Dictionary Tables

The main idea

The AdS/CFT dictionary is not a single table. It is a layered set of identifications between boundary quantum observables and bulk gravitational, stringy, or brane objects. The most compact slogan is

$$Z_{\rm CFT}[J] = Z_{\rm string}\bigl[\text{bulk fields }\to\text{ boundary sources }J\bigr],$$

and in the classical bulk limit,

$$W_{\rm CFT}[J] = \begin{cases} -\,S^{\rm ren}_{E,{\rm bulk}}[\bar\Phi[J]], & \text{Euclidean conventions with } Z=e^{-S_E},\\ S^{\rm ren}_{L,{\rm bulk}}[\bar\Phi[J]], & \text{Lorentzian conventions, up to the usual } i\epsilon \text{ prescription}. \end{cases}$$

Here $\bar\Phi[J]$ is the bulk solution satisfying boundary conditions determined by the source $J$. Correlators are obtained by functional differentiation. One-point functions are obtained from the renormalized radial canonical momenta. Entropies and nonlocal observables are computed by extremizing geometric or brane actions. Thermal states are represented by black-hole or black-brane saddles.

This page collects the practical dictionary in one place. It is meant to be used while doing calculations. The tables are intentionally redundant: in real research, most mistakes happen when one remembers the right slogan but applies it in the wrong normalization, ensemble, or regime.

Conventions used in the tables

Unless explicitly stated otherwise, the boundary theory has dimension $d$ and the bulk has dimension $d+1$. The AdS radius is $L$, the radial coordinate is $z$ near the conformal boundary, and the Poincaré AdS metric is

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right), \qquad z\to 0 \text{ is the boundary}.$$

The Fefferman-Graham form is

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

with

$$g_{\mu\nu}(z,x) = g^{(0)}_{\mu\nu}(x)+z^2g^{(2)}_{\mu\nu}(x)+\cdots+z^d g^{(d)}_{\mu\nu}(x)+\cdots .$$

The source for an operator $\mathcal O$ of dimension $\Delta$ has engineering dimension

$$[J]=d-\Delta,$$

because it appears in the deformation

$$\delta S_{\rm CFT} = \int d^d x\, J(x)\mathcal O(x).$$

A useful warning: the boundary value of a bulk field is not usually the vev. The leading non-normalizable coefficient is usually the source, and the subleading normalizable coefficient usually determines the vev after holographic renormalization. The word “usually” matters: alternate quantization, mixed boundary conditions, gauge constraints, anomalies, and finite counterterms can modify the precise statement.

Master dictionary

Boundary object	Bulk object	Leading relation	Main caveat
Generating functional $Z_{\rm CFT}[J]$	String/gravity partition function with boundary data	$Z_{\rm CFT}[J]=Z_{\rm string}[\Phi\to J]$	Exact statement is stringy and quantum; supergravity is an approximation.
Connected functional $W[J]=\log Z[J]$	Renormalized on-shell action	$W\sim -S_E^{\rm ren}$ in Euclidean signature	Signs depend on Euclidean versus Lorentzian conventions.
Single-trace primary $\mathcal O$	Single-particle bulk field $\Phi$	$J_{\mathcal O}$ is the boundary coefficient of $\Phi$	A local Einstein-like bulk requires large $N$ and a large higher-spin/string gap.
Multi-trace operator $:\mathcal O_1\mathcal O_2:$	Multiparticle bulk state	Dimensions start near $\Delta_1+\Delta_2+2n+\ell$	Interactions generate anomalous dimensions at subleading $1/N$.
Conserved current $J^\mu$	Bulk gauge field $A_M$	$A^{(0)}_\mu$ sources $J^\mu$	Gauge choice and boundary terms determine ensemble.
Stress tensor $T^{\mu\nu}$	Bulk metric $g_{MN}$	$g^{(0)}_{\mu\nu}$ sources $T^{\mu\nu}$	Counterterms and anomalies affect local terms.
Global symmetry group $G$	Bulk gauge symmetry $G$	Boundary global transformations are non-normalizable gauge transformations	A bulk gauge symmetry is a redundancy, not a boundary global symmetry.
Boundary conformal group	Bulk AdS isometry group	$SO(2,d)$ for Lorentzian AdS$_{d+1}$	Boundary conformal frame is not unique.
Thermal state	Euclidean saddle with $S^1_\beta$ or Lorentzian black hole	$T=1/\beta$; $S=A_H/(4G_N)$	Dominant saddle depends on ensemble and boundary topology.
Chemical potential $\mu$	Boundary value of $A_t$	$\mu=A_t(\infty)-A_t(r_h)$	In Euclidean signature, regularity at the thermal circle fixes the horizon gauge.
Charge density $\rho$	Radial electric flux	$\rho\sim \delta S^{\rm ren}/\delta A_t^{(0)}$	Normalization depends on Maxwell coupling and counterterms.
Wilson loop $W(C)$	Fundamental string worldsheet ending on $C$	$\langle W(C)\rangle\sim e^{-S_{\rm NG}^{\rm ren}}$	Perimeter divergences must be subtracted.
Entanglement entropy $S_A$	RT/HRT or QES surface	$S_A={\rm Area}(\gamma_A)/(4G_N)$ at leading order	Quantum corrections add bulk entropy and shift the surface.
Defect CFT	Probe brane or backreacted brane geometry	Defect operators live on brane intersections	Probe limit requires $N_f\ll N$ or analogous suppression.
RG scale	Radial position	Roughly $z\sim 1/E$	Radial evolution is not literally Wilsonian RG without further construction.

Local operators and bulk fields

The most frequently used dictionary entries are local operators and their dual bulk fields.

CFT operator	Dimension/spin	Bulk field	Source	Response/vev	Typical use
Scalar primary $\mathcal O$	$(\Delta,0)$	Scalar $\phi$	Leading coefficient $\phi_{(0)}$	Normalizable coefficient or renormalized momentum	Deformations, condensates, scalar correlators
Conserved current $J^\mu$	$\Delta=d-1$, spin $1$	Gauge field $A_M$	$A^{(0)}_\mu$	$\langle J^\mu\rangle$ from electric flux	Conductivity, charge density, global symmetries
Stress tensor $T^{\mu\nu}$	$\Delta=d$, spin $2$	Metric $g_{MN}$	$g^{(0)}_{\mu\nu}$	$\langle T^{\mu\nu}\rangle$ from Brown-York tensor	Thermodynamics, viscosity, energy density
Fermionic operator $\mathcal O_\psi$	spin $1/2$	Bulk spinor $\psi$	One boundary chirality/component	Conjugate boundary component	Fermion spectral functions, Fermi surfaces
Marginal scalar operator	$\Delta=d$	Massless scalar	Boundary scalar coupling	Vev coefficient plus anomaly terms	Coupling deformations, dilaton/axion sources
Relevant scalar operator	$\Delta<d$	Scalar with $m^2<0$ but above BF bound	Coefficient of $z^{d-\Delta}$	Coefficient of $z^\Delta$	RG flows and domain walls
Irrelevant scalar operator	$\Delta>d$	Massive scalar	Coefficient of $z^{d-\Delta}$ grows near boundary	Vev coefficient	UV completion needed; sources are subtle
Antisymmetric tensor current	depends on form degree	Bulk $p$-form gauge field	Boundary $p$-form source	Higher-form current	Higher-form symmetries, branes
Line operator	nonlocal	String, D-brane, or M-brane	Boundary curve and internal data	Renormalized brane action	Wilson, ’t Hooft, dyonic loops
Surface or higher-dimensional defect	nonlocal	Higher-dimensional brane	Boundary submanifold	Renormalized brane action and fluctuations	Surface operators, defects, interfaces

A single bulk field can encode several boundary quantities once one allows fluctuations around nontrivial states. For example, a bulk gauge field background $A_t(r)$ encodes $\mu$ and $\rho$, while its transverse fluctuation $a_x(r)e^{-i\omega t}$ encodes optical conductivity.

Mass-dimension relations

The mass of a bulk field determines the scaling dimension of the dual operator. The most important case is a scalar:

$$\Delta(\Delta-d)=m^2L^2, \qquad \Delta_\pm=\frac d2\pm \nu, \qquad \nu=\sqrt{\frac{d^2}{4}+m^2L^2}.$$

Standard quantization usually takes $\Delta=\Delta_+$. In the Breitenlohner-Freedman window,

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

one may often choose alternate quantization with $\Delta=\Delta_-$.

Bulk field	Mass-dimension relation	Conserved/massless special case	Boundary interpretation
Scalar $\phi$	$\Delta(\Delta-d)=m^2L^2$	$m^2=0\Rightarrow \Delta=d$ or $0$	Scalar primary or coupling
Massive vector $A_M$	$(\Delta-1)(\Delta-d+1)=m^2L^2$	$m^2=0\Rightarrow \Delta=d-1$	Conserved current
Dirac spinor $\psi$	$\Delta=\frac d2+\lvert mL\rvert$ in standard quantization	$m=0\Rightarrow\Delta=d/2$	Fermionic operator
Massive $p$-form	$(\Delta-p)(\Delta+p-d)=m^2L^2$	Gauge $p$-form gives conserved higher-form current	Higher-form symmetry current
Graviton $h_{MN}$	fixed by diffeomorphism invariance	$\Delta=d$	Stress tensor

Two comments prevent many mistakes. First, a massless scalar in AdS has two formal roots, $\Delta=0$ and $\Delta=d$; the usual standard quantization for a nontrivial scalar operator uses $\Delta=d$. Second, for gauge fields and gravitons, the dimensions of $J^\mu$ and $T^{\mu\nu}$ are fixed by conservation, not by an arbitrary mass parameter.

Near-boundary expansions

Near-boundary expansions encode sources and vevs. The following table suppresses logarithms that appear when dimensions collide with integers or when conformal anomalies are present.

Bulk field	Near-boundary form	Source	Response/vev data
Scalar $\phi$	$\phi=z^{d-\Delta}\phi_{(0)}+\cdots+z^\Delta\phi_{(2\Delta-d)}+\cdots$	$\phi_{(0)}$	$\phi_{(2\Delta-d)}$ plus local terms
Gauge field $A_\mu$	$A_\mu=A_\mu^{(0)}+\cdots+z^{d-2}A_\mu^{(d-2)}+\cdots$	$A^{(0)}_\mu$	Radial electric flux, often proportional to $A_\mu^{(d-2)}$
Metric $g_{\mu\nu}$	$g_{\mu\nu}=g^{(0)}_{\mu\nu}+\cdots+z^d g^{(d)}_{\mu\nu}+\cdots$	$g^{(0)}_{\mu\nu}$	Renormalized Brown-York tensor
Dirac spinor $\psi$	$\psi=z^{d/2-mL}\psi_{(0)}+\cdots+z^{d/2+mL}\psi_{(2mL)}+\cdots$	leading independent spinor component	conjugate component after renormalization
Probe-brane embedding $\theta$	$\theta=z^{d-\Delta}\theta_{(0)}+\cdots+z^\Delta\theta_{(2\Delta-d)}+\cdots$	mass or defect/source parameter	condensate or defect vev

For a scalar in standard quantization, the schematic one-point function is

$$\langle \mathcal O\rangle_J =(2\Delta-d)\mathcal N_\phi\,\phi_{(2\Delta-d)}+\text{local terms},$$

where $\mathcal N_\phi$ is fixed by the normalization of the bulk kinetic term. The phrase “local terms” includes scheme-dependent terms and anomaly-related terms determined by counterterms.

For a Maxwell field with action

$$S_A=-\frac{1}{4g_{d+1}^2}\int d^{d+1}x\sqrt{-g}\,F_{MN}F^{MN},$$

the radial canonical momentum is

$$\Pi^\mu_A =-\frac{1}{g_{d+1}^2}\sqrt{-g}\,F^{z\mu},$$

and the renormalized current is schematically

$$\langle J^\mu\rangle =\Pi^\mu_{A,\rm ren}.$$

For the metric, the stress tensor is obtained from the renormalized action by

$$\langle T^{\mu\nu}\rangle =\frac{2}{\sqrt{-g^{(0)}}}\frac{\delta S^{\rm ren}}{\delta g^{(0)}_{\mu\nu}},$$

up to sign conventions in Lorentzian versus Euclidean signature.

Generating functionals and correlators

Desired CFT quantity	Bulk operation	Formula or prescription	Typical pitfall
One-point function	Vary renormalized on-shell action once	$\langle\mathcal O(x)\rangle_J=\delta W/\delta J(x)$	Forgetting counterterms or factors of $\sqrt g$.
Connected two-point function	Vary twice	$G_c(x,y)=\delta^2 W/\delta J(x)\delta J(y)$	Contact terms are scheme-dependent.
Euclidean correlator	Solve regular Euclidean boundary-value problem	$G_E$ from on-shell action	Not every Euclidean result analytically continues uniquely without specifying contour.
Retarded correlator	Solve Lorentzian problem with infalling horizon condition	$G_R=\Pi_{\rm ren}/\phi_{(0)}$ for linear fields	Outgoing or regular-but-not-infalling conditions give the wrong Green function.
Spectral density	Imaginary part of retarded function	$\rho(\omega,k)=-2\operatorname{Im}G_R(\omega,k)$	Sign conventions vary. State them.
Normal modes	Normalizable solution in horizonless geometry	Source $=0$, regular interior	Normal modes are not the same as black-hole quasinormal modes.
Quasinormal modes	Infalling solution with source $=0$	Poles of $G_R(\omega,k)$	Gauge redundancies require gauge-invariant variables.
Witten diagram	Perturbatively evaluate bulk interactions	Tree diagrams give leading connected large-$N$ correlators	Exchange diagrams require correct cubic couplings and boundary conditions.
OPE coefficient	Compare three-point function normalization	$C_{123}\leftrightarrow$ cubic bulk coupling	Operator normalization must be fixed first.
Double-trace anomalous dimension	Extract from four-point data	Bulk exchange/contact interactions shift dimensions	Mixing is common at finite spin and finite $N$.

In Euclidean signature, with

$$Z[J]=\exp W[J],$$

one often writes

$$\langle \mathcal O(x_1)\cdots \mathcal O(x_n)\rangle_c =\left.\frac{\delta^n W[J]}{\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}.$$

If instead one uses $Z=e^{-S_E}$ and identifies $W=-S_E^{\rm ren}$, the sign is absorbed into the definition of $W$. Do not mix these conventions halfway through a calculation.

Thermodynamic dictionary

Thermal AdS/CFT calculations are saddle comparisons in the Euclidean path integral or horizon calculations in Lorentzian signature.

Boundary thermodynamic quantity	Bulk dual	Formula	Caution
Temperature $T$	Euclidean time circle or surface gravity	$T=1/\beta=\kappa/(2\pi)$	Euclidean smoothness fixes the period only for the contractible thermal circle.
Entropy $S$	Horizon area	$S=A_H/(4G_N)$ for Einstein gravity	Higher-derivative gravity uses Wald-like entropy, not simply area.
Free energy $F$	Renormalized Euclidean action	$F=T S^{\rm ren}_E$	Must compare saddles with the same boundary data.
Energy density $\epsilon$	Boundary stress tensor	$\epsilon=\langle T^{tt}\rangle$	Counterterm subtraction matters.
Pressure $p$	Spatial stress tensor or $-F/V$	For a CFT, $\epsilon=(d-1)p$	Broken conformality changes the equation of state.
Chemical potential $\mu$	Gauge potential difference	$\mu=A_t(\infty)-A_t(r_h)$	Gauge-dependent unless horizon regularity is imposed.
Charge density $\rho$	Electric flux	$\rho=\langle J^t\rangle$	Canonical and grand-canonical ensembles use different boundary terms.
Grand potential $\Omega$	On-shell action with fixed $\mu$	$\Omega=E-TS-\mu Q$	Legendre transform for fixed charge.
Polyakov loop	String worldsheet ending on thermal circle	$\langle P\rangle\sim e^{-S_{\rm string}}$	Contractibility of the thermal circle matters.
Deconfinement transition	Hawking-Page-like transition	$F_{\rm black\ hole}<F_{\rm thermal\ AdS}$	Only sharp at large $N$ in finite volume.

For the planar AdS$_{d+1}$ black brane,

$$ds^2 =\frac{L^2}{z^2}\left[-f(z)dt^2+d\vec x^2+\frac{dz^2}{f(z)}\right], \qquad f(z)=1-\left(\frac{z}{z_h}\right)^d,$$

the temperature and entropy density are

$$T=\frac{d}{4\pi z_h}, \qquad s=\frac{1}{4G_{d+1}}\left(\frac{L}{z_h}\right)^{d-1}.$$

For a conformal plasma, dimensional analysis then gives

$$s\propto C_T T^{d-1}, \qquad p\propto C_T T^d, \qquad \epsilon=(d-1)p.$$

Transport dictionary

Transport coefficients are low-frequency, low-momentum limits of retarded Green functions.

Transport quantity	Kubo formula	Bulk fluctuation	Common limit issue
Electrical conductivity	$\sigma(\omega)=-\frac{i}{\omega}G^R_{J^xJ^x}(\omega,0)$	$a_x(r)e^{-i\omega t}$	At finite density, momentum conservation can create a delta function at $\omega=0$.
DC conductivity	$\sigma_{\rm DC}=\lim_{\omega\to0}\operatorname{Re}\sigma(\omega)$	Horizon electric flux in many models	Need momentum relaxation for finite charge-overlap contribution.
Charge susceptibility	$\chi=\partial\rho/\partial\mu$	Static $A_t$ perturbation	Ensemble and boundary counterterms matter.
Charge diffusion	$\omega=-iD k^2+\cdots$	Coupled longitudinal Maxwell/gravity channel	$D=\sigma/\chi$ only under appropriate hydrodynamic assumptions.
Shear viscosity	$\eta=-\lim_{\omega\to0}\frac{1}{\omega}\operatorname{Im}G^R_{T^{xy}T^{xy}}(\omega,0)$	Transverse graviton $h^x{}_y$	$\eta/s=1/(4\pi)$ assumes two-derivative Einstein gravity.
Sound attenuation	Pole in sound channel	Coupled metric perturbations	Hydrodynamic frame conventions matter.
Thermal conductivity	Heat-current correlator	Coupled metric and gauge perturbations	At finite density, heat and charge transport mix.
Momentum relaxation rate	Width of Drude pole	Translation-breaking scalars, lattices, Q-lattices	Distinguish explicit breaking from spontaneous breaking.
Butterfly velocity $v_B$	OTOC front	Near-horizon shockwave	Not a conventional linear-response coefficient.
Lyapunov exponent $\lambda_L$	OTOC growth	Near-horizon scattering	Einstein black holes saturate $2\pi T$ under standard assumptions.

The membrane-paradigm intuition is that some low-frequency transport coefficients can be computed from horizon data. But this is not a license to ignore the boundary. Horizon data gives the answer only after the correct boundary variational problem and conserved radial flux are identified.

Nonlocal observables and branes

Boundary observable	Bulk object	Leading approximation	Renormalization/subtraction
Fundamental Wilson loop $W(C)$	Fundamental string ending on $C$	$\langle W(C)\rangle\sim e^{-S_{\rm NG}^{\rm ren}}$	Subtract straight-string/perimeter divergence.
’t Hooft loop	D1 string or magnetic dual object	$e^{-S_{\rm D1}^{\rm ren}}$	Depends on duality frame and boundary conditions.
Dyonic loop	$(p,q)$ string	$e^{-S_{(p,q)}^{\rm ren}}$	Sensitive to axio-dilaton and charge lattice.
Antisymmetric Wilson representation	D5-brane with worldvolume flux in AdS$_5\times S^5$	DBI plus WZ saddle	Flux quantization fixes representation data.
Symmetric Wilson representation	D3-brane with worldvolume flux	DBI plus WZ saddle	Valid at large representation rank.
Baryon operator in $SU(N)$-like theory	Wrapped D-brane plus $N$ strings	D5 on $S^5$ for AdS$_5\times S^5$	Gauss law from WZ coupling forces $N$ strings.
Defect CFT	Probe brane with AdS slicing	DBI/WZ action and fluctuations	Probe limit suppresses backreaction.
Meson spectrum	Normal modes on flavor brane	Sturm-Liouville eigenvalue problem	Quark mass and condensate require holographic renormalization.
Entanglement entropy $S_A$	RT/HRT surface	$A/(4G_N)$	Homology constraint and phase transitions are essential.
Quantum entanglement entropy	Quantum extremal surface	$\operatorname{ext}\left[A/(4G_N)+S_{\rm bulk}\right]$	Bulk entropy and area counterterms renormalize together.
Complexity proposals	Maximal volume/action-like quantities	CV, CA, or related prescriptions	Less universal than RT/HRT; normalization is subtle.

For string and brane observables, the boundary conditions include both spacetime data and internal-space data. The half-BPS Maldacena-Wilson loop, for instance, specifies not only a contour $C\subset \mathbb R^4$ but also a coupling to the scalar fields, which is represented by where the string ends on $S^5$.

RG-flow and deformation dictionary

A deformation by a scalar operator is written

$$S_{\rm CFT}\to S_{\rm CFT}+\int d^d x\,\lambda_{\mathcal O}(x)\mathcal O(x).$$

If $\mathcal O$ has dimension $\Delta$, the dual scalar behaves near the boundary as

$$\phi(z,x) =z^{d-\Delta}\lambda_{\mathcal O}(x)+\cdots+z^\Delta v_{\mathcal O}(x)+\cdots .$$

Field-theory concept	Bulk representation	Diagnostic formula	Caution
Relevant deformation	Scalar source with $\Delta<d$	$\phi\sim z^{d-\Delta}\lambda$	Backreaction usually creates a domain wall.
Marginal deformation	Massless scalar source	$\Delta=d$	Exactly marginal only if beta function vanishes.
Irrelevant deformation	Scalar source with $\Delta>d$	Leading mode grows toward boundary	Needs UV completion; hard to impose as a source.
Vev flow	Source set to zero, normalizable mode nonzero	$\lambda=0$, $v\neq0$	Regularity often selects allowed vevs.
Source flow	Nonzero coupling	$\lambda\neq0$	Vev may be induced dynamically.
Holographic beta function	Radial variation of scalar	$\beta(\phi)=d\phi/dA$	Scheme-dependent away from fixed points.
$c$-function	Warp-factor combination	$\mathcal C\propto 1/[A'(r)]^{d-1}$	Monotonicity typically assumes a null energy condition.
Confinement scale	End of geometry or IR wall/cap	mass gap, area law	A discrete spectrum alone is not the same as Wilson-loop confinement.
Chiral symmetry breaking	Brane joining or scalar condensate	$m_q$ source, $\langle\bar q q\rangle$ response	Bottom-up and top-down meanings can differ.

For a domain-wall metric

$$ds^2=dr^2+e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu,$$

the radial coordinate is often interpreted as an energy scale through $E\sim e^{A(r)}$. This is a powerful organizing idea, but one should not confuse coordinate choice with a physical RG scheme.

Entanglement and reconstruction dictionary

Boundary concept	Bulk concept	Leading relation	Important refinement
Region $A$	Boundary domain of dependence $D[A]$	Defines the subregion problem	Use the domain of dependence, not just a time-slice shape.
Entanglement entropy $S_A$	Extremal surface $\gamma_A$	$S_A=A(\gamma_A)/(4G_N)$	Homology condition selects the correct surface.
Time-dependent entropy	HRT surface	Extremal, not necessarily minimal on a chosen time slice	Maximin helps establish properties.
Quantum correction	Bulk entropy across $\gamma_A$	$S_A=A/(4G_N)+S_{\rm bulk}+\cdots$	Use quantum extremal surfaces at higher orders.
Entanglement wedge $\mathcal E[A]$	Bulk domain bounded by $A\cup\gamma_A$	Region reconstructable from $A$ in a code subspace	State and code-subspace dependence matter.
Causal wedge $\mathcal C[A]$	Bulk causal region associated with $D[A]$	Usually smaller than entanglement wedge	Causal wedge is not the full reconstruction wedge.
Modular Hamiltonian $K_A$	Bulk modular Hamiltonian plus area term	JLMS relation	Operator equality holds in an appropriate code subspace.
Relative entropy $S(\rho_A\Vert\sigma_A)$	Bulk relative entropy in wedge	Equality at leading quantum order	Requires matching states in the same code subspace.
Redundant reconstruction	Quantum error correction	Same logical operator on different boundary regions	Bulk operators must be gravitationally dressed.
Island	Part of gravitating region included in radiation wedge	QES generalized entropy saddle	Relevant for evaporating setups or coupled baths.

The RT/HRT entry is one of the cleanest parts of the dictionary, but also one of the easiest to misuse. The surface is not chosen by visual prettiness. It is chosen by extremality, homology, boundary anchoring, and the correct saddle dominance.

Canonical AdS$_5$/CFT$_4$ parameter dictionary

For type IIB string theory on $\mathrm{AdS}_5\times S^5$ dual to four-dimensional $\mathcal N=4$ super Yang-Mills, a common convention is

$$g_{\rm YM}^2=4\pi g_s, \qquad \lambda=g_{\rm YM}^2N=4\pi g_sN, \qquad \frac{L^4}{\alpha'^2}=\lambda.$$

Normalizations differ by factors of $2\pi$ in the literature, especially in the definition of $g_{\rm YM}^2$. Always identify the convention before comparing formulas.

CFT parameter	Bulk/string parameter	Meaning
Rank $N$	Five-form flux through $S^5$	Number of D3-branes; controls degrees of freedom
’t Hooft coupling $\lambda=g_{\rm YM}^2N$	Curvature in string units	$L^4/\alpha'^2=\lambda$
Gauge coupling $g_{\rm YM}$	String coupling $g_s$	$g_s\sim g_{\rm YM}^2$ up to convention
Theta angle $\theta_{\rm YM}$	RR axion $C_0$	Complex coupling maps to axio-dilaton
Central charge $C_T\sim N^2$	$L^3/G_5$	Measures inverse bulk gravitational coupling
Large $N$	Small bulk loop expansion	$G_5/L^3\sim 1/N^2$
Large $\lambda$	Small stringy curvature corrections	$\alpha'/L^2\sim 1/\sqrt\lambda$
Planar limit	Classical string genus expansion	$g_s\sim \lambda/N$ small when $N\gg\lambda$
Supergravity limit	Classical low-energy gravity	$N\gg1$ and $\lambda\gg1$ with appropriate suppression of loops

The hierarchy of approximations is

$$\text{full CFT} \leftrightarrow \text{full string theory} \leftrightarrow \text{classical string theory} \leftrightarrow \text{classical supergravity} \leftrightarrow \text{lower-dimensional Einstein gravity}.$$

Each arrow discards corrections. Do not use a result from the rightmost description unless the discarded corrections are parametrically small for the observable under discussion.

Large-$N$ and bulk perturbation theory

Boundary expansion	Bulk interpretation	Typical scaling
Large-$N$ factorization	Classical bulk limit	Connected normalized correlators are suppressed.
Single-trace operators	Single-particle fields	One bulk field per light single-trace primary.
Multi-trace operators	Multiparticle states	Double-trace dimensions begin near sums of single-trace dimensions.
Planar diagrams	String worldsheet genus zero	Leading in $1/N$.
Nonplanar diagrams	Higher genus worldsheets	Suppressed by powers of $1/N^2$.
$1/N$ corrections	Bulk loops and quantum gravity corrections	Controlled by $G_N/L^{d-1}$.
$1/\lambda$ corrections	Stringy $\alpha'$ corrections	Higher-derivative terms in the bulk effective action.
Large spectral gap	Local bulk EFT	Heavy single-trace operators correspond to stringy states.
No large gap	Higher-spin or stringy bulk	Large $N$ alone does not imply Einstein gravity.

A common large-$N$ normalization is to choose single-trace operators $\mathcal O_i$ such that

$$\langle \mathcal O_i(x)\mathcal O_i(0)\rangle\sim O(1),$$

then connected $n$-point functions often scale as

$$\langle \mathcal O_1\cdots\mathcal O_n\rangle_{\rm conn} \sim N^{2-n}$$

in matrix large-$N$ theories with single-trace operators. Some authors instead normalize operators with explicit powers of $N$. Before comparing OPE coefficients, identify the normalization.

Boundary conditions and ensembles

Boundary conditions are part of the observable. Changing them changes the dual theory, state, or ensemble.

Bulk condition	Boundary meaning	Example	Caution
Dirichlet for scalar leading mode	Fix source	Standard scalar deformation	Vev is determined dynamically.
Alternate quantization	Swap source and response	BF-window scalar	Allowed only in a restricted mass range and with unitarity constraints.
Mixed scalar boundary condition	Multi-trace deformation	$f\mathcal O^2/2$	Requires Legendre transform or boundary term.
Dirichlet for gauge field	Fixed chemical potential/source	Grand-canonical ensemble	Boundary global symmetry is fixed.
Neumann-like gauge boundary condition	Fixed charge or dynamical boundary gauge field in some dimensions	Canonical ensemble	Must add appropriate boundary term.
Fixed boundary metric	CFT on prescribed geometry	Standard AdS/CFT	Integrating over boundary metric changes the theory.
Regular Euclidean interior	Smooth Euclidean saddle	Thermal black-hole cigar	Does not by itself give retarded real-time correlators.
Infalling horizon condition	Retarded response	Lorentzian black brane	Needed for causal dissipative correlators.
Outgoing horizon condition	Advanced response	Time-reversed problem	Wrong for retarded response.
Normalizable at interior	State excitation or mode spectrum	Global AdS normal modes	Horizonless and black-hole interiors differ.
Brane endpoint fixed on boundary	Nonlocal source	Wilson loop contour	Internal-space endpoint may also be part of the source.
Homology condition for RT/HRT	Correct entropy saddle	Thermal entropy contribution	Essential in black-hole backgrounds.

Mixed boundary conditions are especially important. For a scalar with standard source $\alpha$ and response $\beta$, a double-trace deformation

$$\delta S_{\rm CFT}=\frac f2\int d^d x\,\mathcal O^2$$

is implemented by a boundary condition of the schematic form

$$\alpha=f\beta$$

or $\beta=f\alpha$, depending on which quantization and notation are used. This is a place where convention-hunting is not optional.

Ward identities from bulk constraints

Bulk gauge redundancies become boundary Ward identities. These are among the best checks of a holographic calculation.

Bulk symmetry/constraint	Boundary Ward identity	With sources
Gauge invariance of $A_M$	Current conservation	$\nabla_\mu\langle J^\mu\rangle=0$ unless charged sources/anomalies are present
Diffeomorphism invariance	Stress-tensor conservation	$\nabla_\mu\langle T^{\mu\nu}\rangle=F^{\nu}{}_{\mu}\langle J^\mu\rangle+\langle\mathcal O\rangle\nabla^\nu J_{\mathcal O}+\cdots$
Weyl transformations	Trace Ward identity	$\langle T^\mu{}_{\mu}\rangle=(d-\Delta)J\langle\mathcal O\rangle+\mathcal A$
Hamiltonian constraint	RG-like flow equation	Encodes radial evolution of the on-shell action
Momentum constraint	Boundary conservation law	Fixes divergence of Brown-York tensor
Gauss constraint on brane	Charge quantization or string endpoints	Baryon vertex requires attached strings

For a CFT deformed by a scalar source $J(x)$, the trace Ward identity has the schematic form

$$\langle T^\mu{}_{\mu}\rangle =(d-\Delta)J\langle\mathcal O\rangle+\mathcal A,$$

where $\mathcal A$ is the conformal anomaly when present. In odd boundary dimensions and on simple backgrounds, $\mathcal A$ often vanishes; in even dimensions it is generally nonzero.

Common normalization traps

Quantity	Trap	Safer practice
$g_{\rm YM}^2$	Factors of $2\pi$ or $4\pi$ differ	State the convention before using $\lambda$.
$G_5$ versus $G_{10}$	Forgetting compactification volume	Use $1/G_5={\rm Vol}(S^5)/G_{10}$ with radius factors.
Operator normalization	Comparing OPE coefficients across papers	Normalize two-point functions first.
Stress tensor	Sign and factor errors in $2/\sqrt g$ variation	Write the variational definition explicitly.
Current	Maxwell coupling omitted	Keep $1/g_{d+1}^2$ in all formulas.
Euclidean action	Wrong sign in $W=-S_E$	Fix $Z=e^{-S_E}$ or $Z=e^{W}$ at the start.
Fourier transform	Different $2\pi$ conventions	Write the transform convention.
Contact terms	Treating scheme-dependent polynomial terms as physical	Focus on separated points or nonanalytic momentum dependence.
Horizon gauge	Using $A_t(r_h)\neq0$ in Euclidean black holes	Choose a gauge regular at the contractible thermal circle.
Entropy formula	Using area law in higher-derivative gravity	Use Wald or generalized entropy as appropriate.
Probe limit	Forgetting backreaction scale	Check $N_f/N$, brane tension, and charge density scalings.
Bottom-up models	Treating model parameters as top-down predictions	State which parameters are phenomenological inputs.

Minimal working dictionary by problem type

Problem	Boundary input	Bulk setup	Output
Scalar two-point function	Operator dimension $\Delta$, source $\phi_{(0)}$	Free scalar in Euclidean AdS	$\langle\mathcal O(x)\mathcal O(0)\rangle\propto \lvert x\rvert^{-2\Delta}$
Three-point coefficient	Operators $\mathcal O_i$	Cubic coupling $g_{123}\phi_1\phi_2\phi_3$	$C_{123}$ after fixing two-point normalizations
Retarded correlator	Operator and thermal state	Linear fluctuation in black brane with infalling condition	$G_R(\omega,k)$ and spectral density
Conductivity	Current $J^x$	Maxwell fluctuation $a_x(r)$	$\sigma(\omega)$ from $G^R_{J^xJ^x}$
Shear viscosity	Stress tensor $T^{xy}$	Graviton $h^x{}_y(r)$	$\eta$ from Kubo formula
Equation of state	Thermal CFT	Black-brane saddle	$\epsilon$, $p$, $s$, $F$
Heavy-quark potential	Rectangular Wilson loop	U-shaped string worldsheet	$V(R)$ from $S_{\rm NG}^{\rm ren}$
Meson spectrum	Flavor bilinear operators	Probe-brane fluctuations	Discrete normal-mode masses
Entanglement entropy	Region $A$	RT/HRT/QES surface	$S_A$
RG flow	Relevant deformation	Einstein-scalar domain wall	Running vevs, $c$-function, IR geometry
Finite-density phase	$\mu$, $T$, charged operators	Charged black brane plus possible hair	Phase diagram and response functions
Momentum relaxation	Translation-breaking source	Axions, lattices, Q-lattices, massive-gravity-like models	Finite DC transport

A compact “what should I vary?” table

When in doubt, return to the variational problem.

Source varied	Conjugate observable	Bulk canonical object
Scalar source $\phi_{(0)}$	$\langle\mathcal O\rangle$	$\Pi_\phi^{\rm ren}=\delta S^{\rm ren}/\delta\phi_{(0)}$
Gauge source $A^{(0)}_\mu$	$\langle J^\mu\rangle$	$\Pi_A^{\mu,\rm ren}$
Boundary metric $g^{(0)}_{\mu\nu}$	$\langle T^{\mu\nu}\rangle$	Renormalized Brown-York tensor
Fermion source component	$\langle\mathcal O_\psi\rangle$	Conjugate spinor component
Brane embedding source	Defect/flavor condensate	Renormalized brane momentum
Wilson-loop contour	Force or shape response	Variation of string action
Entangling surface shape	Entanglement shape response	Variation of extremal-surface area or generalized entropy
Thermal period $\beta$	Energy/free energy	Variation of Euclidean saddle
Chemical potential $\mu$	Charge density $\rho$	Electric flux

The safest habit is to write the first variation of the renormalized on-shell action:

$$\delta S^{\rm ren}_{\rm os} =\int d^d x\sqrt{|g^{(0)}|}\left( \frac12\langle T^{\mu\nu}\rangle\delta g^{(0)}_{\mu\nu} +\langle J^\mu\rangle\delta A^{(0)}_\mu +\langle\mathcal O\rangle\delta\phi_{(0)} +\cdots \right).$$

This one line fixes many factors and signs, provided your convention for $S_E$ or $S_L$ is stated.

How to use these tables in a real calculation

Use the dictionary in the following order:

1. Identify the observable. Is it a one-point function, a retarded correlator, a transport coefficient, a free energy, a Wilson loop, or an entropy?
2. Identify the source. What boundary data must be held fixed?
3. Identify the state. Vacuum, thermal state, charged state, defect state, time-dependent state, or ensemble?
4. Choose the bulk approximation. Classical supergravity, probe brane, classical string, perturbative Witten diagram, or quantum-corrected QES?
5. Impose the correct boundary and interior conditions. Regular Euclidean, infalling Lorentzian, normalizable, mixed, homology, flux quantization, and so on.
6. Renormalize before differentiating. Bare canonical momenta are usually divergent.
7. Check Ward identities and scaling. A result that violates a Ward identity is almost always wrong.
8. State the regime of validity. Every holographic answer is surrounded by assumptions.

Exercises

Exercise 1: source dimensions

A scalar primary $\mathcal O$ in a $d=4$ CFT has dimension $\Delta=3$. What is the dimension of its source $J$? What is the leading near-boundary power of the dual scalar in standard quantization?

Solution

The deformation is

$$\delta S=\int d^4x\,J\mathcal O,$$

so $[J]=d-\Delta=1$. The standard near-boundary expansion is

$$\phi(z,x)=z^{d-\Delta}\phi_{(0)}(x)+\cdots+z^\Delta\phi_{(2\Delta-d)}(x)+\cdots .$$

For $d=4$ and $\Delta=3$,

$$\phi(z,x)=z\phi_{(0)}(x)+\cdots+z^3\phi_{(2)}(x)+\cdots .$$

The coefficient of $z$ is the source in standard quantization.

Exercise 2: scalar mass from operator dimension

For a scalar operator of dimension $\Delta$ in a $d$-dimensional CFT, show that the dual scalar mass is

$$m^2L^2=\Delta(\Delta-d).$$

Evaluate this for $d=4$, $\Delta=2$. Is the scalar below the BF bound?

Solution

The scalar mass-dimension relation is

$$\Delta(\Delta-d)=m^2L^2.$$

For $d=4$ and $\Delta=2$,

$$m^2L^2=2(2-4)=-4.$$

The BF bound in AdS$_5$ is

$$m^2L^2\ge -\frac{d^2}{4}=-4.$$

Thus $m^2L^2=-4$ saturates the BF bound. It is not below the bound, but logarithmic subtleties may appear because the two roots coincide.

Exercise 3: current from radial flux

Consider a Maxwell field in AdS with action

$$S_A=-\frac{1}{4g_{d+1}^2}\int d^{d+1}x\sqrt{-g}\,F_{MN}F^{MN}.$$

Show that the canonical momentum conjugate to $A_\mu$ along the radial direction is proportional to $\sqrt{-g}F^{z\mu}$.

Solution

Vary the Maxwell action:

$$\delta S_A =-\frac{1}{2g_{d+1}^2}\int d^{d+1}x\sqrt{-g}\,F^{MN}\delta F_{MN}.$$

Using

$$\delta F_{MN}=\nabla_M\delta A_N-\nabla_N\delta A_M,$$

and integrating by parts, the boundary term at fixed radial coordinate is

$$\delta S_A\big|_{\rm bdy} =-\frac{1}{g_{d+1}^2}\int_{z=\epsilon} d^d x\sqrt{-g}\,F^{z\mu}\delta A_\mu,$$

up to orientation conventions. Thus

$$\Pi_A^\mu =-\frac{1}{g_{d+1}^2}\sqrt{-g}\,F^{z\mu}.$$

After adding counterterms and taking the cutoff to the boundary, this becomes the renormalized current expectation value.

Exercise 4: which Green function?

You want the retarded correlator of a scalar operator in a thermal state. Should the bulk field be regular in Euclidean signature, normalizable in global AdS, infalling at the Lorentzian horizon, or outgoing at the Lorentzian horizon?

Solution

For a retarded thermal correlator, use the Lorentzian black-hole or black-brane geometry and impose the infalling condition at the future horizon. The infalling condition implements causal response: the horizon absorbs disturbances rather than emitting them in response to a boundary source.

Regularity in the Euclidean geometry computes Euclidean thermal correlators. Normalizability in global AdS computes normal modes in a horizonless geometry. Outgoing horizon conditions compute the advanced response or a time-reversed problem, not the retarded Green function.

Exercise 5: classify the correction

In AdS$_5$/CFT$_4$, classify each correction as primarily a $1/N$ correction or a $1/\lambda$ correction:

1. A one-loop graviton diagram in AdS.
2. An $R^4$ higher-derivative term in type IIB string theory.
3. A genus-one string worldsheet correction.
4. A finite string length correction to a classical supergravity background.

Solution

1. A one-loop graviton diagram is a bulk quantum correction, so it is primarily a $1/N$ correction.
2. The $R^4$ term is a stringy $\alpha'$ correction, so it is primarily a $1/\lambda$ correction. In AdS$_5\times S^5$, the leading correction scales as a positive power of $\lambda^{-3/2}$ in many observables.
3. A genus-one string worldsheet correction is a string loop correction, so it is controlled by $g_s$ and hence by $1/N$ at fixed or large $\lambda$.
4. A finite string length correction is an $\alpha'$ correction, so it is primarily a $1/\lambda$ correction.

Some effects can mix the two expansions, but this classification captures the leading physical origin.

Exercise 6: build a mini-dictionary

A boundary theory has a conserved $U(1)$ current, is placed at temperature $T$, and is deformed by a relevant scalar operator. You want to compute the DC conductivity at finite density. List the minimal bulk ingredients.

Solution

The minimal ingredients are:

• a bulk metric with a black-hole or black-brane horizon to represent the thermal state;
• a bulk Maxwell field $A_M$ dual to the conserved current;
• a nonzero background $A_t(r)$ with boundary value $\mu$ and radial electric flux $\rho$ to represent finite density;
• a bulk scalar $\phi(r)$ dual to the relevant deformation, with leading boundary coefficient equal to the scalar source;
• a consistent set of coupled background equations for $g_{MN}$, $A_M$, and $\phi$;
• linearized perturbations, typically including $a_x$, $h_{tx}$, and possibly scalar or translation-breaking perturbations;
• infalling horizon conditions for retarded response;
• holographic renormalization to extract $G^R_{J^xJ^x}$;
• if the system has finite density and conserved momentum, a mechanism for momentum relaxation or an explicit treatment of the delta function in the DC conductivity.

Further reading

• S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge Theory Correlators from Non-Critical String Theory.
• E. Witten, Anti De Sitter Space and Holography.
• O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large $N$ Field Theories, String Theory and Gravity.
• K. Skenderis, Lecture Notes on Holographic Renormalization.
• D. T. Son and A. O. Starinets, Minkowski-space Correlators in AdS/CFT: Recipe and Applications.
• N. Iqbal and H. Liu, Universality of the Hydrodynamic Limit in AdS/CFT and the Membrane Paradigm.
• S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT.
• V. E. Hubeny, M. Rangamani, and T. Takayanagi, A Covariant Holographic Entanglement Entropy Proposal.