Dictionary Normalization and Global-Data Audit

An AdS/CFT result is comparable across papers or calculations only after its field normalization, source convention, AdS-radius factors, counterterm scheme, analytic continuation, and global charge data have been translated together. A coefficient can have the correct power law and still be wrong by a sign, a power of $L$, or a charge-unit factor. This page builds two explicit convention systems for scalar and current two-point functions in Euclidean AdS$_{d+1}$, then gives invariant checks for their Lorentzian continuation. Gravity uses the site’s $(+---)$ Lorentzian convention; the Euclidean formulas use a positive-definite metric.

Required background. Radius, Couplings, and the Parameter Map supplies the relations among $L$, $G_N$, gauge couplings, $N$, and the string scale. The GKPW Generating-Functional Relation supplies the source differentiation and on-shell-action prescription. Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form supplies the charge lattice and line-operator data not contained in local correlators.

Helpful background. Basis Translation, Scheme Dependence, and Reproducibility supplies a general method for translating convention-dependent coefficients. Benchmark Provenance and Reproduction supplies reproducibility checks for quoted numerical data.

A scalar coefficient with every convention visible

Use Euclidean Poincaré AdS,

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

and the standard scalar branch $\Delta>d/2$. Convention A is

$$S_E^{(A)}=\frac{\eta_\phi}{2}\int d^{d+1}x\sqrt g \left(g^{MN}\partial_M\phi\partial_N\phi+m^2\phi^2\right),$$

with $\phi=z^{d-\Delta}J+\cdots$, $W[J]=\log Z[J]=-S_{\rm ren}[J]$, and $\langle\mathcal O\rangle=\delta W/\delta J$. The regular bulk-to-boundary kernel normalized to a delta-function source is

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

With the outward normal at the cutoff surface pointing toward decreasing $z$, variation and counterterm subtraction give a positive separated-point two-point function

$$\langle\mathcal O(\mathbf{x})\mathcal O(0)\rangle =\frac{C_{\mathcal O}^{(A)}}{|\mathbf{x}|^{2\Delta}}, \qquad C_{\mathcal O}^{(A)} =\eta_\phi L^{d-1}(2\Delta-d)c_\Delta.$$

The factor $L^{d-1}$ is required by dimensional analysis when $\phi$ is dimensionless and $[\eta_\phi]={\rm length}^{1-d}$. This normalization follows the classic position-space calculation of Freedman et al. 1999, §§2–3. Polynomial momentum terms added by finite counterterms are contact terms; they do not change $C_{\mathcal O}$ at separated points.

The current coefficient and global data

For $d>2$, take

$$S_E[A]=\frac{1}{4g_{d+1}^2}\int d^{d+1}x\sqrt g\,F_{MN}F^{MN},$$

take the boundary value $A_i^{(0)}$ as the source of a current whose unit charge is fixed separately. Gauge invariance gives

$$\langle J_i(\mathbf{x})J_j(0)\rangle =C_J\frac{I_{ij}(\mathbf{x})}{|\mathbf{x}|^{2(d-1)}}, \qquad I_{ij}=\delta_{ij}-2\frac{x_ix_j}{|\mathbf{x}|^2},$$

with

$$C_J=\frac{(d-2)\Gamma(d)} {2\pi^{d/2}\Gamma(d/2)}\, \frac{L^{d-3}}{g_{d+1}^2}.$$

Transversality away from coincident points checks the tensor structure, and reflection positivity checks $C_J>0$. Neither test determines whether the compact group is $SU(k)$, $SU(k)/\mathbb Z_k$, or another global form. The spectrum of genuine Wilson and ’t Hooft lines, large-gauge periodicities, and the allowed charge lattice must be carried as additional data.

First application: translating two convention systems

Define convention B by $\widehat\phi=a\phi$ and $\widehat A=bA$, with $a,b>0$. To describe the same bulk physics,

$$\widehat\eta_\phi=\frac{\eta_\phi}{a^2}, \qquad \frac{1}{\widehat g_{d+1}^{2}}= \frac{1}{b^2g_{d+1}^2}, \qquad \widehat J=aJ, \qquad \widehat A_i^{(0)}=bA_i^{(0)}.$$

Equality of the source couplings gives $\widehat{\mathcal O}=\mathcal O/a$ and $\widehat J_i=J_i/b$. The complete translation is therefore:

Quantity	Convention A	Convention B	Invariant statement
Scalar source/operator	$J$, $\mathcal O$	$aJ$, $\mathcal O/a$	$\int J\mathcal O$
Scalar coefficient	$C_{\mathcal O}$	$C_{\mathcal O}/a^2$	$C_{\mathcal O}J^2$
Gauge source/current	$A^{(0)}_i$, $J_i$	$bA^{(0)}_i$, $J_i/b$	$\int A_i^{(0)}J_i$
Current coefficient	$C_J$	$C_J/b^2$	$C_J(A^{(0)})^2$
Charge unit	$q$	$q/b$ if the connection is rescaled	Holonomy $q\oint A$ and the genuine line spectrum

For a concrete check, take $d=3$, $\Delta=2$, $a=2$, and $b=3$. Since $c_2=1/\pi^2$,

$$C_{\mathcal O}^{(A)}=\frac{\eta_\phi L^2}{\pi^2}, \qquad C_{\mathcal O}^{(B)}=\frac{\eta_\phi L^2}{4\pi^2},$$

while

$$C_J^{(A)}=\frac{2}{\pi^2g_4^2}, \qquad C_J^{(B)}=\frac{2}{9\pi^2g_4^2}.$$

The numerical coefficients differ, but differentiating with respect to the correspondingly rescaled sources gives the same physical response. Rescaling $A$ without rescaling the charge unit would instead change Wilson-line holonomies and would not be a mere convention change.

Adversarial check: signs and radius powers

Two deliberate mistakes expose the most common failures. First, reverse the cutoff-surface normal while leaving $W=-S_{\rm ren}$ and all counterterm signs unchanged. The scalar coefficient changes sign, contradicting reflection positivity for a Hermitian operator. The cure is not an absolute value: restore a consistent normal orientation, Euclidean action sign, and source variation.

Second, omit $L^{d-1}$ from $C_{\mathcal O}$ or $L^{d-3}$ from $C_J$. In dimensionful coordinates the resulting expression carries the wrong units. The same omission also spoils the expected large-$N$ relations $C_T\propto L^{d-1}/G_N$ and $C_J\propto L^{d-3}/g_{d+1}^2$. Dimensional analysis catches the scalar error, while the current Ward identity and positivity catch incorrect tensor signs or longitudinal pieces.

For Lorentzian time-ordered correlators, declare the continuation $\tau=i t$ and the Feynman boundary value $t\mapsto t-i0$. A retarded correlator instead uses the retarded contour and, when a horizon is present, ingoing interior data. Counterterm contact terms may change under scheme translations, but spectral support, causal analyticity, and separated-point coefficients must agree.

Controlled limits, comparison record, and handoff

Before accepting a dictionary calculation, record:

1. the bulk dimension, patch, metric signature, and orientation of every boundary normal;
2. the complete action prefactors, including powers of $L$, $G_N$, and gauge couplings;
3. the near-boundary coefficients, scalar branch, source, and operator normalization;
4. finite counterterms and which quoted quantities are only contact-term invariant;
5. the Euclidean-to-Lorentzian contour, $i0$ prescription, and interior boundary data;
6. compact gauge-group global form, charge unit, large-gauge periodicities, and genuine extended operators.

This record is sufficient to reproduce the two-point examples above and to identify which disagreements are conventions. It is not a substitute for matching higher-point interaction normalizations or for specifying the complete string compactification. Holographic renormalization supplies the systematic counterterm construction de Haro, Solodukhin, and Skenderis 2001, §§4–5 and Skenderis 2002, §§3–4; the next chapter applies it to stress tensors, anomalies, and general asymptotically locally AdS data.

Exercises

Under $\widehat\phi=a\phi$, verify directly from two source derivatives that $C_{\widehat{\mathcal O}}=C_{\mathcal O}/a^2$.

Solution

Since $\widehat J=aJ$, the chain rule gives $\delta/\delta\widehat J=a^{-1}\delta/\delta J$. Applying it twice to the same generating functional yields $\langle\widehat{\mathcal O}\widehat{\mathcal O}\rangle=a^{-2}\langle\mathcal O\mathcal O\rangle$.

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, Sebastián, Sergey N. Solodukhin, and Kostas Skenderis. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. doi:10.1007/s002200100381. arXiv:hep-th/0002230.
• 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:10.1016/S0550-3213(99)00053-X. arXiv:hep-th/9804058.
• Skenderis, Kostas. “Lecture Notes on Holographic Renormalization.” Classical and Quantum Gravity 19 (2002): 5849–5876. doi:10.1088/0264-9381/19/22/306. arXiv:hep-th/0209067.