Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks

TL;DR

This work introduces geodesic Witten diagrams as the holographic duals of conformal blocks, providing a position-space, integration-free method to decompose Witten diagrams into conformal blocks in any dimension. The authors prove, via direct computation and Casimir equations in embedding space, that scalar geodesic diagrams reproduce scalar conformal blocks exactly, and extend the construction to spinning exchanges with explicit Spin-1 and Spin-2 results. They derive a transparent propagator identity that expresses contact and exchange scalar diagrams as sums over geodesic blocks, naturally generating towers of double-trace exchanges and recovering logarithmic terms from anomalous dimensions. The framework generalizes to arbitrary spin, yielding a Casimir-consistent decomposition into blocks, and offers a powerful, geometrically intuitive alternative to Mellin-space methods for holographic CFTs, with clear paths toward loops, higher-point functions, and Virasoro/CFT2 extensions.

Abstract

We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram," which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

Figures (8)

• Figure 1: This is a geodesic Witten diagram in AdS$_{d+1}$, for the exchange of a symmetric traceless spin-$\ell$ tensor with $m^2=\Delta(\Delta-d)-\ell$ in AdS units. Its main feature is that the vertices are integrated over the geodesics connecting the two pairs of boundary points, here drawn as dashed orange lines. This computes the conformal partial wave for the exchange of a CFT$_d$ primary operator of spin $\ell$ and dimension $\Delta$.
• Figure 2: Tree-level four-point Witten diagrams for external scalar operators. On the left is a contact diagram. On the right is an exchange diagram for a symmetric traceless spin-$\ell$ tensor field of dual conformal dimension $\Delta$. Here and throughout this work, orange dots denote vertices integrated over all of AdS.
• Figure 3: The identity \ref{['41a']} obeyed by AdS scalar propagators. The internal line represents bulk-to-bulk propagator for a scalar field of mass $m^2=\Delta_m(\Delta_m-d)$. $a^{12}_m$ and $\Delta_m$ are defined in \ref{['41b']} and \ref{['41c']}, respectively.
• Figure 4: The decomposition of a four-point scalar contact diagram into conformal partial waves disguised as geodesic Witten diagrams. Passage to the second line uses \ref{['41f']}, and passage to the last line uses \ref{['41g']}. The last line captures the infinite set of CFT exchanges of the double-trace operators $[{\cal O}_1{\cal O}_2]_{m,0}$ and $[{\cal O}_3{\cal O}_4]_{n,0}$. We have suppressed OPE coefficients; the exact result is in equation \ref{['41i']}.
• Figure 5: The decomposition of a four-point scalar exchange diagram (upper left) into conformal partial waves, for an exchanged scalar $\phi$ of mass $m^2=\Delta(\Delta-d)$. We have skipped the intermediate steps, which are nearly identical to those of the contact diagram. The term in the upper right captures the single-trace exchange of the scalar operator dual to $\phi$. The second line captures the infinite set of CFT exchanges of the double-trace operators $[{\cal O}_1{\cal O}_2]_{m,0}$ and $[{\cal O}_3{\cal O}_4]_{n,0}$. We have suppressed OPE coefficients; the exact result is in equations \ref{['43d']}--\ref{['43e']}.