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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]
+ Show Author Affiliations
  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 Institute of Technology (CalTech), Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1502431
Grant/Contract Number:  
SC0009988; SC0011632
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); 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.
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Karateev, Denis, Kravchuk, Petr, & Simmons-Duffin, David. Weight shifting operators and conformal blocks. United States. https://doi.org/10.1007/jhep02(2018)081
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Karateev, Denis, Kravchuk, Petr, and Simmons-Duffin, David. Wed . "Weight shifting operators and conformal blocks". United States. https://doi.org/10.1007/jhep02(2018)081. https://www.osti.gov/servlets/purl/1502431.
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@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)},
number = 2,
volume = 2018,
place = {United States},
year = {2018},
month = {2}
}
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https://doi.org/10.1007/jhep02(2018)081
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Long multiplet bootstrap
text, January 2017

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Superintegrability of $d$-Dimensional Conformal Blocks
text, January 2016

Universal Constraints on Conformal Operator Dimensions
text, January 2009

Bounds on 4D Conformal and Superconformal Field Theories
text, January 2010

Central Charge Bounds in 4D Conformal Field Theory
text, January 2010

Bounds in 4D Conformal Field Theories with Global Symmetry
text, January 2010

Solving the 3D Ising Model with the Conformal Bootstrap
text, January 2012

Projectors, Shadows, and Conformal Blocks
text, January 2012

Conformal Regge theory
text, January 2012

The Bootstrap Program for Boundary CFT_d
text, January 2012

Bootstrapping Conformal Field Theories with the Extremal Functional Method
text, January 2012

Bootstrapping the 3d Ising twist defect
text, January 2013

Bootstrapping Mixed Correlators in the 3D Ising Model
text, January 2014

Bounds on OPE Coefficients in 4D Conformal Field Theories
text, January 2014

Bootstrapping phase transitions in QCD and frustrated spin systems
text, January 2014

Conformal correlators of mixed-symmetry tensors
text, January 2014

Deconstructing Conformal Blocks in 4D CFT
preprint, January 2015

Bootstrapping 3D Fermions
text, January 2015

Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
text, January 2015

Scalar-Vector Bootstrap
text, January 2015

Causality Constraints in Conformal Field Theory
text, January 2015

Conformal Collider Physics from the Lightcone Bootstrap
text, January 2015

Towards Bootstrapping QED$_3$
text, January 2016

Seed Conformal Blocks in 4D CFT
text, January 2016

Superintegrability of $d$-dimensional Conformal Blocks
text, January 2016

A Proof of the Conformal Collider Bounds
text, January 2016

Crossing symmetry and Higher spin towers
text, January 2016

Radial expansion for spinning conformal blocks
text, January 2016

A Stereoscopic Look into the Bulk
text, January 2016

Quantum Integrable Systems from Conformal Blocks
text, January 2016

Harmony of Spinning Conformal Blocks
text, January 2016

Analyticity in Spin in Conformal Theories
text, January 2017

Conformal Bootstrap in the Regge Limit
text, January 2017

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    • Journal of High Energy Physics, Vol. 2019, Issue 6
    • DOI: 10.1007/jhep06(2019)026

    Fermion conformal bootstrap in 4d
    journal, June 2019

    • Karateev, Denis; Kravchuk, Petr; Serone, Marco
    • Journal of High Energy Physics, Vol. 2019, Issue 6
    • DOI: 10.1007/jhep06(2019)088

    Beyond a = c : gravitational couplings to matter and the stress tensor OPE
    journal, July 2018

    Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory
    journal, July 2018

    • Isachenkov, Mikhail; Schomerus, Volker
    • Journal of High Energy Physics, Vol. 2018, Issue 7
    • DOI: 10.1007/jhep07(2018)180

    Exact correlation functions in conformal fishnet theory
    journal, August 2019

    • Gromov, Nikolay; Kazakov, Vladimir; Korchemsky, Gregory
    • Journal of High Energy Physics, Vol. 2019, Issue 8
    • DOI: 10.1007/jhep08(2019)123

    Harmonic analysis and mean field theory
    journal, October 2019

    • Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David
    • Journal of High Energy Physics, Vol. 2019, Issue 10
    • DOI: 10.1007/jhep10(2019)217

    Light-ray operators in conformal field theory
    journal, November 2018

    • Kravchuk, Petr; Simmons-Duffin, David
    • Journal of High Energy Physics, Vol. 2018, Issue 11
    • DOI: 10.1007/jhep11(2018)102

    Recursion relation for general 3d blocks
    journal, December 2019

    • Erramilli, Rajeev S.; Iliesiu, Luca V.; Kravchuk, Petr
    • Journal of High Energy Physics, Vol. 2019, Issue 12
    • DOI: 10.1007/jhep12(2019)116

    An introduction to the SYK model
    journal, July 2019

    • Rosenhaus, Vladimir
    • Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 32
    • DOI: 10.1088/1751-8121/ab2ce1

    The conformal bootstrap: Theory, numerical techniques, and applications
    journal, January 2019

    Universality at large transverse spin in defect CFT
    text, January 2018

    • Lemos, Madalena; Liendo, Pedro; Meineri, Marco
    • Deutsches Elektronen-Synchrotron, DESY, Hamburg
    • DOI: 10.3204/pubdb-2018-03901

    A spacetime derivation of the Lorentzian OPE inversion formula
    journal, July 2018

    • Simmons-Duffin, David; Stanford, Douglas; Witten, Edward
    • Journal of High Energy Physics, Vol. 2018, Issue 7
    • DOI: 10.1007/jhep07(2018)085

    Universality at large transverse spin in defect CFT
    journal, September 2018

    • Lemos, Madalena; Liendo, Pedro; Meineri, Marco
    • Journal of High Energy Physics, Vol. 2018, Issue 9
    • DOI: 10.1007/jhep09(2018)091

    Scalar CFTs and their large N limits
    journal, September 2018

    Veneziano amplitude of Vasiliev theory
    journal, October 2018

    • Turiaci, Gustavo J.; Zhiboedov, Alexander
    • Journal of High Energy Physics, Vol. 2018, Issue 10
    • DOI: 10.1007/jhep10(2018)034

    From spinning conformal blocks to matrix Calogero-Sutherland models
    text, January 2018

    From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models
    text, January 2017

    Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory
    text, January 2018

    Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey
    text, January 2017

    Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory
    text, January 2017

    Scalar CFTs and Their Large N Limits
    text, January 2017

    Scalar-Fermion Analytic Bootstrap in 4D
    preprint, January 2017

    Light-ray operators in conformal field theory
    text, January 2018

    The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
    text, January 2018

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