Geodesic
Further Results on a Function Relevant for Conformal Blocks
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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We present further mathematical results on a function appearing in the conformal blocks of four-point correlation functions with arbitrary primary operators. The $H$-function was introduced in a previous article and it has several interesting properties. We prove explicitly the recurrence relation as well as the $D_6$-invariance presented previously. We also demonstrate the proper action of the differential operator used to construct the $H$-function.
Keywords: special functions, conformal field theory. 
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     author = {Vincent Comeau and Jean-Fran\c{c}ois Fortin and Witold Skiba},
     title = {Further {Results} on a {Function} {Relevant} for {Conformal} {Blocks}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a123/}
}
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Vincent Comeau; Jean-François Fortin; Witold Skiba. Further Results on a Function Relevant for Conformal Blocks. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a123/

[1] Beyer W. A., Louck J. D., Stein P. R., “Group theoretical basis of some identities for the generalized hypergeometric series”, J. Math. Phys., 28 (1987), 497–508 | DOI | MR | Zbl

[2] Dirac P. A. M., “Wave equations in conformal space”, Ann. of Math., 37 (1936), 429–442 | DOI | MR

[3] Dobrev V. K., Mack G., Petkova V. B., Petrova S. G., Todorov I. T., “Harmonic analysis: on the $n$-dimensional Lorentz group and its application to conformal quantum field theory”, Lect. Notes Phys., 63, Springer-Verlag, Berlin – Heidelberg, 1977 | DOI | Zbl

[4] Dolan F. A., Osborn H., “Conformal four point functions and the operator product expansion”, Nuclear Phys. B, 599 (2001), 459–496, arXiv: hep-th/0011040 | DOI | MR | Zbl

[5] Dolan F. A., Osborn H., “Conformal partial waves and the operator product expansion”, Nuclear Phys. B, 678 (2004), 491–507, arXiv: hep-th/0309180 | DOI | MR | Zbl

[6] Exton H., “On the system of partial differential equations associated with Appell's function $F_4$”, J. Phys. A: Math. Gen., 28 (1995), 631–641 | DOI | MR | Zbl

[7] Ferrara S., Gatto R., Grillo A. F., “Conformal invariance on the light cone and canonical dimensions”, Nuclear Phys. B, 34 (1971), 349–366 | DOI | MR

[8] Ferrara S., Gatto R., Grillo A. F., Conformal algebra in space-time and operator product expansion, Springer Tracts in Modern Physics, 67, Springer-Verlag, Berlin – Heidelberg, 1973 | DOI | MR

[9] Ferrara S., Grillo A. F., Gatto R., “Manifestly conformal covariant operator-product expansion”, Lett. Nuovo Cimento, 2 (1971), 1363–1369 | DOI | MR

[10] Ferrara S., Grillo A. F., Gatto R., “Manifestly conformal-covariant expansion on the light cone”, Phys. Rev. D, 5 (1972), 3102–3108 | DOI

[11] Ferrara S., Grillo A. F., Gatto R., “Tensor representations of conformal algebra and conformally covariant operator product expansion”, Ann. Physics, 76 (1973), 161–188 | DOI | MR

[12] Ferrara S., Grillo A. F., Parisi G., Gatto R., “The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products”, Lett. Nuovo Cimento, 4 (1972), 115–120 | DOI | MR

[13] Ferrara S., Parisi G., “Conformal covariant correlation functions”, Nuclear Phys. B, 42 (1972), 281–290 | DOI

[14] Fortin J. F., Skiba W., “Conformal bootstrap in embedding space”, Phys. Rev. D, 93 (2016), 105047, 7 pp., arXiv: 1602.05794 | DOI | MR

[15] Fortin J. F., Skiba W., “Conformal differential operator in embedding space and its applications”, J. High Energy Phys., 2019:7 (2019), 093, 19 pp., arXiv: 1612.08672 | DOI | MR

[16] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[17] Horn J., “Ueber die Convergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen”, Math. Ann., 34 (1889), 544–600 | DOI | MR

[18] Isachenkov M., Schomerus V., “Integrability of conformal blocks. Part I. Calogero–Sutherland scattering theory”, J. High Energy Phys., 2018:7 (2018), 180, 66 pp., arXiv: 1711.06609 | DOI | MR | Zbl

[19] Karateev D., Kravchuk P., Simmons-Duffin D., “Weight shifting operators and conformal blocks”, J. High Energy Phys., 2018:2 (2018), 081, 81 pp., arXiv: 1706.07813 | DOI | MR

[20] Mack G., “Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory”, Comm. Math. Phys., 53 (1977), 155–184 | DOI | MR

[21] Mack G., Salam A., “Finite-component field representations of the conformal group”, Ann. Physics, 53 (1969), 174–202 | DOI | MR

[22] Penedones J., Trevisani E., Yamazaki M., “Recursion relations for conformal blocks”, J. High Energy Phys., 2016:9 (2016), 070, 50 pp., arXiv: 1509.00428 | DOI | MR

[23] Poland D., Rychkov S., Vichi A., “The conformal bootstrap: theory, numerical techniques, and applications”, Rev. Modern Phys., 91 (2019), 015002, 74 pp., arXiv: 1805.04405 | DOI | MR

[24] Polyakov A. M., “Non-Hamiltonian approach to conformal quantum field theory”, Sov. Phys. JETP, 39 (1974), 10–18 | MR

[25] Rainville E. D., Special functions, Chelsea Publishing Co., Bronx, N.Y., 1971 | MR | Zbl

[26] Rattazzi R., Rychkov V. S., Tonni E., Vichi A., “Bounding scalar operator dimensions in 4D CFT”, J. High Energy Phys., 2008:12 (2008), 031, 49 pp., arXiv: 0807.0004 | DOI | MR | Zbl

[27] Schomerus V., Sobko E., “From spinning conformal blocks to matrix Calogero–Sutherland models”, J. High Energy Phys., 2018:4 (2018), 052, 29 pp., arXiv: 1711.02022 | DOI | MR

[28] Schomerus V., Sobko E., Isachenkov M., “Harmony of spinning conformal blocks”, J. High Energy Phys., 2017:3 (2017), 085, 23 pp., arXiv: 1612.02479 | DOI | MR

[29] Simmons-Duffin D., “Projectors, shadows, and conformal blocks”, J. High Energy Phys., 2014:4 (2014), 146, 36 pp., arXiv: 1204.3894 | DOI | MR | Zbl

[30] Srivastava H. M., Daoust M. C., “A note on the convergence of Kampé de Fériet's double hypergeometric series”, Math. Nachr., 53 (1972), 151–159 | DOI | MR | Zbl

[31] Srivastava H. M., Karlsson P. W., Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985 | MR | Zbl

[32] Srivastava P., Gupta P., “A note on hypergeometric functions of one and two variables”, Amer. J. Comput. Appl. Math., 3 (2013), 182–185

[33] Weinberg S., “Six-dimensional methods for four-dimensional conformal field theories”, Phys. Rev. D, 82 (2010), 045031, 11 pp., arXiv: 1006.3480 | DOI

[34] Weinberg S., “Six-dimensional methods for four-dimensional conformal field theories. II. Irreducible fields”, Phys. Rev. D, 86 (2012), 085013, 3 pp., arXiv: 1209.4659 | DOI

[35] Zamolodchikov A. B., “Conformal symmetry in two dimensions: an explicit recurrence formula for the conformal partial wave amplitude”, Comm. Math. Phys., 96 (1984), 419–422 | DOI | MR

[36] Zamolodchikov A. B., “Conformal symmetry in two-dimensional space: recursion representation of conformal block”, Theoret. and Math. Phys., 73 (1987), 1088–1093 | DOI | MR