Geodesic Witten diagrams with an external spinning field

TL;DR

This work extends the CPW–GWD correspondence to external spinning fields in AdS/CFT. It derives an explicit CPW for an external spin-1 field and proves its equivalence to the corresponding geodesic Witten diagram amplitude, using embedding formalism and the conformal Casimir equation. The authors then generalize to external symmetric traceless tensor fields of spin $n$ by representing spinning CPWs as scalar GWD seeds acted on with differential operators, and by identifying a covariant bulk three-point interaction that reproduces the CPW tensor structure. These results provide a practical bulk realization of spinning CPWs, with potential impact on the conformal bootstrap calculations and holographic studies of tensor exchanges.

Abstract

We explore AdS/CFT correspondence between geodesic Witten diagrams and conformal blocks (conformal partial waves) with an external symmetric traceless tensor field. We derive an expression for the conformal partial wave with an external spin-1 field and show that this expression is equivalent to the amplitude of the geodesic Witten diagram. We also show the equivalence by using conformal Casimir equation in embedding formalism. Furthermore, we extend the construction of the amplitude of the geodesic Witten diagram to an external arbitrary symmetric traceless tensor field. We show our construction agrees with the known result of the conformal partial waves.

Figures (4)

• Figure 1: Scalar exchange geodesic Witten diagram with an external spin-$n$ field and three external scalar fields. The orange dashed curves are the geodesics $\gamma_{ij}$ between the boundary points ${x_i}$ and $x_{j}$. The blue wavy line represents propagation of the spin-$n$ field and the blue straight lines represent scalar propagation. The interaction vertices are integrated over the points $y$ on the geodesics $\gamma_{ij}$. The amplitude of this diagram is equivalent to the conformal partial wave up to normalization.
• Figure 2: Scalar exchange geodesic Witten diagram with four external scalar fields. Each line has the same meaning as the lines in Figure \ref{['figgwdws']}.
• Figure 3: Euclidean AdS (red hyperboloid) and its conformal boundary (blue light cone) in the embedding space. The blue light ray shows the identification of the boundary points $X^A\sim\lambda X^A$. The black hyperbolic curve displays one choice of the flat section for CFT (the Poincaré section).
• Figure 4: The three point scalar geodesic Witten diagram. The amplitude of this diagram also becomes the form of the scalar three point function in CFT as well as the Witten diagram.