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Casimir recursion relations for general conformal blocks

Kravchuk, Petr (2018) Casimir recursion relations for general conformal blocks. Journal of High Energy Physics, 2018 (2). Art. No. 11. ISSN 1126-6708. doi:10.1007/JHEP02(2018)011.

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We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(d) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The form of this recursion relation is determined by 6j symbols of Spin(d − 1). In particular, it can be written down in closed form in d = 3, d = 4, for seed blocks in general dimensions, or in any other situation when the required 6j symbols can be computed. We work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.

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Kravchuk, Petr0000-0003-0977-3686
Additional Information:© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received: September 23, 2017. Accepted: January 8, 2018. Published: February 2, 2018. I would like to thank Denis Karateev, João Penedones, Fernando Rejón-Barerra, Slava Rychkov, David Simmons-Duffin, Emilio Trevisani, and the participants of Simons Bootstrap Collaboration workshop on numerical bootstrap for valuable discussions. Special thanks to David Simmons-Duffin for comments on the draft. I am grateful to the authors of [76] for making their Mathematica code publicly available. I also thank the Institute for Advanced Study, where part of this work was completed, for hospitality. This work was supported by DOE grant DE-SC0011632.
Group:Walter Burke Institute for Theoretical Physics
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Department of Energy (DOE)DE-SC0011632
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Issue or Number:2
Record Number:CaltechAUTHORS:20170924-172338774
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Official Citation:Kravchuk, P. J. High Energ. Phys. (2018) 2018: 11.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81787
Deposited By: Joy Painter
Deposited On:25 Sep 2017 18:14
Last Modified:15 Nov 2021 19:46

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