Abstract
We derive constraints on two-dimensional conformal field theories with higher spin symmetry due to unitarity, modular invariance, and causality. We focus on CFTs with \( {\mathcal{W}}_N \) symmetry in the “irrational” regime, where c > N − 1 and the theories have an infinite number of higher-spin primaries. The most powerful constraints come from positivity of the Kac matrix, which (unlike the Virasoro case) is non-trivial even when c > N − 1. This places a lower bound on the dimension of any non-vacuum higher-spin primary state, which is linear in the central charge. At large c, this implies that the dual holographic theories of gravity in AdS3, if they exist, have no local, perturbative degrees of freedom in the semi-classical limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].
C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Entanglement Scrambling in 2d Conformal Field Theory, JHEP 09 (2015) 110 [arXiv:1506.03772] [INSPIRE].
A. Bilal, Introduction to W algebras, in proceedings of Spring School on String Theory and Quantum Gravity, Trieste, Italy, April 15-23, 1991, pp. 245-280 [INSPIRE].
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
N. Boulanger, D. Ponomarev, E.D. Skvortsov and M. Taronna, On the uniqueness of higher-spin symmetries in AdS and CFT, Int. J. Mod. Phys. A 28 (2013) 1350162 [arXiv:1305.5180] [INSPIRE].
V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d > 3 dimensions, JHEP 03 (2016) 044 [arXiv:1510.02535] [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
J.D. Qualls and A.D. Shapere, Bounds on Operator Dimensions in 2D Conformal Field Theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE].
N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal Bounds on Charged States in 2d CFT and 3d Gravity, JHEP 08 (2016) 041 [arXiv:1603.09745] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, arXiv:1608.06241 [INSPIRE].
L. Apolo, Bounds on CFTs with \( {\mathcal{W}}_3 \) algebras and AdS 3 higher spin theories, Phys. Rev. D 96 (2017) 086003 [arXiv:1705.10402] [INSPIRE].
N.J. Iles and G.M.T. Watts, Characters of the W 3 algebra, JHEP 02 (2014) 009 [arXiv:1307.3771] [INSPIRE].
N.J. Iles and G.M.T. Watts, Modular properties of characters of the W 3 algebra, JHEP 01 (2016) 089 [arXiv:1411.4039] [INSPIRE].
E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
E. Witten, (2 + 1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons Action for Three-Dimensional anti-de Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
C.-M. Chang and X. Yin, Higher Spin Gravity with Matter in AdS 3 and Its CFT Dual, JHEP 10 (2012) 024 [arXiv:1106.2580] [INSPIRE].
K. Papadodimas and S. Raju, Correlation Functions in Holographic Minimal Models, Nucl. Phys. B 856 (2012) 607 [arXiv:1108.3077] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar and M. Rangamani, The Spectrum of Light States in Large N Minimal Models, JHEP 01 (2014) 116 [arXiv:1310.1744] [INSPIRE].
A. Castro, A. Lepage-Jutier and A. Maloney, Higher Spin Theories in AdS 3 and a Gravitational Exclusion Principle, JHEP 01 (2011) 142 [arXiv:1012.0598] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
S. Ferrara, R. Gatto and A.F. Grillo, Properties of Partial Wave Amplitudes in Conformal Invariant Field Theories, Nuovo Cim. A 26 (1975) 226 [INSPIRE].
V. Fateev and S. Ribault, The Large central charge limit of conformal blocks, JHEP 02 (2012) 001 [arXiv:1109.6764] [INSPIRE].
M.R. Gaberdiel and T. Hartman, Symmetries of Holographic Minimal Models, JHEP 05 (2011) 031 [arXiv:1101.2910] [INSPIRE].
M.R. Gaberdiel, T. Hartman and K. Jin, Higher Spin Black Holes from CFT, JHEP 04 (2012) 103 [arXiv:1203.0015] [INSPIRE].
K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys. C 2 (1991) 787 [INSPIRE].
S. Mizoguchi, Nonunitarity Theorem for the a Type W n Algebra, Phys. Lett. B 231 (1989) 112 [INSPIRE].
G.M.T. Watts, Determinant Formulae for Extended Algebras in Two-dimensional Conformal Field Theory, Nucl. Phys. B 326 (1989) 648 [Erratum ibid. B 336 (1990) 720] [INSPIRE].
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
R. Blumenhagen and E. Plauschinn, Introduction to Conformal Field Theory: With Applications to String Theory, Lect. Notes Phys., Springer Berlin Heidelberg (2009).
S. Mizoguchi, Determinant Formula and Unitarity for the W 3 Algebra, Phys. Lett. B 222 (1989) 226 [INSPIRE].
R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel and R. Varnhagen, W algebras with two and three generators, Nucl. Phys. B 361 (1991) 255 [INSPIRE].
H.G. Kausch and G.M.T. Watts, A Study of W algebras using Jacobi identities, Nucl. Phys. B 354 (1991) 740 [INSPIRE].
C.-J. Zhu, The Complete structure of the nonlinear W 4 and W 5 algebras from quantum Miura transformation, Phys. Lett. B 316 (1993) 264 [hep-th/9306025] [INSPIRE].
K. Hornfeck, W algebras with set of primary fields of dimensions (3, 4, 5) and (3, 4, 5, 6), Nucl. Phys. B 407 (1993) 237 [hep-th/9212104] [INSPIRE].
J.D. Qualls, Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories, arXiv:1508.00548 [INSPIRE].
K. De Vos and P. Van Driel, The Kazhdan-Lusztig conjecture for W algebras, J. Math. Phys. 37 (1996) 3587 [hep-th/9508020] [INSPIRE].
S. Mizoguchi, The Structure of Representation of the W 3 Algebra, Int. J. Mod. Phys. A 6 (1991) 133 [INSPIRE].
L. Borisov, M.B. Halpern and C. Schweigert, Systematic approach to cyclic orbifolds, Int. J. Mod. Phys. A 13 (1998) 125 [hep-th/9701061] [INSPIRE].
R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck and R. Hubel, Coset realization of unifying W algebras, Int. J. Mod. Phys. A 10 (1995) 2367 [hep-th/9406203] [INSPIRE].
P. Bouwknegt, Extended Conformal Algebras from Kac-Moody Algebras, (1988) [INSPIRE].
S. Odake, C = 3d Conformal Algebra With Extended Supersymmetry, Mod. Phys. Lett. A 5 (1990) 561 [INSPIRE].
S. Odake, Character Formulas of an Extended Superconformal Algebra Relevant to String Compactification, Int. J. Mod. Phys. A 5 (1990) 897 [INSPIRE].
S. Odake, Extension of N = 2 Superconformal Algebra and Calabi-Yau Compactification, Mod. Phys. Lett. A 4 (1989) 557 [INSPIRE].
S.L. Shatashvili and C. Vafa, Superstrings and manifold of exceptional holonomy, Selecta Math. 1 (1995) 347 [hep-th/9407025] [INSPIRE].
D. Gepner and B. Noyvert, Unitary representations of SW (3/2, 2) superconformal algebra, Nucl. Phys. B 610 (2001) 545 [hep-th/0101116] [INSPIRE].
N. Benjamin, S.M. Harrison, S. Kachru, N.M. Paquette and D. Whalen, On the elliptic genera of manifolds of Spin(7) holonomy, Annales Henri Poincaré 17 (2016) 2663 [arXiv:1412.2804] [INSPIRE].
S. Mallwitz, On SW minimal models and N = 1 supersymmetric quantum Toda field theories, Int. J. Mod. Phys. A 10 (1995) 977 [hep-th/9405025] [INSPIRE].
B. Noyvert, Unitary minimal models of SW (3/2, 3/2, 2) superconformal algebra and manifolds of G 2 holonomy, JHEP 03 (2002) 030 [hep-th/0201198] [INSPIRE].
J. de Boer, A. Naqvi and A. Shomer, The Topological G 2 string, Adv. Theor. Math. Phys. 12 (2008) 243 [hep-th/0506211] [INSPIRE].
Open Access
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1707.07717
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Afkhami-Jeddi, N., Colville, K., Hartman, T. et al. Constraints on higher spin CFT2. J. High Energ. Phys. 2018, 92 (2018). https://doi.org/10.1007/JHEP05(2018)092
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2018)092