The Minimal AdS/CFT Dictionary for Black Holes

The black-hole part of the AdS/CFT dictionary is a map between boundary data and bulk boundary conditions, not a slogan that every bulk quantity is automatically an observable. A bulk field identifies a boundary source, the renormalized radial momentum identifies an expectation value, and the Euclidean or Lorentzian state preparation determines which solution is used. For thermal black holes this minimal package is enough to relate temperature, chemical potential, entropy, and response without importing the whole subject of conformal field theory.

Required background. Gauge Theory and String-Theory Preparation supplies large-N counting, operator/source couplings, and string-scale control used below.

Helpful background. Quantum Gravity frames the semiclassical expansion, Quantum Information distinguishes states from observables, Black-Hole Geometry and Mechanics supplies horizon thermodynamics, and Quantum Fields in Curved Spacetime clarifies state and boundary-condition dependence.

Boundary sources and radial data

Choose the Euclidean convention

$$Z_{\mathrm{CFT}}[J] \equiv \left\langle \exp\!\left(\int d^dx\sqrt{\gamma_{(0)}}\,J\mathcal O\right) \right\rangle.$$

At leading bulk saddle, the renormalized generating functional is

$$Z_{\mathrm{CFT}}[J] = \exp\!\left(-I_{\mathrm{ren}}[\phi_{\mathrm{cl}};J]\right), \qquad \langle\mathcal O(x)\rangle_J =- \frac{1}{\sqrt{\gamma_{(0)}}} \frac{\delta I_{\mathrm{ren}}}{\delta J(x)}.$$

Here $\phi_{\mathrm{cl}}$ is the bulk solution satisfying the chosen interior condition and asymptotic boundary condition, while $\gamma_{(0)}$ is a representative of the boundary conformal metric. This statement follows from the saddle approximation to the bulk path integral; it is not an exact equality at finite $N$ unless the complete quantum bulk functional is meant. The original prescription and its use for correlators are given in Gubser, Klebanov, and Polyakov 1998, §§2–3, pp. 108–112 and Witten 1998, §§2–3, pp. 258–266.

For a scalar in Fefferman–Graham coordinate $z$, the two independent asymptotic behaviors are

$$\phi(z,x) =z^{d-\Delta}\!\left[J(x)+\cdots\right] +z^\Delta\!\left[A(x)+\cdots\right], \qquad m^2L^2=\Delta(\Delta-d).$$

This displays the generic distinct-power case. In standard quantization $\Delta=\Delta_+=d/2+\nu$, with $\nu=\sqrt{d^2/4+m^2L^2}$; $J$ is the source and $A$, with its action-dependent prefactor and local counterterms, determines $\langle\mathcal O\rangle$. For $0\lt\nu\lt1$, both standard and alternate scalar quantizations are available. At the BF bound $\nu=0$, the independent behaviors instead involve $z^{d/2}$ and $z^{d/2}\log z$; resonant expansions can contain other logarithms. The endpoints must not be inferred by simply relabeling two powers Klebanov and Witten 1999, §§2.1–2.2.

For example, a scalar with $m^2L^2=-2$ in AdS$_4$ has $\Delta_-=1$ and $\Delta_+=2$. The same bulk differential equation therefore permits two different source/response assignments, defining different boundary theories. One must specify the quantization and variational problem before comparing their black-hole responses.

Thermodynamic entries in the dictionary

A stationary AdS black hole prepares a thermal or grand-canonical state only after the boundary geometry and ensemble are fixed. Smooth Euclidean time has period $\beta=T^{-1}$; the boundary value of a regular bulk gauge field supplies a chemical potential; the canonical charges follow from the renormalized boundary stress tensor and current. At leading saddle order,

$$I_E=\beta\bigl(E-TS-\Omega_iJ_i-\mu_aQ_a\bigr), \qquad F_{\Omega,\mu}=\frac{I_E}{\beta}.$$

The gauge is consequential: regularity sets the contraction of the gauge potential with the horizon generator to zero at the Euclidean tip, so the chemical potential is a potential difference between the boundary and horizon. Likewise, angular velocity is measured relative to a declared nonrotating frame at infinity. Mixing these conventions changes the apparent first law.

The parametric entries are equally important. For familiar large-N examples, $L^{d-1}/G_{d+1}$ scales like the number of boundary degrees of freedom, bulk loops are suppressed by inverse powers of that number, and higher-derivative string corrections are suppressed only when the appropriate coupling makes $L^2/\alpha'$ large. Large $N$ and strong coupling are independent limits.

A scalar check and a Ward check

The scalar falloffs provide a direct consistency test. Substitute $\phi=z^p$ into the Klein–Gordon equation near the AdS boundary. The indicial equation is $p(p-d)=m^2L^2$, whose roots are $p=d-\Delta$ and $p=\Delta$. The source has dimension $d-\Delta$, exactly what is required for the deformation $\int d^dx\sqrt{\gamma_{(0)}}\,J\mathcal O$ to be dimensionless.

A second check comes from boundary gauge invariance. If $A^{(0)}_i$ sources a conserved current, invariance of $I_{\mathrm{ren}}$ under $A^{(0)}_i\mapsto A^{(0)}_i+\partial_i\lambda$ gives $\nabla_i\langle J^i\rangle=0$, up to a declared anomaly. This tests the counterterms and normalization without assuming a black-hole equation of state.

Limits of the minimal map

The map does not say that a thermal density matrix and an individual heavy pure state have identical fine-grained observables. It also does not determine an exact finite-N interior, remove state dependence, or make an ensemble average into a single microscopic theory. Those questions require the later pages on heavy states and finite-N corrections. The next step is to see how boundary thermal data select a Euclidean black-hole saddle.

Exercises

Derive the two scalar falloffs and explain which extra datum is needed before calling either coefficient the source.

Solution

Near $z=0$, insert $\phi=z^p$ into $(\Box-m^2)\phi=0$ and retain the radial indicial terms. They give $p(p-d)-m^2L^2=0$, so $p=d-\Delta$ or $p=\Delta$. Because normalizability depends on the mass and the chosen variational problem, the powers alone do not select the source. One must specify standard, alternative, or mixed quantization and the counterterm scheme. As a check, $[J]=d-\Delta$ makes $\int J\mathcal O$ dimensionless.

References

• 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.
• Klebanov, Igor R., and Edward Witten. “AdS/CFT Correspondence and Symmetry Breaking.” Nuclear Physics B 556 (1999): 89–114. DOI. Open PDF.
• Witten, Edward. “Anti de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. DOI.