratios may be normalization-independent, but individual $C_\mathcal O$, $C_J$, and $C_T$ values are not.

Exercises

Exercise 1: Signs in the Euclidean source convention

Suppose

$$Z[J] = \left\langle e^{\int J\mathcal O}\right\rangle, \qquad W[J]=\log Z[J],$$

and the classical Euclidean bulk saddle gives

$$Z[J]\simeq e^{-S^E_{\mathrm{ren}}[J]}.$$

Show that

$$\langle\mathcal O(x)\rangle_J = -\frac{1}{\sqrt{g_{(0)}(x)}} \frac{\delta S^E_{\mathrm{ren}}}{\delta J(x)}.$$

Solution

By definition,

$$W[J]=\log Z[J].$$

The saddle relation gives

$$W[J]=-S^E_{\mathrm{ren}}[J].$$

The source convention gives

$$\langle\mathcal O(x)\rangle_J = \frac{1}{\sqrt{g_{(0)}(x)}} \frac{\delta W}{\delta J(x)}.$$

Combining the two formulas yields

$$\langle\mathcal O(x)\rangle_J = -\frac{1}{\sqrt{g_{(0)}(x)}} \frac{\delta S^E_{\mathrm{ren}}}{\delta J(x)}.$$

If one instead defines the Euclidean source through $S_E\to S_E+\int J\mathcal O$, the sign changes because the source appears in $Z$ as $e^{-\int J\mathcal O}$.

Exercise 2: On-shell scalar action

For

$$S_\phi = \frac12\int_M d^{d+1}x\sqrt g\, \left(g^{ab}\partial_a\phi\partial_b\phi+m^2\phi^2\right),$$

show that on a solution of $(\nabla^2-m^2)\phi=0$, the action reduces to

$$S_{\mathrm{os}} = \frac12\int_{\partial M}d^d x\sqrt\gamma\,\phi n^a\partial_a\phi.$$

Solution

Use

$$\nabla_a(\phi\nabla^a\phi) = (\nabla_a\phi)(\nabla^a\phi)+\phi\nabla^2\phi.$$

Therefore

$$(\nabla\phi)^2 = \nabla_a(\phi\nabla^a\phi)-\phi\nabla^2\phi.$$

Substitute into the action:

$$S_\phi = \frac12\int_M\sqrt g\,\phi(-\nabla^2+m^2)\phi + \frac12\int_{\partial M}\sqrt\gamma\,\phi n^a\partial_a\phi.$$

On a classical solution, the bulk term vanishes, leaving the boundary term.

Exercise 3: Correlator from a quadratic generating functional

Assume the renormalized Euclidean on-shell action has the nonlocal quadratic part

$$S^E_{\mathrm{ren}}[J] = -\frac12\int\frac{d^d p}{(2\pi)^d}\,J(p)G(p)J(-p).$$

Using $W=-S^E_{\mathrm{ren}}$, compute the connected two-point function.

Solution

The generating functional is

$$W[J] = \frac12\int\frac{d^d p}{(2\pi)^d}\,J(p)G(p)J(-p).$$

Differentiating once gives

$$\frac{\delta W}{\delta J(-p)}=G(p)J(p),$$

assuming $G(p)=G(-p)$ and suppressing standard momentum-space delta functions. Differentiating again gives

$$\frac{\delta^2 W}{\delta J(p)\delta J(-p)}=G(p).$$

Thus

$$\langle\mathcal O(p)\mathcal O(-p)\rangle_c=G(p),$$

plus possible contact terms from finite local counterterms.

Exercise 4: Finite counterterms and contact terms

Add the finite counterterm

$$\Delta S_{\mathrm{ren}} = \frac c2\int d^d x\,J(x)(-\partial^2)J(x)$$

on a flat boundary. Show that it shifts the momentum-space two-point function by a polynomial in $p^2$. What does this mean in position space?

Solution

Fourier transforming gives

$$\Delta S_{\mathrm{ren}} = \frac c2\int\frac{d^d p}{(2\pi)^d}\,J(p)p^2J(-p).$$

Since $W=-S^E_{\mathrm{ren}}$ in the convention of this page, the contribution to the connected two-point function is

$$\Delta G(p)=-c p^2,$$

up to the overall sign convention for the source. The important point is not the sign but the analytic dependence on momentum: $p^2$ is a polynomial.

In position space, a polynomial in $p$ corresponds to derivatives of a delta function. Thus this finite counterterm changes only contact terms, not the separated-point correlator for $x\ne0$.

Exercise 5: Which coefficient is the source?

Let a scalar mass lie in the window

$$-\frac{d^2}{4}<m^2L^2<-\frac{d^2}{4}+1.$$

The two possible dimensions are

$$\Delta_+=\frac d2+\nu, \qquad \Delta_-=\frac d2-\nu, \qquad 0<\nu<1.$$

Explain why there can be two admissible quantizations and why the generating functionals are related by a Legendre transform.

Solution

In this mass range both near-boundary falloffs are normalizable. Therefore the variational problem is not forced to hold fixed only the slower falloff. One may choose standard quantization, in which the coefficient of the $z^{\Delta_-}$ falloff is the source and the operator has dimension $\Delta_+$. Or one may choose alternate quantization, in which the conjugate coefficient is treated as the source and the operator has dimension $\Delta_-$.

Changing which variable is held fixed is precisely what a Legendre transform does. If $W_+[J]$ is the standard-quantization generating functional, the alternate-quantization generating functional is obtained by transforming from the source variable to its conjugate response variable, up to local terms and normalization conventions.

Exercise 6: Diagnose a holographic two-point computation

A student solves a scalar wave equation in Euclidean AdS, substitutes the solution into the bulk action, obtains a divergent answer proportional to $\epsilon^{-2\nu}$, differentiates it twice, and concludes that the CFT two-point function is cutoff-dependent. What went wrong?

Solution

The student differentiated the regulated on-shell action rather than the renormalized on-shell action. The divergence is expected: it comes from the infinite volume near the AdS boundary and corresponds to UV divergences in the source-dependent CFT generating functional.

The correct procedure is to add local covariant counterterms on the cutoff surface, define

$$S_{\mathrm{ren}} = \lim_{\epsilon\to0}(S_{\mathrm{reg}}+S_{\mathrm{ct}}),$$

and then differentiate $W=-S^E_{\mathrm{ren}}$. The remaining nonlocal part gives the physical separated-point two-point function. Finite local counterterms may still shift contact terms.

Further reading

• S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge Theory Correlators from Non-Critical String Theory”. The original GKPW correlator prescription from the supergravity side.
• E. Witten, “Anti De Sitter Space And Holography”. The complementary formulation emphasizing boundary behavior, operator dimensions, and CFT observables.
• K. Skenderis, “Lecture Notes on Holographic Renormalization”. The standard technical reference for counterterms, one-point functions, Ward identities, and anomalies.
• I. R. Klebanov and E. Witten, “AdS/CFT Correspondence and Symmetry Breaking”. A classic discussion of vevs, alternate quantization, and Legendre transforms.