How to Compute a Holographic Observable

The main idea

A holographic calculation is not a spell of the form “write down a black hole and read off the answer.” It is a controlled boundary-value problem whose output is a CFT observable.

The exact statement is a statement about quantum theories:

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

Almost every practical calculation in this course uses a semiclassical limit:

$$Z_{\rm string}[J] \simeq \exp\!\left(-S_{\rm bulk}^{\rm ren}[\Phi_{\rm cl};J]\right),$$

or its Lorentzian real-time version. The job is then to solve a bulk variational problem with the right asymptotic boundary conditions, the right interior or horizon conditions, the right counterterms, and the right normalization. The observable is obtained by differentiating, extremizing, or evaluating the renormalized bulk answer.

A good holographic calculation therefore has the following skeleton:

$$\boxed{ \text{CFT observable} \longrightarrow \text{bulk dual object} \longrightarrow \text{boundary value problem} \longrightarrow \text{renormalized on-shell quantity} \longrightarrow \text{CFT answer} }$$

The skeleton is simple. The craft lies in the details.

A practical workflow for holographic calculations. The central line is the computation; the side boxes are the checks that prevent many wrong answers. The same logic applies to local correlators, thermal free energies, Wilson loops, transport coefficients, probe-brane observables, and RT/HRT entanglement entropy.

This page is deliberately procedural. It is the page to consult when you know what you want to compute but are unsure how to turn the dictionary into a reliable calculation.

Step 1: identify the boundary observable

Start on the field-theory side. Do not start by guessing a metric.

Ask what the actual observable is:

CFT object	Typical notation	Bulk object
partition function	$Z[J]$	string/gravity partition function with boundary sources
one-point function	$\langle \mathcal O\rangle_J$	normalizable coefficient or radial canonical momentum
Euclidean correlator	$\langle \mathcal O(x)\mathcal O(0)\rangle$	Euclidean boundary-value problem
retarded correlator	$G_R(\omega,k)$	Lorentzian problem with infalling horizon condition
current response	$\sigma(\omega)$, $\chi$, $D$	Maxwell perturbations and radial electric flux
stress-tensor response	$\eta$, sound modes	metric perturbations and Brown-York tensor
Wilson loop	$\langle W(C)\rangle$	fundamental string worldsheet ending on $C$
entanglement entropy	$S_A$	extremal surface or quantum extremal surface
defect observable	defect one-point data, meson spectra	probe-brane embedding and fluctuations
thermal free energy	$F(T,\mu)$	renormalized Euclidean action of a saddle

The precise definition matters. For example, the words “conductivity” could mean optical conductivity at zero momentum,

$$\sigma(\omega) = \frac{1}{i\omega}G^R_{J_xJ_x}(\omega,k=0),$$

DC conductivity,

$$\sigma_{\rm DC} = \lim_{\omega\to 0}\sigma(\omega),$$

or an incoherent conductivity with the momentum-overlap contribution removed. These are different observables, and in a translationally invariant finite-density system the distinction is not cosmetic: the ordinary DC conductivity contains a delta function because momentum cannot decay.

Similarly, the phrase “entropy” could mean thermal entropy, coarse-grained black-hole entropy, fine-grained entanglement entropy, generalized entropy, or the entropy density of a black brane. The right bulk calculation depends on which one you mean.

Step 2: specify the state, ensemble, and sources

A CFT observable is not just an operator. It is an operator evaluated in a state or ensemble, possibly with sources turned on.

Common choices are:

boundary situation	bulk saddle or condition
vacuum on $\mathbb R^{1,d-1}$	Poincaré AdS
vacuum on $\mathbb R\times S^{d-1}$	global AdS
thermal state on $\mathbb R^{d-1}$	planar AdS black brane
thermal state on $S^{d-1}$	thermal AdS or global AdS-Schwarzschild
finite chemical potential	charged AdS black hole or black brane
relevant deformation	scalar source and backreacted domain wall
expectation-value flow	normalizable scalar profile with vanishing source
defect CFT	probe brane or interface geometry
real-time response	Lorentzian geometry with causal boundary conditions

For a single-trace scalar operator, a source deformation has the schematic form

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

The bulk scalar near the boundary behaves as

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

where $\phi_{(0)}$ is proportional to the source in standard quantization. A common mistake is to compute a beautiful bulk solution and only afterward ask whether the non-normalizable coefficient was actually zero. That changes the CFT problem.

For a conserved current,

$$A_\mu(z,x) = A^{(0)}_\mu(x)+\cdots,$$

and $A^{(0)}_\mu$ sources $J^\mu$. A chemical potential is boundary data for $A_t$, but it is also a choice of ensemble and gauge. For a static charged black hole one usually imposes regularity at the Euclidean horizon, which sets the contractible thermal-cycle holonomy in the correct gauge.

For the stress tensor,

$$g_{\mu\nu}(z,x) \sim \frac{L^2}{z^2} \left(g^{(0)}_{\mu\nu}(x)+\cdots\right),$$

and $g^{(0)}_{\mu\nu}$ is the background metric of the CFT. Varying it gives $\langle T^{\mu\nu}\rangle$ and stress-tensor correlators.

Step 3: choose the bulk dual and approximation

The dictionary is not only a list of fields. It is also a statement about approximation schemes.

For the canonical duality,

$$\mathcal N=4\ {\rm SYM}_{4} \quad \longleftrightarrow \quad \text{type IIB string theory on } \mathrm{AdS}_5\times S^5,$$

the main parameters are

$$\lambda=g_{\rm YM}^2N, \qquad \frac{L^4}{\alpha'^2}=\lambda, \qquad g_s\sim \frac{\lambda}{N}, \qquad \frac{L^3}{G_5}\sim N^2.$$

Thus the hierarchy is:

approximation	field-theory meaning	bulk meaning
exact AdS/CFT	exact finite-$N$ CFT	full quantum string theory
large $N$	factorization, planar expansion	suppressed bulk loops
large $\lambda$	strong coupling, large gap	suppressed stringy $\alpha'$ corrections
classical supergravity	$N\gg 1$ and $\lambda\gg 1$	two-derivative gravity plus light fields
probe limit	$N_f/N\to 0$ or small backreaction	brane or matter fields on fixed geometry
derivative expansion	long wavelength	hydrodynamics or fluid/gravity
semiclassical worldsheet	$\sqrt\lambda\gg 1$	saddle-point string worldsheet

A calculation should state its approximation explicitly. The answer

$$\frac{\eta}{s}=\frac{1}{4\pi}$$

is not a theorem about every quantum field theory. It is the leading classical two-derivative Einstein-gravity answer. Higher-derivative terms, finite coupling, anisotropy, nonminimal matter couplings, or broken assumptions can change it.

Likewise, the RT formula

$$S_A=\frac{\mathrm{Area}(\gamma_A)}{4G_N}$$

is the leading large-$N$ classical answer. At the next order one must use generalized entropy,

$$S_A = \underset{X\sim A}{\mathrm{ext}}\left[ \frac{\mathrm{Area}(X)}{4G_N} +S_{\rm bulk}(\Sigma_X) +\cdots \right].$$

The “bulk dual” of an observable is therefore a pair:

$$\boxed{ \text{bulk object} \;+\; \text{regime of validity}. }$$

Step 4: formulate the variational problem

For local operators, the basic object is the renormalized on-shell action. Consider a bulk field $\Phi$ with action

$$S[\Phi] = \int_{\mathcal M} d^{d+1}x\,\mathcal L(\Phi,\partial\Phi) +S_{\rm bdy}.$$

On shell, the variation reduces to boundary terms:

$$\delta S_{\rm os} = \int_{\partial\mathcal M} d^d x\, \Pi\,\delta \Phi + \text{terms from metric or gauge variations},$$

where $\Pi$ is the radial canonical momentum. After adding counterterms and taking the cutoff to the boundary, the renormalized momentum gives the one-point function.

For a scalar in standard quantization,

$$\langle \mathcal O(x)\rangle_J = \frac{1}{\sqrt{|g_{(0)}|}} \frac{\delta S_{\rm ren}}{\delta \phi_{(0)}(x)},$$

up to the sign convention chosen in the Euclidean generating functional. The sign is not universal because authors define

$$Z[J]=\left\langle e^{+\int J\mathcal O}\right\rangle, \qquad \text{or} \qquad Z[J]=\left\langle e^{-\int J\mathcal O}\right\rangle.$$

The physics is independent of this bookkeeping, but mixing conventions halfway through a calculation is poison.

For gauge fields and metrics, the corresponding definitions are usually written as

$$\langle J^\mu\rangle = \frac{1}{\sqrt{|g_{(0)}|}} \frac{\delta S_{\rm ren}}{\delta A^{(0)}_\mu},$$

and

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

again with sign conventions depending on Euclidean versus Lorentzian definitions.

For nonlocal observables, the variational problem changes:

observable	variational problem
Wilson loop $W(C)$	minimize or extremize the string action with worldsheet boundary $C$
baryon vertex	wrapped brane plus attached strings, with worldvolume Gauss law
defect free energy	probe-brane DBI/WZ action plus counterterms
RT/HRT entropy	extremize area subject to anchoring and homology
QES entropy	extremize generalized entropy, not just area
quasinormal spectrum	solve homogeneous linearized equations with source-free boundary data and infalling horizon behavior

The unifying principle is not “differentiate the action” but solve the correct variational problem with the correct boundary conditions.

Step 5: impose boundary and interior conditions

Boundary conditions are part of the observable. They are not an afterthought.

Asymptotic AdS boundary

Near a Fefferman-Graham boundary,

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

the cutoff surface $z=\epsilon$ regulates the gravitational IR divergence, which is the CFT UV divergence. Sources are specified by leading coefficients at this boundary. Vevs are extracted from subleading coefficients after renormalization.

Euclidean interiors

In Euclidean black-hole geometries, regularity at the origin of the thermal cigar fixes the period of Euclidean time. For a nonextremal horizon,

$$ds^2 \simeq \rho^2 \kappa^2 d\tau^2+d\rho^2+\text{transverse directions},$$

smoothness requires

$$\tau\sim \tau+\frac{2\pi}{\kappa}, \qquad T=\frac{\kappa}{2\pi}.$$

Gauge fields must also be regular on the contractible thermal circle. This is why one often chooses a gauge in which $A_\tau$ vanishes at the Euclidean horizon.

Lorentzian horizons

For retarded correlators at finite temperature, impose infalling boundary conditions at the future horizon. Near the horizon,

$$\phi(r)\sim (r-r_h)^{-i\omega/(4\pi T)}$$

in Schwarzschild-like coordinates. Equivalently, the solution should be regular in ingoing Eddington-Finkelstein coordinates. The outgoing solution gives the wrong causal Green function.

Normalizable modes

For spectra and quasinormal modes, one usually imposes vanishing source at the boundary. In the simplest scalar case, this means setting the non-normalizable coefficient to zero. A nontrivial solution then exists only for special frequencies or masses.

This is the logic behind normal modes in global AdS, glueball spectra in hard-wall models, meson spectra on probe branes, and quasinormal-mode poles of thermal retarded correlators.

Step 6: solve the bulk equations

There are four common solution regimes.

Analytic exact solutions

Some backgrounds and probes are exactly solvable: pure AdS scalar wave equations, planar black-brane thermodynamics, geodesics in AdS$_3$, simple RT surfaces, and near-boundary expansions.

Exact examples are valuable because they expose normalization and boundary-condition issues without numerical noise.

Perturbation theory

Many useful calculations are perturbative. Examples include:

• small source expansions around AdS,
• linear response around a black brane,
• hydrodynamic expansions in $\omega/T$ and $k/T$,
• near-boundary recursive Fefferman-Graham expansions,
• probe-brane fluctuations around a classical embedding,
• $1/\Delta$ geodesic corrections for heavy operators.

For a fluctuation $\varphi$ around a background,

$$\Phi=\Phi_{\rm bg}+\varepsilon\varphi+O(\varepsilon^2),$$

the quadratic on-shell action gives two-point functions, while cubic and quartic terms give higher-point functions through Witten diagrams.

Matched asymptotics

Finite-density AdS$_2$ throat calculations often use matched asymptotic expansions. One solves the near-horizon IR problem, solves the UV problem, and matches them in an overlap region. The resulting Green function often has the structure

$$G_R(\omega,k) = \frac{b_+(k)+b_-(k)\mathcal G_{\rm IR}(\omega,k)} {a_+(k)+a_-(k)\mathcal G_{\rm IR}(\omega,k)},$$

where $\mathcal G_{\rm IR}$ is controlled by the AdS$_2$ scaling exponent. This expression is not magic; it is just linear ODE matching.

Numerical boundary-value problems

For backreacted RG flows, inhomogeneous lattices, holographic superconductors, nontrivial probe embeddings, and many real-time problems, numerical methods are unavoidable. The same workflow still applies:

$$\text{choose ansatz} \to \text{fix gauge} \to \text{impose boundary expansions} \to \text{solve} \to \text{verify constraints and convergence}.$$

For static gravitational boundary-value problems, the Einstein-DeTurck method is often useful. It replaces the Einstein equation by a gauge-fixed elliptic problem and then requires checking that the DeTurck vector vanishes in the solution.

Step 7: renormalize

The bare on-shell action almost always diverges near the AdS boundary. Holographic renormalization removes these divergences by adding local covariant counterterms on the cutoff surface.

The regulated action has the form

$$S_{\rm reg}[\epsilon] = S_{\rm bulk}\big|_{z\ge \epsilon} +S_{\rm GH}\big|_{z=\epsilon} +S_{\rm other\ bdy}\big|_{z=\epsilon}.$$

The renormalized action is

$$S_{\rm ren} = \lim_{\epsilon\to 0} \left(S_{\rm reg}[\epsilon]+S_{\rm ct}[\epsilon]\right).$$

The counterterm action is local in the induced fields at the cutoff surface:

$$S_{\rm ct} = \int_{z=\epsilon} d^d x\sqrt{|\gamma|}\, \left( \frac{c_0}{L} +c_1 L R[\gamma] +c_\phi \phi^2 +\cdots \right).$$

The ellipsis includes curvature terms, matter terms, logarithmic counterterms in even boundary dimension, finite scheme-dependent terms, and terms required for a well-posed variational principle.

Two lessons are crucial.

First, counterterms are not optional decorations. Without them, functional derivatives of the action are usually divergent or scheme-confused.

Second, counterterms determine contact terms and scheme-dependent local pieces. They do not change separated-point nonlocal correlators, pole locations, entropy density, or other genuinely nonlocal data, except when finite counterterms represent physical choices of scheme or ensemble.

For a scalar, after renormalization one often finds schematically

$$\langle \mathcal O\rangle = (2\Delta-d)\phi_{(2\Delta-d)} + \text{local terms in sources},$$

where the coefficient depends on normalization of the bulk scalar action. The phrase “up to local terms” is not a license to ignore contact terms; it means one should state what data are scheme-independent.

Step 8: extract the observable

Different observables require different extraction rules.

One-point functions

For a scalar source,

$$\langle \mathcal O\rangle_J = \frac{1}{\sqrt{|g_{(0)}|}} \frac{\delta S_{\rm ren}}{\delta \phi_{(0)}}.$$

For currents and stress tensors,

$$\langle J^\mu\rangle = \frac{1}{\sqrt{|g_{(0)}|}} \frac{\delta S_{\rm ren}}{\delta A^{(0)}_\mu}, \qquad \langle T^{\mu\nu}\rangle = \frac{2}{\sqrt{|g_{(0)}|}} \frac{\delta S_{\rm ren}}{\delta g^{(0)}_{\mu\nu}}.$$

Euclidean correlators

Connected correlators come from differentiating

$$W[J]=\log Z[J] \simeq -S_{\rm E}^{\rm ren}[J]$$

in Euclidean signature, modulo the chosen source-sign convention.

A scalar two-point function is obtained by solving the linearized equation with source $\phi_{(0)}$, substituting into the quadratic on-shell action, and differentiating twice:

$$G_E(x,y) = \frac{\delta^2 W}{\delta \phi_{(0)}(x)\delta \phi_{(0)}(y)}.$$

Retarded correlators

For a linearized Lorentzian fluctuation $\varphi(r;\omega,k)$, the retarded Green function is read from the ratio of renormalized response to source, with infalling horizon conditions:

$$G_R(\omega,k) = \frac{\Pi_{\rm ren}(\omega,k)}{\varphi_{(0)}(\omega,k)} \bigg|_{\rm infalling},$$

more precisely after accounting for mixing, indices, counterterms, and normalization. For coupled fields this ratio becomes a matrix.

Free energy and thermodynamics

For a Euclidean saddle,

$$Z(\beta,\mu) \simeq \exp\!\left(-I_E^{\rm ren}\right), \qquad \Omega(T,\mu)=T I_E^{\rm ren}.$$

Then

$$S=-\left(\frac{\partial \Omega}{\partial T}\right)_\mu, \qquad Q=-\left(\frac{\partial \Omega}{\partial \mu}\right)_T,$$

in the grand-canonical ensemble. In the canonical ensemble, one performs the appropriate Legendre transform.

Wilson loops

At leading large $\lambda$,

$$\langle W(C)\rangle \sim \exp\!\left(-S_{\rm NG}^{\rm ren}[\Sigma_C]\right),$$

where $\partial\Sigma_C=C$ at the boundary. For a rectangular loop of time length $\mathcal T$ and spatial separation $R$,

$$\langle W(\mathcal T,R)\rangle \sim \exp[-\mathcal T V(R)], \qquad \mathcal T\to\infty.$$

The straight-string self-energy subtraction is part of the renormalization of the heavy external quarks.

Entanglement entropy

At leading classical order,

$$S_A = \frac{\mathrm{Area}(\gamma_A)}{4G_N},$$

where $\gamma_A$ is anchored on $\partial A$ and satisfies the homology condition. At quantum order,

$$S_A = \underset{X\sim A}{\mathrm{ext}}\left[ \frac{\mathrm{Area}(X)}{4G_N}+S_{\rm bulk}(\Sigma_X)+\cdots \right].$$

The symbol $\cdots$ is doing real work: it includes higher-derivative entropy functionals and renormalization subtleties.

Step 9: take limits in the right order

Holographic observables often depend on noncommuting limits. Order matters.

Examples:

problem	dangerous limits
DC conductivity	$\omega\to 0$ versus $k\to 0$; momentum conservation versus relaxation
hydrodynamic poles	$\omega,k\ll T$ with $\omega/k$ fixed or not fixed
extremal black holes	$T\to 0$ versus $\omega\to 0$
quasinormal modes	ingoing horizon condition before source-free boundary condition
entanglement transitions	large $N$ saddle competition before $1/N$ smoothing
Wilson loops	$\mathcal T\to\infty$ before extracting $V(R)$
Euclidean-to-Lorentzian continuation	analytic continuation after choosing the correct thermal contour
probe limit	$N_f/N\to0$ before backreaction effects are interpreted

A compact way to state the rule is:

$$\boxed{ \text{The observable defines the limits, not the other way around.} }$$

For instance, the diffusion constant is defined from a hydrodynamic pole,

$$\omega(k)=-iDk^2+O(k^4),$$

not from an arbitrary independent small-$\omega$, small-$k$ limit. The shear viscosity is defined by

$$\eta = -\lim_{\omega\to0}\frac{1}{\omega}\operatorname{Im}G^R_{T^{xy}T^{xy}}(\omega,\mathbf 0),$$

where the spatial momentum is set to zero before the frequency limit in the usual Kubo formula.

Step 10: validate the answer

A holographic result should be checked from several angles.

Symmetry checks

Does the answer transform correctly under conformal symmetry, gauge symmetry, rotations, translations, parity, time reversal, or supersymmetry? A scalar two-point function in a CFT must scale as

$$\langle \mathcal O(x)\mathcal O(0)\rangle \propto \frac{1}{|x|^{2\Delta}}.$$

A stress tensor one-point function in a homogeneous thermal CFT must obey the conformal equation of state,

$$\epsilon=(d-1)p,$$

unless sources or anomalies break conformal invariance.

Ward identities

Gauge and diffeomorphism constraints in the bulk become Ward identities on the boundary. With no explicit symmetry-breaking source,

$$\nabla_\mu \langle J^\mu\rangle=0, \qquad \nabla_\mu \langle T^{\mu\nu}\rangle=0.$$

With sources, the Ward identities acquire source terms. For a scalar source $J$,

$$\nabla_\mu \langle T^{\mu\nu}\rangle = \langle \mathcal O\rangle\nabla^\nu J +F^{\nu}{}_{\mu}\langle J^\mu\rangle +\cdots.$$

The trace Ward identity has the schematic form

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

where $\mathcal A$ is the conformal anomaly when present.

Thermodynamic checks

For a black brane, verify the first law,

$$d\epsilon=T ds+\mu d\rho,$$

and the correct relation between pressure and grand potential,

$$p=-\Omega/V.$$

In a conformal plasma, verify scaling with temperature. For example, in $d$ boundary dimensions,

$$s\propto T^{d-1}, \qquad \epsilon\propto T^d.$$

Causality and positivity checks

Retarded Green functions must be analytic in the upper half of the complex $\omega$ plane. Spectral densities should satisfy positivity conditions for physical operators, such as

$$\rho(\omega,\mathbf k) = -2\operatorname{Im}G_R(\omega,\mathbf k)$$

with the appropriate sign convention and support properties. Higher-derivative models must be checked for causality, ghosts, and consistency with known CFT positivity constraints.

Dimensional checks

If the only scale is $T$, dimensional analysis is powerful. A conductivity in a $d$-dimensional CFT has dimension $d-3$. A viscosity has dimension $d-1$. A charge density has dimension $d-1$. A scalar one-point function has dimension $\Delta$.

Many wrong formulas fail this test before any physics is needed.

Limit checks

Does the answer reduce to a known result in a controlled limit?

Examples:

• Pure AdS correlators should reproduce conformal two- and three-point structures.
• Black-brane thermodynamics should reproduce area-law entropy.
• Hydrodynamic correlators should have diffusion or sound poles.
• The Wilson-loop potential in a conformal theory should scale as $1/R$.
• RT surfaces for intervals in AdS$_3$ should reproduce $S=(c/3)\log(\ell/\epsilon)$.

Good holographic practice is not merely solving equations. It is solving equations and then trying hard to disprove your result.

Worked template A: scalar one-point function in a deformed CFT

Suppose a CFT is deformed by a scalar source $J(x)$ for a single-trace scalar operator $\mathcal O$.

Boundary problem

Choose a bulk scalar with mass

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

and impose near-boundary behavior

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

If $J$ is constant and the deformation backreacts significantly, solve the coupled Einstein-scalar system. If $J$ is infinitesimal, solve the scalar equation on the undeformed background.

Renormalization

Add counterterms, including a scalar term of the schematic form

$$S_{\rm ct}\supset \frac{d-\Delta}{2L} \int_{z=\epsilon} d^d x\sqrt{|\gamma|}\,\phi^2 +\cdots.$$

The precise coefficient and additional terms depend on conventions, derivative terms, and resonances.

Extraction

Read

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

up to the normalization of the bulk scalar kinetic term.

Checks

Verify the trace Ward identity,

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

and make sure any claimed spontaneous vev has $J=0$.

Worked template B: retarded correlator and conductivity

Suppose you want the optical conductivity of a homogeneous thermal state.

Boundary observable

At zero spatial momentum,

$$\sigma(\omega) = \frac{1}{i\omega}G^R_{J_xJ_x}(\omega,0).$$

Bulk problem

Use a Maxwell fluctuation

$$A_x(r,t)=a_x(r)e^{-i\omega t}$$

on the relevant black-brane background. The quadratic Maxwell action has radial canonical momentum

$$\Pi^x(r,\omega) = -\sqrt{-g}\,Z(\phi)F^{rx},$$

where $Z(\phi)$ is a possible gauge kinetic function.

Boundary conditions

Impose infalling behavior at the horizon and source normalization at the boundary:

$$a_x(r)\to a_x^{(0)}+\cdots.$$

Extraction

The retarded Green function is

$$G_R(\omega) = \lim_{r\to\infty} \frac{\Pi^x_{\rm ren}(r,\omega)} {a_x(r,\omega)}.$$

Then use the Kubo formula for $\sigma(\omega)$.

Checks

At finite charge density, ask whether $a_x$ mixes with metric perturbations. If translations are unbroken, expect a delta function in $\operatorname{Re}\sigma(\omega)$ from momentum conservation. If you obtain a finite ordinary DC conductivity without momentum relaxation, something is probably missing.

Worked template C: a holographic Wilson loop

Suppose you want the potential between a heavy external quark and antiquark separated by $R$.

Boundary observable

Use a rectangular Wilson loop $C$ with temporal extent $\mathcal T$:

$$\langle W(C)\rangle \sim \exp[-\mathcal T V(R)], \qquad \mathcal T\to\infty.$$

Bulk object

Find a classical string worldsheet ending on the rectangle at the boundary. The leading action is

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

Renormalization

Subtract the self-energy of two isolated straight strings, or equivalently add the appropriate boundary counterterm for the open string.

Extraction

Compute

$$V(R) = \lim_{\mathcal T\to\infty} \frac{S_{\rm NG}^{\rm ren}}{\mathcal T}.$$

Checks

In pure AdS$_5$, conformal invariance requires

$$V(R)\propto -\frac{\sqrt\lambda}{R}.$$

In a confining geometry, the large-$R$ behavior can become

$$V(R)\sim T_{\rm QCD}R,$$

if the string-frame geometry satisfies the confinement criterion.

Worked template D: entanglement entropy

Suppose you want the entanglement entropy of a spatial region $A$ in a large-$N$ holographic CFT.

Boundary observable

Define

$$S_A=-\operatorname{Tr}\rho_A\log\rho_A, \qquad \rho_A=\operatorname{Tr}_{\bar A}\rho.$$

Bulk problem

At leading classical order, find a codimension-two extremal surface $\gamma_A$ such that

$$\partial \gamma_A=\partial A$$

and $\gamma_A$ is homologous to $A$.

Extraction

Compute

$$S_A=\frac{\mathrm{Area}(\gamma_A)}{4G_N}.$$

At quantum order, replace area by generalized entropy and extremize over quantum extremal surfaces.

Checks

The answer should satisfy entropy inequalities at leading classical order under the appropriate conditions. In AdS$_3$/CFT$_2$, for an interval of length $\ell$ in vacuum,

$$S_A=\frac{c}{3}\log\frac{\ell}{\epsilon}.$$

If you miss the homology constraint, you will get wrong answers for thermal states and black holes.

A compact dictionary of extraction rules

task	solve	impose	extract
scalar one-point	nonlinear or linear scalar equation	source $\phi_{(0)}$ and regular IR	$\delta S_{\rm ren}/\delta\phi_{(0)}$
Euclidean two-point	linear Euclidean fluctuation	prescribed boundary source, regular interior	second derivative of $S_E^{\rm ren}$
retarded two-point	linear Lorentzian fluctuation	source at boundary, infalling at horizon	response/source ratio
quasinormal modes	homogeneous fluctuation equation	source-free boundary, infalling horizon	discrete complex frequencies
conductivity	Maxwell or coupled Maxwell-metric perturbations	electric source, infalling horizon	$\sigma=G_R/(i\omega)$
stress tensor	asymptotic metric expansion	fixed boundary metric	Brown-York tensor plus counterterms
thermal entropy	black-hole saddle	smooth Euclidean cigar	area law or $-\partial F/\partial T$
Wilson loop	string worldsheet	boundary contour $C$	$e^{-S_{\rm NG}^{\rm ren}}$
meson spectrum	probe-brane fluctuation	normalizable boundary data, regular brane interior	normal-mode eigenvalues
entanglement entropy	extremal surface	anchoring and homology	area or generalized entropy

Common mistakes

Mistake 1: confusing source and vev

For a scalar, the leading and subleading coefficients have different interpretations. In standard quantization,

$$\phi_{(0)}\quad \text{is the source}, \qquad \phi_{(2\Delta-d)}\quad \text{is related to the vev}.$$

A solution with a nonzero non-normalizable mode is not a spontaneous state. It is a deformed theory.

Mistake 2: using Euclidean regularity for a retarded correlator

Euclidean regularity computes Euclidean correlators. Retarded correlators require Lorentzian causal boundary conditions, usually infalling behavior at a future horizon.

Mistake 3: omitting counterterms

The divergent pieces are not harmless. They affect the variational principle, one-point functions, Ward identities, and contact terms.

Mistake 4: forgetting operator mixing

At finite density or in symmetry-broken phases, Maxwell, scalar, and metric perturbations can mix. Reading a Green function from a single decoupled-looking equation may miss constraints or contact terms.

Mistake 5: treating a bottom-up model as a top-down theorem

A phenomenological action can be useful, but its predictions are only as reliable as the assumptions built into the model: field content, potentials, gauge kinetic functions, IR boundary conditions, and higher-derivative terms.

Mistake 6: ignoring ensemble dependence

Fixing $A_t^{(0)}$ computes the grand-canonical ensemble. Fixing charge density requires a Legendre transform or different boundary term. The same bulk solution can represent different thermodynamic ensembles depending on boundary conditions.

Mistake 7: trusting a numerical solution without residual checks

For numerical gravity, check constraints, convergence, boundary falloffs, regularity, gauge conditions, and thermodynamic identities. A smooth-looking plot is not a solution.

A research-level checklist

Before presenting a holographic computation, be able to answer the following questions.

Observable

What exact CFT quantity is being computed? Is it Euclidean, Lorentzian, thermal, entanglement, transport, a spectrum, or a nonlocal observable?

Dictionary

Which bulk field, brane, surface, or geometry is dual to it? What is the normalization of the source and the operator?

State

What boundary state or ensemble is being used? Vacuum, thermal, finite density, deformed, defect, time-dependent, or mixed?

Approximation

Which limits are assumed? Large $N$, large $\lambda$, probe limit, derivative expansion, small source, classical gravity, or linear response?

Boundary conditions

What data are fixed at the AdS boundary? What regularity or causal condition is imposed in the interior?

Renormalization

Which counterterms are included? What terms are scheme-dependent? Are finite counterterms physically relevant?

Extraction

Is the observable a derivative of the on-shell action, a response/source ratio, an extremal area, a worldsheet action, a pole, or an eigenvalue?

Checks

Which Ward identities, thermodynamic identities, positivity constraints, scaling laws, and limiting cases are satisfied?

This checklist is not bureaucratic. It is how one prevents “holographic numerology,” where a bulk model produces a number but the boundary meaning is unclear.

Exercises

Exercise 1: classify the observable

For each boundary question below, identify the bulk object and the extraction rule.

1. What is $\langle \mathcal O\rangle$ in a CFT deformed by $J\mathcal O$?
2. What is the retarded current correlator $G^R_{J_xJ_x}(\omega,0)$ in a thermal state?
3. What is the heavy quark-antiquark potential $V(R)$?
4. What is the entropy of a region $A$ in a large-$N$ vacuum state?

Solution

1. Use the bulk scalar dual to $\mathcal O$. Fix the non-normalizable coefficient $\phi_{(0)}=J$, solve with regular interior conditions, renormalize the on-shell action, and compute $\delta S_{\rm ren}/\delta\phi_{(0)}$.

2. Use the bulk gauge field dual to $J^\mu$. Turn on a linearized fluctuation $A_x(r)e^{-i\omega t}$, impose an infalling horizon condition, extract the renormalized radial momentum over the boundary source, and read $G_R$.

3. Use a classical fundamental string worldsheet ending on a rectangular Wilson loop. Compute $S_{\rm NG}^{\rm ren}$ and take $V(R)=\lim_{\mathcal T\to\infty}S_{\rm NG}^{\rm ren}/\mathcal T$.

4. Use the RT/HRT prescription. Find the extremal surface anchored on $\partial A$ and homologous to $A$, then compute $S_A=\mathrm{Area}/(4G_N)$ at leading classical order.

Exercise 2: source or vev?

A scalar in AdS$_{d+1}$ has near-boundary expansion

$$\phi(z)=z^{d-\Delta}a+z^\Delta b+\cdots, \qquad \Delta>\frac d2.$$

In standard quantization, which coefficient corresponds to the source? What condition should be imposed for a spontaneous expectation value with no explicit source?

Solution

In standard quantization, the coefficient $a$ of the non-normalizable mode is the source, up to normalization. The coefficient $b$ is related to the expectation value, again up to normalization and possible local terms.

A spontaneous expectation value with no explicit source requires

$$a=0, \qquad b\ne 0.$$

One must also check regularity in the interior and, in a backreacted solution, the gravitational constraints and Ward identities.

Exercise 3: why the horizon condition matters

Near a nonextremal black-brane horizon, a scalar fluctuation behaves as

$$\phi(r)\sim A(r-r_h)^{-i\omega/(4\pi T)} + B(r-r_h)^{+i\omega/(4\pi T)}.$$

Which coefficient should be set to zero for a retarded Green function in Schwarzschild-like coordinates?

Solution

For a retarded Green function, impose infalling boundary conditions at the future horizon. With the convention shown, the infalling mode is

$$(r-r_h)^{-i\omega/(4\pi T)},$$

so one sets

$$B=0.$$

Equivalently, the solution should be regular in ingoing Eddington-Finkelstein coordinates. The outgoing mode corresponds to the wrong causal prescription.

Exercise 4: conductivity and momentum conservation

A translationally invariant finite-density CFT has charge density $\rho\ne0$ and momentum density $P^i$. Explain why the ordinary DC conductivity is infinite even if the system is strongly interacting.

Solution

An electric field accelerates the charge density. If translations are exact, momentum cannot decay. Since the electric current generally overlaps with momentum at finite charge density, the current has a component protected from relaxation. This produces a delta function in $\operatorname{Re}\sigma(\omega)$ and a pole in $\operatorname{Im}\sigma(\omega)$.

To obtain finite ordinary DC conductivity, one must relax momentum, for example by lattices, disorder, axions, Q-lattices, or explicit translation-breaking boundary conditions. Alternatively, one can study an incoherent conductivity in a current orthogonal to momentum.

Exercise 5: finite counterterms

Suppose a scalar two-point function in momentum space contains

$$G(k)=A k^{2\Delta-d}+B k^2+C.$$

Assume $Bk^2+C$ are analytic local terms allowed by finite counterterms. Which part is scheme-independent at separated points?

Solution

The nonanalytic term

$$A k^{2\Delta-d}$$

is the scheme-independent separated-point contribution, assuming $2\Delta-d$ is not an even integer that produces logarithmic subtleties. The analytic polynomial terms $Bk^2+C$ Fourier transform to contact terms and derivatives of contact terms. They can be shifted by finite local counterterms.

Exercise 6: design a computation

You want to compute the shear viscosity of a strongly coupled plasma dual to a two-derivative Einstein black brane. Write the minimal sequence of steps.

Solution

A minimal sequence is:

1. Identify the Kubo formula

$$\eta=-\lim_{\omega\to0}\frac{1}{\omega}\operatorname{Im}G^R_{T^{xy}T^{xy}}(\omega,0).$$

2. Turn on the metric fluctuation $h^x{}_y(r)e^{-i\omega t}$ around the black-brane background.

3. Expand the Einstein-Hilbert action to quadratic order. For two-derivative Einstein gravity, this transverse graviton behaves like a minimally coupled massless scalar.

4. Impose infalling boundary conditions at the horizon and fix the boundary source $h^x{}_y{}^{(0)}$.

5. Evaluate the renormalized canonical momentum and compute the retarded Green function.

6. Take the Kubo limit and divide by the entropy density $s=A/(4G_N V)$.

7. For Einstein gravity, obtain

$$\frac{\eta}{s}=\frac{1}{4\pi}.$$

One should state that this result assumes classical two-derivative Einstein gravity and can receive higher-derivative or finite-coupling corrections.

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.
• V. E. Hubeny, M. Rangamani, and T. Takayanagi, A Covariant Holographic Entanglement Entropy Proposal.