A multipoint conformal block chain in d dimensions
Sarthak Parikh
Abstract
A bstract 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 3 F 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.