The AdS/CFT correspondence identifies a conformal field theory with quantum gravity on an asymptotically anti-de Sitter spacetime. Boundary sources couple to CFT operators and equal boundary values of bulk fields.
A Euclidean conformal transformation pulls the flat metric back to a position-dependent positive multiple of itself. Its infinitesimal generator obeys the conformal Killing equation.
Three scalar primaries have a conformally fixed position dependence, leaving one theory-dependent coefficient .
A Witten diagram is a position-space perturbative diagram in anti-de Sitter spacetime whose external bulk-to-boundary propagators end at boundary operator insertions.
In a large-rank gauge theory, a single-trace operator is dual at leading order to a single bulk field, while multi-trace operators correspond to multiparticle bulk states.
The holographic dictionary identifies leading near-boundary bulk-field coefficients with CFT sources and subleading normalizable coefficients with operator expectation values.
A scalar field of mass in behaves near the conformal boundary as , where . The leading coefficient is the source and the normalizable coefficient determines the operator expectation value.
For a scalar field of mass on of radius , the scaling dimension of the dual scalar primary obeys . 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 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 .
For a static holographic state and boundary region ,where is the least-area bulk surface homologous to and anchored on .
For a round disk of radius in the vacuum of a holographic conformal field theory on flat space, the static Poincare-AdS Ryu--Takayanagi surface is the hemisphere .
The Brown--Henneaux central charge is the central charge of the two-dimensional conformal field theory dual to Einstein gravity on :
For a unit spatial circle, the vacuum energy of a two-dimensional conformal field theory is . 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 .
A Hubeny–Rangamani–Takayanagi surface is the covariant extremal-area surface homologous to a boundary region. Its area divided by 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.
Linearizing the von Neumann entropy around a reference state gives . Around a thermal state this becomes .
Holographic mutual information is nonnegative because the disconnected union is an admissible surface for , 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.
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The AdS/CFT correspondence, also known as the Anti-de Sitter/Conformal Field Theory correspondence, is a theoretical framework in theoretical physics that relates two seemingly different types of physical theories. Specifically, it suggests a relationship between a type of string theory formulated on a certain geometric space known as Anti-de Sitter (AdS) space and a conformal field theory (CFT) defined on the boundary of that space.