Gauge-gravity duality

Gauge/gravity duality relates a quantum field theory without dynamical gravity to a gravitational or string theory in one additional emergent dimension.

A conformal field theory is a quantum field theory invariant under conformal transformations. Local operators are organized into conformal families labelled by scaling dimension and spin.

A conformal cross-ratio is a dimensionless combination of point separations invariant under conformal transformations. Four generic points in dimension greater than two have two independent cross-ratios,

The scaling dimension of a local operator is its eigenvalue under dilations. Under $x\mapsto \lambda x$, a scalar operator of dimension $\Delta$ transforms by $\mathcal{O}(x)\mapsto {\lambda }^{-\Delta }\mathcal{O}(\lambda x)$.

An exactly marginal operator has scaling dimension equal to the spacetime dimension and a deformation coupling whose beta function vanishes to all orders. Varying that coupling moves along a family of conformal field theories without introducing a scale.

The conformal algebra in $d$ Euclidean dimensions is generated by translations $P_a$, rotations $M_{ab}$, dilations $D$, and special conformal transformations $K_a$.

A conformal primary operator is annihilated at the origin by every special conformal generator. Acting with translation generators produces its conformal descendants.

A scalar primary operator transforms without a spin rotation and is characterized by its scaling dimension. Conformal symmetry fixes its two- and three-point functions up to constants, while a four-point function can depend on conformal cross-ratios.

A conformal descendant is obtained by applying one or more translation generators to a conformal primary operator. Its scaling dimension exceeds that of the primary by a nonnegative integer.

Positivity of descendant norms constrains the scaling dimension of a primary operator. For a nonidentity scalar primary in $d\ge 2$, unitarity requires $\Delta \ge (d-2)/2$.

The state–operator correspondence maps a local operator of scaling dimension $\Delta$ to a state on $S^{d-1}\times \mathbb{R}$ whose cylinder energy is $\Delta /r$ for sphere radius $r$.

For a primary operator, $T(z)\mathcal{O}(0)\sim h\mathcal{O}(0)/z^2+\partial \mathcal{O}(0)/z$. Taking the residue against $z^{n+1}$ gives the weight for $n=0$, the derivative descendant for $n=-1$, and zero for every $n>0$. Conversely the absence of positive-mode actions excludes higher poles in this operator product and gives the primary transformation law. The antiholomorphic argument is identical. Thus primary states and primary operators correspond under radial quantization.

If the only light field in AdS4 is the graviton, the dual CFT3 spectrum begins with the vacuum at dimension zero, the spin-two stress tensor at dimension three, and its non-null level-one descendants of spins two and three at dimension four.

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.

Conformal invariance fixes the two-point function of identical scalar primaries of dimension $\Delta$ to $C_2/|x_{12}{|}^{2\Delta }$.

Three scalar primaries have a conformally fixed position dependence, leaving one theory-dependent coefficient $C_{123}$.

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 $d{s}^2=L^2[d{Z}^2/Z^2+g_{ij}(Z,x)d{x}^{i}d{x}^j/Z^2]$.

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.

ABJM theory is a three-dimensional supersymmetric Chern–Simons matter theory with a large-$N$ holographic dual. In its M-theory regime the bulk Newton constant scales as $G_N\sim N^{-3/2}$ in AdS units.