AdS/CFT Dictionary

The AdS/CFT correspondence—also called gauge/gravity duality or holography—is the most concrete realization of the idea that a quantum theory of gravity can be equivalent to a non-gravitational quantum field theory in one lower dimension. In its cleanest form, it relates a quantum gravity or string theory on an asymptotically Anti-de Sitter spacetime to a conformal field theory living on the conformal boundary.

The phrase AdS/CFT dictionary means the map between the two descriptions:

$$\text{bulk data} \quad \longleftrightarrow \quad \text{boundary data}.$$

Bulk fields map to boundary operators, boundary values of bulk fields map to sources, radial evolution maps to scale dependence, black holes map to thermal states, and geometric areas map to entropies. The exact statement is a quantum equivalence. Most formulas used in calculations are controlled limits of that equivalence, especially the large-$N$, large-gap, classical-gravity regime.

Use this page as a reference

This page is a compact dictionary, not a derivation. For derivations, see the Foundations pages on the statement of the correspondence, the GKP/Witten prescription, fields in AdS, and holographic renormalization.

Conventions and normalizations

The following conventions are used unless stated otherwise.

Symbol or convention	Meaning
$d$	boundary spacetime dimension
$d+1$	bulk spacetime dimension
$M,N=0,\ldots,d$	bulk indices
$\mu,\nu=0,\ldots,d-1$	boundary indices
$L$	AdS radius
$G_{d+1}$	Newton constant of the effective $(d+1)$-dimensional bulk theory
$z$	Poincaré radial coordinate, with boundary at $z\to0$
$r$	alternative radial coordinate, often with boundary at $r\to\infty$
$g^{(0)}_{\mu\nu}$	boundary metric representative; only its conformal class is physical in a CFT
$\phi_{(0)}$	source for the operator dual to a bulk field $\phi$
$\langle\mathcal O\rangle$	renormalized response or expectation value

Spacetime metrics are written in Lorentzian mostly-plus signature,

$$\eta_{\mu\nu}=\mathrm{diag}(-,+,\ldots,+),$$

unless a Euclidean metric is explicitly displayed. Partition functions and on-shell actions are often written in Euclidean signature, where

$$Z\sim e^{-I_E}.$$

The Wick rotation is schematically $t=-i\tau$, so $\eta_{\mu\nu}\to\delta_{\mu\nu}$ in flat space. Lorentzian real-time correlators require additional causal prescriptions; they are not obtained by blindly replacing $\tau$ by $it$.

For the gravitational normalization, the Lorentzian Einstein action is taken to be

$$S_{\mathrm{grav}} = \frac{1}{16\pi G_{d+1}} \int_M d^{d+1}x\sqrt{-g}\left(R+\frac{d(d-1)}{L^2}\right) + \frac{1}{8\pi G_{d+1}} \int_{\partial M} d^d x\sqrt{-\gamma}\,K +S_{\mathrm{ct}}+S_{\mathrm{matter}}.$$

Equivalently, the cosmological constant is

$$\Lambda=-\frac{d(d-1)}{2L^2},$$

and pure AdS obeys

$$R_{MN}=-\frac{d}{L^2}g_{MN}, \qquad R=-\frac{d(d+1)}{L^2}.$$

Here $\gamma_{\mu\nu}$ is the induced metric on a cutoff surface and $K$ is the trace of its extrinsic curvature. In Euclidean signature one usually works with the Euclidean action $I_E$, including the corresponding Gibbons-Hawking-York term and counterterms.

For QFT sources, this page uses

$$Z_{\mathrm{CFT}}[J] = \left\langle \exp\!\left(\int d^dx\sqrt{g_{(0)}}\,J(x)\mathcal O(x)\right) \right\rangle, \qquad W[J]=\log Z[J].$$

In the classical Euclidean gravity approximation,

$$W[J]\approx -I_{\mathrm{ren,on-shell}}[J].$$

Therefore

$$\langle\mathcal O(x)\rangle_J = \frac{1}{\sqrt{g_{(0)}}}\frac{\delta W}{\delta J(x)} = -\frac{1}{\sqrt{g_{(0)}}}\frac{\delta I_{\mathrm{ren,on-shell}}}{\delta J(x)}.$$

If one instead writes the Euclidean QFT deformation as $S_E\to S_E+\int J\mathcal O$, all source signs are reversed. This is one of the most common sign-convention traps in holographic calculations.

Quick reference

Bulk side	Boundary side	Careful statement
$\mathrm{AdS}_{d+1}$ isometries	conformal group	$SO(2,d)$ in Lorentzian signature
conformal boundary	spacetime of the CFT	metric defined up to Weyl rescaling
radial position $z$	RG or energy scale $\mu$	roughly $\mu\sim1/z$, with gauge/scheme caveats
bulk field $\phi$	local operator $\mathcal O$	usually single-trace at large $N$
leading boundary coefficient $\phi_{(0)}$	source for $\mathcal O$	fixed boundary condition in standard quantization
normalizable coefficient	vev or state data	precise vev from varying $I_{\mathrm{ren}}$
bulk metric $g_{MN}$	stress tensor $T_{\mu\nu}$	$g^{(0)}_{\mu\nu}$ sources $T^{\mu\nu}$
bulk gauge field $A_M$	conserved current $J^\mu$	$A^{(0)}_\mu$ sources $J^\mu$
boundary value $A^{(0)}_t$	chemical potential $\mu$	gauge-invariantly, $\mu=A_t(\infty)-A_t(r_h)$ in black-hole backgrounds
bulk gauge symmetry	global symmetry of CFT	boundary gauge field is normally a nondynamical source
bulk diffeomorphisms	stress-tensor Ward identities	radial constraints encode conservation and trace relations
classical saddle	leading large-$N$ physics	tree-level bulk computation
bulk loops	$1/N$ corrections	quantum gravity corrections
$\alpha'$ or higher-derivative terms	finite-coupling / finite-gap corrections	stringy corrections
AdS black hole or black brane	thermal state	horizon area gives entropy at leading order
infalling horizon condition	retarded correlator	Lorentzian causal prescription
quasinormal modes	poles of retarded correlators	thermal relaxation and response
string worldsheet ending on boundary loop	Wilson loop	$\langle W(C)\rangle\sim e^{-S_{\mathrm{string}}}$ at strong coupling
extremal codimension-2 surface	entanglement entropy	RT/HRT at leading classical order
quantum extremal surface	quantum-corrected entropy	generalized entropy extremization

Claim status: what is exact, what is approximate?

Level	Bulk statement	Boundary statement	Status
exact duality	full quantum string/M-theory with specified asymptotics	complete large-$N$ quantum theory with sources	exact in known top-down examples, though often not rigorously proven in a mathematical sense
semiclassical bulk	bulk path integral dominated by saddles	leading large-$N$ generating functional	controlled when $G_{d+1}/L^{d-1}\ll1$
classical string theory	string loops suppressed but string scale may matter	planar large-$N$ theory at finite coupling	requires $N\gg1$, not necessarily Einstein gravity
classical Einstein gravity	two-derivative gravity plus light fields	large-$N$, strong coupling, sparse spectrum / large gap	strongest computational regime
bottom-up holography	effective gravitational model chosen by symmetries and fields	phenomenological model of strongly coupled dynamics	useful but not automatically a UV-complete CFT dual

This distinction is essential. Classical gravity is not AdS/CFT itself; it is a powerful limit of the full correspondence.

Geometry and symmetry

AdS spacetime

A useful definition of Lorentzian $\mathrm{AdS}_{d+1}$ is the hyperboloid

$$-X_{-1}^2-X_0^2+X_1^2+\cdots+X_d^2=-L^2$$

embedded in $\mathbb R^{2,d}$. This makes the isometry group manifest:

$$\mathrm{Isom}(\mathrm{AdS}_{d+1})=SO(2,d),$$

matching the conformal group of a $d$-dimensional Lorentzian CFT.

In Lorentzian Poincaré coordinates,

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

The conformal boundary is at $z\to0$. The induced metric diverges, but after multiplying by $z^2/L^2$, one obtains the boundary representative $\eta_{\mu\nu}dx^\mu dx^\nu$.

In Euclidean Poincaré coordinates,

$$ds_E^2 = \frac{L^2}{z^2}\left(dz^2+\delta_{ij}dx^i dx^j\right).$$

Global Lorentzian AdS can be written as

$$ds^2 =L^2\left[-(1+r^2)dt^2+\frac{dr^2}{1+r^2}+r^2d\Omega_{d-1}^2\right], \qquad r\ge0.$$

Its boundary is conformal to

$$\mathbb R_t\times S^{d-1}.$$

This is why global AdS naturally describes the CFT on the cylinder.

Fefferman-Graham expansion

Near the boundary of an asymptotically locally AdS spacetime, one can often choose Fefferman-Graham coordinates:

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

The near-boundary expansion has the schematic form

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

The leading coefficient $g^{(0)}_{\mu\nu}$ is the boundary metric source for $T_{\mu\nu}$. The important point is that, for pure Einstein gravity, the coefficients below order $z^d$ are locally determined by $g^{(0)}_{\mu\nu}$ and other sources. The dynamical stress-tensor data first appears at order $z^d$.

For flat boundary metric and no additional sources, the standard result is

$$\langle T_{\mu\nu}\rangle = \frac{dL^{d-1}}{16\pi G_{d+1}}\,g^{(d)}_{\mu\nu}.$$

More generally,

$$\langle T_{\mu\nu}\rangle = \frac{dL^{d-1}}{16\pi G_{d+1}}\,g^{(d)}_{\mu\nu} +X_{\mu\nu}[g^{(0)},\text{sources}],$$

where $X_{\mu\nu}$ is a local, scheme-dependent functional determined by holographic renormalization. For even boundary dimension $d$, logarithmic terms encode the Weyl anomaly.

Do not read $g^{(2)}$ as the stress tensor

Except in special low-dimensional cases, $g^{(2)}_{\mu\nu}$ is not the stress-tensor vev. It is locally fixed by the boundary geometry. The nonlocal, state-dependent stress-tensor data is contained in $g^{(d)}_{\mu\nu}$, up to local anomaly and counterterm contributions.

Symmetry dictionary

The kinematic match is

$$SO(2,d)_{\mathrm{AdS}} \cong \mathrm{Conf}(\mathbb R^{1,d-1}).$$

Additional bulk gauge fields encode boundary global symmetries. For a bulk gauge field $A_M$,

$$A^{(0)}_\mu \quad\longleftrightarrow\quad \text{source for a conserved current }J^\mu.$$

The gauge redundancy in the bulk is not a global symmetry acting on physical bulk states. Rather, it is the bulk description of a global symmetry of the boundary theory. If the boundary gauge field is made dynamical, the boundary theory itself has been changed.

Anomalies and topological terms

Boundary anomalies are encoded in bulk terms whose gauge or Weyl variation reduces to a boundary contribution. Examples include:

• Chern-Simons terms for global-symmetry anomalies,
• logarithmic counterterms for Weyl anomalies,
• mixed gauge-gravitational Chern-Simons terms for mixed anomalies.

Thus anomaly matching is one of the cleanest checks of the dictionary.

Radial direction and renormalization group

In Poincaré AdS, the boundary is at $z=0$. Near the boundary,

$$\text{small }z \quad\longleftrightarrow\quad \text{UV physics},$$

while moving deeper into the bulk corresponds roughly to flowing toward the IR:

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

A radial cutoff $z=\epsilon$ acts as a UV regulator in the boundary theory. Removing the cutoff requires adding local counterterms on the cutoff surface. This is holographic renormalization.

The slogan

$$\text{radial evolution}\sim\text{RG flow}$$

is powerful but not literal in every gauge. A local statement about the radial coordinate can depend on the choice of bulk coordinates. The precise, gauge-invariant content is encoded in asymptotic expansions, radial Hamilton-Jacobi equations, Ward identities, and boundary observables.

Field/operator map

The central entry of the dictionary is:

$$\boxed{\text{bulk field }\phi \quad\longleftrightarrow\quad \text{boundary operator }\mathcal O.}$$

For a single-trace primary operator $\mathcal O$, the dual bulk field has matching spin and global quantum numbers. In the large-$N$ regime, single-trace operators behave like single-particle bulk fields; multi-trace operators behave like multi-particle states or modified boundary conditions.

Scalar field: source and vev

For a scalar field of mass $m$ in $\mathrm{AdS}_{d+1}$,

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

In standard quantization, $\Delta=\Delta_+$. The near-boundary expansion is best written as

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

The first independent coefficient is the source:

$$\phi_{(0)}(x) \quad\longleftrightarrow\quad \text{source for }\mathcal O(x).$$

The second independent coefficient is the response. With the scalar Euclidean bulk action normalized as

$$I_\phi = \frac{\mathcal N_\phi}{2} \int d^{d+1}x\sqrt{g}\left[(\nabla\phi)^2+m^2\phi^2\right],$$

the renormalized one-point function has the standard form

$$\boxed{ \langle\mathcal O(x)\rangle = \mathcal N_\phi(2\Delta-d)\,\phi_{(2\Delta-d)}(x) +\langle\mathcal O(x)\rangle_{\mathrm{local}}. }$$

The local term is fixed by counterterms and vanishes in many simple flat-boundary examples with no sources. The safest definition is always variational:

$$\langle\mathcal O(x)\rangle = -\frac{1}{\sqrt{g_{(0)}}} \frac{\delta I_{\mathrm{ren}}}{\delta\phi_{(0)}(x)}.$$

Logarithmic terms

When the two radial exponents differ by an integer, especially when

$$\Delta-\frac d2\in\mathbb Z_{\ge0},$$

the expansion can contain logarithmic terms such as

$$z^\Delta\log z\,\psi_{(2\Delta-d)}(x).$$

These terms are not optional decoration: they are tied to anomalies, scheme dependence, and local counterterms. They appear frequently for stress tensors, conserved currents, and marginal operators.

BF bound and alternate quantization

Stability in AdS requires the Breitenlohner-Freedman bound

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

When

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

both falloffs can be normalizable. One may then choose alternate quantization, in which the roles of source and response are exchanged. This corresponds to the operator dimension $\Delta_-$. Double-trace deformations can interpolate between the alternate and standard quantizations.

Spin and dimension table

Bulk field	Boundary operator	Dimension relation
scalar $\phi$	scalar primary $\mathcal O$	$m^2L^2=\Delta(\Delta-d)$
massless vector $A_M$	conserved current $J^\mu$	$\Delta=d-1$
massive vector $A_M$	nonconserved vector operator	$m^2L^2=(\Delta-1)(\Delta+1-d)$
Dirac fermion $\psi$	spinor operator $\mathcal O_\psi$	$\Delta_+=\frac d2+
massless graviton $h_{MN}$	stress tensor $T_{\mu\nu}$	$\Delta=d$
$p$-form field	$p$-form current/operator	$m^2L^2=(\Delta-p)(\Delta+p-d)$

For fermions, the first-order Dirac equation means that one fixes only half of the boundary spinor components as sources. Alternate quantization is possible in an appropriate mass window, commonly $|mL|<1/2$.

Relevant, marginal, and irrelevant deformations

Perturbing the boundary theory by

$$S_{\mathrm{CFT}}\to S_{\mathrm{CFT}}+\int d^dx\sqrt{g_{(0)}}\,g\,\mathcal O$$

gives the dimension statement below. With the source convention used earlier, this deformation corresponds to $J=-g$ in Euclidean signature; the sign does not affect the scaling classification.

$$[g]=d-\Delta.$$

Thus:

Type	Condition	Bulk interpretation
relevant	$\Delta<d$	non-normalizable scalar source grows toward the IR; often produces domain-wall RG flow
marginal	$\Delta=d$	massless scalar or modulus; logarithms may appear quantum mechanically
irrelevant	$\Delta>d$	source dominates near the boundary; UV completion must be handled carefully

Single-trace and multi-trace operators

In a large-$N$ gauge theory,

$$\mathcal O_{\mathrm{single}}\sim\mathrm{Tr}(\cdots)$$

usually maps to a single-particle bulk field. Products such as

$$\mathcal O^2, \qquad \mathcal O_1\mathcal O_2,$$

map to multi-particle states or to changes in boundary conditions.

A double-trace deformation

$$S_{\mathrm{CFT}}\to S_{\mathrm{CFT}}+\frac f2\int d^dx\sqrt{g_{(0)}}\,\mathcal O^2$$

is implemented, at leading large $N$, by mixed boundary conditions relating source and response. As with single-trace sources, signs depend on whether the deformation is written in the action or in the exponent of the generating functional. Schematically,

$$\phi_{\mathrm{response}}\sim f\,\phi_{\mathrm{source}},$$

with the exact proportionality fixed by normalization and quantization convention.

GKPW prescription

The Gubser-Klebanov-Polyakov/Witten prescription is the operational heart of the dictionary:

$$\boxed{ Z_{\mathrm{CFT}}[\text{sources}] = Z_{\mathrm{bulk}}[\text{asymptotic boundary data}]. }$$

For a scalar operator,

$$Z_{\mathrm{CFT}}[\phi_{(0)}] = Z_{\mathrm{bulk}}\!\left[\phi(z,x)\sim z^{d-\Delta}\phi_{(0)}(x)\right].$$

At leading order in the classical Euclidean saddle approximation,

$$Z_{\mathrm{bulk}}[\phi_{(0)}] \approx \exp\left[-I_{\mathrm{ren,on-shell}}[\phi_{(0)}]\right],$$

so

$$W_{\mathrm{CFT}}[\phi_{(0)}] \equiv \log Z_{\mathrm{CFT}}[\phi_{(0)}] \approx -I_{\mathrm{ren,on-shell}}[\phi_{(0)}].$$

Connected correlators are obtained by differentiating $W$:

$$\langle\mathcal O(x_1)\cdots\mathcal O(x_n)\rangle_c = \frac{1}{\sqrt{g_{(0)}(x_1)}\cdots\sqrt{g_{(0)}(x_n)}} \frac{\delta^n W}{\delta\phi_{(0)}(x_1)\cdots\delta\phi_{(0)}(x_n)}\bigg|_{\phi_{(0)}=0}.$$

With $W=-I_{\mathrm{ren}}$, this is

$$\langle\mathcal O(x_1)\cdots\mathcal O(x_n)\rangle_c = - \frac{1}{\sqrt{g_{(0)}(x_1)}\cdots\sqrt{g_{(0)}(x_n)}} \frac{\delta^n I_{\mathrm{ren}}}{\delta\phi_{(0)}(x_1)\cdots\delta\phi_{(0)}(x_n)}\bigg|_{\phi_{(0)}=0},$$

up to contact terms, operator mixing, and source-dependent disconnected pieces. The sign does not alternate as $(-1)^n$ in this source convention.

One-point functions of standard sources

Using covariant boundary metric variation,

$$\delta W = \int d^dx\sqrt{g_{(0)}}\left( \frac12\langle T^{\mu\nu}\rangle\delta g^{(0)}_{\mu\nu} +\langle J^\mu\rangle\delta A^{(0)}_\mu +\langle\mathcal O\rangle\delta\phi_{(0)} \right).$$

Thus, in the classical gravity approximation,

$$\langle\mathcal O\rangle =-\frac{1}{\sqrt{g_{(0)}}}\frac{\delta I_{\mathrm{ren}}}{\delta\phi_{(0)}} ,$$ $$\langle J^\mu\rangle =-\frac{1}{\sqrt{g_{(0)}}}\frac{\delta I_{\mathrm{ren}}}{\delta A^{(0)}_\mu} ,$$ $$\langle T^{\mu\nu}\rangle =-\frac{2}{\sqrt{g_{(0)}}}\frac{\delta I_{\mathrm{ren}}}{\delta g^{(0)}_{\mu\nu}} .$$

If one varies with respect to $g^{\mu\nu}$ rather than $g_{\mu\nu}$, the displayed stress-tensor sign changes in the usual way.

Lorentzian real-time prescription

Euclidean regularity computes Euclidean correlators. Lorentzian correlators require a choice of state and contour.

The retarded Green function is

$$G^R_{\mathcal O\mathcal O}(t,\mathbf x) = -i\theta(t)\langle[\mathcal O(t,\mathbf x),\mathcal O(0)]\rangle.$$

In a black-hole or black-brane background, the holographic prescription for $G^R$ imposes infalling boundary conditions at the future horizon. Poles of the resulting retarded correlator are quasinormal modes of the bulk perturbation.

Different correlators require different prescriptions:

Boundary correlator	Bulk condition
Euclidean vacuum correlator	regularity in Euclidean AdS
thermal Euclidean correlator	smoothness on Euclidean black-hole cigar
retarded correlator	infalling at future horizon
advanced correlator	outgoing at future horizon
Schwinger-Keldysh correlators	real-time contour with doubled fields/saddles

Holographic renormalization

On-shell AdS actions diverge because the boundary is at infinite proper distance. Holographic renormalization gives finite variational data.

The standard procedure is:

1. introduce a cutoff surface $z=\epsilon$;
2. evaluate the bulk action plus boundary terms on $z\ge\epsilon$;
3. add local counterterms on the cutoff surface;
4. take $\epsilon\to0$;
5. vary the resulting finite functional with respect to sources.

Symbolically,

$$I_{\mathrm{ren}} = \lim_{\epsilon\to0}\left( I_{\mathrm{bulk}}^{z\ge\epsilon} +I_{\mathrm{GHY}}^{z=\epsilon} +I_{\mathrm{ct}}^{z=\epsilon} \right).$$

Counterterms are local functionals of the induced fields at $z=\epsilon$. Finite local counterterms change contact terms and scheme-dependent one-point functions, but they do not change separated-point nonlocal correlators.

Ward identities

Bulk constraints imply boundary Ward identities. In the presence of scalar sources $\phi_{(0)}^I$ and background gauge fields $A^{(0)}_\mu$, the schematic identities are

$$\nabla_\mu\langle J^\mu_a\rangle =\mathcal A_a,$$ $$\nabla_\mu\langle T^{\mu}{}_{\nu}\rangle = F^{(0)a}_{\nu\mu}\langle J^\mu_a\rangle +\sum_I \langle\mathcal O_I\rangle\nabla_\nu\phi^I_{(0)} +\mathcal A_\nu,$$

and

$$\langle T^\mu{}_{\mu}\rangle = \mathcal A_{\mathrm{Weyl}} +\sum_I (d-\Delta_I)\phi^I_{(0)}\langle\mathcal O_I\rangle +\cdots .$$

The anomaly terms vanish in simple flat examples without anomalous sources, but they are essential in curved backgrounds and even boundary dimensions.

Parameter maps and regimes of validity

The two most important expansions are:

$$\frac{G_{d+1}}{L^{d-1}}\ll1 \qquad\text{and}\qquad \frac{\ell_s}{L}\ll1.$$

The first suppresses bulk quantum loops. The second suppresses stringy higher-derivative corrections.

In a large-$N$ holographic CFT, one usually has

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

for matrix-like adjoint degrees of freedom. In vector-like models the scaling can differ.

A simple Einstein-gravity dual typically requires:

• large $N$ or large central charge,
• a sparse low-dimension single-trace spectrum,
• a large gap to higher-spin/stringy operators,
• a consistent set of boundary conditions and counterterms,
• a controlled state or ensemble.

Large $N$ alone is not enough. A weakly curved local Einstein bulk is a special corner of holography.

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

The original and best-studied example is

$$\text{type IIB string theory on }\mathrm{AdS}_5\times S^5 \quad\longleftrightarrow\quad \mathcal N=4\;SU(N)\;\text{super-Yang-Mills in four dimensions}.$$

With common conventions,

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

The regimes are:

Boundary regime	Bulk regime
finite $N$, finite $\lambda$	full quantum string theory
$N\gg1$, finite $\lambda$	classical string theory, generally stringy
$N\gg1$, $\lambda\gg1$	classical type IIB supergravity
$1/N$ corrections	bulk quantum loops
$1/\lambda$ corrections	$\alpha'$ corrections

The central charges are

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

at large $N$. In the exact $SU(N)$ gauge theory, $a=c=(N^2-1)/4$.

AdS$_3$/CFT$_2$ and Brown-Henneaux

Three-dimensional AdS gravity is special because asymptotic symmetries enhance to two copies of the Virasoro algebra. For Einstein gravity on $\mathrm{AdS}_3$,

$$\boxed{ c=\frac{3L}{2G_3} }$$

is the Brown-Henneaux central charge.

For a rotating BTZ black hole with mass $M$ and angular momentum $J$, the CFT weights obey

$$h-\frac{c}{24}=\frac12(ML+J), \qquad \bar h-\frac{c}{24}=\frac12(ML-J).$$

For $J=0$,

$$\Delta-\frac{c}{12}=ML, \qquad \Delta=h+\bar h.$$

The Cardy formula then reproduces the BTZ entropy,

$$S_{\mathrm{BTZ}} =2\pi\left[ \sqrt{\frac{c}{6}\left(h-\frac{c}{24}\right)} + \sqrt{\frac{c}{6}\left(\bar h-\frac{c}{24}\right)} \right] =\frac{A}{4G_3}.$$

Other standard top-down examples

Bulk background	Boundary theory	Comment
type IIB on $\mathrm{AdS}_5\times S^5$	4d $\mathcal N=4$ SYM	canonical example
M-theory on $\mathrm{AdS}_4\times S^7$ or quotients	3d M2-brane CFTs, including ABJM variants	important for AdS$_4$/CFT$_3$
M-theory on $\mathrm{AdS}_7\times S^4$	6d $(2,0)$ theory	no ordinary Lagrangian description in general
type IIB on $\mathrm{AdS}_3\times S^3\times M_4$	2d D1-D5 CFT	black-hole microstates, AdS$_3$/CFT$_2$
less symmetric compactifications	quiver, defect, flavor, or RG-flow theories	often more realistic but harder to control

States, saddles, and thermodynamics

A state or ensemble in the CFT is represented by a choice of bulk state, geometry, or saddle. Common entries are:

Boundary object	Bulk object
CFT vacuum on $\mathbb R^{1,d-1}$	Poincaré AdS
CFT vacuum on $\mathbb R\times S^{d-1}$	global AdS
thermal state	AdS black hole or black brane
finite chemical potential	charged AdS black hole / bulk electric flux
relevant deformation	scalar profile and domain-wall geometry
confining-like phase	geometry caps off or has a mass gap
heavy operator/state	massive particle, black hole, or backreacted geometry depending on $\Delta$

Operator-state correspondence

A primary operator of dimension $\Delta$ creates an energy eigenstate on the cylinder:

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

for a boundary sphere of radius $R$. On the bulk side, this is global AdS energy.

Thermal physics

A thermal CFT partition function is

$$Z(\beta)=\mathrm{Tr}\,e^{-\beta H}.$$

In Euclidean gravity, the dominant saddle has a Euclidean time circle of period

$$\beta=\frac1T.$$

At leading classical order,

$$F=-T\log Z\approx T I_{E,\mathrm{ren,on-shell}}.$$

For an Einstein-gravity black hole,

$$S_{\mathrm{thermal}}=\frac{\mathrm{Area}(\mathcal H)}{4G_{d+1}}.$$

Higher-derivative gravity replaces area by Wald entropy or its appropriate generalization.

Finite density

A conserved current $J^\mu$ allows a chemical potential $\mu$ for charge $Q$. In the bulk,

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

in a simple gauge. In a black-hole background the gauge-invariant statement is

$$\mu=A_t(\infty)-A_t(r_h),$$

and regularity often sets $A_t(r_h)=0$ in Euclidean signature. Charge density is extracted from the radial electric flux, equivalently from

$$\langle J^t\rangle =-\frac{1}{\sqrt{g_{(0)}}}\frac{\delta I_{\mathrm{ren}}}{\delta A^{(0)}_t}.$$

Nonlocal and extended observables

Wilson loops

In a gauge theory, a Wilson loop is schematically

$$W(C)=\frac1N\mathrm{Tr}\,\mathcal P\exp\left(i\oint_C A_\mu dx^\mu\right).$$

In $\mathcal N=4$ SYM, the supersymmetric Wilson loop also couples to scalars. Holographically, a fundamental Wilson loop maps to a fundamental string worldsheet ending on the contour $C$ at the AdS boundary:

$$\langle W(C)\rangle \sim \exp[-S_{\mathrm{string}}(C)].$$

At strong coupling the leading term is often the Nambu-Goto action

$$S_{\mathrm{NG}} =\frac{1}{2\pi\alpha'}\int d^2\sigma\sqrt{\det h_{ab}}.$$

Higher representations can be described by D-branes in appropriate regimes.

Probe branes and flavor

Fields in the adjoint representation are naturally part of the closed-string/gravity sector. Fundamental matter is often introduced by adding flavor branes. In the probe limit,

$$N_f\ll N,$$

flavor branes propagate on a fixed background and compute meson spectra, flavor currents, conductivities, and defect observables. Backreaction becomes important when $N_f/N$ is not negligible.

Entanglement entropy: RT, HRT, and QES

For a boundary spatial region $A$, the static Ryu-Takayanagi formula is

$$S_A = \frac{\mathrm{Area}(\gamma_A)}{4G_{d+1}}, \qquad \partial\gamma_A=\partial A,$$

where $\gamma_A$ is the minimal bulk codimension-2 surface homologous to $A$.

For time-dependent states, the Hubeny-Rangamani-Takayanagi prescription replaces the minimal surface by an extremal surface:

$$S_A = \frac{\mathrm{Area}(X_A)}{4G_{d+1}}.$$

At the next order in bulk quantum corrections, one uses generalized entropy:

$$S_{\mathrm{gen}}(X) = \frac{\mathrm{Area}(X)}{4G_{d+1}} +S_{\mathrm{bulk}}(\Sigma_X)+\cdots .$$

The quantum extremal surface prescription is

$$S_A = \min_{X}\,\mathrm{ext}_{X}\,S_{\mathrm{gen}}(X), \qquad \partial X=\partial A.$$

In black-hole evaporation and related setups, the same generalized-entropy logic leads to island formulas.

Applied holography and bottom-up models

A bottom-up holographic model begins with an effective bulk action chosen to capture symmetries, conserved quantities, relevant operators, and desired IR behavior. Examples include Einstein-Maxwell theory, Einstein-Maxwell-dilaton models, axion models for momentum relaxation, and Einstein-Maxwell-scalar models for holographic superconductors.

Bottom-up models can be powerful, but they should be read with the following labels:

Label	Meaning
top-down	derived from a known string/M-theory construction or consistent truncation
consistent truncation	every solution of the lower-dimensional theory uplifts to a solution of the higher-dimensional theory
effective model	designed to describe a controlled sector or phenomenon, not necessarily a complete UV dual
phenomenological model	useful for mechanisms and scaling, but not automatically an exact dual of a known QFT

A useful bottom-up model should state:

• field content and symmetries,
• boundary conditions and ensemble,
• UV asymptotics,
• IR geometry or horizon behavior,
• counterterms and variational principle,
• regime of validity,
• which observables are robust and which are model-dependent.

Common pitfalls

“The boundary is a wall inside AdS.”

No. The boundary is a conformal boundary at infinity. It is not a material shell at finite proper distance.

“The radial direction is literally the RG scale.”

Only approximately and with caveats. The relation is sharpest near the boundary and in gauge-invariant observables. Local radial statements can be coordinate dependent.

“Classical gravity is the full duality.”

No. Classical gravity is a limit. The full duality involves quantum gravity or string/M-theory.

“Every CFT has a simple Einstein dual.”

No. A simple local Einstein bulk requires special large-$N$ and spectral-gap properties.

“The source is the value of the field at a finite cutoff.”

Not quite. At finite $z=\epsilon$, the field contains both source and response. The source is defined by the coefficient in the asymptotic expansion after holographic renormalization.

“Normalizable means vev, always and immediately.”

Normalizable modes often encode state or response data, but the renormalized one-point function is defined by varying $I_{\mathrm{ren}}$. Local terms, anomalies, alternate quantization, and operator mixing can modify the naive coefficient.

“Euclidean and Lorentzian correlators are the same calculation.”

No. Euclidean regularity and Lorentzian infalling boundary conditions compute different analytic objects, though they can be related by analytic continuation in controlled situations.

“The $z^2$ Fefferman-Graham coefficient is the stress tensor.”

Generally false. In Einstein gravity the stress tensor is encoded in $g^{(d)}_{\mu\nu}$ plus local terms, not generically in $g^{(2)}_{\mu\nu}$.

“A bottom-up action guarantees a healthy CFT.”

No. A bulk effective action is not automatically UV complete. It may still be useful as a controlled phenomenological model.

Glossary

AdS radius $L$

The curvature scale of AdS. Curvatures scale as $R\sim -1/L^2$.

Alternate quantization

A choice of boundary condition, available in certain mass windows, in which the usual source and response roles are exchanged.

Asymptotically AdS

A spacetime whose metric approaches AdS near the conformal boundary, allowing a boundary conformal structure and holographic source data.

BF bound

The stability bound for scalars in $\mathrm{AdS}_{d+1}$:

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

Brown-York tensor

The quasi-local stress tensor obtained by varying the renormalized gravitational action with respect to the boundary metric. After counterterms, it gives the holographic CFT stress tensor.

Bulk-to-boundary propagator

The kernel that builds a bulk solution from a boundary source. It is the basic ingredient for tree-level Witten diagrams.

Chemical potential

The source for a conserved charge. Holographically it is the boundary value of a bulk gauge potential, modulo gauge choices and horizon regularity.

Conformal boundary

The boundary obtained after stripping off the divergent AdS conformal factor. The CFT lives on this conformal class, not on a unique metric.

Double-trace deformation

A deformation by an operator such as $\mathcal O^2$. At leading large $N$, it is represented by mixed boundary conditions for the dual bulk field.

Fefferman-Graham coordinates

A near-boundary gauge in which

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

Useful for holographic renormalization.

GKPW prescription

The equality of the CFT generating functional and the bulk partition function with matching boundary data:

$$Z_{\mathrm{CFT}}[\phi_{(0)}]=Z_{\mathrm{bulk}}[\phi\to\phi_{(0)}].$$

Holographic renormalization

The procedure of adding local boundary counterterms to obtain a finite on-shell action and finite one-point functions.

Large-$N$ factorization

The property that connected correlators of suitably normalized single-trace operators are suppressed at large $N$. This is the CFT origin of weakly interacting bulk fields.

Normalizable and non-normalizable modes

Near-boundary falloffs of bulk fields. In standard quantization, the leading non-normalizable coefficient is the source and the normalizable coefficient is related to the response or state.

Quasinormal mode

A bulk fluctuation satisfying source-free boundary behavior and infalling horizon behavior. In holography, quasinormal frequencies are poles of retarded correlators.

RT/HRT surface

The bulk codimension-2 minimal or extremal surface used to compute boundary entanglement entropy at leading classical order.

Single-trace operator

An operator such as $\mathrm{Tr}(F^2)$ in a large-$N$ gauge theory. It usually maps to a single-particle bulk field.

Witten diagram

An AdS perturbation-theory diagram. Tree-level Witten diagrams compute leading large-$N$ CFT correlators; bulk loops compute $1/N$ corrections.

Further reading

Original and foundational papers

• J. Maldacena, The Large $N$ Limit of Superconformal Field Theories and Supergravity.
• 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.

Holographic renormalization and correlators

• S. de Haro, K. Skenderis, and S. N. Solodukhin, Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.
• K. Skenderis, Lecture Notes on Holographic Renormalization.
• D. T. Son and A. O. Starinets, Minkowski-Space Correlators in AdS/CFT Correspondence.

Standard textbooks and lecture notes

• M. Ammon and J. Erdmenger, Gauge/Gravity Duality: Foundations and Applications.
• M. Natsuume, AdS/CFT Duality User Guide, arXiv:1409.3575.
• H. Nastase, Introduction to the AdS/CFT Correspondence.
• J. Polchinski, Introduction to Gauge/Gravity Duality, arXiv:1010.6134.
• H. Liu, String Theory and Holographic Duality, MIT OpenCourseWare lecture notes.

Entanglement, black holes, and quantum information

• 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.
• T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum Corrections to Holographic Entanglement Entropy.
• N. Engelhardt and A. C. Wall, Quantum Extremal Surfaces.
• M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, arXiv:1609.01287.

Applications and quantum matter

• S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Building a Holographic Superconductor.
• S. A. Hartnoll, Lectures on Holographic Methods for Condensed Matter Physics.
• J. McGreevy, Holographic Duality with a View Toward Many-Body Physics.
• S. A. Hartnoll, A. Lucas, and S. Sachdev, Holographic Quantum Matter.