Abstract
We explicitly construct the holographic dual configuration for the four dimensional \( \mathcal{N} \) = 4 superconformal block containing half-BPS scalar primary operators by considering its full AdS5 × S5 dual geometry. We extend the embedding space formalism and the related Harmonic analysis to general d-dimensional sphere Sd, and obtain precisely the R-symmetry contribution to the half-BPS scalar superconformal blocks, which we refer as “R-symmetry block”. We also observe that the R-symmetry quadratic Casimir operator can be mapped to BC2 Calogero-Sutherland system Hamiltonian, such that R-symmetry block is in terms identified as its bound state wave function.
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References
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional Conformal Blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP 07 (2018) 180 [arXiv:1711.06609] [INSPIRE].
I. Buric, V. Schomerus and E. Sobko, Superconformal Blocks: General Theory, JHEP 01 (2020) 159 [arXiv:1904.04852] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N , Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
G. Arutyunov and S. Frolov, Some cubic couplings in type IIB supergravity on AdS5 × S5 and three point functions in SYM(4) at large N , Phys. Rev. D 61 (2000) 064009 [hep-th/9907085] [INSPIRE].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS /CFT correspondence, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2001): Strings, Branes and EXTRA Dimensions, pp. 3–158, [hep-th/0201253] [INSPIRE].
M. Nirschl and H. Osborn, Superconformal Ward identities and their solution, Nucl. Phys. B 711 (2005) 409 [hep-th/0407060] [INSPIRE].
F.A. Dolan, L. Gallot and E. Sokatchev, On four-point functions of 1/2-BPS operators in general dimensions, JHEP 09 (2004) 056 [hep-th/0405180] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
L.I. Uruchurtu, Next-next-to-extremal Four Point Functions of N = 4 1/2 BPS Operators in the AdS/CFT Correspondence, JHEP 08 (2011) 133 [arXiv:1106.0630] [INSPIRE].
A. Bissi and T. Łukowski, Revisiting \( \mathcal{N} \) = 4 superconformal blocks, JHEP 02 (2016) 115 [arXiv:1508.02391] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial wave expansions for N = 4 chiral four point functions, Annals Phys. 321 (2006) 581 [hep-th/0412335] [INSPIRE].
R. Doobary and P. Heslop, Superconformal partial waves in Grassmannian field theories, JHEP 12 (2015) 159 [arXiv:1508.03611] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D¿= 3 Dimensions, SpringerBriefs in Physics (2016), DOI [arXiv:1601.05000] [INSPIRE].
V. Bargmann and I.T. Todorov, Spaces of Analytic Functions on a Complex Cone as Carries for the Symmetric Tensor Representations of SO(N), J. Math. Phys. 18 (1977) 1141 [INSPIRE].
http://functions.wolfram.com/HypergeometricFunctions/GegenbauerC3General/17/02/02/.
M.S. Costa, V. Gonçalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
H.-Y. Chen, E.-J. Kuo and H. Kyono, Towards Spinning Mellin Amplitudes, Nucl. Phys. B 931 (2018) 291 [arXiv:1712.07991] [INSPIRE].
H. Exton, On the system of partial differential equations associated with Appell’s function F4 , J. Phys. A 28 (1995) 631.
S. Ferrara and E. Sokatchev, Universal properties of superconformal OPEs for 1/2 BPS operators in 3 ≤ D ≤ 6, New J. Phys. 4 (2002) 2 [hep-th/0110174] [INSPIRE].
C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
P.J. Heslop, Aspects of superconformal field theories in six dimensions, JHEP 07 (2004) 056 [hep-th/0405245] [INSPIRE].
L. Rastelli and X. Zhou, Holographic Four-Point Functions in the (2, 0) Theory, JHEP 06 (2018) 087 [arXiv:1712.02788] [INSPIRE].
H.-Y. Chen and J.D. Qualls, Quantum Integrable Systems from Conformal Blocks, Phys. Rev. D 95 (2017) 106011 [arXiv:1605.05105] [INSPIRE].
G.J. Heckman and E.M. Opdam, Root systems and hypergeometric functions. I, Compos. Math. 64 (1987) 329.
G.J. Heckman, Root systems and hypergeometric functions. II, Comp. Math. 64 (1987) 353.
E.M. Opdam, Root systems and hypergeometric functions. III, Comp. Math. 67 (1988) 21.
E.M. Opdam, Root systems and hypergeometric functions. IV, Comp. Math. 67 (1988) 191.
M. Pérez-Saborid, The coordinate-free approach to spherical harmonics, arXiv:0806.3367.
A. Rubin and C.R. Ordonez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. 25 (1984) 2888.
C. Sleight, Interactions in Higher-Spin Gravity: a Holographic Perspective, J. Phys. A 50 (2017) 383001 [arXiv:1610.01318] [INSPIRE].
J.-F. Fortin and W. Skiba, New methods for conformal correlation functions, JHEP 06 (2020) 028 [arXiv:1905.00434] [INSPIRE].
H.-Y. Chen and H. Kyono, On conformal blocks, crossing kernels and multi-variable hypergeometric functions, JHEP 10 (2019) 149 [arXiv:1906.03135] [INSPIRE].
F.A. Dolan and H. Osborn, Implications of N = 1 superconformal symmetry for chiral fields, Nucl. Phys. B 593 (2001) 599 [hep-th/0006098] [INSPIRE].
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Chen, HY., Sakamoto, Ji. Superconformal block from holographic geometry. J. High Energ. Phys. 2020, 28 (2020). https://doi.org/10.1007/JHEP07(2020)028
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DOI: https://doi.org/10.1007/JHEP07(2020)028