Dictionary Tables and Common Normalizations

This page is a reference sheet for the conventions used throughout the holographic quantum matter course. It is deliberately more table-heavy than the previous pages. The goal is not to replace the derivations; the goal is to make it easy to translate between papers, check dimensions, and catch normalization mistakes before they infect a calculation.

The moral is simple:

$$\text{a holographic formula is only meaningful after its conventions are fixed.}$$

For example, the statement “$A_t=\mu+\rho r^{d-2}+\cdots$” is not a universal equation. It depends on the radial coordinate, the Maxwell normalization, the sign convention for $F_{rt}$, and whether the boundary has $d$ or $d+1$ spacetime dimensions in the author’s notation. The physics is invariant, but the coefficient called $\rho$ may differ by factors of $g_F^2$, $2\kappa^2$, $L$, $d-2$, $4\pi$, or a sign. This page makes those dependencies explicit.

Throughout the page, I use the following default convention unless stated otherwise:

$$\text{boundary spacetime dimension}=d, \qquad \text{boundary spatial dimension}=d_s=d-1, \qquad \text{bulk dimension}=d+1.$$

The boundary is often at $r=0$. Many papers instead put the boundary at $R\to\infty$. A large fraction of normalization confusion is just this coordinate choice wearing a fake mustache.

Coordinate conventions

Two common radial coordinates

For asymptotically $AdS_{d+1}$ in Poincaré form, two common radial conventions are

$$ds^2=\frac{L^2}{r^2}\left(dr^2+\eta_{\mu\nu}dx^\mu dx^\nu\right), \qquad r\to 0,$$

and

$$ds^2=L^2\frac{dR^2}{R^2}+R^2\eta_{\mu\nu}dx^\mu dx^\nu, \qquad R\to\infty.$$

They are related by $R=L/r$ if the boundary coordinates are kept fixed up to powers of $L$. In practice, many authors set $L=1$ and write $R=1/r$.

Item	Boundary at $r\to 0$	Boundary at $R\to\infty$
UV boundary	small $r$	large $R$
Deep IR/horizon	larger $r$	smaller $R$
Energy scale	$E\sim 1/r$	$E\sim R$
Relevant deformation grows toward	IR	IR
Normalizable scalar mode	$r^\Delta$	$R^{-\Delta}$
Non-normalizable scalar mode	$r^{d-\Delta}$	$R^{\Delta-d}$
Black-brane horizon	$r=r_h$	$R=R_h$

A useful rule of thumb is

$$r\partial_r=-R\partial_R.$$

So an exponent that increases toward the boundary in one convention decreases in the other.

Fefferman—Graham gauge

For holographic renormalization, Fefferman—Graham coordinates are often the cleanest:

$$ds^2 = L^2\left(\frac{d\rho^2}{4\rho^2}+\frac{1}{\rho}g_{\mu\nu}(x,\rho)dx^\mu dx^\nu\right), \qquad \rho\to0.$$

The boundary metric is

$$g_{\mu\nu}(x,\rho)=g_{(0)\mu\nu}(x)+\rho g_{(2)\mu\nu}(x)+\cdots+\rho^{d/2}g_{(d)\mu\nu}(x)+\cdots.$$

When $d$ is even, logarithmic terms can appear:

$$g_{\mu\nu}(x,\rho) = \cdots+\rho^{d/2}\left(g_{(d)\mu\nu}(x)+\log\rho\,h_{(d)\mu\nu}(x)\right)+\cdots.$$

The coefficient $h_{(d)\mu\nu}$ is tied to conformal anomalies. In a numerical gauge that is not Fefferman—Graham, the same data are still present, but one must either transform asymptotically or use covariant counterterms directly in the chosen gauge.

Bulk fields and boundary operators

The most basic dictionary is source/response. A boundary source couples to a boundary operator by

$$\delta S_{\rm QFT} = \int d^d x\sqrt{-g_{(0)}}\,J_I O^I.$$

The corresponding bulk field $\Phi^I$ has an asymptotic mode fixed by $J_I$ and another mode determining $\langle O^I\rangle$. At large $N$ and strong coupling, the classical on-shell action gives

$$Z_{\rm QFT}[J_I] \simeq \exp\left(iS_{\rm ren}[\Phi^I_{\rm cl};J_I]\right),$$

and therefore

$$\langle O_I(x)\rangle = \frac{1}{\sqrt{-g_{(0)}}} \frac{\delta S_{\rm ren}}{\delta J_I(x)}.$$

Field/operator table

Bulk object	Boundary object	Source	Response	Common use in HQM
Metric $g_{ab}$	Stress tensor $T^{\mu\nu}$	Boundary metric $g_{(0)\mu\nu}$	$\langle T^{\mu\nu}\rangle$	Thermodynamics, viscosity, momentum relaxation
Maxwell field $A_a$	Conserved current $J^\mu$	Boundary gauge field $A^{(0)}_\mu$	$\langle J^\mu\rangle$	Charge density, conductivity, Hall response
Scalar $\phi$	Scalar operator $O$	Leading mode $\phi_{(0)}$	Subleading normalizable data	Relevant deformations, order parameters, lattices
Charged scalar $\Psi$	Charged order parameter $O_\Psi$	Source for $O_\Psi$	Condensate $\langle O_\Psi\rangle$	Holographic superfluids/superconductors
Spinor $\psi$	Fermionic operator $\mathcal O_\psi$	Non-normalizable spinor component	Normalizable spinor component	Spectral functions, Fermi surfaces
Axion scalars $\psi_I$	Neutral scalar operators $O_I$	$\psi_I^{(0)}=kx_I$	$\langle O_I\rangle$	Homogeneous momentum relaxation
Probe-brane embedding $X(r)$	Flavor mass/operator pair	Quark/flavor mass	Condensate	Meson melting, flavor response
DBI worldvolume gauge field	Flavor current $J_f^\mu$	Flavor chemical potential or electric field	Flavor density/current	Nonlinear transport, probe charge sector
Bulk $p$-form	Higher-form current/operator	Boundary $p$-form source	Higher-form current	Magnetic flux, generalized symmetries
Chern—Simons term	Anomaly/contact structure	Background gauge or metric fields	Anomalous currents	CME, CVE, Hall terms

The phrase “source” always means a boundary condition in the variational problem. The phrase “response” means a derivative of the renormalized action. Reading off a coefficient from a near-boundary expansion is only shorthand for this variational statement.

Dimensions and large-$N$ normalization

A typical two-derivative bulk action is written as

$$S = \frac{1}{2\kappa^2}\int d^{d+1}x\sqrt{-g}\left[ R-2\Lambda -\frac{Z(\phi)}{4}F_{ab}F^{ab} -\frac{1}{2}(\partial\phi)^2 -V_{\rm int}(\phi) +\cdots \right] +S_{\rm bdy}.$$

The gravitational normalization is

$$2\kappa^2=16\pi G_{N,d+1}.$$

In many top-down examples,

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

so classical gravity computes the leading large-$N$ answer. If the Maxwell term is written instead as

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

then $1/g_F^2$ carries the current two-point normalization. In the combined convention above, the effective gauge coupling is $g_F^2\sim 2\kappa^2/Z(0)$ near the boundary.

Dimensional bookkeeping

With $\hbar=c=k_B=1$, all quantities are powers of energy. In $d$ boundary spacetime dimensions:

Quantity	Symbol	Dimension
Spacetime coordinate	$x^\mu$	$-1$
Temperature	$T$	$1$
Frequency	$\omega$	$1$
Momentum	$k$	$1$
Chemical potential	$\mu$	$1$
Entropy density	$s$	$d_s$
Charge density	$\rho$	$d_s$
Energy density	$\epsilon$	$d$
Pressure	$p$	$d$
Current	$J^\mu$	$d-1$
Stress tensor	$T^{\mu\nu}$	$d$
Scalar operator	$O$	$\Delta$
Source for $O$	$J_O$	$d-\Delta$
Electric conductivity	$\sigma$	$d-3$
Shear viscosity	$\eta$	$d_s$
Diffusion constant	$D$	$-1$
Momentum relaxation rate	$\Gamma$	$1$

In $d=3$ boundary spacetime dimensions, electric conductivity is dimensionless. This is one reason $AdS_4/CFT_3$ Maxwell theory is such a clean playground for quantum-critical conductivity.

Scalar operators

For a scalar field in $AdS_{d+1}$,

$$S_\phi=-\frac{1}{2}\int d^{d+1}x\sqrt{-g}\left[(\partial\phi)^2+m^2\phi^2+\cdots\right],$$

the relation between the bulk mass and the boundary scaling dimension is

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

Equivalently,

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

The Breitenlohner—Freedman bound is

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

Near the boundary $r\to0$,

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

In standard quantization,

$$\phi_{(0)} \quad\text{is the source}, \qquad \phi_{(2\Delta-d)} \quad\text{determines } \langle O\rangle.$$

For a free scalar with canonical normalization and no mixing, the leading nonlocal part of the vev is often summarized as

$$\langle O\rangle \sim (2\Delta-d)\phi_{(2\Delta-d)}+\text{local terms},$$

but the local terms are not optional. They are fixed by counterterms and by the finite renormalization scheme.

Standard versus alternative quantization

If

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

both modes can be normalizable. Then one may choose alternative quantization, in which the operator dimension is $\Delta_-$ rather than $\Delta_+$. In that case the roles of source and response are exchanged by a Legendre transform.

Quantization	Operator dimension	Source	Response
Standard	$\Delta_+$	coefficient of $r^{d-\Delta_+}=r^{\Delta_-}$	coefficient of $r^{\Delta_+}$
Alternative	$\Delta_-$	coefficient of $r^{\Delta_+}$	coefficient of $r^{\Delta_-}$
Mixed	set by double-trace deformation	relation between the two coefficients	conjugate combination

Double-trace deformations impose mixed boundary conditions. Schematically, adding

$$\delta S_{\rm QFT}=\frac{f}{2}\int d^d x\,O^2$$

changes the boundary condition from “source equals coefficient” to “source equals coefficient plus $f$ times response,” with the precise statement depending on the quantization convention.

Logarithms and anomalies

When $2\Delta-d$ is an integer, near-boundary expansions can contain logarithms:

$$\phi(r,x)=r^{d-\Delta}\left[\phi_{(0)}(x)+\cdots+r^{2\Delta-d}\left(\phi_{(2\Delta-d)}(x)+\log r\,\tilde\phi_{(2\Delta-d)}(x)\right)+\cdots\right].$$

Logs are not a nuisance to be deleted. They encode conformal anomalies, explicit running, or contact-term data. Dropping them usually breaks Ward identities.

Gauge fields, chemical potential, and charge density

For a boundary global $U(1)$ current $J^\mu$, the dual bulk field is a gauge field $A_a$. Near the boundary,

$$A_\mu(r,x)=A_\mu^{(0)}(x)+\cdots+A_\mu^{(1)}(x)r^{d-2}+\cdots,$$

with possible logarithms in special dimensions. The source is the background gauge field $A_\mu^{(0)}$. The vev is obtained from the canonical radial momentum:

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

For a Maxwell action

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

the unrenormalized radial momentum is

$$\Pi^\mu_A = -\frac{1}{g_F^2}\sqrt{-g}\,F^{r\mu}.$$

The charge density is therefore a radial electric flux:

$$\rho=\langle J^t\rangle=\lim_{r\to0}\Pi_A^t+\text{counterterm contributions}.$$

At finite density, one often writes

$$A=A_t(r)dt, \qquad A_t(r)=\mu+c_\rho\rho\,r^{d-2}+\cdots,$$

where $c_\rho$ is convention-dependent. If the boundary has $d$ spacetime dimensions, the power is $d-2$ for a standard Maxwell field in $AdS_{d+1}$. If a paper uses $d$ for the number of spatial dimensions, the same power may be written as $d-1$.

Chemical potential

The gauge-invariant chemical potential is a potential difference between the boundary and the horizon or IR endpoint:

$$\mu = A_t(r\to0)-A_t(r_h),$$

in a static gauge. For a smooth Euclidean black hole, the thermal circle shrinks at the horizon, so regularity usually sets

$$A_t(r_h)=0.$$

This is a gauge choice plus a regularity condition, not a new equation of motion.

Gauge-field choices that often get confused

Choice	Meaning	Common trap
Dirichlet $A_\mu^{(0)}$ fixed	Global current coupled to background gauge field	Treating the boundary gauge field as dynamical
Neumann electric flux fixed	Fixed charge density ensemble	Forgetting the Legendre transform
Mixed boundary condition	Double-trace/current-current deformation or dynamical boundary gauge field	Comparing conductivity without specifying boundary photon dynamics
$A_t(r_h)=0$	Regular gauge at Euclidean horizon	Mistaking $A_t(0)$ for gauge-invariant $\mu$ in a gauge with $A_t(r_h)\neq0$
Chern—Simons term present	Anomaly/contact contribution	Missing Bardeen counterterms or consistent/covariant current distinction

Metric, stress tensor, and thermodynamics

The boundary metric sources the stress tensor:

$$\delta S_{\rm QFT} = \frac{1}{2}\int d^d x\sqrt{-g_{(0)}}\,T^{\mu\nu}\delta g_{(0)\mu\nu}.$$

The holographic stress tensor is

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

In a homogeneous isotropic equilibrium state on flat space,

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

with $d_s$ copies of $p$. With mostly-plus metric $\eta_{\mu\nu}=\operatorname{diag}(-1,1,\ldots,1)$,

$$T^{tt}=\epsilon, \qquad T^{ij}=p\delta^{ij}.$$

For a conformal state with no anomaly on flat space,

$$T^\mu_{\ \mu}=0 \quad\Longrightarrow\quad \epsilon=d_s p.$$

The thermodynamic relations in the grand canonical ensemble are

$$dp=s\,dT+\rho\,d\mu,$$ $$\epsilon+p=sT+\mu\rho,$$

and

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

The combination

$$\chi_{PP}=\epsilon+p$$

is the momentum susceptibility of a relativistic homogeneous fluid. It appears constantly in metallic transport.

Heat current conventions

At finite density, the heat current is not the energy current. The standard convention is

$$Q^i=T^{ti}-\mu J^i.$$

This is the current conjugate to a temperature gradient in the thermoelectric matrix. Some authors write the energy current as $J_E^i=T^{ti}$ and reserve $Q^i$ for heat. Mixing them is a classic way to get wrong factors of $\mu$.

Current	Symbol	Definition	Couples naturally to
Charge current	$J^i$	$U(1)$ current	Electric field $E_i$
Energy current	$J_E^i$	$T^{ti}$	Metric perturbation $h_{ti}$
Heat current	$Q^i$	$T^{ti}-\mu J^i$	$-\nabla_i T/T$
Momentum density	$P^i$	$T^{ti}$ in relativistic theories	Velocity/source for boosts

In nonrelativistic systems, energy current and momentum density are no longer identical. In relativistic holographic fluids, they are the same component of the stress tensor, but the heat current still differs from them at finite chemical potential.

Ensembles and boundary terms

The same bulk solution can represent different thermodynamic ensembles depending on boundary terms.

Boundary data held fixed	Boundary condition	Thermodynamic potential	Example
$T,\mu$	Fix boundary metric and $A_t^{(0)}$	Grand potential $\Omega(T,\mu)$	RN-AdS black brane in grand canonical ensemble
$T,\rho$	Fix electric flux	Helmholtz free energy $F(T,\rho)$	Legendre-transformed Maxwell boundary term
$s,\rho$	Fix horizon area and flux	Energy $E(s,\rho)$	Microcanonical discussion
Source $\phi_{(0)}$	Dirichlet scalar	Generating functional as function of source	Explicit deformation
Vev-like scalar coefficient	Neumann/alternative	Legendre-transformed functional	Alternative quantization
Relation among modes	Mixed	Deformed theory	Double-trace deformation

For Maxwell fields, the canonical ensemble is obtained by adding a boundary term that Legendre transforms from $A_\mu$ to its radial momentum. For scalar fields in alternative quantization, the Legendre transform exchanges the leading and subleading modes. For gravity, changing the boundary metric boundary condition is more drastic: it can make boundary gravity dynamical.

Retarded Green functions

The retarded correlator is

$$G^R_{AB}(t,\vec x) = -i\theta(t)\langle[O_A(t,\vec x),O_B(0,\vec0)]\rangle.$$

Fourier conventions vary. In this course, perturbations are usually written as

$$\delta\Phi(r,t,\vec x)=\delta\Phi(r)e^{-i\omega t+i\vec k\cdot\vec x}.$$

With this convention, an infalling mode near a nonextremal horizon behaves as

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

when $r\to r_h$ and the boundary is at $r=0$. If instead the perturbation is written as $e^{+i\omega t}$, the sign of the exponent changes. This is not physics; it is Fourier bookkeeping.

The spectral density is

$$\rho_{AB}(\omega,k) = -2\operatorname{Im}G^R_{AB}(\omega,k)$$

for $A=B$ in the convention above. Some communities define $+2\operatorname{Im}G^R$ instead. Always check positivity: for a bosonic operator, the spectral weight at positive frequency should be nonnegative in the corresponding convention.

Boundary conditions for correlators and modes

Object	UV condition	IR/horizon condition	Output
Retarded Green function	Turn on source(s), read response(s)	Infalling at future horizon	$G^R_{AB}(\omega,k)$
Quasinormal mode	Source-free/normalizable	Infalling at future horizon	Poles $\omega_n(k)$
Euclidean correlator	Boundary source at Matsubara frequency	Regular at Euclidean tip	$G_E(i\omega_n,k)$
Static susceptibility	Static source	Regular horizon/interior	$\chi$
Normal mode in horizonless geometry	Normalizable	Regular in interior	Discrete spectrum
Instability threshold	Source-free static mode	Regular	Critical temperature or wavevector

A quasinormal mode is not a source response computation with a source accidentally set to zero at the end. It is an eigenvalue problem: source-free UV boundary condition plus infalling IR boundary condition.

Transport conventions

Electric conductivity

Linear response defines

$$J_i(\omega)=\sigma_{ij}(\omega)E_j(\omega).$$

With $E_j=-\partial_t A_j$ at zero spatial momentum and the $e^{-i\omega t}$ convention,

$$E_j=i\omega A_j.$$

If the current response is

$$\delta\langle J_i\rangle=G^R_{J_iJ_j}(\omega,0)A_j,$$

then

$$\sigma_{ij}(\omega)=\frac{1}{i\omega}G^R_{J_iJ_j}(\omega,0),$$

up to contact/diamagnetic terms. Equivalently,

$$\operatorname{Re}\sigma_{ij}(\omega) = -\frac{1}{\omega}\operatorname{Im}G^R_{J_iJ_j}(\omega,0)$$

for the convention above.

Thermoelectric matrix

A common convention is

$$\begin{pmatrix} J^i\\ Q^i \end{pmatrix} = \begin{pmatrix} \sigma^{ij} & T\alpha^{ij}\\ T\alpha^{ij} & T\bar\kappa^{ij} \end{pmatrix} \begin{pmatrix} E_j\\ -\nabla_j T/T \end{pmatrix}.$$

Here $\bar\kappa$ is the thermal conductivity at zero electric field. The experimentally common open-circuit thermal conductivity $\kappa$ is measured at $J=0$:

$$\kappa = \bar\kappa-T\alpha\sigma^{-1}\alpha$$

in matrix notation. In isotropic systems this reduces to

$$\kappa=\bar\kappa-\frac{T\alpha^2}{\sigma}.$$

If you compare thermal conductivities across papers, check whether the author reports $\kappa$ or $\bar\kappa$.

Kubo formula table

The following formulas assume isotropy, zero background magnetic field unless stated otherwise, and the $e^{-i\omega t}$ convention.

Coefficient	Kubo formula	Notes
Electric conductivity	$\sigma=\displaystyle\lim_{\omega\to0}\frac{1}{i\omega}G^R_{J_xJ_x}(\omega,0)$	Include contact terms where required
Charge diffusion	$D=\sigma/\chi$	$\chi=(\partial\rho/\partial\mu)_T$
Shear viscosity	$\eta=-\displaystyle\lim_{\omega\to0}\frac{1}{\omega}\operatorname{Im}G^R_{T_{xy}T_{xy}}(\omega,0)$	For two-derivative isotropic Einstein gravity, $\eta/s=1/(4\pi)$
Bulk viscosity	$\zeta$ from trace stress correlator	Vanishes in conformal theories on flat space
Momentum relaxation rate	Drude width $\Gamma$ or memory matrix	Requires translation breaking
Thermoelectric coefficient	$\alpha$ from mixed $J_xQ_x$ response	Watch heat-current definition
Thermal conductivity at $E=0$	$\bar\kappa$ from $Q_xQ_x$ response	Not the same as open-circuit $\kappa$
Hall conductivity	$\sigma_{xy}=\displaystyle\lim_{\omega\to0}\frac{1}{i\omega}G^R_{J_xJ_y}$	Magnetization/contact terms can matter

At finite density with exact translations, $\sigma_{\rm dc}$ is generally infinite because current overlaps with conserved momentum. A finite horizon conductivity does not by itself remove the delta function from momentum conservation.

Clean metallic hydrodynamics

A useful schematic formula is

$$\sigma(\omega) = \sigma_Q+\frac{\rho^2}{\epsilon+p}\frac{i}{\omega+i0^+}$$

in the translation-invariant limit. With weak momentum relaxation,

$$\sigma(\omega) = \sigma_Q+\frac{\rho^2}{\epsilon+p}\frac{1}{\Gamma-i\omega}.$$

Here $\sigma_Q$ is the incoherent or pair-creation conductivity of the fluid, and the second term is controlled by the slow relaxation of momentum. In a quasiparticle metal, the Drude peak comes from quasiparticle momentum relaxation. In a holographic incoherent metal, a Drude-like peak can arise from hydrodynamics even when quasiparticles do not exist.

The momentum-orthogonal current can be normalized as

$$J_{\rm inc}^i = (\epsilon+p)J^i-\rho T^{ti}.$$

Multiplying $J_{\rm inc}$ by an overall constant changes the normalization of its conductivity but not the physics. The defining property is its zero static overlap with momentum.

Hydrodynamic notation

Hydrodynamics is an effective theory of conserved densities and Goldstone modes. In a relativistic charged fluid, a common Landau-frame constitutive relation is

$$T^{\mu\nu}=\epsilon u^\mu u^\nu+p\Delta^{\mu\nu}+\tau^{\mu\nu},$$ $$J^\mu=\rho u^\mu+\nu^\mu,$$

where

$$\Delta^{\mu\nu}=g^{\mu\nu}+u^\mu u^\nu, \qquad \nu^\mu u_\mu=0, \qquad \tau^{\mu\nu}u_\nu=0.$$

Landau frame fixes $u^\mu$ by the energy flow. Eckart frame fixes it by the charge flow. The frame choice changes intermediate definitions of $u^\mu$, $\nu^\mu$, and $\tau^{\mu\nu}$, but physical correlators and transport coefficients are frame-invariant when computed consistently.

Hydrodynamic poles

Channel	Conserved data	Small-$k$ pole	Transport data
Charge diffusion at $\rho=0$	$\rho$	$\omega=-iDk^2+\cdots$	$D=\sigma/\chi$
Shear momentum diffusion	transverse momentum	$\omega=-iD_\eta k^2+\cdots$	$D_\eta=\eta/(\epsilon+p)$
Sound	energy and longitudinal momentum	$\omega=\pm v_s k-i\Gamma_s k^2/2+\cdots$	$v_s^2=(\partial p/\partial\epsilon)$, $\eta,\zeta$
Momentum relaxation	momentum	$\omega=-i\Gamma+\cdots$	Drude width
Superfluid phase	Goldstone plus conserved densities	first/second sound	superfluid density, phase relaxation
Pinned density wave	phason	$\omega\approx\pm\omega_0-i\Omega/2$	pinning frequency, phase relaxation

For a conformal relativistic fluid on flat space,

$$v_s^2=\frac{1}{d_s}.$$

Scaling geometries

A common hyperscaling-violating metric convention is

$$ds^2 = r^{-2(d_s-\theta)/d_s}\left( -r^{-2(z-1)}dt^2+dr^2+d\vec x^2 \right),$$

with $r\to0$ interpreted as the UV in this convention. Under scaling,

$$t\to\lambda^z t, \qquad \vec x\to\lambda \vec x, \qquad r\to\lambda r, \qquad ds\to\lambda^{\theta/d_s}ds.$$

The effective spatial dimensionality is

$$d_{\rm eff}=d_s-\theta.$$

The entropy density scales as

$$s\sim T^{(d_s-\theta)/z}.$$

Exponent	Meaning	Diagnostic
$z$	Dynamical critical exponent	$\omega\sim k^z$
$\theta$	Hyperscaling violation exponent	$s\sim T^{(d_s-\theta)/z}$
$d_s-\theta$	Effective spatial dimension	Thermal and entanglement scaling
$\Phi$ or $\zeta$	Anomalous charge-density exponent, notation varies	Charge/transport scaling
$\nu_k$	IR scaling exponent in $AdS_2$ or semi-local regions	$G^R_{\rm IR}\sim\omega^{2\nu_k}$

The symbols for the charge exponent are especially nonstandard. Some authors use $\Phi$, some use $\zeta$, and some absorb it into the gauge coupling $Z(\phi)$. Always read the scaling transformation of $A_t$, not just the symbol.

Null energy constraints

For the metric convention above, the null energy condition often gives inequalities of the schematic form

$$(d_s-\theta)\left(d_s(z-1)-\theta\right)\ge0,$$ $$(z-1)(d_s+z-\theta)\ge0.$$

These formulas are convention-sensitive. They are useful for sanity checks, not a substitute for deriving the constraints in the metric convention being used.

Fermion conventions

For a Dirac fermion in $AdS_{d+1}$,

$$S_\psi = \int d^{d+1}x\sqrt{-g}\,i\bar\psi\left(\Gamma^aD_a-m\right)\psi+\cdots.$$

The dual fermionic operator has dimension

$$\Delta_\psi=\frac{d}{2}+mL$$

in standard quantization, for the usual mass range. Near the boundary, the spinor decomposes into eigenspaces of the radial gamma matrix:

$$\psi=\psi_+ + \psi_- , \qquad \Gamma^r\psi_\pm=\pm\psi_\pm.$$

The source and response are the two independent boundary spinor components. Which component is called source depends on the sign of $m$ and on gamma-matrix conventions, so the invariant statement is this: impose the boundary condition that fixes half of the spinor components, then obtain the conjugate half from the on-shell variation.

A common holographic Fermi-surface Green function near $k_F$ is

$$G_R(\omega,k) \simeq \frac{h_1}{k-k_F-\omega/v_F-h_2\omega^{2\nu_{k_F}}}.$$

Symbol	Meaning	Warning
$mL$	Bulk spinor mass	Fixes UV dimension, not the Fermi momentum by itself
$q$	Bulk spinor charge	Controls coupling to $A_t$ and IR exponent
$k_F$	Momentum of normalizable zero mode	Model-dependent; not automatically a Luttinger count
$\nu_k$	IR exponent	Determines low-energy self-energy in $AdS_2$ matching
$A(\omega,k)$	Spectral function	Often $A=-2\operatorname{Im}\operatorname{Tr}G_R$
Dipole coupling $p$	Pauli coupling to $F_{ab}$	Can create zeros/pseudogaps; not universal

A sharp peak in a large-$N$ fermion spectral function is a pole of a probe operator correlator. It is not automatically an electron quasiparticle of a microscopic material.

Probe branes and DBI notation

Probe branes add flavor degrees of freedom in the limit $N_f\ll N_c$. The DBI action is commonly written

$$S_{\rm DBI} = -T_q\int d^{q+1}\xi\,e^{-\Phi} \sqrt{-\det\left(P[g+B]_{ab}+2\pi\alpha' F_{ab}\right)}.$$

The worldvolume gauge field is dual to a flavor current. Its radial electric displacement is the flavor density:

$$d = \frac{\delta \mathcal L_{\rm DBI}}{\delta A_t'}.$$

Because DBI is nonlinear, one must distinguish the closed-string metric $g_{ab}$ from the open-string metric controlling fluctuations around a background field strength. Conductivity calculations in DBI systems often depend on the worldvolume horizon, not just the background spacetime horizon.

DBI quantity	Boundary meaning	Common trap
Brane separation/asymptotic embedding	Flavor mass	Coordinate-dependent normalization
Normalizable embedding coefficient	Flavor condensate	Requires counterterms
$A_t$	Flavor chemical potential	Only potential difference is gauge-invariant
Electric displacement $d$	Flavor density	Fixed-density ensemble uses Legendre transform
Worldvolume horizon	Effective dissipation for flavor fluctuations	May differ from background horizon in driven states
$2\pi\alpha'F$	Dimensionless DBI field strength	Forgetting $2\pi\alpha'$ changes nonlinear scales

Magnetic fields, Hall response, and anomalies

In $2+1$ boundary dimensions, a background magnetic field is introduced by

$$F_{xy}=B.$$

The conductivity tensor is

$$\sigma_{ij} = \begin{pmatrix} \sigma_{xx} & \sigma_{xy}\\ -\sigma_{xy} & \sigma_{xx} \end{pmatrix}$$

in an isotropic parity-breaking state. It is often useful to use circular components

$$\sigma_\pm=\sigma_{xx}\pm i\sigma_{xy}.$$

Magnetization currents do not represent net transport through the sample. In inhomogeneous or finite-magnetic-field backgrounds, transport currents may require subtracting local magnetization curls:

$$J^i_{\rm transport}=J^i_{\rm total}-\epsilon^{ij}\partial_j M.$$

For heat currents, there is an analogous energy magnetization subtraction. This is one of the most common sources of disagreement between naive holographic currents and transport coefficients.

Anomaly conventions

A five-dimensional Chern—Simons term may be written schematically as

$$S_{\rm CS}=\kappa_{\rm CS}\int A\wedge F\wedge F.$$

It induces a four-dimensional boundary anomaly of the form

$$\nabla_\mu J^\mu\propto \kappa_{\rm CS}\,E\cdot B.$$

The exact coefficient depends on whether the current is consistent or covariant and on which Bardeen counterterms have been added. When comparing chiral magnetic or chiral vortical coefficients, write down the anomaly polynomial or the Chern—Simons normalization first.

Entanglement, chaos, and diffusion notation

For Einstein gravity, the Ryu—Takayanagi prescription for a static region $A$ is

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

where $\gamma_A$ is the bulk minimal surface homologous to $A$. In time-dependent settings, replace the minimal surface by the HRT extremal surface.

Chaos is often diagnosed by an out-of-time-order correlator with early-time behavior

$$C(t,\vec x)\sim \exp\left[\lambda_L\left(t-\frac{|\vec x|}{v_B}\right)\right].$$

The maximal Einstein-gravity value is

$$\lambda_L=2\pi T$$

in units with $\hbar=k_B=1$. With constants restored,

$$\lambda_L\le \frac{2\pi k_B T}{\hbar}.$$

Relations such as

$$D\sim \frac{v_B^2}{T}$$

are model-dependent scaling relations, not universal identities. The coefficient and even the relevant diffusion constant depend on the IR fixed point and on whether charge, energy, or momentum is the slow mode.

Common notation translations

Boundary dimension notation

This course	Some papers	Meaning
$d$	$D-1$	Boundary spacetime dimension
$d_s=d-1$	$d$	Boundary spatial dimension
$d+1$	$D$	Bulk spacetime dimension
$AdS_{d+1}$	$AdS_D$	Bulk anti-de Sitter spacetime

This is especially important for finite density. If the boundary has $d$ spacetime dimensions, a Maxwell field has $A_t=\mu+\cdots+\rho r^{d-2}+\cdots$. If the author calls the number of boundary spatial dimensions $d$, the same formula becomes $A_t=\mu+\cdots+\rho r^{d-1}+\cdots$.

Metric signature and Fourier convention

Convention	Choice in this course	Alternative
Metric signature	mostly plus $(-,+,+,\ldots)$	mostly minus $(+,-,-,\ldots)$
Fourier mode	$e^{-i\omega t+i kx}$	$e^{+i\omega t-i kx}$
Retarded horizon condition	infalling $\sim(r_h-r)^{-i\omega/4\pi T}$	exponent sign flips with Fourier convention
Spectral function	$\rho=-2\operatorname{Im}G^R$	$+2\operatorname{Im}G^R$ in some texts
Conductivity	$\sigma=G^R/(i\omega)$	$\sigma=-G^R/(i\omega)$ if source convention differs

The safest way to compare signs is to compute something positive: entropy density, susceptibility, or positive-frequency spectral weight.

Restoring constants

Most holographic calculations set

$$\hbar=c=k_B=1.$$

To restore constants in thermal/transport statements:

Natural-units expression	With constants restored
$T$ as an energy	$k_B T$
$\omega/T$	$\hbar\omega/(k_B T)$
Planckian time $\tau\sim 1/T$	$\tau\sim\hbar/(k_B T)$
Chaos bound $\lambda_L\le2\pi T$	$\lambda_L\le2\pi k_B T/\hbar$
Chemical potential $\mu/T$	$\mu/(k_B T)$
Conductivity in $2+1$d	multiply dimensionless result by $e^2/h$ for electric units, if $J$ is electric charge current

For condensed-matter comparison, one must also decide whether the holographic $U(1)$ charge is normalized as particle number, electric charge, or some emergent conserved charge. The conversion to SI units is not universal; it depends on that identification.

Common pitfalls

Pitfall 1: confusing source-free with zero source

A spontaneous condensate means the source coefficient vanishes while the response coefficient is nonzero:

$$\phi_{(0)}=0, \qquad \langle O\rangle\ne0.$$

It does not mean the whole bulk field vanishes near the boundary. In a broken phase, the normalizable mode is precisely the order parameter.

Pitfall 2: forgetting contact terms

Conductivity, Hall response, stress-tensor correlators, and anomaly-induced currents can all contain contact terms. These are invisible if one only stares at the imaginary part of a numerical fluctuation ratio. They are fixed by the renormalized action, not by the second-order ODE alone.

Pitfall 3: treating $\bar\kappa$ as $\kappa$

Holographic horizon formulae often naturally give $\bar\kappa$, the heat conductivity at zero electric field. Experiments often report $\kappa$, the heat conductivity at zero electric current. They differ by

$$\kappa=\bar\kappa-T\alpha\sigma^{-1}\alpha.$$

Pitfall 4: using $\eta/s=1/(4\pi)$ outside its domain

The universal value holds for isotropic two-derivative Einstein gravity in the classical limit. Higher-derivative terms, anisotropy, explicit translation breaking in tensor channels, or finite-coupling corrections can change the answer.

Pitfall 5: overinterpreting a bottom-up normalization

A bottom-up action can compute robust dimensionless structures: pole motion, scaling exponents, ratios protected by symmetry, and hydrodynamic constraints. Absolute magnitudes require a normalization of $G_N$, $g_F$, operator normalizations, and the map between the model’s $U(1)$ and the experimental charge.

Pitfall 6: calling every finite DC conductivity “incoherent”

A finite DC conductivity can arise from momentum relaxation, pair creation, probe-sector dissipation, horizon absorption, disorder, or explicit lattices. “Incoherent” should mean that the measured current has negligible overlap with long-lived momentum, not merely that the system is strongly coupled.

A compact dictionary

The following table is the fastest way to orient a new computation.

Boundary question	Bulk operation	Minimal warning
What is the entropy density?	Horizon area density $s=A_h/(4G_NV)$	Extremal horizons may have residual entropy
What is the charge density?	Radial electric flux	Normalization depends on Maxwell coupling
What is the chemical potential?	Boundary-to-horizon potential difference	$A_t$ itself is gauge-dependent
What is the condensate?	Normalizable scalar mode with source set to zero	Counterterms and alternative quantization matter
What is $G^R$?	Linearized fluctuation with infalling horizon condition	Use renormalized canonical momenta
What are QNMs?	Source-free infalling eigenmodes	Remove pure gauge modes
What is $\sigma_{\rm dc}$?	Kubo formula or horizon DC method	Clean finite-density systems have momentum delta functions
What is $\eta$?	Metric tensor-channel fluctuation	Universality has assumptions
What is a Fermi surface?	Normalizable bulk spinor zero mode	Probe Fermi surfaces need not exhaust charge
What is an anomaly?	Chern—Simons inflow plus boundary counterterms	Consistent versus covariant currents differ
What is an explicit lattice?	Spatially dependent source	Do not confuse with spontaneous modulation
What is a density wave?	Source-free finite-$k$ normalizable mode	Sliding mode gives infinite conductivity unless pinned

Exercises

Exercise 1: translate radial conventions

A paper using coordinate $R\to\infty$ writes a scalar as

$$\phi(R,x)=a(x)R^{\Delta-d}+b(x)R^{-\Delta}+\cdots.$$

Rewrite this expansion in the $r\to0$ convention with $R=1/r$. Identify the source in standard quantization.

Solution

Using $R=1/r$,

$$R^{\Delta-d}=r^{d-\Delta}, \qquad R^{-\Delta}=r^\Delta.$$

Therefore

$$\phi(r,x)=a(x)r^{d-\Delta}+b(x)r^\Delta+\cdots.$$

In standard quantization, $a(x)$ is the source and $b(x)$ determines the vev, up to the factor $2\Delta-d$ and local counterterm contributions.

Exercise 2: scalar dimension from a bulk mass

For a scalar in $AdS_4$, suppose

$$m^2L^2=-2.$$

Find $\Delta_\pm$. Is alternative quantization allowed?

Solution

Here $d=3$. The dimension formula is

$$m^2L^2=\Delta(\Delta-d)=\Delta(\Delta-3).$$

So

$$\Delta(\Delta-3)=-2 \quad\Longrightarrow\quad \Delta^2-3\Delta+2=0.$$

Thus

$$\Delta_+=2, \qquad \Delta_-=1.$$

The alternative-quantization window is

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

For $d=3$, this is

$$-\frac{9}{4}<m^2L^2<-\frac{5}{4}.$$

Since $-2$ lies in this interval, alternative quantization is allowed.

Exercise 3: density as electric flux

Consider Maxwell theory in a fixed asymptotically $AdS_{d+1}$ background with

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

Show that the boundary charge density is proportional to the radial electric flux.

Solution

Vary the action:

$$\delta S_A = -\frac{1}{2g_F^2}\int \sqrt{-g}\,F^{ab}\delta F_{ab}.$$

Using $\delta F_{ab}=\partial_a\delta A_b-\partial_b\delta A_a$ and integrating by parts gives a bulk equation plus a boundary term:

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

Therefore the canonical radial momentum is

$$\Pi_A^\mu=-\frac{1}{g_F^2}\sqrt{-g}\,F^{r\mu}.$$

The source is $A_\mu^{(0)}$, so

$$\langle J^\mu\rangle = \lim_{r\to0}\Pi_A^\mu+\text{counterterms}.$$

For $\mu=t$ this is the charge density:

$$\rho=\langle J^t\rangle.$$

Thus density is radial electric flux, with normalization set by $g_F^2$ and counterterms.

Exercise 4: $\bar\kappa$ versus $\kappa$

Suppose an isotropic holographic model gives

$$\sigma=4, \qquad \alpha=3, \qquad \bar\kappa=20, \qquad T=2.$$

Compute the open-circuit thermal conductivity $\kappa$.

Solution

For an isotropic system,

$$\kappa=\bar\kappa-\frac{T\alpha^2}{\sigma}.$$

Substitute the data:

$$\kappa=20-\frac{2\times 3^2}{4}=20-\frac{18}{4}=20-4.5=15.5.$$

So

$$\kappa=\frac{31}{2}.$$

Exercise 5: dimensionality of conductivity

Using $J_i=\sigma E_i$, show that $[\sigma]=d-3$ in $d$ boundary spacetime dimensions.

Solution

The current has dimension

$$[J_i]=d-1.$$

The gauge potential $A_\mu$ has dimension $1$ because it couples as $\int d^d x\,A_\mu J^\mu$, so

$$[A_\mu]+[J^\mu]=d \quad\Longrightarrow\quad [A_\mu]=1.$$

The electric field has one derivative acting on $A$:

$$[E_i]=2.$$

Therefore

$$[\sigma]=[J_i]-[E_i]=(d-1)-2=d-3.$$

For $d=3$, the conductivity is dimensionless.

Exercise 6: identify a clean-limit delta function

A translationally invariant relativistic charged fluid has $\rho\ne0$ and $\epsilon+p<\infty$. Explain why the real part of $\sigma(\omega)$ contains a delta function at $\omega=0$.

Solution

At finite density, the electric current overlaps with momentum. In a relativistic fluid,

$$J^i=\rho v^i+\cdots, \qquad P^i=T^{ti}=(\epsilon+p)v^i+\cdots.$$

Therefore the current contains a piece proportional to conserved momentum:

$$J^i\supset \frac{\rho}{\epsilon+p}P^i.$$

If translations are exact, $P^i$ does not decay. A small electric field accelerates the conserved momentum indefinitely, producing an infinite DC response. In frequency space, the hydrodynamic conductivity contains

$$\sigma(\omega)\supset \frac{\rho^2}{\epsilon+p}\frac{i}{\omega+i0^+}.$$

Using

$$\frac{i}{\omega+i0^+}=\pi\delta(\omega)+i\,\mathcal P\frac{1}{\omega},$$

the real part contains a delta function. This is a consequence of momentum conservation, not quasiparticles.

Exercise 7: entropy scaling in a hyperscaling-violating geometry

A scaling phase in $d_s=2$ spatial dimensions has $z=3/2$ and $\theta=1$. What is the low-temperature entropy scaling?

Solution

The entropy density scales as

$$s\sim T^{(d_s-\theta)/z}.$$

With $d_s=2$, $\theta=1$, and $z=3/2$,

$$\frac{d_s-\theta}{z} = \frac{2-1}{3/2} = \frac{2}{3}.$$

Thus

$$s\sim T^{2/3}.$$

Further reading

For the standard field/operator map, holographic renormalization, and correlation-function normalization, see the AdS/CFT textbook treatments by Ammon—Erdmenger, Natsuume, Năstase, and the classic large-$N$/AdS review by Aharony—Gubser—Maldacena—Ooguri—Oz. For holographic quantum matter conventions, especially transport, finite density, $AdS_2$ exponents, probe fermions, incoherent currents, and memory-matrix notation, use Hartnoll—Lucas—Sachdev as the main reference spine. For entanglement conventions, use Rangamani—Takayanagi.