The GKPW Generating-Functional Dictionary

The GKPW dictionary identifies a boundary generating functional with a bulk path integral subject to specified asymptotic sources and state data. In the semiclassical regime, that path integral is approximated by a renormalized on-shell action. Functional derivatives then yield connected correlators. Boundary conditions, counterterm scheme, contour, and source normalization are part of the equality; the saddle formula is not an exact proof of duality.

Required background. Bulk fields and boundary operators fixes the source falloff, and scalar two- and three-point functions fixes the conformal target. Helpful background. Exact statements, saddle expansions, and conditional derivations separates the proposed exact path-integral identity from its large-$N$ evaluation.

The Euclidean statement and its approximation

Use positive-definite Euclidean AdS$_{d+1}$ with Poincaré metric

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

Choose the boundary source convention

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

The dictionary proposes

$$Z_{\mathrm{CFT}}[J] =Z_{\mathrm{bulk}}\!\left[\phi\sim z^{d-\Delta}J\right].$$

This source-dependent equality is the prescription of Gubser, Klebanov, and Polyakov 1998, pp. 109–112 and Witten 1998, §§2–3.

When the bulk is weakly coupled and one saddle dominates,

$$\log Z_{\mathrm{CFT}}[J] =-S_{E,\mathrm{ren}}[\phi_{\mathrm{cl}};J] +O(G_{d+1}^0)+O(\alpha'/L^2).$$

The displayed remainder notation separates loop and higher-derivative corrections only schematically; a concrete top-down model fixes their powers. If several saddles contribute, the logarithm is not the action of whichever saddle is most convenient.

Connected correlators follow from

$$\left\langle\mathcal O(x_1)\cdots\mathcal O(x_n)\right\rangle_{\mathrm{conn}} =\left.\frac{\delta^n\log Z[J]} {\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}.$$

With the chosen source sign, the one-point function at saddle level is $-\delta S_{E,\mathrm{ren}}/\delta J$. A source convention with $e^{-\int J\mathcal O}$ reverses this intermediate sign; separated-point positivity and Ward identities provide the translation check.

First application: differentiating the scalar saddle twice

Take

$$S_E=\frac{\mathcal N_\phi}{2}\int\sqrt g \left[(\nabla\phi)^2+m^2\phi^2\right].$$

The regular solution with boundary source $J$ is

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

where

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

for the standard nonexceptional normalization. Integrating the action by parts leaves a boundary term. On the cutoff region $z\geq\epsilon$, the outward unit normal points toward decreasing $z$. Adding local counterterms and taking $\epsilon\to0$ gives the nonlocal quadratic functional

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

in the present source-sign convention, with

$$C_{\mathcal O} =\mathcal N_\phi L^{d-1}(2\Delta-d) \frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\Delta-d/2)}.$$

Here $\mathcal N_\phi$ is the coefficient in dimensionful bulk coordinates; if a convention absorbs $L^{d-1}$ into a dimensionless kinetic prefactor, the product $\mathcal N_\phi L^{d-1}$ is what must be compared.

Twice differentiating $-S_{E,\mathrm{ren}}$ recovers

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

away from coincidence. This is the first controlled application of the dictionary and matches the classic calculation of Freedman et al. 1999, §§2–3.

Adversarial check: contact terms and contours

A finite local counterterm such as $c\int J\Box^kJ$ changes derivatives of delta functions but cannot change the separated-point power law. It may change a scheme-dependent one-point function in a background source. Therefore “the correlator changed” must specify whether the statement concerns separated points, contact terms, or an integrated observable sensitive to contact terms.

Lorentzian GKPW needs more data. A Feynman contour gives time-ordered correlators with a prescribed $i0$; infalling horizon data give retarded response in appropriate states; Euclidean caps prepare bra and ket data. Changing the contour can alter poles’ boundary values and the state while leaving the Euclidean differential equation unchanged. The real-time construction of Skenderis and van Rees 2009, §§3–4 makes these gluing data explicit.

The strongest invariant result under finite local counterterms is the separated-point structure and nonlocal momentum dependence. Under a contour change even that analytic boundary value can change, so the licensed statement must name the ordering and state.

Controlled limits and handoff

The displayed equality is the leading renormalized saddle for a specified source, branch, and Euclidean contour; finite-$N$ bulk loops and nonperturbative saddles are not included. Currents, Stress Tensor, and Bulk Gauge and Metric Fields extends source differentiation to constrained fields, and Euclidean Preparation and Lorentzian State Dictionaries supplies state-preparing caps and real-time orderings.

Exercise

Why does adding $c\int J^2$ not change the two-point function at $x\neq y$?

Solution

Two functional derivatives of $c\int J^2$ produce a term proportional to $\delta^{(d)}(x-y)$. It has support only at coincidence. The coefficient of $\lvert x-y\rvert^{-2\Delta}$ at separated points is unchanged, although integrated observables and Ward identities with contact terms may record the scheme shift.

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

• 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. arXiv. DOI.
• 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. arXiv. DOI.
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples.” Journal of High Energy Physics 2009, 085 (2009). arXiv. DOI.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. arXiv. DOI.