Common Normalizations and Conventions

Why this page exists

Many wrong holographic answers are not conceptually wrong. They are off by a sign, a factor of $2$, a factor of $L$, a factor of $2\pi$, or a hidden choice of source convention.

That sounds harmless until one compares a viscosity, a charge susceptibility, a stress tensor, a central charge, or a thermodynamic potential across papers. Then the gremlins come out wearing little $\kappa_{d+1}^2$ hats.

This page is a normalization manual for the rest of the course. It does not try to impose the only possible convention. Instead, it fixes a coherent set of conventions and explains how to translate when a paper uses another one.

The guiding rule is simple:

A holographic result is not fully specified until the action, boundary term, source normalization, operator normalization, Fourier convention, and variational convention have all been stated.

Throughout this page, the bulk dimension is $d+1$ and the boundary dimension is $d$. The AdS radius is $L$. We usually set

$$\hbar = c = k_B = 1,$$

but we keep $L$, $G_{d+1}$, and gauge couplings visible when they carry useful information.

Basic spacetime and index conventions

Our default Lorentzian boundary metric has mostly-plus signature,

$$\eta_{\mu\nu} = \mathrm{diag}(-1,+1,\ldots,+1),$$

with Greek indices $\mu,\nu=0,1,\ldots,d-1$ and bulk Latin indices $a,b=0,1,\ldots,d$.

The Poincare AdS$_{d+1}$ metric is

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

In an $r$ coordinate, the same metric is often written as

$$ds^2 = \frac{r^2}{L^2}\eta_{\mu\nu}dx^\mu dx^\nu + \frac{L^2}{r^2}dr^2, \qquad r\to \infty \quad \text{at the boundary},$$

with

$$z = \frac{L^2}{r}.$$

When $L=1$, this becomes the familiar $z=1/r$.

For pure AdS$_{d+1}$ we use

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

The Einstein equation following from the action below is

$$R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=8\pi G_{d+1}T_{ab}^{\mathrm{matter}}.$$

Radial-coordinate translation table

Quantity	$z$ coordinate	$r$ coordinate
Boundary	$z\to 0$	$r\to\infty$
UV cutoff	$z=\epsilon$	$r=R=L^2/\epsilon$
Boundary scale	$\mu_{\mathrm{RG}}\sim 1/z$	$\mu_{\mathrm{RG}}\sim r/L^2$
Scalar source mode	$z^{d-\Delta}J(x)$	$r^{\Delta-d}L^{2(d-\Delta)}J(x)$
Scalar response mode	$z^\Delta A(x)$	$r^{-\Delta}L^{2\Delta}A(x)$
Black-brane horizon	$z=z_h$	$r=r_h=L^2/z_h$

Do not mix expansions in $r$ and $z$ without transforming the powers. A common error is to say that the source mode is $r^{d-\Delta}$; with the above $r$ coordinate it is $r^{\Delta-d}$.

Gravitational action conventions

Our Lorentzian Einstein-Hilbert action is

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

Here $\gamma_{ij}$ is the induced metric at the regulated boundary, $K_{ij}$ is the extrinsic curvature, and $K=\gamma^{ij}K_{ij}$. The counterterm action $S_{\mathrm{ct}}$ is local in the induced boundary fields and removes divergences as the cutoff is taken to the boundary.

The Euclidean action is related by $S_L=iI_E$ after Wick rotation, but in practice one should write it independently:

$$I_{\mathrm{grav}} = -\frac{1}{16\pi G_{d+1}} \int_{\mathcal M_E} d^{d+1}x\sqrt{g}\,(R-2\Lambda) - \frac{1}{8\pi G_{d+1}} \int_{\partial\mathcal M_E}d^d x\sqrt{\gamma}\,K + I_{\mathrm{ct}}.$$

With this convention the Euclidean saddle contributes

$$Z_{\mathrm{bulk}}\simeq e^{-I_E^{\mathrm{ren}}}.$$

For a thermal saddle with boundary Euclidean time period $\beta$, the grand potential is

$$\Omega = T I_E^{\mathrm{ren}}, \qquad T=\beta^{-1},$$

provided the action is evaluated with grand-canonical boundary conditions. If instead charge is fixed, a boundary Legendre transform is required.

$G$, $\kappa$, and annoying factors of two

Many papers use

$$2\kappa_{d+1}^2 = 16\pi G_{d+1}, \qquad \kappa_{d+1}^2 = 8\pi G_{d+1}.$$

Then

$$\frac{1}{16\pi G_{d+1}} = \frac{1}{2\kappa_{d+1}^2}.$$

So the following two forms are identical:

$$S_{\mathrm{grav}} = \frac{1}{16\pi G_{d+1}}\int \sqrt{-g}\,(R-2\Lambda) = \frac{1}{2\kappa_{d+1}^2}\int \sqrt{-g}\,(R-2\Lambda).$$

Before comparing formulas, always check whether the author writes $1/(2\kappa^2)$ or $1/\kappa^2$ in front of the action.

Source and generating-functional conventions

In Euclidean QFT we use

$$Z_E[J] = \left\langle \exp\left(\int d^d x\sqrt{g_{(0)}}\,J(x)\mathcal O(x)\right) \right\rangle = \exp W_E[J].$$

Equivalently, the Euclidean action is deformed as

$$I_E \to I_E - \int d^d x\sqrt{g_{(0)}}\,J\mathcal O.$$

Then

$$\langle \mathcal O(x)\rangle_J = \frac{1}{\sqrt{g_{(0)}}}\frac{\delta W_E}{\delta J(x)}.$$

The Euclidean GKPW relation in the semiclassical gravity limit is

$$W_E[J] = -I_E^{\mathrm{ren}}[\phi_{\mathrm{cl}};\phi_{(0)}=J].$$

Therefore

$$\langle \mathcal O(x)\rangle_J = -\frac{1}{\sqrt{g_{(0)}}} \frac{\delta I_E^{\mathrm{ren}}}{\delta J(x)}.$$

Some authors instead define $Z_E[J]=\langle e^{-\int J\mathcal O}\rangle$. Their one-point function differs by a sign. The physics does not change, but the sign of a quoted one-point function or two-point kernel may.

In Lorentzian signature, one often writes

$$Z_L[J] = \left\langle \exp\left(i\int d^d x\sqrt{-g_{(0)}}\,J\mathcal O\right) \right\rangle = \exp(iW_L[J]),$$

so that

$$\langle \mathcal O(x)\rangle_J = \frac{1}{\sqrt{-g_{(0)}}} \frac{\delta W_L}{\delta J(x)}.$$

The rule of thumb is:

$$\text{Euclidean: } W_E=-I_E^{\mathrm{ren}}, \qquad \text{Lorentzian: } W_L=S_L^{\mathrm{ren}}.$$

But real-time correlators require more than simply replacing $I_E$ by $-iS_L$. The choice of interior boundary condition selects the Green function.

Scalar-field normalization

For a minimally coupled scalar in Lorentzian signature, use

$$S_\phi = -\frac{\mathcal N_\phi}{2} \int d^{d+1}x\sqrt{-g} \left(g^{ab}\partial_a\phi\partial_b\phi+m^2\phi^2\right).$$

The Euclidean version is

$$I_\phi = \frac{\mathcal N_\phi}{2} \int d^{d+1}x\sqrt{g} \left(g^{ab}\partial_a\phi\partial_b\phi+m^2\phi^2\right).$$

The constant $\mathcal N_\phi$ is not cosmetic. It sets the normalization of the dual operator. In a top-down compactification it is determined by dimensional reduction. In a bottom-up model it is a choice that must be fixed by matching a CFT normalization, a susceptibility, a central charge, or some experimental input.

The mass-dimension relation is

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

For standard quantization, the near-boundary expansion in $z$ is schematically

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

where

$$\phi_{(0)}(x)=J(x).$$

When $2\Delta-d$ is an integer, logarithms may appear. They encode conformal anomalies or contact-term ambiguities.

For separated points and no logarithmic subtlety, the renormalized one-point function has the form

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

The phrase “local terms” matters. They are contact terms in correlators and can be changed by finite counterterms. They are invisible at separated points but visible in momentum-space polynomials.

Bulk-to-boundary normalization

A common Euclidean bulk-to-boundary propagator is

$$K_\Delta(z,x;x') = C_\Delta \left(\frac{z}{z^2+\lvert x-x'\rvert^2}\right)^\Delta,$$

with

$$C_\Delta = \frac{\Gamma(\Delta)}{ \pi^{d/2}\Gamma\left(\Delta-\frac d2\right)}.$$

This normalization is chosen so that, distributionally,

$$\lim_{z\to 0}z^{\Delta-d}K_\Delta(z,x;x')=\delta^{(d)}(x-x').$$

With the scalar action above, the separated-point two-point function is proportional to

$$\langle \mathcal O(x)\mathcal O(0)\rangle \propto \mathcal N_\phi(2\Delta-d)C_\Delta \frac{1}{\lvert x\rvert^{2\Delta}}.$$

The proportionality becomes an equality only after fixing all conventions: Euclidean source sign, scalar action normalization, counterterm scheme, and normalization of $\mathcal O$.

Alternate quantization and Legendre transforms

When

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

both roots

$$\Delta_+=\frac d2+\nu, \qquad \Delta_-=\frac d2-\nu$$

are allowed by unitarity. Standard quantization uses $\Delta_+$; alternate quantization uses $\Delta_-$.

In standard quantization,

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

and $\phi_{(0)}$ is the source.

In alternate quantization, the roles of source and response are exchanged by a Legendre transform. One convenient schematic statement is

$$W_-[\widetilde J] = W_+[J]+\int d^d x\,J\widetilde J, \qquad \widetilde J\sim \phi_{(2\nu)}.$$

Do not quote a dimension from the mass alone without specifying the quantization.

Gauge-field normalization

For a bulk $U(1)$ gauge field, use

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

The boundary value $A_\mu^{(0)}$ sources a conserved current:

$$A_\mu^{(0)} \leftrightarrow J^\mu.$$

The variation of the on-shell Lorentzian action contains

$$\delta S_A^{\mathrm{os}} = \int_{\partial\mathcal M} d^d x\, \Pi^\mu\delta A_\mu,$$

where the radial canonical momentum is

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

in a $z$-radial coordinate with outward normal toward $z=0$. Depending on whether one writes the boundary term with the outward unit normal explicitly, this sign may be absorbed into the definition of $\Pi^\mu$.

Our practical convention is

$$\langle J^\mu\rangle = \frac{1}{\sqrt{-g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta A_\mu^{(0)}}.$$

For a Maxwell field in asymptotically AdS$_{d+1}$ with $d>2$,

$$A_\mu(z,x) = A_\mu^{(0)}(x) + \cdots + z^{d-2}A_\mu^{(d-2)}(x) + \cdots,$$

and roughly

$$\langle J^\mu\rangle \sim \frac{L^{d-3}}{g_{d+1}^2}(d-2)A^{(d-2)\mu} + \text{local terms}.$$

The exact sign and local terms depend on radial coordinate, boundary metric, and counterterms. The scaling with $1/g_{d+1}^2$ is invariant.

Chemical potential and horizon gauge

At finite density,

$$\mu = A_t(z\to 0)-A_t(z_h).$$

For a Euclidean black hole, regularity at the contractible thermal circle usually requires

$$A_t(z_h)=0$$

in a smooth gauge. Then $\mu=A_t^{(0)}$. This is not just a decorative gauge choice: the chemical potential is the gauge-invariant Wilson line around the thermal circle. A constant shift in $A_t$ is physical only after the horizon regularity condition and boundary ensemble have been specified.

Metric-source and stress-tensor normalization

The boundary metric sources the stress tensor:

$$\delta S_{\mathrm{CFT}} = \frac12 \int d^d x\sqrt{-g_{(0)}}\, T^{\mu\nu}\delta g_{(0)\mu\nu}.$$

Therefore our Lorentzian convention is

$$\langle T^{\mu\nu}\rangle = \frac{2}{\sqrt{-g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta g_{(0)\mu\nu}}.$$

In Euclidean signature, with $W_E=-I_E^{\mathrm{ren}}$,

$$\langle T^{ij}\rangle_E = -\frac{2}{\sqrt{g_{(0)}}} \frac{\delta I_E^{\mathrm{ren}}}{\delta g_{(0)ij}}.$$

For gravity, the unrenormalized Brown-York tensor has the schematic form

$$T_{ij}^{\mathrm{BY}} = \frac{1}{8\pi G_{d+1}} \left(K_{ij}-K\gamma_{ij}\right),$$

with counterterm contributions added:

$$T_{ij}^{\mathrm{ren}} = \frac{1}{8\pi G_{d+1}} \left(K_{ij}-K\gamma_{ij}+\cdots_{\mathrm{ct}}\right).$$

Signs can change if the outward normal or $K_{ij}$ convention differs. The invariant check is that pure Poincare AdS has vanishing stress tensor in the flat-boundary vacuum after counterterms, while global AdS can have a Casimir energy when the boundary is $S^{d-1}\times \mathbb R$.

Fefferman-Graham stress tensor

In Fefferman-Graham gauge,

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

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 coefficient $g_{(d)\mu\nu}$ contains the state-dependent stress tensor, but the exact formula includes local curvature terms when $g_{(0)}$ is curved and anomaly terms when $d$ is even. On a flat boundary, the relation simplifies schematically to

$$\langle T_{\mu\nu}\rangle \sim \frac{dL^{d-1}}{16\pi G_{d+1}}g_{(d)\mu\nu}.$$

Again, the exact statement depends on the normalization of the FG expansion. Use this as a scaling check, not as a substitute for a renormalized variation.

Boundary conditions: Dirichlet, Neumann, mixed

For a scalar field in standard quantization, Dirichlet boundary conditions fix the source:

$$\delta \phi_{(0)}=0.$$

Neumann or alternate quantization fixes the response-like variable. Mixed boundary conditions implement multi-trace deformations.

A double-trace deformation

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

is implemented holographically by a mixed boundary condition of the schematic form

$$\phi_{(2\Delta-d)}=f\phi_{(0)}$$

up to normalization and local terms. More precisely, the equation relates the renormalized canonical momentum to $f\phi_{(0)}$.

For a gauge field, Dirichlet boundary conditions fix $A_\mu^{(0)}$ and describe a global symmetry current in the boundary theory. Neumann or mixed conditions can make the boundary gauge field dynamical in suitable dimensions. This is a different boundary theory, not a harmless alternative description of the same one.

For gravity, fixing $g_{(0)\mu\nu}$ means the boundary metric is nondynamical. Allowing it to fluctuate corresponds to coupling the CFT to dynamical gravity, which is again a different problem.

Fourier-transform conventions

For Lorentzian real-time calculations, we use

$$\phi(t,\mathbf x,z) = \int\frac{d\omega\,d^{d-1}k}{(2\pi)^d} \,e^{-i\omega t+i\mathbf k\cdot\mathbf x} \phi(z;\omega,\mathbf k).$$

Thus

$$\partial_t\to -i\omega, \qquad \partial_i\to ik_i.$$

The retarded Green function is

$$G_R^{AB}(t,\mathbf x) = -i\theta(t) \langle [A(t,\mathbf x),B(0,\mathbf 0)]\rangle,$$

and its Fourier transform is

$$G_R^{AB}(\omega,\mathbf k) = \int dt\,d^{d-1}x\, e^{i\omega t-i\mathbf k\cdot\mathbf x} G_R^{AB}(t,\mathbf x).$$

With this convention, stable quasinormal-mode poles of $G_R$ lie in the lower half of the complex $\omega$ plane:

$$\mathrm{Im}\,\omega_n<0,$$

because time dependence is $e^{-i\omega t}$.

The spectral density convention used here is

$$\rho_{AB}(\omega,\mathbf k) = -2\,\mathrm{Im}\,G_R^{AB}(\omega,\mathbf k)$$

for bosonic Hermitian operators when $A=B$ and with the above Fourier convention. Some communities use $+2\mathrm{Im}G_R$ instead. Before comparing plots, check the sign.

In Euclidean signature, Matsubara frequencies are

$$\omega_n=2\pi nT \quad \text{for bosons}, \qquad \omega_n=(2n+1)\pi T \quad \text{for fermions}.$$

Analytic continuation to a retarded correlator is usually

$$i\omega_n\to \omega+i0^+,$$

but this compact rule hides assumptions about analyticity and operator ordering. In holography, the equivalent Lorentzian statement is that one imposes infalling boundary conditions at the future horizon.

Real-time holography and horizon signs

Near a nonextremal horizon, introduce the tortoise coordinate $r_*$ such that the metric locally has

$$ds^2\sim -f(r)dt^2+\frac{dr^2}{f(r)}, \qquad r_*\sim \frac{1}{4\pi T}\log(r-r_h).$$

With time dependence $e^{-i\omega t}$, the two local behaviors are

$$e^{-i\omega(t+r_*)} \quad \text{infalling}, \qquad e^{-i\omega(t-r_*)} \quad \text{outgoing}.$$

Equivalently,

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

in Schwarzschild-like coordinates. In ingoing Eddington-Finkelstein coordinates,

$$v=t+r_*,$$

an infalling solution is regular as $e^{-i\omega v}$.

If you use $e^{+i\omega t}$ instead, the exponent flips. This is one of the most common sign traps in real-time holography.

Kubo formulas in our conventions

The basic transport coefficients are defined by low-frequency, zero-momentum limits of retarded functions.

For an electric current,

$$\sigma = -\lim_{\omega\to 0}\frac{1}{\omega} \mathrm{Im}\,G_R^{J_xJ_x}(\omega,\mathbf k=0),$$

assuming no delta-function contribution has been left implicit. At finite charge density in a translationally invariant system, momentum conservation usually produces a delta function in $\mathrm{Re}\,\sigma(\omega)$, so one must separate coherent, incoherent, and momentum-relaxing pieces.

For shear viscosity,

$$\eta = -\lim_{\omega\to 0}\frac{1}{\omega} \mathrm{Im}\,G_R^{T_{xy}T_{xy}}(\omega,\mathbf k=0).$$

For bulk viscosity,

$$\zeta = -\frac{1}{d-1} \lim_{\omega\to 0}\frac{1}{\omega} \mathrm{Im}\,G_R^{T^i{}_iT^j{}_j}(\omega,\mathbf k=0),$$

with indices over spatial directions and with convention-dependent normalization factors in nonconformal theories. In a conformal theory,

$$\zeta=0.$$

For a diffusion pole,

$$\omega=-iDk^2+\cdots.$$

In a simple charge-diffusion problem with no momentum mixing,

$$D=\frac{\sigma}{\chi}, \qquad \chi=\frac{\partial \rho}{\partial \mu}.$$

At finite density, this formula must be used with care because charge transport can mix with momentum and heat transport.

Thermodynamic conventions

For the grand-canonical ensemble,

$$Z(\beta,\mu) = \mathrm{Tr}\,e^{-\beta(H-\mu Q)}, \qquad \Omega=-T\log Z.$$

Holographically,

$$\Omega=T I_E^{\mathrm{ren}}$$

when the Euclidean action is evaluated with fixed boundary chemical potential.

The pressure is

$$p=-\frac{\Omega}{V},$$

and the entropy and charge densities are

$$s=-\left(\frac{\partial \Omega/V}{\partial T}\right)_\mu, \qquad \rho=-\left(\frac{\partial \Omega/V}{\partial \mu}\right)_T.$$

For a homogeneous isotropic state,

$$\langle T^\mu{}_{\nu}\rangle =\mathrm{diag}(-\epsilon,p,\ldots,p).$$

Conformal invariance implies

$$-\epsilon+(d-1)p=0, \qquad \epsilon=(d-1)p.$$

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

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

one has

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

For AdS$_5$/CFT$_4$ in the classical supergravity limit,

$$s=\frac{\pi^2}{2}N^2T^3, \qquad p=\frac{\pi^2}{8}N^2T^4, \qquad \epsilon=\frac{3\pi^2}{8}N^2T^4.$$

These formulas assume the standard AdS$_5\times S^5$ normalization with $a=c=N^2/4$ at large $N$.

AdS$_5$/CFT$_4$ parameter dictionary

For $N$ D3-branes and the convention

$$\mathrm{tr}(T^aT^b)=\frac12\delta^{ab},$$

we use

$$g_{\mathrm{YM}}^2=4\pi g_s, \qquad \lambda=g_{\mathrm{YM}}^2N=4\pi g_sN.$$

The radius/flux relation is

$$L^4=4\pi g_sN\alpha'^2,$$

or

$$\frac{L^4}{\alpha'^2}=\lambda.$$

The ten-dimensional Newton constant is

$$G_{10}=8\pi^6g_s^2\alpha'^4,$$

and since

$$\mathrm{Vol}(S^5)=\pi^3L^5,$$

one has

$$G_5=\frac{G_{10}}{\pi^3L^5}.$$

Combining these gives

$$\frac{L^3}{G_5}=\frac{2N^2}{\pi},$$

and therefore

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

at leading large $N$.

Some authors use $g_{\mathrm{YM}}^2=2\pi g_s$. This usually reflects a different normalization of the gauge generators or of the Yang-Mills action. The invariant relation is the physical one: large $\lambda$ means $L^2/\alpha'\gg 1$, and large $N$ means $L^3/G_5\gg 1$.

Trace normalizations in gauge theory

For $SU(N)$ generators in the fundamental representation, the most common high-energy convention is

$$\mathrm{tr}(T^aT^b)=\frac12\delta^{ab}.$$

Then

$$F_{\mu\nu}=F_{\mu\nu}^aT^a,$$

and the Yang-Mills kinetic term is often written as

$$S_{\mathrm{YM}} =-\frac{1}{2g_{\mathrm{YM}}^2} \int d^4x\,\mathrm{tr}(F_{\mu\nu}F^{\mu\nu}) = -\frac{1}{4g_{\mathrm{YM}}^2} \int d^4x\,F_{\mu\nu}^aF^{a\mu\nu}.$$

If instead one uses generators normalized by

$$\mathrm{Tr}(T^aT^b)=\delta^{ab},$$

then factors of $2$ move between the trace and the coupling. This affects the displayed relation between $g_s$ and $g_{\mathrm{YM}}$, but not physical observables once the operator normalization is fixed.

For single-trace operators, specify whether

$$\mathcal O=\mathrm{tr}(X^2)$$

or

$$\mathcal O=\frac{1}{N}\mathrm{tr}(X^2)$$

or

$$\mathcal O=\frac{1}{\sqrt N}\mathrm{tr}(X^2)$$

is being used. The $N$-scaling of correlators changes accordingly.

A useful CFT normalization for large-$N$ discussions is to choose single-trace operators so that

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

which implies connected higher-point functions scale as

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

for matrix large-$N$ theories with a classical gravity dual. But many supergravity computations naturally produce unnormalized operators with two-point functions of order $N^2$. Both conventions are fine if stated.

Wilson-loop and string-action conventions

The fundamental string action in Euclidean signature is

$$I_{\mathrm{NG}} = \frac{1}{2\pi\alpha'} \int d^2\sigma\sqrt{\det h_{\alpha\beta}},$$

where

$$h_{\alpha\beta}=G_{MN}\partial_\alpha X^M\partial_\beta X^N.$$

The holographic Wilson-loop saddle is

$$\langle W(C)\rangle \sim \exp\left(-I_{\mathrm{NG}}^{\mathrm{ren}}[C]\right).$$

In AdS$_5\times S^5$,

$$\frac{L^2}{\alpha'}=\sqrt\lambda,$$

so the string action scales as

$$I_{\mathrm{NG}}\sim \sqrt\lambda.$$

For a rectangular loop of temporal length $\mathcal T$ and spatial separation $R$,

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

in Euclidean signature. In Lorentzian signature the phase convention differs, but the extracted static energy is the same after continuation.

The bare string area has a perimeter divergence from the infinite mass of an external quark. The standard heavy-quark potential subtracts two straight strings:

$$V_{Q\bar Q}(R) = \frac{1}{\mathcal T} \left(I_{\mathrm{U\text{-}shape}}-2I_{\mathrm{straight}}\right).$$

Changing the subtraction changes the additive constant in the potential, not the force.

Entanglement conventions

For Einstein gravity, the classical RT/HRT formula is

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

where $\gamma_A$ is the codimension-two extremal surface homologous to $A$.

For quantum corrections, use the generalized entropy

$$S_{\mathrm{gen}}(\mathcal X) = \frac{\mathrm{Area}(\mathcal X)}{4G_{d+1}} + S_{\mathrm{bulk}}(\Sigma_A) + \text{counterterms},$$

and extremize over candidate quantum extremal surfaces:

$$S_A = \min_{\mathrm{QES}}\,\mathrm{ext}\, S_{\mathrm{gen}}.$$

Do not compare the area term and the bulk entropy term without specifying the cutoff and renormalization of $G_{d+1}$. The UV divergences of $S_{\mathrm{bulk}}$ are absorbed into gravitational couplings, just as UV divergences in the on-shell action are absorbed by holographic counterterms.

Central charges and two-point normalizations

The symbol $c$ is treacherous. In two-dimensional CFT it usually means the Virasoro central charge. In four-dimensional CFT, $a$ and $c$ are Weyl-anomaly coefficients. In arbitrary dimension, $C_T$ is the coefficient of the stress-tensor two-point function.

For AdS$_3$/CFT$_2$,

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

For AdS$_5$/CFT$_4$ with pure Einstein gravity,

$$a=c=\frac{\pi L^3}{8G_5}.$$

For general AdS$_{d+1}$ Einstein gravity,

$$C_T\propto \frac{L^{d-1}}{G_{d+1}},$$

but the coefficient of proportionality depends on the precise definition of the tensor structure in

$$\langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle = \frac{C_T}{\lvert x\rvert^{2d}} \mathcal I_{\mu\nu,\rho\sigma}(x).$$

Never compare $C_T$ values across papers without checking the normalization of $\mathcal I_{\mu\nu,\rho\sigma}$.

The same warning applies to current two-point functions:

$$\langle J_\mu(x)J_\nu(0)\rangle = \frac{C_J}{\lvert x\rvert^{2(d-1)}} I_{\mu\nu}(x),$$

where $C_J$ depends on both the normalization of the current and the normalization of the bulk gauge kinetic term.

Counterterms, contact terms, and scheme dependence

Holographic renormalization removes cutoff divergences with local counterterms. Finite local counterterms are still allowed when they respect the symmetries. Therefore:

• separated-point correlators are usually scheme independent;
• momentum-space correlators can differ by polynomials in $\omega$ and $k$;
• one-point functions in curved or sourced backgrounds can differ by local terms;
• anomalies are scheme independent up to standard trivial-anomaly ambiguities;
• transport coefficients extracted from dissipative imaginary parts are usually insensitive to local real contact terms.

For example, adding a finite counterterm

$$\Delta S_{\mathrm{ct}} =\frac{\alpha}{2}\int d^d x\sqrt{-g_{(0)}}\,J^2$$

changes the two-point function by

$$\Delta G(k)=\alpha,$$

a contact term in position space. It does not change the separated-point power law.

For gauge fields in even boundary dimensions, logarithmic counterterms can introduce scale dependence. This is not a failure of holography; it is the ordinary renormalization of a current correlator in momentum space.

A practical translation checklist

When reading or writing a holographic calculation, identify the following before comparing numbers:

1. Signature: Lorentzian mostly-plus, mostly-minus, or Euclidean?
2. Radial coordinate: boundary at $z=0$ or $r=\infty$?
3. AdS radius: set to $L=1$ or kept explicit?
4. Gravitational coupling: $G_{d+1}$, $\kappa_{d+1}^2$, or an overall $\mathcal N$?
5. Gauge coupling: is the Maxwell term $-F^2/(4g^2)$ or $-ZF^2/4$?
6. Scalar normalization: is there an overall $\mathcal N_\phi$?
7. Source sign: does $Z[J]$ contain $e^{+\int J\mathcal O}$ or $e^{-\int J\mathcal O}$?
8. Fourier convention: $e^{-i\omega t}$ or $e^{+i\omega t}$?
9. Horizon condition: infalling, outgoing, regular Euclidean, or normalizable global mode?
10. Ensemble: fixed source, fixed vev, fixed chemical potential, or fixed charge?
11. Counterterm scheme: minimal subtraction, supersymmetric counterterms, or finite matching terms?
12. Operator normalization: canonical CFT two-point normalization, large-$N$ normalized operator, or top-down supergravity normalization?

This checklist is painfully mundane. It also saves entire afternoons.

Common mistake catalog

Mistake 1: Treating $L=1$ as dimensionless physics

Setting $L=1$ is a coordinate convention. Restoring $L$ is necessary for checking dimensions. For example,

$$\frac{L^2}{\alpha'}=\sqrt\lambda$$

is meaningful; $1/\alpha'=\sqrt\lambda$ is only meaningful after setting $L=1$.

Mistake 2: Forgetting the Euclidean minus sign

If

$$Z_E\simeq e^{-I_E^{\mathrm{ren}}},$$

then

$$W_E=-I_E^{\mathrm{ren}}.$$

A one-point function from $W_E$ carries the corresponding minus sign if the source convention is $e^{+\int J\mathcal O}$.

Mistake 3: Confusing normalizable with response in every situation

For standard scalar quantization, the normalizable coefficient is the response. In alternate quantization, the roles are exchanged. For gauge fields and gravitons, constraints and gauge redundancies add further subtleties.

Mistake 4: Comparing $C_T$ without comparing $T_{\mu\nu}$

A rescaling

$$T_{\mu\nu}\to \alpha T_{\mu\nu}$$

rescales $C_T$ by $\alpha^2$. The physically meaningful comparison requires the same stress-tensor definition, including the same source coupling

$$\frac12\int T^{\mu\nu}\delta g_{\mu\nu}^{(0)}.$$

Mistake 5: Identifying $A_t^{(0)}$ with $\mu$ without horizon regularity

At finite temperature, the chemical potential is a Wilson-line-like difference between boundary and horizon. In the regular Euclidean gauge, $A_t(z_h)=0$, so $A_t^{(0)}=\mu$. Without that gauge choice, $A_t^{(0)}$ alone is not gauge invariant.

Mistake 6: Taking limits in the wrong order

Hydrodynamic transport coefficients require small $\omega$ and $k$ limits in a specified order. Static susceptibilities, optical conductivities, and diffusion constants are not interchangeable limits of the same function.

Mistake 7: Calling every finite counterterm an error

Finite counterterms represent scheme choices. They can change contact terms and local one-point contributions. They cannot change universal separated-point data, dissipative transport, or properly defined anomalies.

Worked mini-examples

Example 1: Restoring $L$ in the black-brane entropy density

Suppose a paper sets $L=1$ and writes

$$s=\frac{1}{4G_{d+1}z_h^{d-1}}.$$

The metric before setting $L=1$ is

$$ds^2=\frac{L^2}{z^2}\left(-fdt^2+d\mathbf x^2+\frac{dz^2}{f}\right).$$

The area density at the horizon is

$$\sqrt{\det g_{ij}}=\left(\frac{L}{z_h}\right)^{d-1}.$$

Therefore

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

The missing factor is $L^{d-1}$.

Example 2: Converting a scalar expansion from $z$ to $r$

Start with

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

Using $z=L^2/r$ gives

$$\phi(r,x) =\left(\frac{L^2}{r}\right)^{d-\Delta}J(x) + \left(\frac{L^2}{r}\right)^\Delta A(x)+\cdots.$$

Thus

$$\phi(r,x)=L^{2(d-\Delta)}r^{\Delta-d}J(x)+L^{2\Delta}r^{-\Delta}A(x)+\cdots.$$

If the author absorbs powers of $L$ into $J$ and $A$, the powers of $r$ remain the essential diagnostic.

Example 3: Why $\eta/s$ forgets $G_{d+1}$

In two-derivative Einstein gravity, shear viscosity scales like

$$\eta\sim \frac{1}{G_{d+1}}\left(\frac{L}{z_h}\right)^{d-1},$$

while entropy density is

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

The same area-density and Newton-constant factors appear in both, leaving

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

The cancellation is not a statement that normalizations do not matter. It is a special universality of two-derivative horizon dynamics.

Exercises

Exercise 1: Euclidean source sign

Use the convention

$$Z_E[J]=\left\langle e^{\int J\mathcal O}\right\rangle=e^{W_E[J]}, \qquad W_E[J]=-I_E^{\mathrm{ren}}[J].$$

Show that

$$\langle \mathcal O(x)\rangle_J =-\frac{1}{\sqrt{g_{(0)}}} \frac{\delta I_E^{\mathrm{ren}}}{\delta J(x)}.$$

How would the formula change if the source deformation were instead $e^{-\int J\mathcal O}$?

Solution

By definition,

$$\langle \mathcal O(x)\rangle_J = \frac{1}{\sqrt{g_{(0)}}}\frac{\delta W_E}{\delta J(x)}.$$

Using $W_E=-I_E^{\mathrm{ren}}$ gives

$$\langle \mathcal O(x)\rangle_J =-\frac{1}{\sqrt{g_{(0)}}} \frac{\delta I_E^{\mathrm{ren}}}{\delta J(x)}.$$

If instead

$$Z_E[J]=\left\langle e^{-\int J\mathcal O}\right\rangle,$$

then differentiating $W_E$ gives

$$\frac{1}{\sqrt{g_{(0)}}}\frac{\delta W_E}{\delta J(x)} =-\langle \mathcal O(x)\rangle_J.$$

So with the same $W_E=-I_E^{\mathrm{ren}}$ one finds

$$\langle \mathcal O(x)\rangle_J =\frac{1}{\sqrt{g_{(0)}}} \frac{\delta I_E^{\mathrm{ren}}}{\delta J(x)}.$$

The sign flip is purely conventional, but it must be tracked.

Exercise 2: $r$ versus $z$ scalar falloffs

A scalar in AdS$_{d+1}$ has standard-quantization expansion

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

Convert this expansion to the coordinate $r=L^2/z$. Identify the source and response powers of $r$.

Solution

Substitute $z=L^2/r$:

$$\phi(r,x) =\left(\frac{L^2}{r}\right)^{d-\Delta}J(x) + \left(\frac{L^2}{r}\right)^\Delta A(x)+\cdots.$$

Therefore

$$\phi(r,x) =L^{2(d-\Delta)}r^{\Delta-d}J(x) + L^{2\Delta}r^{-\Delta}A(x)+\cdots.$$

The source mode scales as $r^{\Delta-d}$ and the response mode scales as $r^{-\Delta}$. If $L=1$, these become $r^{\Delta-d}$ and $r^{-\Delta}$.

Exercise 3: Operator normalization from the scalar action

Suppose two bottom-up models use the same scalar mass and background, but one has

$$I_\phi=\frac12\int\sqrt g\left[(\partial\phi)^2+m^2\phi^2\right]$$

and the other has

$$I_\phi=\frac{\mathcal N}{2}\int\sqrt g\left[(\partial\phi)^2+m^2\phi^2\right].$$

How are the separated-point two-point functions related if both identify the same boundary coefficient $J$ as the source?

Solution

The classical solution is the same because the overall factor $\mathcal N$ does not change the linear equation of motion. But the on-shell action is multiplied by $\mathcal N$. Since the two-point function is obtained by differentiating the on-shell action twice with respect to $J$, it is also multiplied by $\mathcal N$:

$$\langle \mathcal O(x)\mathcal O(0)\rangle_{\mathcal N} = \mathcal N \langle \mathcal O(x)\mathcal O(0)\rangle_{\mathcal N=1}$$

up to contact terms and source-sign conventions. Equivalently, changing $\mathcal N$ changes the normalization of the dual operator.

Exercise 4: Recovering $a=c=N^2/4$ from $L^3/G_5$

Use

$$G_{10}=8\pi^6g_s^2\alpha'^4, \qquad \mathrm{Vol}(S^5)=\pi^3L^5, \qquad L^4=4\pi g_sN\alpha'^2.$$

Show that

$$\frac{L^3}{G_5}=\frac{2N^2}{\pi},$$

where $G_5=G_{10}/\mathrm{Vol}(S^5)$. Then show that

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

Solution

First reduce the Newton constant:

$$G_5=\frac{G_{10}}{\pi^3L^5} =\frac{8\pi^6g_s^2\alpha'^4}{\pi^3L^5} =\frac{8\pi^3g_s^2\alpha'^4}{L^5}.$$

Therefore

$$\frac{L^3}{G_5} =\frac{L^8}{8\pi^3g_s^2\alpha'^4}.$$

Using

$$L^4=4\pi g_sN\alpha'^2$$

gives

$$L^8=16\pi^2g_s^2N^2\alpha'^4.$$

Thus

$$\frac{L^3}{G_5} = \frac{16\pi^2g_s^2N^2\alpha'^4}{8\pi^3g_s^2\alpha'^4} = \frac{2N^2}{\pi}.$$

Finally,

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

This is the leading large-$N$ result; for $SU(N)$ rather than $U(N)$ the exact free-field value involves $N^2-1$.

Exercise 5: Infalling exponent

Near a nonextremal horizon, suppose

$$f(r)\simeq 4\pi T(r-r_h), \qquad r_*\simeq \frac{1}{4\pi T}\log(r-r_h).$$

With time dependence $e^{-i\omega t}$, show that an infalling mode behaves as

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

Solution

An infalling wave is regular in the ingoing coordinate

$$v=t+r_*.$$

Therefore its spacetime dependence is

$$e^{-i\omega v}=e^{-i\omega t}e^{-i\omega r_*}.$$

Using

$$r_*\simeq \frac{1}{4\pi T}\log(r-r_h),$$

we obtain

$$e^{-i\omega r_*} \simeq \exp\left[-\frac{i\omega}{4\pi T}\log(r-r_h)\right] = (r-r_h)^{-i\omega/(4\pi T)}.$$

Thus the radial part behaves as claimed. With the opposite Fourier convention $e^{+i\omega t}$, the sign of the exponent would be reversed.

Further reading

• K. Skenderis, “Lecture Notes on Holographic Renormalization”. The standard practical reference for counterterms, one-point functions, Ward identities, and anomalies.
• S. de Haro, K. Skenderis, and S. N. Solodukhin, “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence”. The systematic Fefferman-Graham reconstruction and holographic stress-tensor framework.
• V. Balasubramanian and P. Kraus, “A Stress Tensor for Anti-de Sitter Gravity”. The classic Brown-York plus counterterms stress-tensor prescription.
• D. T. Son and A. O. Starinets, “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications”. The canonical reference for retarded correlators from infalling boundary conditions.
• O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, “Large $N$ Field Theories, String Theory and Gravity”. The broad normalization and parameter-dictionary reference for the original AdS/CFT correspondence.