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. [1] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk threepoint 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 geodesic Witten diagram and the shadow operator formalism.
Highlights
Both approaches focus on the properties and computation of correlation functions of primary operators
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 [1] to arbitrary spin
In the context of the AdS/CFT correspondence, the four-point function can be computed by summing bulk Witten diagrams with the four points xi located on the AdS boundary and internal vertices integrated over the entire AdS geometry
Summary
It has been understood since the work of Dirac, that the d-dimensional conformal group, SO(d + 1, 1), is quite powerful in fixing local correlation functions. Two-point functions of (normalized) primary scalar operators are completely fixed, and their three-point functions are determined up to a constant: O(P1)O(P2) O1(P1)O2(P2)O3(P3). It is useful to decompose higher point functions in terms of structures which are invariants of the conformal symmetry. Hij = −tr (CiCj) = −2[(Ui · Uj)(Pi · Pj) − (Ui · Pj)(Uj · Pi)] This is written in terms of the useful intermediate structure, CiAB = UiAPiB − UiBPiA. Denoting the independent structures by VI (P1, U1; P2, U2; P3, U3), we can write the general expression for a three point function of spinning operators as the sum. As in the scalar case, we can decompose the four-point function of spinning operators in terms of the three-point coefficients, which specify the dynamical data, and the spinning conformal blocks, which encapsulate the kinematics. The technology of geodesic Witten diagrams was first developed for scalar four-point functions in [1], and we review this construction below
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