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Title: Weight shifting operators and conformal blocks

Abstract

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for “seed conformal blocks” in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including deriving recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/“crossing kernels”) of the conformal group.

Authors:
 [1];  [2];  [3]
  1. International School for Advanced Studies (SISSA) and Italian National Inst. of Nuclear Physics (INFN), Trfieste (Italy)
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States). Water Burke Inst. for Theoretical Physics
  3. California Inst. of Technology (CalTech), Pasadena, CA (United States). Water Burke Inst. for Theoretical Physics; Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
Publication Date:
Research Org.:
Institute for Advanced Study, Princeton, NJ (United States); California Inst. of Technology, Pasadena (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
OSTI Identifier:
1502431
Grant/Contract Number:  
SC0009988; SC0011632
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2018; Journal Issue: 2; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal and W Symmetry; Conformal Field Theory; Space-Time Symmetries

Citation Formats

Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Weight shifting operators and conformal blocks. United States: N. p., 2018. Web. doi:10.1007/jhep02(2018)081.
Karateev, Denis, Kravchuk, Petr, & Simmons-Duffin, David. Weight shifting operators and conformal blocks. United States. doi:10.1007/jhep02(2018)081.
Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Wed . "Weight shifting operators and conformal blocks". United States. doi:10.1007/jhep02(2018)081. https://www.osti.gov/servlets/purl/1502431.
@article{osti_1502431,
title = {Weight shifting operators and conformal blocks},
author = {Karateev, Denis and Kravchuk, Petr and Simmons-Duffin, David},
abstractNote = {We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for “seed conformal blocks” in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including deriving recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/“crossing kernels”) of the conformal group.},
doi = {10.1007/jhep02(2018)081},
journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 2,
volume = 2018,
place = {United States},
year = {2018},
month = {2}
}

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Cited by: 28 works
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