The GKPW Generating-Functional Dictionary

The GKPW relation turns a complete AdS boundary-value problem into a boundary generating functional. For a specified dual pair, source assignment, state, and integration contour, it equates the normalized CFT functional of sources with the bulk quantum path integral carrying the matching asymptotic data. In a weakly coupled bulk, a renormalized classical saddle makes that relation calculable. This page derives the scalar two-point coefficient—including its sign and normalization—and then separates contact-term freedom, competing saddles, quantization choices, and Lorentzian contours. The dictionary is conditional on the duality; neither a successful saddle calculation nor the dictionary itself proves a complete equivalence.

Required background. Bulk Fields and Boundary Operators fixes the scalar branch and source falloff. Scalar Two- and Three-Point Functions fixes the conformal target. Helpful background. Exact Statements, Saddle Expansions, and Conditional Derivations distinguishes a proposed exact identity from its semiclassical evaluation.

Boundary sources as asymptotic data

Work first in positive-definite Euclidean Poincaré AdS$_{d+1}$,

$$\mathrm ds_E^2=\frac{L^2}{z^2} \left(\mathrm dz^2+\mathrm d\mathbf x^2\right), \qquad z>0,$$

with a real scalar whose dimensionful action is

$$S_E[\phi] :=\frac{\mathcal N_\phi}{2} \int_M\mathrm d^{d+1}x\sqrt g\, \left[(\nabla\phi)^2+m^2\phi^2\right], \qquad \mathcal N_\phi>0.$$

Use standard quantization with

$$m^2L^2=\Delta(\Delta-d), \qquad \Delta>\frac d2,$$

and exclude the coalescing Breitenlohner–Freedman case $\Delta=d/2$ from the calculation below. Near $z=0$, a solution has the form

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

$J$ is held fixed and has scaling dimension $d-\Delta$; $A$ is the state- and interior-dependent response. Source-local powers of $z$ and, at resonant values $\Delta-d/2\in\mathbb Z_{>0}$, logarithms have been suppressed. They matter for counterterms and contact terms but do not replace the nonlocal coefficient $A$.

On the flat boundary, choose the source sign

$$Z_{\mathrm{CFT}}[J] :=\left\langle \exp\!\left(\int\mathrm d^d x\,J\mathcal O\right) \right\rangle_0, \qquad Z_{\mathrm{CFT}}[0]=1,$$

where the subscript indicates a normalized vacuum expectation value. On a curved boundary, the source integral also contains $\sqrt{g_{(0)}}$. The bulk functional must therefore be normalized in the same way. If $\mathcal C$ denotes the bulk integration cycle and its Euclidean interior or state condition, the proposed dictionary is

$$Z_{\mathrm{CFT}}[J] =\frac{Z_{\mathrm{bulk}}[J;\mathcal C]} {Z_{\mathrm{bulk}}[0;\mathcal C]},$$

with

$$Z_{\mathrm{bulk}}[J;\mathcal C] :=\int_{\mathcal C\,; \,\phi\sim z^{d-\Delta}J} \mathcal D\Phi\,e^{-S_{E,\mathrm{bulk}}[\Phi]}.$$

$\Phi$ denotes all bulk fields, not just $\phi$, and all unshown boundary sources, global sectors, and state data are held fixed. For the free scalar calculation below, $\mathcal C$ is the stable real Gaussian contour compatible with the stated Dirichlet boundary condition, and interior regularity selects the Euclidean vacuum solution. In quantum gravity the integration cycle can be subtler and is part of the definition, not an afterthought. A unitary Lorentzian interpretation additionally requires the resulting Euclidean correlators to satisfy reflection positivity.

The original prescriptions identify boundary sources with asymptotic bulk fields and the boundary functional with the corresponding bulk functional Gubser, Klebanov, and Polyakov 1998, printed p. 4, eq. (12), immediately before §2.1, Open PDF and Witten 1998, §2.3, printed pp. 9–10, eqs. (2.11)–(2.13), Open PDF.

The four objects most often conflated are summarized below. On a narrow screen, scroll horizontally without shrinking the text.

Functionals that appear in the GKPW dictionary

Object	Expression	Meaning
Normalized boundary functional	ZCFT[J]	Generates all source moments, including disconnected products
Full bulk functional	Zbulk[J]/Zbulk[0]	The quantum object proposed to equal the CFT functional for the declared dual pair
Connected functional	W[J] := log ZCFT[J]	Generates connected correlators, or cumulants
Classical evaluation	Wtree[J] = −SE,ren[J] + SE,ren[0]	The leading result only when one renormalized saddle dominates

The logarithm is essential. Functional differentiation gives

$$\left\langle \mathcal O(x_1)\cdots\mathcal O(x_n) \right\rangle_{J,\mathrm{conn}} :=\frac{\delta^n W[J]} {\delta J(x_1)\cdots\delta J(x_n)}.$$

At $J=0$ this is the vacuum connected correlator. At nonzero $J$ it is a connected correlator in the sourced background. Connected Correlators and Cumulants derives the same logarithm rule without holography.

The bulk saddle and its corrections

A saddle expansion of the bulk path integral is schematically

$$Z_{\mathrm{bulk}}[J;\mathcal C] \simeq \sum_{\alpha\in\mathcal C} e^{-S_{E,\mathrm{ren}}^{(\alpha)}[J]} Z_{\alpha,\text{1-loop}}[J] \left(1+\delta_{\alpha,\mathrm{higher}}+\cdots\right).$$

The label $\alpha$ runs over saddles on the declared integration cycle. Gauge fixing, ghosts, zero modes, and collective coordinates are included schematically in the one-loop factor. If one saddle $\alpha_*$ dominates in a neighborhood of $J=0$, then

$$W_{\mathrm{tree}}[J] =-S_{E,\mathrm{ren}}^{(\alpha_*)}[J] +S_{E,\mathrm{ren}}^{(\alpha_*)}[0].$$

The subtraction is the bulk counterpart of $Z_{\mathrm{CFT}}[0]=1$. Three independent controls are needed:

• bulk loops: in an Einstein regime the dimensionless loop parameter scales as $G_{d+1}/L^{d-1}$, while the classical action scales as $L^{d-1}/G_{d+1}$;
• higher derivatives: a term with additional derivatives is suppressed by a model-dependent power of $\ell_{\mathrm{UV}}/L$, where $\ell_{\mathrm{UV}}$ may be a string, Kaluza–Klein, or other microscopic length;
• other saddles: a second saddle with action gap $\Delta S_E>0$ contributes relatively as $e^{-\Delta S_E}$ and cannot be discarded near degeneracy.

There is no universal additive $O(\alpha'/L^2)$ correction: both the first allowed operator and its power depend on the bulk theory. Likewise, $G_{d+1}$ by itself is dimensionful and is not an expansion parameter. At a saddle crossing, $W$ is the logarithm of the saddle sum, not minus the action of whichever geometry is easiest to evaluate.

Renormalized variation reads the response

Put a cutoff at $z=\epsilon$. The outward unit normal of the region $z\geq\epsilon$ points toward decreasing $z$,

$$n=-\frac zL\,\partial_z.$$

On shell, integration by parts reduces the scalar action to

$$S_{E,\mathrm{reg}} =\frac{\mathcal N_\phi}{2} \int_{z=\epsilon}\mathrm d^d x\sqrt\gamma\, \phi\,n^M\partial_M\phi.$$

Its divergent terms are local functionals of the cutoff field. Add local covariant counterterms and remove the regulator,

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

For the source sign chosen above, the renormalized first variation is

$$\delta S_{E,\mathrm{ren}} =-\int\mathrm d^d x\, \left[ \mathcal N_\phi L^{d-1}(2\Delta-d)A(x) +\mathcal F_{\mathrm{local}}[J](x) \right]\delta J(x).$$

Consequently,

$$\langle\mathcal O(x)\rangle_J =\frac{\delta W}{\delta J(x)} =\mathcal N_\phi L^{d-1}(2\Delta-d)A(x) +\mathcal F_{\mathrm{local}}[J](x)$$

at tree level. The response factor and local ambiguity follow from the renormalized variational problem de Haro, Skenderis, and Solodukhin 2001, §5.1, especially eqs. (5.9)–(5.11), Open PDF. A convention with $e^{-\int J\mathcal O}$ reverses the intermediate one-point sign. The invariant check is that the source term, response normalization, separated-point correlator, and Ward identities are translated together.

The local term depends on finite counterterms and, at resonance, on the renormalization scale. It contributes only contact distributions to correlators around the vacuum. The scalar holographic-renormalization page derives the full counterterm recursion and logarithmic cases.

The scalar two-point function

Interior regularity in Euclidean Poincaré AdS fixes the free solution linearly in $J$,

$$\phi(z,x)=\int\mathrm d^d y\, K_\Delta(z,x;y)J(y),$$

where, for standard quantization with $\Delta>d/2$,

$$K_\Delta(z,x;y) =c_\Delta \left( \frac{z}{z^2+\lvert x-y\rvert^2} \right)^\Delta, \qquad c_\Delta:= \frac{\Gamma(\Delta)} {\pi^{d/2}\Gamma(\Delta-d/2)}.$$

Pointwise at $x\neq y$, this kernel begins as $z^\Delta$, so why does its convolution produce the slower source falloff? Set $y=x+zu$. Then

$$\int\mathrm d^d y\,K_\Delta(z,x;y)J(y) =z^{d-\Delta}c_\Delta \int\mathrm d^d u\, \frac{J(x+zu)}{(1+u^2)^\Delta}.$$

The integral identity

$$\int\mathrm d^d u\,(1+u^2)^{-\Delta} =\pi^{d/2} \frac{\Gamma(\Delta-d/2)}{\Gamma(\Delta)} =\frac1{c_\Delta}$$

therefore gives

$$\lim_{z\to0}z^{\Delta-d}K_\Delta(z,x;y) =\delta^{(d)}(x-y)$$

as a distribution. The same kernel fixes the nonlocal part of the fast coefficient,

$$A_{\mathrm{nonlocal}}(x)=c_\Delta \int\mathrm d^d y\, \left[ \frac1{\lvert x-y\rvert^{2\Delta}} \right]_{\!R}J(y).$$

Because $\Delta>d/2$, the separated-point kernel is not locally integrable at $y=x$. The symbol $[\cdots]_R$ therefore denotes a renormalized distribution that agrees with $\lvert x-y\rvert^{-2\Delta}$ away from coincidence. Two allowed extensions differ only by derivatives of delta functions, which shift the local response $\mathcal F_{\mathrm{local}}[J]$ and the local action below. The two kernel limits are complementary: the shrinking region $\lvert x-y\rvert\sim z$ produces the source, while fixed separated points determine the scheme-independent nonlocal response.

Substitution into the renormalized boundary term yields the nonlocal quadratic action

$$S_{E,\mathrm{ren}}^{(2)}[J] =-\frac{\mathcal N_\phi L^{d-1}}2 (2\Delta-d) \int\mathrm d^d x\,J(x)A_{\mathrm{nonlocal}}(x) +S_{\mathrm{local}}[J].$$

Equivalently,

$$S_{E,\mathrm{ren}}^{(2)}[J] =-\frac12 \int\mathrm d^d x\,\mathrm d^d y\, J(x)C_{\mathcal O} \left[ \frac1{\lvert x-y\rvert^{2\Delta}} \right]_{\!R}J(y) +S_{\mathrm{local}}[J],$$

with

$$C_{\mathcal O} :=\mathcal N_\phi L^{d-1}(2\Delta-d)c_\Delta.$$

Since $W_{\mathrm{tree}}=-S_{E,\mathrm{ren}}+\text{constant}$, two derivatives give

$$\langle\mathcal O(x)\mathcal O(y)\rangle =\frac{C_{\mathcal O}} {\lvert x-y\rvert^{2\Delta}}, \qquad x\neq y.$$

The factor $1/2$ in the quadratic functional cancels because the symmetric kernel can be differentiated in either source slot. The power law has the required scale weight $2\Delta$. For $\mathcal N_\phi>0$ and $\Delta>d/2$, $C_{\mathcal O}>0$, as reflection positivity requires for a Hermitian scalar. The factor $2\Delta-d$ is essential: the careful finite-cutoff calculation corrects a tempting but wrong boundary-limit ordering Freedman et al. 1999, §2, eqs. (11)–(17), and appendix, especially eqs. (88)–(95), Open PDF.

As a numerical check, in $d=3$ with $\Delta=2$,

$$c_2=\frac1{\pi^2}, \qquad C_{\mathcal O}=\frac{\mathcal N_\phi L^2}{\pi^2}.$$

If a convention absorbs $L^{d-1}$ into the kinetic prefactor, compare the dimensionless product $\mathcal N_\phi L^{d-1}$ rather than either symbol separately. Bulk-to-Boundary and Bulk-to-Bulk Propagators develops the propagator construction beyond this first normalization check.

Higher derivatives of the functional

For a free scalar, the saddle solution is linear in $J$ and $W$ is quadratic, so connected correlators with $n>2$ vanish at tree level. Bulk interactions make the classical solution nonlinear in the source. Substituting that solution into $S_{E,\mathrm{ren}}$ produces cubic and higher powers of $J$; differentiating them gives connected boundary correlators. At tree level these terms are organized by contact and exchange Witten diagrams. Double- and higher-trace conformal families already occur in generalized-free data and tree-level diagrams; bulk loops supply further $1/N$ corrections, including corrections to their dimensions and OPE coefficients.

This statement is organizational, not a calculation of every diagram. Contact Witten Diagrams and Exchange Witten Diagrams and Conformal Blocks evaluate those contributions. Source differentiation for constrained gauge and metric fields is developed by Currents, Stress Tensor, and Bulk Gauge and Metric Fields.

Contact terms, branches, saddles, and contours

Several operations that are sometimes called “changing the prescription” have physically different effects. On a narrow screen, scroll horizontally without shrinking the text.

Distinct changes to boundary data and bulk evaluation

Change	What changes	Controlled statement that survives
Add a finite local counterterm ½∫ J P(□)J	Polynomial momentum terms and derivatives of delta functions; some one-point functions in nonzero backgrounds	The separated-point power law and other nonlocal Euclidean data
Pass to alternate quantization	Which asymptotic coefficient is the source and the operator dimension; the full quantum functional changes polarization, while its saddle limit is Legendre transformed	The same bulk mass and the requirement of a well-posed, flux-preserving variational problem
Retain an additional saddle	Generally the nonlocal functional, phase structure, and finite-N smoothing near a crossing	Only the full declared saddle sum, not one selected action
Change the real-time contour or state	Pole boundary values, operator ordering, homogeneous normalizable data, and possibly the response itself	The local bulk equations and ultraviolet counterterm structure

A finite local counterterm is a scheme change. Alternate quantization is not. In the window $0<\nu:=\Delta_+-d/2<1$, the standard and alternate on-shell generating functionals are related by a Legendre transform at semiclassical large $N$ and describe different source assignments Klebanov and Witten 1999, §2, printed pp. 8–12, Open PDF. For the full quantum partition function, the corresponding change of boundary polarization is a functional Fourier, or canonical, transform; stationary phase reduces it to the Legendre transform. The BF endpoint and upper endpoint require separate logarithmic or singleton analyses, so one must not substitute them blindly into the open-window formula.

Euclidean GKPW computes Euclidean correlators in the state selected by the Euclidean filling and integration cycle. Lorentzian data require more. A Feynman contour yields time-ordered correlators with a specified $i0$ prescription; a Schwinger–Keldysh contour produces causal and Wightman combinations; Euclidean caps prepare bra and ket data. The bulk fills the complete contour and obeys matching conditions at its Euclidean–Lorentzian corners Skenderis and van Rees 2009, §§2.1–3.2, especially §§3.2.4–3.2.5, and §4, Open PDF. In a thermal black-hole background, infalling horizon data select the retarded response rather than a generic time-ordered one Son and Starinets 2002, §3, Open PDF.

Thus the strongest scheme-independent Euclidean result in the scalar example is the separated-point kernel and its normalization. A contour change can alter its Lorentzian analytic boundary value, and a branch or saddle change can alter the nonlocal functional itself. Every correlator claim must therefore name the source branch, state, contour, saddle treatment, and approximation order.

Domain of the result and where to continue

The explicit derivation assumes a free scalar on fixed Euclidean Poincaré AdS, standard quantization, $\Delta>d/2$, interior regularity, a positive kinetic coefficient, and separated boundary points. Resonant values with $\Delta-d/2\in\mathbb Z_{>0}$ add source-local logarithms and scale dependence; the coalescing BF case needs a logarithmic source convention. For an irrelevant operator with $\Delta>d$, a finite source grows toward the ultraviolet and can spoil the ordinary asymptotically AdS expansion once interactions and backreaction are included. Such sources should be treated perturbatively unless additional ultraviolet data justify more.

The proposed equality of normalized full functionals is conceptually distinct from the one-saddle formula used to evaluate it. Bulk loops, higher-derivative terms, additional saddles, and nonperturbative sectors lie beyond the displayed tree result. The page has also not established that an arbitrary CFT admits a bulk dual or that matching this two-point function determines the global theory.

Generating Functionals and Source Differentiation gives the general QFT construction. Boundary Conditions, Alternate Quantization, and Deformations changes the source assignment within the admissible window. Scalar Counterterms and the Renormalized On-Shell Action supplies the detailed regulator removal. Euclidean Preparation and Lorentzian State Dictionaries supplies state-preparing caps and contour orderings. The Witten-diagram chapter evaluates interacting correlators rather than redefining the dictionary.

Common pitfalls

Differentiating $Z$ and calling the answer connected. Derivatives of $Z$ contain disconnected products. Differentiate $W=\log Z$ to obtain cumulants.

Using the bare on-shell action. The cutoff action diverges. Source variation is meaningful only after local counterterms have made the variational problem finite.

Taking the boundary limit too early. Pointwise and distributional limits of $K_\Delta$ answer different questions. Dropping terms before imposing finite-cutoff boundary data can miss the factor $2\Delta-d$.

Treating a branch change as a finite scheme change. Local counterterms alter contact data. At the semiclassical saddle, a Legendre transform exchanges source and response and generally changes the nonlocal boundary theory; at the full quantum level the corresponding operation is a functional Fourier transform.

Letting the source choose a Lorentzian state. Timelike AdS evolution also needs normalizable state data or a contour prescription. A source falloff alone does not select Feynman, retarded, or Wightman response.

Replacing a saddle sum by a preferred geometry. Saddle dominance is an estimate with an action gap. Near competing saddles, the logarithm of the sum is the relevant functional.

Exercises

Exercise 1 — Connected differentiation. Show that the second derivative of $W=\log Z$ is connected even when $\langle\mathcal O\rangle_J\neq0$.

Solution to Exercise 1

The first derivative is

$$\frac{\delta W}{\delta J(x)} =\frac1Z\frac{\delta Z}{\delta J(x)}.$$

Differentiating again gives

$$\frac{\delta^2W}{\delta J(x)\delta J(y)} =\frac1Z\frac{\delta^2Z}{\delta J(x)\delta J(y)} -\frac1{Z^2} \frac{\delta Z}{\delta J(x)} \frac{\delta Z}{\delta J(y)}.$$

The first term is $\langle\mathcal O(x)\mathcal O(y)\rangle_J$ and the second subtracts $\langle\mathcal O(x)\rangle_J\langle\mathcal O(y)\rangle_J$. The result is the connected two-point function.

Exercise 2 — Kernel normalization. Verify the distributional normalization of $K_\Delta$ against $J\in C_c^\infty(\mathbb R^d)$ and then evaluate $C_{\mathcal O}$ for $d=3$, $\Delta=2$.

Solution to Exercise 2

With $y=x+zu$ and $J\in C_c^\infty(\mathbb R^d)$,

$$z^{\Delta-d}\int\mathrm d^d y\, K_\Delta(z,x;y)J(y) =c_\Delta\int\mathrm d^d u\, \frac{J(x+zu)}{(1+u^2)^\Delta}.$$

Dominated convergence applies for $\Delta>d/2$. Taking $z\to0$ and using

$$c_\Delta\int\mathrm d^d u\,(1+u^2)^{-\Delta}=1$$

returns $J(x)$, which proves the delta-function limit. For $d=3$ and $\Delta=2$,

$$c_2= \frac{\Gamma(2)} {\pi^{3/2}\Gamma(1/2)} =\frac1{\pi^2}, \qquad 2\Delta-d=1.$$

Therefore $C_{\mathcal O}=\mathcal N_\phi L^2/\pi^2$.

Exercise 3 — A finite counterterm. Let $k\in\mathbb Z_{\geq0}$ and use the Euclidean convention $\Box:=\delta^{ij}\partial_i\partial_j$. Add the finite counterterm

$$S_{\mathrm{fin}}[J] =\frac c2\int\mathrm d^d x\, J\Box^kJ.$$

What does it do to the two-point function?

Solution to Exercise 3

Because $W_{\mathrm{tree}}=-S_{E,\mathrm{ren}}$, the counterterm shifts the two-point kernel by

$$\Delta\langle\mathcal O(x)\mathcal O(y)\rangle =-c\,\Box_x^k\delta^{(d)}(x-y),$$

The shift is supported at coincidence and is a polynomial in momentum space. It cannot change $C_{\mathcal O}/\lvert x-y\rvert^{2\Delta}$ for $x\neq y$.

Exercise 4 — Competing saddles. Suppose two Euclidean saddles contribute with $S_2=S_1+\Delta S$, where $\Delta S>0$. Find both the relative correction to $Z$ and the additive correction to $W=\log Z$ made by retaining the second saddle.

Solution to Exercise 4

Ignoring one-loop factors for clarity,

$$Z=e^{-S_1}+e^{-S_2} =e^{-S_1}\left(1+e^{-\Delta S}\right).$$

Hence

$$\log Z=-S_1+\log\!\left(1+e^{-\Delta S}\right).$$

Relative to the first-saddle contribution, the correction to $Z$ is $e^{-\Delta S}$. The additive correction to $W$ is $\log(1+e^{-\Delta S})$, which is approximately $e^{-\Delta S}$ only when $\Delta S\gg1$. At a crossing, $\Delta S\to0$ and the one-saddle approximation fails even though each individual saddle is classical.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• de Haro, Sebastian, Kostas Skenderis, and Sergey N. Solodukhin. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. DOI. Open PDF.
• Freedman, Daniel Z., Samir D. Mathur, Alec Matusis, and Leonardo Rastelli. “Correlation Functions in the CFT$_d$/AdS$_{d+1}$ Correspondence.” Nuclear Physics B 546 (1999): 96–118. DOI. Open PDF.
• Gubser, Steven S., Igor R. Klebanov, and Alexander M. Polyakov. “Gauge Theory Correlators from Non-Critical String Theory.” Physics Letters B 428 (1998): 105–114. DOI. Open PDF.
• Klebanov, Igor R., and Edward Witten. “AdS/CFT Correspondence and Symmetry Breaking.” Nuclear Physics B 556 (1999): 89–114. DOI. Open PDF.
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples.” Journal of High Energy Physics 05 (2009): 085. DOI. Open PDF.
• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 09 (2002): 042. DOI. Open PDF.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. DOI. Open PDF.