Casimir recursion relations for general conformal blocks

Kravchuk, Petr

doi:10.1007/jhep02(2018)011

Casimir recursion relations for general conformal blocks

Journal Article · Fri Feb 02 00:00:00 EST 2018 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/jhep02(2018)011· OSTI ID:1499658

Kravchuk, Petr ^[1]

California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics; California Inst. of Technology, Pasadena (United States)

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. As a result, we work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.

View Accepted Manuscript (DOE)

Research Organization:: California Inst. of Technology, Pasadena, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

Grant/Contract Number:: SC0011632

OSTI ID:: 1499658

Alternate ID(s):: OSTI ID: 22739013

Journal Information:: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 2 Vol. 2018; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

References (65)

Racah’s method for general subalgebra chains: Coupling coefficients of SO(5) in canonical and physical bases Caprio, M. A.; Sviratcheva, K. D.; McCoy, A. E. Journal of Mathematical Physics, Vol. 51, Issue 9 https://doi.org/10.1063/1.3445529	journal	September 2010
Bootstrapping Conformal Field Theories with the Extremal Functional Method El-Showk, Sheer; Paulos, Miguel F. Physical Review Letters, Vol. 111, Issue 24 https://doi.org/10.1103/physrevlett.111.241601	journal	December 2013
The superconformal bootstrap for structure constants Alday, Luis F.; Bissi, Agnese arXiv https://doi.org/10.48550/arxiv.1310.3757	text	January 2013
Representation of Lie Groups and Special Functions Volume 3: Classical and Quantum Groups and Special Functions Vilenkin, N. Ja.; Klimyk, A. U. Mathematics and its Applications https://doi.org/10.1007/978-94-017-2881-2	book	January 1992
Conformal symmetry in two-dimensional space: Recursion representation of conformal block Zamolodchikov, Al. B. Theoretical and Mathematical Physics, Vol. 73, Issue 1 https://doi.org/10.1007/BF01022967	journal	October 1987
Improved bounds for CFT’s with global symmetries Vichi, Alessandro Journal of High Energy Physics, Vol. 2012, Issue 1 https://doi.org/10.1007/JHEP01(2012)162	journal	January 2012
General three-point functions in 4D CFT Elkhidir, Emtinan; Karateev, Denis; Serone, Marco Journal of High Energy Physics, Vol. 2015, Issue 1 https://doi.org/10.1007/JHEP01(2015)133	journal	January 2015
Bootstrapping N = 2 $$ \mathcal{N}=2 $$ chiral correlators Lemos, Madalena; Liendo, Pedro Journal of High Energy Physics, Vol. 2016, Issue 1 https://doi.org/10.1007/JHEP01(2016)025	journal	January 2016
Accidental symmetries and the conformal bootstrap Chester, Shai M.; Giombi, Simone; Iliesiu, Luca V. Journal of High Energy Physics, Vol. 2016, Issue 1 https://doi.org/10.1007/JHEP01(2016)110	journal	January 2016
Scalar-vector bootstrap Rejon-Barrera, Fernando; Robbins, Daniel Journal of High Energy Physics, Vol. 2016, Issue 1 https://doi.org/10.1007/JHEP01(2016)139	journal	January 2016
Seed conformal blocks in 4D CFT Echeverri, Alejandro Castedo; Elkhidir, Emtinan; Karateev, Denis Journal of High Energy Physics, Vol. 2016, Issue 2 https://doi.org/10.1007/JHEP02(2016)183	journal	February 2016
Bootstrapping the 3d Ising twist defect Gaiotto, Davide; Mazac, Dalimil; Paulos, Miguel F. Journal of High Energy Physics, Vol. 2014, Issue 3 https://doi.org/10.1007/JHEP03(2014)100	journal	March 2014
Bootstrapping 3D fermions Iliesiu, Luca; Kos, Filip; Poland, David Journal of High Energy Physics, Vol. 2016, Issue 3 https://doi.org/10.1007/JHEP03(2016)120	journal	March 2016
The N = 2 $$ \mathcal{N}=2 $$ superconformal bootstrap Beem, Christopher; Lemos, Madalena; Liendo, Pedro Journal of High Energy Physics, Vol. 2016, Issue 3 https://doi.org/10.1007/JHEP03(2016)183	journal	March 2016
Harmony of spinning conformal blocks Schomerus, Volker; Sobko, Evgeny; Isachenkov, Mikhail Journal of High Energy Physics, Vol. 2017, Issue 3 https://doi.org/10.1007/JHEP03(2017)085	journal	March 2017
The lightcone bootstrap and the spectrum of the 3d Ising CFT Simmons-Duffin, David Journal of High Energy Physics, Vol. 2017, Issue 3 https://doi.org/10.1007/JHEP03(2017)086	journal	March 2017
Projectors, shadows, and conformal blocks Simmons-Duffin, David Journal of High Energy Physics, Vol. 2014, Issue 4 https://doi.org/10.1007/JHEP04(2014)146	journal	April 2014
Bootstrapping N = 3 $$ \mathcal{N}=3 $$ superconformal theories Lemos, Madalena; Liendo, Pedro; Meneghelli, Carlo Journal of High Energy Physics, Vol. 2017, Issue 4 https://doi.org/10.1007/JHEP04(2017)032	journal	April 2017
The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points Gliozzi, Ferdinando; Guerrieri, Andrea L.; Petkou, Anastasios C. Journal of High Energy Physics, Vol. 2017, Issue 4 https://doi.org/10.1007/JHEP04(2017)056	journal	April 2017
Bootstrapping mixed correlators in the five dimensional critical O(N) models Li, Zhijin; Su, Ning Journal of High Energy Physics, Vol. 2017, Issue 4 https://doi.org/10.1007/JHEP04(2017)098	journal	April 2017
Bounds on 4D conformal and superconformal field theories Poland, David; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2011, Issue 5 https://doi.org/10.1007/JHEP05(2011)017	journal	May 2011
Carving out the space of 4D CFTs Poland, David; Simmons-Duffin, David; Vichi, Alessandro Journal of High Energy Physics, Vol. 2012, Issue 5 https://doi.org/10.1007/JHEP05(2012)110	journal	May 2012
Bootstrapping O(N ) vector models with four supercharges in 3 ≤ d ≤ 4 Chester, Shai M.; Iliesiu, Luca V.; Pufu, Silviu S. Journal of High Energy Physics, Vol. 2016, Issue 5 https://doi.org/10.1007/JHEP05(2016)103	journal	May 2016
A Mellin space approach to the conformal bootstrap Gopakumar, Rajesh; Kaviraj, Apratim; Sen, Kallol Journal of High Energy Physics, Vol. 2017, Issue 5 https://doi.org/10.1007/JHEP05(2017)027	journal	May 2017
(2, 2) superconformal bootstrap in two dimensions Lin, Ying-Hsuan; Shao, Shu-Heng; Wang, Yifan Journal of High Energy Physics, Vol. 2017, Issue 5 https://doi.org/10.1007/JHEP05(2017)112	journal	May 2017
N $$ \mathcal{N} $$ = 4 superconformal bootstrap of the K3 CFT Lin, Ying-Hsuan; Shao, Shu-Heng; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2017, Issue 5 https://doi.org/10.1007/JHEP05(2017)126	journal	May 2017
Bootstrapping the O(N ) vector models Kos, Filip; Poland, David; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2014, Issue 6 https://doi.org/10.1007/JHEP06(2014)091	journal	June 2014
A semidefinite program solver for the conformal bootstrap Simmons-Duffin, David Journal of High Energy Physics, Vol. 2015, Issue 6 https://doi.org/10.1007/JHEP06(2015)174	journal	June 2015
More on analytic bootstrap for O(N) models Dey, Parijat; Kaviraj, Apratim; Sen, Kallol Journal of High Energy Physics, Vol. 2016, Issue 6 https://doi.org/10.1007/JHEP06(2016)136	journal	June 2016
Cut-touching linear functionals in the conformal bootstrap Qiao, Jiaxin; Rychkov, Slava Journal of High Energy Physics, Vol. 2017, Issue 6 https://doi.org/10.1007/JHEP06(2017)076	journal	June 2017
The bootstrap program for boundary CFT d Liendo, Pedro; Rastelli, Leonardo; van Rees, Balt C. Journal of High Energy Physics, Vol. 2013, Issue 7 https://doi.org/10.1007/JHEP07(2013)113	journal	July 2013
Projectors and seed conformal blocks for traceless mixed-symmetry tensors Costa, Miguel S.; Hansen, Tobias; Penedones, João Journal of High Energy Physics, Vol. 2016, Issue 7 https://doi.org/10.1007/JHEP07(2016)018	journal	July 2016
Bootstrap bound for conformal multi-flavor QCD on lattice Nakayama, Yu Journal of High Energy Physics, Vol. 2016, Issue 7 https://doi.org/10.1007/JHEP07(2016)038	journal	July 2016
Radial expansion for spinning conformal blocks Costa, Miguel S.; Hansen, Tobias; Penedones, João Journal of High Energy Physics, Vol. 2016, Issue 7 https://doi.org/10.1007/JHEP07(2016)057	journal	July 2016
Bootstrapping mixed correlators in 4D N $$ \mathcal{N} $$ = 1 SCFTs Li, Daliang; Meltzer, David; Stergiou, Andreas Journal of High Energy Physics, Vol. 2017, Issue 7 https://doi.org/10.1007/JHEP07(2017)029	journal	July 2017
Diagonal limit for conformal blocks in d dimensions Hogervorst, Matthijs; Osborn, Hugh; Rychkov, Slava Journal of High Energy Physics, Vol. 2013, Issue 8 https://doi.org/10.1007/JHEP08(2013)014	journal	August 2013
Deconstructing conformal blocks in 4D CFT Echeverri, Alejandro Castedo; Elkhidir, Emtinan; Karateev, Denis Journal of High Energy Physics, Vol. 2015, Issue 8 https://doi.org/10.1007/JHEP08(2015)101	journal	August 2015
Towards bootstrapping QED3 Chester, Shai M.; Pufu, Silviu S. Journal of High Energy Physics, Vol. 2016, Issue 8 https://doi.org/10.1007/JHEP08(2016)019	journal	August 2016
Precision islands in the Ising and O(N ) models Kos, Filip; Poland, David; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2016, Issue 8 https://doi.org/10.1007/JHEP08(2016)036	journal	August 2016
Carving out the end of the world or (superconformal bootstrap in six dimensions) Chang, Chi-Ming; Lin, Ying-Hsuan Journal of High Energy Physics, Vol. 2017, Issue 8 https://doi.org/10.1007/JHEP08(2017)128	journal	August 2017
The N = 8 $$ \mathcal{N}=8 $$ superconformal bootstrap in three dimensions Chester, Shai M.; Lee, Jaehoon; Pufu, Silviu S. Journal of High Energy Physics, Vol. 2014, Issue 9 https://doi.org/10.1007/JHEP09(2014)143	journal	September 2014
The superconformal bootstrap for structure constants Alday, Luis F.; Bissi, Agnese Journal of High Energy Physics, Vol. 2014, Issue 9 https://doi.org/10.1007/JHEP09(2014)144	journal	September 2014
Recursion relations for conformal blocks Penedones, João; Trevisani, Emilio; Yamazaki, Masahito Journal of High Energy Physics, Vol. 2016, Issue 9 https://doi.org/10.1007/JHEP09(2016)070	journal	September 2016
Bounds on OPE coefficients in 4D Conformal Field Theories Caracciolo, Francesco; Echeverri, Alejandro Castedo; von Harling, Benedict Journal of High Energy Physics, Vol. 2014, Issue 10 https://doi.org/10.1007/JHEP10(2014)020	journal	October 2014
On the four-point function of the stress-energy tensors in a CFT Dymarsky, Anatoly Journal of High Energy Physics, Vol. 2015, Issue 10 https://doi.org/10.1007/JHEP10(2015)075	journal	October 2015
Comments on determinant formulas for general CFTs Yamazaki, Masahito Journal of High Energy Physics, Vol. 2016, Issue 10 https://doi.org/10.1007/JHEP10(2016)035	journal	October 2016
Long multiplet bootstrap Cornagliotto, Martina; Lemos, Madalena; Schomerus, Volker Journal of High Energy Physics, Vol. 2017, Issue 10 https://doi.org/10.1007/JHEP10(2017)119	journal	October 2017
Spinning conformal blocks Costa, Miguel S.; Penedones, João; Poland, David Journal of High Energy Physics, Vol. 2011, Issue 11 https://doi.org/10.1007/JHEP11(2011)154	journal	November 2011
Bootstrapping mixed correlators in the 3D Ising model Kos, Filip; Poland, David; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2014, Issue 11 https://doi.org/10.1007/JHEP11(2014)109	journal	November 2014
Bootstrapping the O(N) archipelago Kos, Filip; Poland, David; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2015, Issue 11 https://doi.org/10.1007/JHEP11(2015)106	journal	November 2015
Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$ c -Minimization and Precise Critical Exponents El-Showk, Sheer; Paulos, Miguel F.; Poland, David Journal of Statistical Physics, Vol. 157, Issue 4-5 https://doi.org/10.1007/s10955-014-1042-7	journal	June 2014
Tensor representations of conformal algebra and conformally covariant operator product expansion Ferrara, S.; Grillo, A. F.; Gatto, R. Annals of Physics, Vol. 76, Issue 1 https://doi.org/10.1016/0003-4916(73)90446-6	journal	March 1973
Duality in quantum field theory Mack, G. Nuclear Physics B, Vol. 118, Issue 5 https://doi.org/10.1016/0550-3213(77)90238-3	journal	February 1977
Conformal four point functions and the operator product expansion Dolan, F. A.; Osborn, H. Nuclear Physics B, Vol. 599, Issue 1-2 https://doi.org/10.1016/S0550-3213(01)00013-X	journal	April 2001
Conformal partial waves and the operator product expansion Dolan, F. A.; Osborn, H. Nuclear Physics B, Vol. 678, Issue 1-2 https://doi.org/10.1016/j.nuclphysb.2003.11.016	journal	February 2004
Five dimensional O(N)-symmetric CFTs from conformal bootstrap Nakayama, Yu; Ohtsuki, Tomoki Physics Letters B, Vol. 734 https://doi.org/10.1016/j.physletb.2014.05.058	journal	June 2014
Wigner Coefficients for the R ₄ Group and Some Applications Biedenharn, L. C. Journal of Mathematical Physics, Vol. 2, Issue 3 https://doi.org/10.1063/1.1703728	journal	May 1961
Wigner coefficients for a semisimple Lie group and the matrix elements of the O( n ) generators Gould, M. D. Journal of Mathematical Physics, Vol. 22, Issue 11 https://doi.org/10.1063/1.524820	journal	November 1981
Multiplicity‐free Wigner coefficients for semisimple Lie groups. I. The U( n ) pattern calculus Gould, M. D. Journal of Mathematical Physics, Vol. 27, Issue 8 https://doi.org/10.1063/1.527013	journal	August 1986
Multiplicity‐free Wigner coefficients for semisimple Lie groups. II. A pattern calculus for O( n ) Gould, M. D. Journal of Mathematical Physics, Vol. 27, Issue 8 https://doi.org/10.1063/1.527014	journal	August 1986
Reduction Factors for O( N )⊃O( N -1) Hongzhou, Sun; Qizhi, Han; Mei, Zhang Communications in Theoretical Physics, Vol. 30, Issue 4 https://doi.org/10.1088/0253-6102/30/4/541	journal	December 1998
Bounding scalar operator dimensions in 4 D CFT Rattazzi, Riccardo; Rychkov, Vyacheslav S.; Tonni, Erik Journal of High Energy Physics, Vol. 2008, Issue 12 https://doi.org/10.1088/1126-6708/2008/12/031	journal	December 2008
Constraints on Conformal Field Theories in Diverse Dimensions from the Bootstrap Mechanism Gliozzi, Ferdinando Physical Review Letters, Vol. 111, Issue 16 https://doi.org/10.1103/PhysRevLett.111.161602	journal	October 2013
Bootstrapping Conformal Field Theories with the Extremal Functional Method El-Showk, Sheer; Paulos, Miguel F. Physical Review Letters, Vol. 111, Issue 24 https://doi.org/10.1103/PhysRevLett.111.241601	journal	December 2013
PyCFTBoot: A Flexible Interface for the Conformal Bootstrap Behan, Connor Communications in Computational Physics, Vol. 22, Issue 1 https://doi.org/10.4208/cicp.OA-2016-0107	journal	May 2017

Cited By (22)

Multipoint conformal blocks in the comb channel Rosenhaus, Vladimir Journal of High Energy Physics, Vol. 2019, Issue 2 https://doi.org/10.1007/jhep02(2019)142	journal	February 2019
Mellin amplitudes for fermionic conformal correlators Faller, Josua; Sarkar, Sourav; Verma, Mritunjay Journal of High Energy Physics, Vol. 2018, Issue 3 https://doi.org/10.1007/jhep03(2018)106	journal	March 2018
Spheres, charges, instantons, and bootstrap: A five-dimensional odyssey Chang, Chi-Ming; Fluder, Martin; Lin, Ying-Hsuan Journal of High Energy Physics, Vol. 2018, Issue 3 https://doi.org/10.1007/jhep03(2018)123	journal	March 2018
From spinning conformal blocks to matrix Calogero-Sutherland models Schomerus, Volker; Sobko, Evgeny Journal of High Energy Physics, Vol. 2018, Issue 4 https://doi.org/10.1007/jhep04(2018)052	journal	April 2018
Scalar-fermion analytic bootstrap in 4D Elkhidir, Emtinan; Karateev, Denis Journal of High Energy Physics, Vol. 2019, Issue 6 https://doi.org/10.1007/jhep06(2019)026	journal	June 2019
Fermion conformal bootstrap in 4d Karateev, Denis; Kravchuk, Petr; Serone, Marco Journal of High Energy Physics, Vol. 2019, Issue 6 https://doi.org/10.1007/jhep06(2019)088	journal	June 2019
A spacetime derivation of the Lorentzian OPE inversion formula Simmons-Duffin, David; Stanford, Douglas; Witten, Edward Journal of High Energy Physics, Vol. 2018, Issue 7 https://doi.org/10.1007/jhep07(2018)085	journal	July 2018
Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory Isachenkov, Mikhail; Schomerus, Volker Journal of High Energy Physics, Vol. 2018, Issue 7 https://doi.org/10.1007/jhep07(2018)180	journal	July 2018
Veneziano amplitude of Vasiliev theory Turiaci, Gustavo J.; Zhiboedov, Alexander Journal of High Energy Physics, Vol. 2018, Issue 10 https://doi.org/10.1007/jhep10(2018)034	journal	October 2018
The conformal bootstrap at finite temperature Iliesiu, Luca; Koloğlu, Murat; Mahajan, Raghu Journal of High Energy Physics, Vol. 2018, Issue 10 https://doi.org/10.1007/jhep10(2018)070	journal	October 2018
Harmonic analysis and mean field theory Karateev, Denis; Kravchuk, Petr; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2019, Issue 10 https://doi.org/10.1007/jhep10(2019)217	journal	October 2019
Recursion relation for general 3d blocks Erramilli, Rajeev S.; Iliesiu, Luca V.; Kravchuk, Petr Journal of High Energy Physics, Vol. 2019, Issue 12 https://doi.org/10.1007/jhep12(2019)116	journal	December 2019
The conformal bootstrap: Theory, numerical techniques, and applications Poland, David; Rychkov, Slava; Vichi, Alessandro Reviews of Modern Physics, Vol. 91, Issue 1 https://doi.org/10.1103/revmodphys.91.015002	journal	January 2019
Canonical relativistic formulation to calculate the Casimir force four-vector on anisotropic conductor polarizable and magnetizable media Amooshahi, Majid Modern Physics Letters A, Vol. 34, Issue 30 https://doi.org/10.1142/s021773231950247x	journal	September 2019
From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models Schomerus, Volker; Sobko, Evgeny Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2017-11881	text	January 2017
From spinning conformal blocks to matrix Calogero-Sutherland models Schomerus, Volker; Sobko, Evgeny Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2018-02255	text	January 2018
Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory Isachenkov, Mikhail; Schomerus, Volker Deutsches Elektronen-Synchrotron, DESY, Hamburg https://doi.org/10.3204/pubdb-2018-03900	text	January 2018
Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey Chang, Chi-Ming; Fluder, Martin; Lin, Ying-Hsuan arXiv https://doi.org/10.48550/arxiv.1710.08418	text	January 2017
Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory Isachenkov, Mikhail; Schomerus, Volker arXiv https://doi.org/10.48550/arxiv.1711.06609	text	January 2017
Scalar-Fermion Analytic Bootstrap in 4D Elkhidir, Emtinan; Karateev, Denis arXiv https://doi.org/10.48550/arxiv.1712.01554	preprint	January 2017
The Conformal Bootstrap at Finite Temperature Iliesiu, Luca; Koloğlu, Murat; Mahajan, Raghu arXiv https://doi.org/10.48550/arxiv.1802.10266	text	January 2018
The Conformal Bootstrap: Theory, Numerical Techniques, and Applications Poland, David; Rychkov, Slava; Vichi, Alessandro arXiv https://doi.org/10.48550/arxiv.1805.04405	text	January 2018

Similar Records

Weight shifting operators and conformal blocks

Journal Article · Tue Feb 13 19:00:00 EST 2018 · Journal of High Energy Physics (Online) · OSTI ID:1502431

Recursion relation for general 3d blocks

Journal Article · Sun Dec 15 19:00:00 EST 2019 · Journal of High Energy Physics (Online) · OSTI ID:1596664

Recursion relations for 5-point conformal blocks

Journal Article · Thu Sep 30 20:00:00 EDT 2021 · Journal of High Energy Physics (Online) · OSTI ID:1976565

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Conformal Field Theory
Conformal and W Symmetry

Casimir recursion relations for general conformal blocks

Citation Formats

References (65)

Cited By (22)

Similar Records

Related Subjects