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.
In flat dimensions, an infinitesimal conformal transformation generated by satisfies .
Conformal invariance fixes the two-point function of identical scalar primaries of dimension to .
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.
Fefferman--Graham coordinates put an asymptotically anti-de Sitter metric in the form .
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.

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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.