Holographic dictionary

The holographic dictionary identifies leading near-boundary bulk-field coefficients with CFT sources and subleading normalizable coefficients with operator expectation values.

Fefferman--Graham coordinates put an asymptotically anti-de Sitter metric in the form $d{s}^2=L^2[d{Z}^2/Z^2+g_{ij}(Z,x)d{x}^{i}d{x}^j/Z^2]$.

A scalar field of mass $m$ in ${\mathrm{AdS}}_{d+1}$ behaves near the conformal boundary as $\varphi =z^{d-\Delta }J+z^{\Delta }A+\cdots$, where $m^2=\Delta (\Delta -d)$. The leading coefficient is the source and the normalizable coefficient determines the operator expectation value.

For a scalar field of mass $m$ on ${\mathrm{AdS}}_{d+1}$ of radius $L$, the scaling dimension of the dual scalar primary obeys $m^2{L}^2=\Delta (\Delta -d)$. The two roots correspond to the two independent near-boundary falloffs.

When both scalar falloffs in anti-de Sitter spacetime are normalizable, alternative quantization exchanges which near-boundary coefficient is treated as the source and which as the expectation value. In AdS4, a field with $m^2=-2$ may be dual to a scalar operator of dimension one.

Holographic renormalization subtracts local boundary counterterms from the regulated on-shell bulk action. It gives finite correlation functions and identifies the renormalized normalizable coefficient with an operator expectation value.

At leading semiclassical order, holographic entanglement entropy is the area of a bulk extremal surface anchored on the boundary of the chosen region, divided by $4G_N$.

For a static holographic state and boundary region $A$,where ${\gamma }_A$ is the least-area bulk surface homologous to $A$ and anchored on $\partial A$.

For a round disk of radius $R$ in the vacuum of a holographic conformal field theory on flat space, the static Poincare-AdS Ryu--Takayanagi surface is the hemisphere $r^2+z^2=R^2$.

The Brown--Henneaux central charge is the central charge of the two-dimensional conformal field theory dual to Einstein gravity on ${\mathrm{AdS}}_3$:

For a unit spatial circle, the vacuum energy of a two-dimensional conformal field theory is $E_0=-c/12$. It arises from the anomalous Schwarzian shift when the plane vacuum is mapped to the cylinder and is dual to the renormalized energy of global ${\mathrm{AdS}}_3$.

A Hubeny–Rangamani–Takayanagi surface is the covariant extremal-area surface homologous to a boundary region. Its area divided by $4G_N$ gives leading semiclassical entanglement entropy in a time-dependent holographic spacetime.

A unitary acting on only one factor of a thermofield double state preserves the spectrum of either reduced density matrix and hence preserves left--right entanglement. A bulk shell created on one boundary can enlarge the late event horizon while the HRT surface remains at the earlier bifurcation surface.

For two copies of a quantum system,Tracing out either copy produces the thermal density operator $e^{-\beta H}/Z$.

The modular Hamiltonian of a density operator is $K=-\log \rho$. For a Gibbs state, $K=\beta H+\log Z$.

Linearizing the von Neumann entropy around a reference state gives $\delta S=\mathrm{Tr}(\delta \rho K)$. Around a thermal state this becomes $\delta S=\beta \delta ⟨H⟩$.

Holographic mutual information is nonnegative because the disconnected union ${\gamma }_A\cup {\gamma }_B$ is an admissible surface for $A\cup B$, whose minimizing surface can have no larger area.

As two boundary regions separate, the minimizing surface for their union can switch from connected to disconnected. At leading order this produces a continuous entropy with a kink and makes holographic mutual information vanish beyond a critical separation.

For a bulk gauge field, the leading boundary value of its time component is the dual chemical potential, while the normalizable radial coefficient determines the charge density.

At nonzero charge density, a translationally invariant holographic state has infinite D.C. conductivity because an electric field continuously injects conserved momentum. Momentum relaxation broadens the zero-frequency delta function and makes the D.C. value finite.