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A multipoint conformal block chain in \(d\) dimensions. (English) Zbl 1437.83116

Summary: 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.

MSC:

83E05 Geometrodynamics and the holographic principle
81T55 Casimir effect in quantum field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T35 Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.)

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