Gauge-gravity duality

Gauge-gravity duality relates a quantum field theory description to a description involving dynamical gravity, usually with additional string degrees of freedom. In the best-understood AdS/CFT correspondence, the field theory lives on the spacetime boundary and the gravitational description has an extra radial direction, possibly together with compact internal dimensions.

The two descriptions assign different variables to the same observables: a field-theory local field operator corresponds to a bulk field, and a boundary correlation function can be computed from a bulk response. A weakly curved classical bulk may describe strongly coupled field-theory dynamics when the required large-rank and coupling limits hold. The utility is therefore calculational as well as conceptual, but the existence of a field theory does not automatically guarantee a simple classical gravitational dual.

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 local field operator specifies how it rescales under a dilation. To avoid mixing active and passive conventions, write a scalar transformation as $x^{\prime }=\lambda x$, ${\mathcal{O}}^{\prime }(x^{\prime })={\lambda }^{-\Delta }\mathcal{O}(x)$, where $\Delta$ is the scaling dimension.

Consequently, an identical-scalar two-point function in a conformal field theory has distance dependence $|x-y{|}^{-2\Delta }$ up to normalization. The classical dimension follows from the action's units; quantum effects can add an anomalous dimension. Scaling dimensions also classify deformations: coupling an operator of dimension $\Delta$ in $d$ dimensions gives a coupling of dimension $d-\Delta$. Whether that coupling grows or shrinks under coarse graining controls its relevance near a renormalization-group fixed point.

An exactly marginal operator has scaling dimension equal to the spacetime dimension and a deformation coupling whose renormalization-group 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.

A single-trace operator contains one trace over matrix-valued gauge group indices. For adjoint matrix fields, $\mathrm{Tr}({\Phi }^2)$ is a simple example, whereas $\mathrm{Tr}({\Phi }^2)\mathrm{Tr}({\Phi }^3)$ is a multi-trace product. The trace makes these expressions invariant under simultaneous conjugation by the gauge group.

In a suitable large-rank holographic theory, normalized single-trace operators correspond at leading order to single-particle bulk excitations, while multi-trace products create multiparticle states. This is an additional dynamical interpretation, rather than the definition of taking one trace. Mixing and finite-rank trace identities qualify the separation beyond the ideal large-rank limit. The distinction organizes correlation function counting and the bulk field content in AdS/CFT correspondence.

The holographic dictionary maps observables and parameters between a boundary quantum field theory and a bulk gravitational description. In the ordinary scalar Dirichlet prescription, one near-boundary coefficient acts as the source for a boundary local field operator, while another independent coefficient is related to its expectation value after holographic renormalization.

The AdS scalar mass--dimension relation connects the bulk mass to the boundary scaling dimension. Solving the bulk field equation with a specified boundary source then determines the response and hence boundary correlation functions. Source and response assignments depend on boundary conditions: alternative quantization, logarithmic terms and local counterterms require care. They cannot always be identified just by calling every leading coefficient a source and every subleading coefficient an expectation value.

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

For a free scalar field of mass $m$ in Anti-de Sitter spacetime of dimension $d+1$ and radius $L$, use coordinates withThe conformal boundary is at $z=0$. The Klein-Gordon equation gives the radial leading balance $z^2{\varphi }_{zz}-(d-1)z{\varphi }_z-m^2{L}^2\varphi \simeq 0$. A power law $z^{\Delta }$ therefore obeys the AdS scalar mass--dimension relationFor distinct roots and generic nonresonant parameters,In standard quantization, $J$ is the source of the dual local field operator; $A$ determines its expectation value after holographic renormalization, with a normalization fixed by the bulk action. ^[ref]

The Breitenlohner-Freedman bound $m^2{L}^2\ge -d^2/4$ makes $\nu$ real. At $\nu =0$ the roots coincide and a second solution involves $z^{d/2}\log z$. Resonant expansions can also contain logarithms. In the appropriate mass window, Alternative quantization of an AdS scalar field changes which coefficient acts as source. ^[ref]

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 two-dimensional conformal field theory on a spatial circle of circumference $L$, with propagation speed $v$ and left/right central charges $c_L,c_R$, the plane-vacuum sector has Casimir energyWhen $c_L=c_R=c$, the unit-radius convention $L=2\pi$, $\hslash =v=1$ gives $E_0=-c/12$. A circle of unit circumference instead gives $E_0=-\pi c/6$ in these units. Specifying radius versus circumference prevents a factor-of-$2\pi$ ambiguity.

The exponential map from a cylinder to the complex plane shifts the stress-energy tensor by its Schwarzian derivative term. Equivalently, the cylinder Hamiltonian is $H=(2\pi \hslash v/L)[L_0+{\bar{L}}_0-(c_L+c_R)/24]$. Setting both plane-vacuum weights to zero yields the formula. Other sectors can have different lowest weights. In AdS/CFT correspondence, the unit-radius result matches the renormalized vacuum energy of global three-dimensional Anti-de Sitter spacetime.^[ref]

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.

Holographic D.C. conductivity describes the zero-frequency electric response of a boundary system using a bulk gauge field. In a clean normal state at nonzero charge density, the electric current generally overlaps with conserved momentum. A constant electric field then continuously accelerates that momentum, producing a zero-frequency delta function rather than an ordinary finite D.C. response.

Introducing relaxation of momentum can broaden this contribution and yield finite conductivity when no other protected dissipationless channel remains. That qualification matters: a superfluid can retain a delta function even when translations are broken. One must specify the phase, conserved quantities and frequency limit before interpreting a bulk calculation as a measured D.C. conductivity.

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.