A multipoint conformal block chain in $d$ dimensions

TL;DR

This paper solves the long-standing problem of explicit d-dimensional multipoint global conformal blocks in the comb channel for scalar external and exchanged operators. By combining AdS geodesic-diagram techniques with conformal Casimir equations, it first constructs holographic representations for the (n+2)-point block and then, via a double-OPE limit, derives a convergent power-series expansion in conformal cross-ratios whose coefficients are purely Pochhammer products and ${}_3F_2$ functions. The authors prove the series representation is consistent with OPE limits and Casimir constraints, with explicit checks for low n and analytic/numerical support for higher n. This framework generalizes known results (n=4,5) to arbitrary n, offering a practical route toward higher-point conformal bootstrap and potential Feynman-like rules for blocks. The work also discusses convergence improvements and connections to p-adic AdS/CFT, signaling broad applicability in holographic CFT analyses.

Abstract

Conformal blocks play a central role in CFTs as the basic, theory-independent building blocks. However, only limited results are available concerning multipoint blocks associated with the global conformal group. In this paper, we systematically work out the $d$-dimensional $n$-point global conformal blocks (for arbitrary $d$ and $n$) for external and exchanged scalar operators in the so-called comb channel. We use kinematic aspects of holography and previously worked out higher-point AdS propagator identities to first obtain the geodesic diagram representation for the $(n+2)$-point block. Subsequently, upon taking a particular double-OPE limit, we obtain an explicit power series expansion for the $n$-point block expressed in terms of powers of conformal cross-ratios. Interestingly, the expansion coefficient is written entirely in terms of Pochhammer symbols and $(n-4)$ factors of the generalized hypergeometric function ${}_3F_2$, for which we provide a holographic explanation. This generalizes the results previously obtained in the literature for $n=4, 5$. We verify the results explicitly in embedding space using conformal Casimir equations.

Figures (2)

• Figure 1: The comb channel $n$-point global conformal block for external scalar operators ${\cal O}_1(x_1), \ldots, {\cal O}_n(x_n)$ with conformal dimensions $\Delta_1,\ldots,\Delta_n$ and insertion coordinates $x_1,\ldots,x_n$ respectively, and exchanged scalar operators ${\cal O}_{\delta_1},\ldots,{\cal O}_{\delta_{n-3}}$ with conformal dimensions $\Delta_{\delta_1},\ldots,\Delta_{\delta_{n-3}}$, respectively. When there is no scope for confusion, we will often abbreviate it as $W^{(n)}(x_i)$ or simply $W^{(n)}$.
• Figure 2: How to read comb-channel geodesic bulk diagrams. Geodesic bulk diagrams (also referred to as geodesic Witten diagrams) are AdS Feynman diagrams except with all bulk integrations restricted to boundary-anchored geodesics. Throughout this paper, boundary-anchored geodesics over which bulk points are to be integrated will be shown as red-dashed lines. In the diagram above, they represent the geodesics $\gamma_{12}$ and $\gamma_{56}$ joining $x_1$ to $x_2$ and $x_5$ to $x_6$, respectively. Bulk-to-boundary propagators $\hat{K}_\Delta(x,z)$ will be shown with solid blue lines, and whenever the conformal dimension $\Delta$ associated with it is not clear from the figure, it will be mentioned explicitly. For example, in the six-point geodesic diagram above, the four bulk-to-boundary propagators incident on bulk points to be integrated over boundary anchored geodesics are associated with the conformal dimensions $\Delta_i$ of the operator insertions ${\cal O}_i$ as marked. The remaining four bulk-to-boundary propagators emanating from the coordinates $x_3$ and $x_4$ have conformal dimensions as displayed next to the blue lines. Unless stated otherwise, the operator ${\cal O}_i$ is understood to be located at boundary coordinate $x_i$. Solid black lines will refer to purely boundary contractions; for example in the diagram above the solid black line joining $x_3$ to $x_4$ corresponds to a factor of $(x_{34}^2)^{-\Delta_D}$. Finally dotted black lines will stand for factors of chordal distance $(\xi(w_1,w_2)/2)^{\Delta}$ where $\xi(w_1,w_2)^{-1} = \cosh \sigma(w_1,w_2)$ where $\sigma(w_1,w_2)$ is the geodesic distance between bulk points $w_1$ and $w_2$. We will be using the same propagator normalizations as in ref. Parikh:2019ygo; see in particular Parikh:2019ygo for the normalization of the bulk-to-boundary propagator as well as the relation between the bulk-to-bulk propagator and the chordal distance factor above.