Witten Diagram: AdS/CFT Correlators

• Witten diagrams are AdS analogues of Feynman diagrams that compute CFT correlation functions using bulk-to-boundary propagators, bulk interactions, and bulk-to-bulk propagators.
• Geodesic Witten diagrams restrict interaction vertices to boundary-anchored geodesics, isolating individual conformal blocks or partial waves through Casimir equations and OPE boundary conditions.
• The framework extends to spinning, fermionic, antisymmetric, and mixed-symmetry fields and helps decompose exchange diagrams into single-trace and double-trace conformal families.

A Witten diagram is an AdS analogue of a Feynman diagram used to compute conformal-field-theory correlation functions through the AdS/CFT correspondence. External CFT operators are represented by bulk-to-boundary propagators, bulk interaction vertices are integrated over Euclidean $\mathrm{AdS}_{d+1}$, and internal fields are represented by bulk-to-bulk propagators. At large $N$, equivalently at small bulk Newton constant $G_N\sim N^{-2}$, the AdS path integral is evaluated semiclassically by summing such diagrams. A geodesic Witten diagram is a related object in which interaction vertices are restricted to bulk geodesics anchored at selected boundary pairs; after suitable normalization and choice of OPE branch, it represents an individual conformal block or conformal partial wave rather than the full correlator (Hijano et al., 2015).

1. AdS/CFT definition and geometric ingredients

In Poincaré coordinates, Euclidean $\mathrm{AdS}_{d+1}$ has metric

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$

with unit AdS radius. A scalar bulk field of mass $m$ is dual to a boundary primary of dimension $\Delta$ satisfying

$m^2=\Delta(\Delta-d).$

The two formal solutions are $\Delta$ and $d-\Delta$; standard AdS/CFT quantization selects the solution associated with the chosen boundary condition.

The fundamental scalar propagators are the bulk-to-boundary propagator

$N$0

up to normalization, and the bulk-to-bulk propagator

$N$1

where $N$2 is the geodesic distance. In terms of

$N$3

one may equivalently write

$N$4

A local quartic interaction integrated over one bulk point gives a contact Witten diagram. For a nonderivative interaction $N$5,

$N$6

where

$N$7

This is the AdS $N$8-function. Derivative contact interactions produce shifted $N$9-functions and polynomial Mellin amplitudes.

An ordinary scalar exchange Witten diagram is

$G_N\sim N^{-2}$0

For spin $G_N\sim N^{-2}$1, the internal propagator is a bitensor whose indices are contracted with tensor structures and derivatives supplied by the cubic vertices. The two bulk interaction points are integrated over all of AdS, making exchange diagrams substantially more difficult than contact diagrams.

2. Conformal blocks and the geodesic construction

A scalar four-point function can be decomposed into conformal partial waves,

$G_N\sim N^{-2}$2

where

$G_N\sim N^{-2}$3

A conformal block contains a primary and all descendants in one conformal family. A conformal partial wave is naturally produced by the shadow formalism and generally contains both a block and its shadow:

$G_N\sim N^{-2}$4

Let $G_N\sim N^{-2}$5 and $G_N\sim N^{-2}$6 be the bulk geodesics joining the boundary points $G_N\sim N^{-2}$7 and $G_N\sim N^{-2}$8. The scalar geodesic Witten diagram is

$G_N\sim N^{-2}$9

The geodesic between embedding-space boundary points $\mathrm{AdS}_{d+1}$0 and $\mathrm{AdS}_{d+1}$1 has the parametrization

$\mathrm{AdS}_{d+1}$2

In Poincaré coordinates it is a semicircle, with

$\mathrm{AdS}_{d+1}$3

For scalar external operators, the geodesic diagram satisfies

$\mathrm{AdS}_{d+1}$4

where

$\mathrm{AdS}_{d+1}$5

and $\mathrm{AdS}_{d+1}$6 is defined analogously. Removing the external-position prefactor from the partial wave gives the conformal block. This relation is exact for arbitrary spacetime dimension, external dimensions, and exchanged dimension; it is not a large-dimension or saddle-point approximation (Hijano et al., 2015).

The geometric distinction is therefore:

• Ordinary exchange diagram: vertices integrated over all of AdS and containing single-trace and double-trace conformal families.
• Geodesic exchange diagram: vertices integrated only along boundary-anchored geodesics and representing one exchanged conformal family after the appropriate normalization and branch selection.

3. The Casimir characterization

The central proof uses the conformal quadratic Casimir. For an exchanged primary of dimension $\mathrm{AdS}_{d+1}$7 and spin $\mathrm{AdS}_{d+1}$8,

$\mathrm{AdS}_{d+1}$9

in one sign convention, or

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$0

in the opposite convention.

A conformal partial wave satisfies

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$1

Consider the part of a geodesic diagram supported on $ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$2:

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$3

Invariance under simultaneous AdS and conformal transformations gives

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$4

For a scalar bulk function,

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$5

Away from coincidence, the bulk-to-bulk propagator obeys the homogeneous equation

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$6

Consequently,

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$7

The geodesic restriction is essential. In an ordinary exchange diagram, the two bulk points can coincide. Acting with the bulk Laplacian on the propagator then produces its delta-function source, so the resulting Casimir equation is inhomogeneous and generates contact contributions. An arbitrary curve would also introduce geometric data not determined solely by the boundary points. A geodesic is the natural conformally invariant curve selected by its two endpoints.

The Casimir equation alone admits both the direct and shadow solutions. The OPE asymptotics select the desired branch:

$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$8

Thus the Casimir eigenvalue together with the direct-channel boundary behavior uniquely identifies the geodesic diagram with the conformal block or partial wave.

4. Spin, tensor structures, and generalized representations

For a spin-$ds^2=\frac{du^2+dx^i dx^i}{u^2}, \qquad u>0,$9 exchanged CFT primary, the dual AdS field is a symmetric traceless rank-$m$0 tensor. Its equation of motion can be written as

$m$1

together with

$m$2

The corresponding geodesic diagram pulls the tensor propagator back onto the geodesic tangent vectors:

$m$3

It obeys the same Casimir equation as the spin-$m$4 partial wave.

The construction extends to external symmetric traceless tensors. Boundary polarizations $m$5 package tensor indices, with

$m$6

Bulk polarizations $m$7 satisfy

$m$8

The spinning bulk-to-boundary propagator is

$m$9

For arbitrary external spins $\Delta$0 and exchanged spin $\Delta$1, independent three-point tensor structures are indexed by integers describing derivative contractions and direct index contractions. A preferred boundary basis is built from

$\Delta$2

where

$\Delta$3

The paper "Spinning Geodesic Witten Diagrams" constructs bulk cubic vertices in one-to-one correspondence with a preferred basis of all symmetric-traceless boundary three-point structures. The resulting two-geodesic integrals represent spinning conformal blocks for arbitrary integer external spins and exchanged spin (Dyer et al., 2017). The construction is kinematic: the bulk vertices need not arise from a consistent interacting higher-spin theory. For multiple boundary tensor structures, the geodesic diagrams form a corresponding multiplet of conformal partial waves.

The same framework has been extended to additional representations:

• External spinning fields: scalar exchange with one or more external symmetric traceless fields can be generated by spinning propagators, derivative couplings, or embedding-space differential operators (Nishida et al., 2016).
• Antisymmetric exchange: $\Delta$4-form fields are represented using Grassmann-odd polarizations. Their geodesic diagrams obey the Casimir equation with eigenvalue

$\Delta$5

and reproduce partial waves for antisymmetric exchange (Tamaoka, 2017).

• Fermionic exchange: Dirac spinors require a modified embedding-space bulk constraint, and their geodesic exchange diagrams reproduce spin-$\Delta$6 conformal partial waves. The fermionic propagator decomposes into chiral components corresponding to independent spinor structures (Nishida et al., 2018).
• Mixed-symmetry fields: Grassmann-odd and Grassmann-even auxiliary variables provide a natural extension to hooks and other mixed-symmetry representations, although general bulk-to-bulk propagators are not constructed explicitly (Tamaoka, 2017).

5. Shadow formalism, split representations, and Witten-diagram decomposition

The shadow of a primary of dimension $\Delta$7 has dimension

$\Delta$8

and the same spin. Harmonic functions in AdS contain both branches:

$\Delta$9

Their split representations factorize the bulk harmonic function through an integral over a boundary point:

$m^2=\Delta(\Delta-d).$0

After inserting a split representation into a geodesic diagram, the two geodesics become three-point geodesic diagrams joined by a boundary shadow integral. This is the direct AdS realization of the CFT shadow construction. The harmonic-function diagram produces a direct-minus-shadow combination, while the physical bulk-to-bulk propagator and the desired OPE asymptotics select the direct block.

The same mechanism decomposes ordinary exchange Witten diagrams. A scalar product of bulk-to-boundary propagators admits a geodesic expansion,

$m^2=\Delta(\Delta-d).$1

with

$m^2=\Delta(\Delta-d).$2

and

$m^2=\Delta(\Delta-d).$3

The dimensions $m^2=\Delta(\Delta-d).$4 are the leading dimensions of scalar double-trace operators

$m^2=\Delta(\Delta-d).$5

For a scalar contact diagram, applying this identity to both pairs of external legs and using the bulk resolvent identity gives a sum of geodesic Witten diagrams containing only double-trace blocks. For scalar exchange,

$m^2=\Delta(\Delta-d).$6

The first term is the exchanged single-trace block; the remaining towers are double-trace blocks. Contact diagrams contain only double-trace conformal families, whereas exchange diagrams contain a single-trace family together with double-trace families (Hijano et al., 2015).

For exchanged spin $m^2=\Delta(\Delta-d).$7, the split representation contains harmonic functions of spins $m^2=\Delta(\Delta-d).$8. Consequently, scalar-external exchange diagrams contain double-trace towers of spins up to $m^2=\Delta(\Delta-d).$9. Apparent spurious poles in lower-spin terms cancel after summing the complete split representation (1702.08818).

Ordinary spinning exchange diagrams can be treated systematically by first diagonalizing the map between local AdS cubic couplings and CFT three-point structures, then inserting the harmonic split representation and identifying the resulting products of three-point amplitudes with spinning conformal partial waves. The physical single-trace pole comes from the exchanged bulk field; double-trace poles arise from the AdS integrations and from lower-spin or off-shell sectors (Sleight et al., 2017).

When two double-trace masses coincide, resolvent denominators vanish and difference quotients become derivatives with respect to the exchanged mass or dimension:

$\Delta$0

On the CFT side,

$\Delta$1

so logarithms in tree-level Witten diagrams encode anomalous dimensions of double-trace operators.

6. Applications, extensions, and computational methods

Geodesic Witten diagrams have applications beyond four-point scalar correlators. In two-dimensional CFTs, global torus blocks can be represented by single-winding Witten diagrams in thermal AdS. For a torus one-point block, the one-winding bulk propagator obeys the same global $\Delta$2 Casimir equation as the CFT block, and the low-temperature condition

$\Delta$3

selects the desired solution. In the semiclassical limit, conformal blocks exponentiate into networks of bulk geodesics whose vertices obey force-balance conditions. A Chern–Simons description gives an equivalent Wilson-line network, with a line wrapping the thermal cycle implementing $\Delta$4 (Kraus et al., 2017).

In holographic boundary and defect CFTs, weighted geodesic or X-ray transforms extract radial modes in an $\Delta$5-sliced bulk geometry. The resulting $\Delta$6 bulk-to-bulk propagators reproduce boundary-channel conformal blocks. Ambient-channel blocks arise from geodesic diagrams connected to deformation sources. This provides an equivalence between boundary-channel mode decompositions and ambient-channel source-insertion diagrams in fully backreacted holographic boundary geometries (Karch et al., 2017).

For entanglement entropy of two disjoint intervals, the identity geodesic diagram reproduces the Ryu–Takayanagi contribution. Higher conformal blocks correspond to stripped geodesic diagrams in which external bulk-to-boundary legs are removed, leaving bulk-to-bulk propagation between the two RT geodesics. These diagrams represent bulk entanglement contributions at order $\Delta$7, while replica-induced backreaction supplies additional quantum corrections (Prudenziati, 2019).

Geodesic diagrams also furnish computational bases for recursion and dimensional reduction. The AdS equation of motion implies that a conformal Casimir acting on an exchange diagram produces a finite sum of contact diagrams. In crossed-channel decompositions, this produces finite-difference relations among double-trace coefficients. In one dimension the recursion is three-term and is determined by the lowest-dimension double-trace coefficients; in higher dimensions it is five-term, and the seed data lie along the entire minimal-twist line (Zhou, 2018).

The same transfer principle applies to dimensional-reduction identities. A finite five-term relation connects exchange Witten diagrams in dimensions $\Delta$8 and $\Delta$9, paralleling the corresponding conformal-block identity associated with Parisi–Sourlas dimensional reduction:

$d-\Delta$0

The relation depends on compatible contact-term conventions because spinning exchange diagrams are defined only up to contact diagrams (Zhou, 2020).

Differential representations provide another approach. Bulk covariant derivatives can be replaced by conformal generators acting on contact diagrams, and a Casimir cut removes an exchanged single-trace propagator by converting it into a delta function. In Mellin space, the resulting differential equations become finite-difference equations. This method has been used to compute scalar six- and eight-point amplitudes mediated by gluons and scalars (Li et al., 2023). A related operator-valued-integral formalism extends differential representations beyond tree level by using split representations and spectral integration over auxiliary boundary points. It has produced explicit three-point bubble and triangle results, including

$d-\Delta$1

in $d-\Delta$2, and triangle results for $d-\Delta$3 (Herderschee, 2021).

7. Scope, limitations, and conceptual status

A geodesic Witten diagram is not itself a complete holographic correlator. An ordinary Witten diagram includes all conformal families compatible with the bulk interactions: a single-trace exchange family, double-trace towers, descendants, and contact contributions. A geodesic diagram isolates the kinematic contribution of one conformal representation after the correct branch and normalization are chosen.

Several qualifications are intrinsic to the construction. Normalizations depend on bulk-to-boundary and bulk-to-bulk propagators, CFT two- and three-point functions, shadow projectors, and bulk interactions. For general spinning operators, multiple three-point tensor structures require matrix-valued dictionaries between boundary structures and bulk vertices. Mixed-symmetry representations require additional auxiliary variables and propagator technology. Intersecting geodesics and contact terms require separate treatment in Casimir arguments.

Geodesic restriction can also change the appropriate bulk interaction. A cubic vertex that is nonzero after integration over all of AdS may vanish when naively restricted to a particular geodesic. Integration by parts can produce boundary terms, and the geodesic arrangement breaks the cyclic symmetry of an ordinary cubic vertex. Consequently, decomposing a spinning Witten diagram into geodesic constituents requires checking each vertex and geodesic assignment separately (1702.08818).

The representation is primarily kinematic. It depends on conformal symmetry, bulk equations of motion, geodesic geometry, and OPE boundary conditions, but it does not require a conventional gravitational dual. Its central correspondence is

$d-\Delta$4

Within its stated assumptions, this correspondence converts conformal-block decompositions of tree-level Witten diagrams from difficult bulk integrations into geometric integrals, propagator identities, Casimir equations, split representations, and algebraic or recursive manipulations.