A multipoint conformal block chain in d dimensions

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 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.

On the Virasoro six-point identity block and chaos

“…As mentioned earlier, the CPW contains the block of interest as well as a shadow block. The six-point global blocks are obtained from the CPWs by performing a monodromy projection, which we denote abstractly as follows: 14) and will provide the details of this procedure in section 2.3, however, let us give a rough intuition of the procedure now. In the next section, following [32], we will give Feynman rules for computing CPWs with stress tensor exchanges.…”

“…It is straightforward to express the coordinate dependence of the five-point blocks in (2.45) in terms of the two independent combinations of cross-ratios out of (z, u, v) which remain finite in the respective OPE limit. 9 We refer the reader to [5,14] for more details on OPE limits of higher-point blocks. Symmetric presentation of the star channel block.…”

“…It was discussed more generally in [9], where comb channel blocks were determined explicitly in terms of generalizations of hypergeometric functions. See also [6,8,10,11,47] for physical applications and [5,7,[12][13][14][15] for discussions in the context of Witten diagrams and AdS/CFT.…”

“…, where the first two factors are the same as for the star channel, and the "reflection" Z 14 We illustrate these symmetries schematically in figure 5.…”

“…We will compare the star channel with the comb channel (see figure 1), which is characterized by a different OPE channel topology [5][6][7][8][9][10][11][12][13][14][15]. The comb channel is not a pure stress tensor block as it involves the exchange of an internal operator that is not a descendant of the identity.…”