Spinning Geodesic Witten Diagrams

TL;DR

This work extends the geodesic Witten diagram approach to spinning operators, providing a pair-of-geodesics integral representation for four-point conformal blocks of symmetric traceless fields with arbitrary spin. By constructing a bulk basis of three-point vertices and establishing a 1:1 mapping to boundary tensor structures, the authors connect spinning blocks to both bulk cubic couplings and the shadow formalism via the split representation and monodromy projection. They prove that the spinning geodesic Witten diagrams satisfy the conformal Casimir equation and exhibit correct short-distance behavior, and they show how these bulk constructions reproduce the shadow integral results, offering a bulk-centric viewpoint on conformal blocks. The paper also sketches hybrid and single-geodesic formulations, highlighting practical benefits for calculating spinning blocks and insights into bulk geometry's role in encoding conformal data, with potential implications for geodesic OPE blocks and stress-tensor exchanges.

Abstract

We present an expression for the four-point conformal blocks of symmetric traceless operators of arbitrary spin as an integral over a pair of geodesics in Anti-de Sitter space, generalizing the geodesic Witten diagram formalism of Hijano et al [arXiv:1508.00501] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk three-point interaction vertices which give rise to all possible boundary three point structures. We highlight a direct connection between the representation of the conformal block as a geodesic Witten diagram and the shadow operator formalism.

Figures (8)

• Figure 1: A geodesic Witten diagram (GWD) for a scalar conformal block. The GWD consists of a regular exchange Witten diagram, where the interaction vertices are restricted to lie on the geodesics $\gamma_{12}$,$\gamma_{34}$ connecting the boundary operators.
• Figure 2: The geodesic Witten diagram (GWD) for the harmonic function. On the LHS, the bulk propagator in the regular GWD has been replaced by a bulk harmonic function, represented by the double line. On the RHS, an equivalent representation for the bulk harmonic function is given in the 'split representation' where bulk-to-boundary operators for the bulk field and its shadow are integrated over the boundary, as in Eq. \ref{['simpsplitrep']}.
• Figure 3: The geodesic Witten diagram is recovered from acting on the bulk harmonic with a suitable projector. This is equivalent to acting with a projector on the shadow-operator representation of the harmonic function, and demonstrates the connection between the conformal block and the GWD.
• Figure 4: The geodesic witten diagram for two scalar and one vector operator. The vertex $V_{A}(X_{\sigma})$ is integrated over the geodesic $\gamma_{12}$.
• Figure 5: A geodesic witten diagram for a spinning conformal block. The vertices $V_{I}$ and $V_{J}$ are integrated over the geodesics $\gamma_{12}$ and $\gamma_{34}$ respectively.