AdS/CFT Dictionary

The AdS/CFT correspondence (gauge/gravity duality) is the most concrete realization of holography: a quantum theory of gravity on an asymptotically Anti–de Sitter spacetime is equivalent to a conformal field theory (CFT) living on its conformal boundary. The phrase “AdS/CFT dictionary” refers to the precise map between bulk data (fields, couplings, boundary conditions, saddles) and boundary data (operators, sources, correlation functions, states). The duality is exact at the level of partition functions, but many formulas below are used in the semiclassical regime where the bulk is well approximated by classical gravity (typically large $N$ and strong ’t Hooft coupling).

Notation and conventions

• Bulk spacetime dimension: $d{+}1$.
• Boundary spacetime dimension: $d$.
• Bulk indices: $M,N=0,\dots,d$.
• Boundary indices: $\mu,\nu=0,\dots,d{-}1$.
• AdS radius: $L$.
• Poincaré radial coordinate: $z$ with the boundary at $z\to 0$.
• The “boundary metric” $g^{(0)}_{\mu\nu}$ is defined only up to Weyl rescaling (a conformal class).
• “Source” vs “vev” is defined by the near-boundary expansion of bulk fields (see Field/operator map).
• Unless stated, formulas are written in Euclidean signature, where $Z\sim e^{-S_E}$. In Lorentzian signature, $S_E\to iS$ and real-time correlators require additional choices of boundary conditions (e.g. infalling at horizons for retarded correlators).

Quick reference

Bulk (AdS)	Boundary (CFT)	Dictionary entry
AdS isometries	conformal group	$SO(d,2)$
radial position $z$	RG scale $\mu$	$z\sim 1/\mu$
bulk field $\Phi$	local operator $\mathcal{O}$	source $\phi_0$
bulk gauge field $A_M$	conserved current $J^\mu$	$A_\mu^{(0)}$
bulk gauge potential $A_t$	chemical potential $\mu$	$A_t^{(0)}=\mu$
bulk metric $g_{MN}$	stress tensor $T_{\mu\nu}$	$g^{(0)}_{\mu\nu}$
classical bulk saddle	CFT state/ensemble	vacuum, thermal
AdS black hole	thermal state	$T$, $S$
string worldsheet	Wilson loop $W(C)$	$S_{\mathrm{NG}}$
extremal surface	entanglement entropy $S_A$	RT/HRT
bulk loops	$1/N$ effects	quantum gravity
$\alpha'$ corrections	$1/\sqrt{\lambda}$ effects	stringy

Entries are expanded below.

Geometry and symmetry

AdS spacetime

A convenient form of the AdS$_{d+1}$ metric is the Poincaré patch:

$$ds^2 = \frac{L^2}{z^2}\Bigl(dz^2 + \eta_{\mu\nu}\,dx^\mu dx^\nu\Bigr),\qquad z>0.$$

The conformal boundary is at $z\to 0$. Up to the divergent conformal factor $L^2/z^2$, the induced boundary metric is $\eta_{\mu\nu}$.

Global AdS makes the timelike boundary manifest; one common form is

$$ds^2 = L^2\left(-(1+r^2)dt^2 + \frac{dr^2}{1+r^2} + r^2 d\Omega_{d-1}^2\right),\qquad r\ge 0.$$

In global coordinates the boundary is conformal to $\mathbb{R}\times S^{d-1}$.

Fefferman–Graham form (near-boundary)

For general asymptotically AdS solutions it is convenient to use Fefferman–Graham coordinates, in which the metric takes the form

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

with an asymptotic expansion $g_{\mu\nu}(z,x)=g^{(0)}_{\mu\nu}(x)+z^2 g^{(2)}_{\mu\nu}(x)+\cdots$. The leading term $g^{(0)}_{\mu\nu}$ is the boundary metric source for $T_{\mu\nu}$; subleading coefficients encode renormalized one-point functions.

Symmetry match

The AdS$_{d+1}$ isometry group is $SO(d,2)$, which is also the conformal symmetry group of a $d$-dimensional CFT. This symmetry match is the kinematic backbone of the correspondence.

A core structural rule is:

• Bulk gauge symmetries correspond to global symmetries in the boundary CFT.

For example, a bulk $U(1)$ gauge field is dual to a conserved current $J^\mu$ of a global $U(1)$ symmetry in the CFT.

Anomalies and topological terms

Global-symmetry ’t Hooft anomalies in the CFT are encoded by topological terms in the bulk effective action (often Chern–Simons terms for bulk gauge fields). Under a bulk gauge transformation, the variation of these terms reduces to a boundary term that reproduces the CFT anomaly. Similarly, Weyl anomalies are reflected in logarithmic divergences of the regulated bulk on-shell action and are captured systematically by holographic renormalization.

Radial direction and renormalization group

The holographic coordinate behaves like an energy scale:

• $z\to 0$ (near the boundary) corresponds to the UV of the CFT.
• increasing $z$ corresponds to flowing toward the IR.

A useful heuristic is

$$z\sim \frac{1}{\mu},$$

where $\mu$ is an energy scale in the boundary theory.

Introducing a radial cutoff $z\ge \epsilon$ is the bulk dual of a UV regulator. Removing the cutoff $\epsilon\to 0$ requires renormalization; in the bulk this is implemented by holographic renormalization (local counterterms at $z=\epsilon$) and yields finite, scheme-dependent correlators consistent with CFT Ward identities.

Field/operator map

The central entry of the dictionary is the field/operator correspondence:

• each (single-trace) primary operator $\mathcal{O}(x)$ in the CFT corresponds to a bulk field $\Phi(x,z)$,
• quantum numbers (spin, global charges) match on both sides,
• the near-boundary behavior of $\Phi$ encodes the source and expectation value of $\mathcal{O}$.

Near-boundary expansion: source vs vev

For a scalar bulk field $\Phi$ dual to a scalar operator $\mathcal{O}$ of dimension $\Delta$, one finds asymptotically

$$\Phi(x,z) = z^{d-\Delta}\bigl(\phi_0(x) + \cdots\bigr) + z^{\Delta}\bigl(\phi_1(x) + \cdots\bigr).$$

• $\phi_0(x)$ is the source that couples to $\mathcal{O}$ in the boundary action.
• $\phi_1(x)$ is proportional (after renormalization) to the vev $\langle\mathcal{O}(x)\rangle$.

The terminology “non-normalizable mode” (source) vs “normalizable mode” (state/response) is common, especially in the semiclassical bulk limit.

Mass–dimension relation and the BF bound

For a scalar in AdS$_{d+1}$, the bulk mass $m$ and boundary dimension $\Delta$ are related by

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

Stability requires the Breitenlohner–Freedman (BF) bound

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

If $-\tfrac{d^2}{4} < m^2L^2 < -\tfrac{d^2}{4}+1$, both $\Delta_+$ and $\Delta_-$ quantizations can be consistent (“alternate quantization”), with the choice corresponding to which mode is treated as the source.

Relevant, marginal, and irrelevant deformations

Perturbing a CFT by a local operator is written schematically as

$$S_{\mathrm{CFT}}\;\to\;S_{\mathrm{CFT}}+\int d^d x\, g\,\mathcal{O}(x).$$

If $\mathcal{O}$ has dimension $\Delta$, then the coupling $g$ has engineering dimension $[g]=d-\Delta$.

• Relevant: $\Delta<d$ (the deformation grows in the IR).
• Marginal: $\Delta=d$.
• Irrelevant: $\Delta>d$.

In the bulk, turning on $g$ corresponds to imposing a nonzero source $\phi_0\sim g$ for the dual field. Relevant deformations typically produce holographic RG-flow geometries (domain walls) supported by bulk scalar profiles.

Spin and conserved operators

Canonical examples:

• bulk gauge field $A_M$ $\leftrightarrow$ conserved current $J^\mu$ with $\Delta=d-1$,
• bulk metric $g_{MN}$ $\leftrightarrow$ stress tensor $T_{\mu\nu}$ with $\Delta=d$.

More generally, bulk fields of spin $s$ map to boundary operators of the same spin (in the appropriate representation of $SO(d,2)$), subject to conservation/shortening conditions.

Single-trace and multi-trace

In large-$N$ gauge theories, single-trace operators (schematically $\mathrm{Tr}(\cdots)$) typically map to single-particle bulk fields. Multi-trace deformations and operators correspond to multi-particle bulk states and, in many situations, to modified boundary conditions for the corresponding bulk fields.

A standard example is a double-trace deformation,

$$S_{\mathrm{CFT}}\;\to\;S_{\mathrm{CFT}}+\frac{f}{2}\int d^d x\,\mathcal{O}(x)^2.$$

At leading order in large $N$ this deformation is implemented in the bulk by mixed boundary conditions that relate the near-boundary coefficients (schematically, $\phi_1\propto f\,\phi_0$, with proportionality fixed by conventions). When alternate quantization is available, such deformations can drive flows between the $\Delta_-$ and $\Delta_+$ boundary conditions.

GKPW: partition functions, sources, and correlators

The Gubser–Klebanov–Polyakov–Witten (GKPW) relation identifies the bulk partition function with the CFT generating functional:

$$Z_{\mathrm{bulk}}[\{\phi_{0,i}\}] \,=\, Z_{\mathrm{CFT}}[\{\phi_{0,i}\}] \,=\, \left\langle \exp\left(\sum_i\int d^d x\,\phi_{0,i}(x)\,\mathcal{O}_i(x)\right)\right\rangle.$$

In the semiclassical bulk limit,

$$Z_{\mathrm{bulk}}[\{\phi_0\}]\approx \exp\bigl(-S_{\mathrm{ren}}[\phi_{\mathrm{cl}}(\phi_0)]\bigr),$$

where $S_{\mathrm{ren}}$ is the renormalized on-shell action evaluated on the classical solution with boundary data $\phi_0$.

One-point functions

Varying the renormalized on-shell action gives renormalized expectation values. For a scalar source $\phi_0$,

$$\langle \mathcal{O}(x)\rangle_{\phi_0} = \frac{1}{\sqrt{g^{(0)}}}\frac{\delta\log Z_{\mathrm{CFT}}}{\delta\phi_0(x)} = -\frac{1}{\sqrt{g^{(0)}}}\frac{\delta S_{\mathrm{ren}}}{\delta\phi_0(x)}.$$

Similarly, for the stress tensor and a conserved current,

$$\langle T^{\mu\nu}(x)\rangle = \frac{2}{\sqrt{g^{(0)}}}\frac{\delta\log Z_{\mathrm{CFT}}}{\delta g^{(0)}_{\mu\nu}(x)} = -\frac{2}{\sqrt{g^{(0)}}}\frac{\delta S_{\mathrm{ren}}}{\delta g^{(0)}_{\mu\nu}(x)},$$ $$\langle J^{\mu}(x)\rangle = \frac{1}{\sqrt{g^{(0)}}}\frac{\delta\log Z_{\mathrm{CFT}}}{\delta A^{(0)}_{\mu}(x)} = -\frac{1}{\sqrt{g^{(0)}}}\frac{\delta S_{\mathrm{ren}}}{\delta A^{(0)}_{\mu}(x)}.$$

$n$-point functions and Witten diagrams

Connected correlators are obtained by further functional differentiation with respect to sources. In the semiclassical bulk limit one often writes, for the connected part,

$$\langle \mathcal{O}_{i_1}(x_1)\cdots \mathcal{O}_{i_n}(x_n)\rangle_c = (-1)^n\frac{\delta^n S_{\mathrm{ren}}}{\delta \phi_{0,i_1}(x_1)\cdots \delta \phi_{0,i_n}(x_n)}\Bigg|_{\phi_0=0}.$$

At leading order in large $N$, these correlators are computed by tree-level Witten diagrams in the bulk; bulk loop diagrams correspond to $1/N$ corrections.

Lorentzian real-time correlators (brief)

In Lorentzian signature the dictionary remains valid, but extracting specific real-time Green’s functions requires additional input. For thermal states, retarded correlators are obtained by choosing infalling (regular) boundary conditions for bulk fields at the future horizon; other correlators correspond to different $i\epsilon$ prescriptions and time contours. The guiding principle is that the bulk boundary conditions implement the causal/analytic structure of the desired CFT correlator.

Holographic renormalization

Bulk actions evaluated on asymptotically AdS solutions diverge because the near-boundary volume is infinite. A standard procedure is:

1. introduce a cutoff surface $z=\epsilon$,
2. add local counterterms on that surface (built from induced fields and geometry),
3. take $\epsilon\to 0$ after combining bulk and counterterm contributions.

The result is a finite functional $S_{\mathrm{ren}}$ whose variations satisfy the expected CFT Ward identities. Logarithmic divergences encode conformal (Weyl) anomalies when present.

Parameter map and semiclassical limits

A convenient way to organize “when classical gravity works” is via two expansions:

• $1/N$ expansion: controls bulk quantum (loop) effects.
• $1/\lambda$ (or $\alpha'$) expansion: controls stringy higher-derivative effects.

In many holographic CFTs,

$$\frac{L^{d-1}}{G_{N}^{(d+1)}}\sim N^2,$$

so that large $N$ corresponds to small Newton coupling in AdS units.

When does classical gravity apply?

Not every CFT admits a simple weakly-curved AdS dual. A useful set of sufficient conditions (heuristic, but widely used) for an effective description by local Einstein gravity coupled to a small number of light fields is:

• Large $N$: suppresses bulk quantum loops ($G_N$ small in AdS units).
• Strong effective coupling (often large $\lambda$): suppresses stringy higher-derivative corrections ($\alpha'$ effects).
• A sparse low-dimension single-trace spectrum: a large gap between the stress tensor (and a few conserved currents) and the next single-trace operators. This corresponds to a separation between light supergravity fields and heavy string states.

When these conditions fail, the dictionary still exists, but the bulk description may require full string theory (many light fields, strong curvature) rather than classical Einstein gravity.

Canonical example: $\mathrm{AdS}_5\times S^5$ and $\mathcal{N}{=}4$ SYM

For type IIB string theory on $\mathrm{AdS}_5\times S^5$ dual to $\mathcal{N}{=}4$ $SU(N)$ super Yang–Mills,

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

Thus:

• large $N$ $\Rightarrow$ $g_s\ll 1$ (string loops suppressed),
• large $\lambda$ $\Rightarrow$ $L\gg \ell_s$ (curvature small in string units).

In this example the 4d central charges satisfy

$$a=c=\frac{\pi L^3}{8G_5}\approx \frac{N^2}{4}$$

at large $N$.

States, saddles, and thermodynamics

A useful dictionary entry is “state $\leftrightarrow$ geometry”:

• the CFT vacuum $\leftrightarrow$ pure AdS,
• a thermal state $\leftrightarrow$ an AdS black hole (or black brane),
• deformed CFTs / RG flows $\leftrightarrow$ bulk domain-wall geometries supported by scalars.

Operator/state correspondence

Placing the CFT on the cylinder $\mathbb{R}\times S^{d-1}$ (as suggested by global AdS) makes the operator/state correspondence explicit: a local primary operator $\mathcal{O}$ of scaling dimension $\Delta$ creates an energy eigenstate on $S^{d-1}$ with

$$E=\frac{\Delta}{R}$$

when the sphere has radius $R$ (equivalently, $E=\Delta$ on a unit-radius sphere). On the bulk side, this matches the fact that global AdS energy is quantized and the AdS Hamiltonian generates boundary time translations.

Temperature and free energy

In Euclidean signature, a thermal state corresponds to compact Euclidean time with period $\beta=1/T$. The free energy is

$$F=-T\log Z_{\mathrm{CFT}}\approx T\,I_E^{\text{on-shell}},$$

where $I_E$ is the renormalized Euclidean on-shell action of the dominant bulk saddle.

Black hole entropy

For a semiclassical bulk black hole with horizon $\mathcal{H}$,

$$S_{\mathrm{th}}=\frac{\mathrm{Area}(\mathcal{H})}{4G_N},$$

matching the thermal entropy of the dual CFT state.

Conserved charge and chemical potential

If the CFT has a conserved current $J^\mu$, one can study a grand-canonical ensemble with chemical potential $\mu$ for the associated charge. In the bulk, $\mu$ is the boundary value of the time component of the dual gauge field:

$$A_t^{(0)}=\mu.$$

At finite temperature and finite charge density, the dominant bulk saddles are typically charged AdS black holes/black branes (e.g. Reissner–Nordström–AdS), with bulk electric flux encoding the CFT charge density.

Nonlocal observables

Wilson loops

A Wilson loop along a closed contour $C$ is

$$W(C)=\frac{1}{N}\,\mathrm{Tr}\,\mathcal{P}\exp\left(i\oint_C A_\mu\,dx^\mu\right).$$

In holographic large-$N$ theories, fundamental Wilson loops map to fundamental strings whose worldsheet ends on $C$ at the AdS boundary. At strong coupling,

$$\langle W(C)\rangle\sim e^{-S_{\mathrm{string}}[C]},\qquad S_{\mathrm{NG}}=\frac{1}{2\pi\alpha'}\int d^2\sigma\,\sqrt{\det h},$$

where $S_{\mathrm{NG}}$ is the Nambu–Goto action of the minimal worldsheet with induced metric $h$.

Entanglement entropy: RT/HRT

For a spatial region $A$ in the boundary CFT, the Ryu–Takayanagi prescription (static case) states

$$S_A = \frac{\mathrm{Area}(\gamma_A)}{4G_N},\qquad \partial\gamma_A=\partial A.$$

In time-dependent settings, $\gamma_A$ is replaced by the Hubeny–Rangamani–Takayanagi (HRT) extremal surface.

Quantum corrections: generalized entropy and QES

Beyond classical gravity, entanglement entropy receives bulk quantum corrections. A compact modern statement is the “quantum extremal surface” (QES) formula:

$$S_A = \min_{X}\,\Bigl[\frac{\mathrm{Area}(X)}{4G_N}+S_{\mathrm{bulk}}(\Sigma_X)\Bigr], \qquad \partial X=\partial A\ \text{and}\ X\ \text{is extremal for }S_{\mathrm{gen}}.$$

where $X$ is a codimension-2 surface anchored on $\partial A$, $\Sigma_X$ is the bulk region between $A$ and $X$, and $S_{\mathrm{bulk}}$ is the bulk entanglement entropy across $X$. (Equivalently: extremize the generalized entropy $S_{\mathrm{gen}}$ over such surfaces and then choose the minimal saddle.)

Glossary

AdS radius $L$

The curvature scale of AdS. Curvatures scale as $R\sim -1/L^2$. Classical gravity typically requires $L$ large in Planck units.

Asymptotically AdS

A spacetime whose metric approaches AdS near the boundary up to a Weyl factor. The boundary data defines the sources of the dual CFT.

Boundary metric $g^{(0)}_{\mu\nu}$

The leading term in the near-boundary metric (in Fefferman–Graham form). Acts as the source for the CFT stress tensor.

Bulk

The $(d{+}1)$-dimensional gravitational (or string) spacetime. Local bulk fields encode nonlocal, gauge-invariant boundary data.

Bulk-to-boundary propagator

The solution kernel $K(z,x;x')$ that takes a boundary source at $x'$ to a bulk field profile. It is the basic building block of Witten diagrams.

Conformal dimension $\Delta$

The scaling weight of a CFT operator. For scalars, $m^2L^2=\Delta(\Delta-d)$ in AdS$_{d+1}$.

Conformal boundary

The boundary at infinity of AdS after removing an overall divergent Weyl factor. The CFT is defined on this boundary (more precisely, on its conformal class).

Breitenlohner–Freedman bound

The stability bound for scalar fields in AdS$_{d+1}$:

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

Above this bound, linearized scalar fluctuations do not grow exponentially in time.

Chemical potential $\mu$

In a grand-canonical ensemble, the source for a conserved charge. Holographically, it is identified with the boundary value of the time component of the dual bulk gauge field, $A_t^{(0)}=\mu$.

’t Hooft coupling $\lambda$

For a gauge theory with coupling $g_{\mathrm{YM}}$ and gauge group $SU(N)$, $\lambda=g_{\mathrm{YM}}^2N$. In many AdS/CFT examples, large $\lambda$ corresponds to small curvature in string units ($\alpha'$ corrections suppressed).

Quantum extremal surface (QES)

The codimension-2 surface $X$ that extremizes the generalized entropy

$$S_{\mathrm{gen}}(X)=\frac{\mathrm{Area}(X)}{4G_N}+S_{\mathrm{bulk}}(\Sigma_X).$$

Evaluating $S_{\mathrm{gen}}$ on the minimal (among extrema) QES computes boundary entanglement entropy beyond the classical RT/HRT limit.

GKPW prescription

The identification $Z_{\mathrm{bulk}}[\phi_0]=Z_{\mathrm{CFT}}[\phi_0]$, with $\phi_0$ both the boundary value of a bulk field and the source for the dual operator.

Holographic renormalization

The procedure that renders $S_{\text{on-shell}}$ finite by introducing a cutoff, adding local counterterms, and removing the cutoff. Defines renormalized correlators.

Large-$N$ limit

The limit in which $N\to\infty$ with ’t Hooft coupling fixed. Bulk quantum corrections are suppressed by powers of $1/N$.

Normalizable / non-normalizable modes

The two independent near-boundary falloffs of a bulk field. In standard quantization, the non-normalizable mode is the source and the normalizable mode encodes the state/vev.

Ryu–Takayanagi / HRT

Geometric prescriptions for entanglement entropy in holography via minimal (RT) or extremal (HRT) bulk surfaces.

Single-trace operator

In large-$N$ gauge theories, an operator built from a single trace, e.g. $\mathrm{Tr}(F^2)$. Typically dual to a single bulk field (single-particle mode).

Witten diagram

The AdS analogue of a Feynman diagram. Tree-level Witten diagrams compute leading large-$N$ correlators; bulk loops give $1/N$ corrections.