A multipoint conformal block chain in d dimensions
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Springer
Received: December 17, 2019 Accepted: May 5, 2020 Published: May 25, 2020
Sarthak Parikh Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, U.S.A.
E-mail: [email protected] 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 F2 , 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. Keywords: Conformal Field Theory, AdS-CFT Correspondence, Conformal and W Symmetry ArXiv ePrint: 1911.09190 Dedicated to the memory of Steven S. Gubser.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)120
JHEP05(2020)120
A multipoint conformal block chain in d dimensions
Contents 1 Introduction
1 5 6 10 11
3 Multipoint block in the comb channel 3.1 Holographic dual of the (n + 2)-point block and its double-OPE limit 3.2 OPE limit of the n-point block 3.3 Conformal Casimir check
14 14 17 19
4 Discussion
26
A Obtaining the six-point holographic dual
30
B Five- and seven-point examples B.1 Double-OPE limit and the five-point block
35 37
C Further technical details C.1 Proof of (3.15) C.2 Proof of (3.32) C.3 Proof of (3.37)
38 38 39 40
1
Introduction
Conformal field theories (CFTs) are important for several reasons — they serve as salient guideposts in the space of quantum field theories, describe a variety of critical phenomena, and help elucidate aspects of quantum gravity via the AdS/CFT correspondence [1–3]. Conformal blocks play a central role in CFTs. They are the basic kinematic building blocks of local observables, encoding the contribution of primary operators (and all their descendants) to any given correlation function. Given a d-dimensional CFT (more precisely the dynamical data in the form of the spectrum of all primary operators and the operator product expansion (OPE) coefficients between them), the knowledge of d-dimensional conformal blocks permits the explicit writing of all possible correlators. Conversely, the conformal bootstrap program [4–6] (see also the recent revi
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