Abstract
We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal field theories with a global abelian symmetry. We find that the bound scales with c and provide examples that parametrically saturate this bound. We also prove that any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [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].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [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].
N. Benjamin, S. Kachru, C.A. Keller and N.M. Paquette, Emergent space-time and the supersymmetric index, JHEP 05 (2016) 158 [arXiv:1512.00010] [INSPIRE].
N. Benjamin, M.C.N. Cheng, S. Kachru, G.W. Moore and N.M. Paquette, Elliptic Genera and 3d Gravity, arXiv:1503.04800 [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
J.D. Qualls, Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories, arXiv:1508.00548 [INSPIRE].
J.D. Qualls, Universal Bounds in Even-Spin CFTs, JHEP 12 (2015) 001 [arXiv:1412.0383] [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].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
D. Harlow, Wormholes, Emergent Gauge Fields and the Weak Gravity Conjecture, JHEP 01 (2016) 122 [arXiv:1510.07911] [INSPIRE].
B. Heidenreich, M. Reece and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140 [arXiv:1509.06374] [INSPIRE].
Y. Nakayama and Y. Nomura, Weak gravity conjecture in the AdS/CFT correspondence, Phys. Rev. D 92 (2015) 126006 [arXiv:1509.01647] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
C. Cheung and G.N. Remmen, Infrared Consistency and the Weak Gravity Conjecture, JHEP 12 (2014) 087 [arXiv:1407.7865] [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys. 115 (1988) 649 [INSPIRE].
E.P. Verlinde and H.L. Verlinde, Chiral Bosonization, Determinants and the String Partition Function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].
E. Kiritsis, String theory in a nutshell, Princeton University Press (2007).
L. Álvarez-Gaumé, G.W. Moore and C. Vafa, Theta Functions, Modular Invariance and Strings, Commun. Math. Phys. 106 (1986) 1 [INSPIRE].
T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].
B. Rasof, The initial- and final-value theorems in laplace transform theory, J. Franklin Inst. 274 (1962) 165.
W.A. Stein, Modular Forms, A Computational Approach, Graduate Studies in Mathematics, American Mathematical Society (2007) and online at http://wstein.org/books/modform/modform/.
P. Jenkins and J. Rouse, Bounds for Coefficients of Cusp Forms and Extremal Lattices, Bull. London Math. Soc. 43 (2011) 927 [arXiv:1012.5991].
J. Leech, Notes on sphere packings, Can. J. Math. 19 (1967) 251.
G. Nebe, Some cyclo-quaternionic lattices, J. Algebra 199 (1998) 472.
G. Nebe, An even unimodular 72-dimensional lattice of minimum 8, J. Reine Angew. Math. 673 (2012) 237 [arXiv:1008.2862].
M. Caselle and K.S. Narain, A New Approach to the Construction of Conformal Field Theories, Nucl. Phys. B 323 (1989) 673 [INSPIRE].
L. Dolan, P. Goddard and P. Montague, Conformal Field Theory of Twisted Vertex Operators, Nucl. Phys. B 338 (1990) 529 [INSPIRE].
N. Benjamin, E. Dyer, A.L. Fitzpatrick, A. Maloney and E. Perlmutter, Small Black Holes and Near-Extremal CFTs, arXiv:1603.08524 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
G. Moore, Trieste Lectures on Mathematical Aspects of Supersymmetric Black Holes, http://www.physics.rutgers.edu/~gmoore/TriesteLectures March28 2008.pdf.
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].
M.-A. Fiset and J. Walcher, Bounding the Heat Trace of a Calabi-Yau Manifold, JHEP 09 (2015) 124 [arXiv:1506.08407] [INSPIRE].
P. Kraus, Lectures on black holes and the AdS 3 /CF T 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].
N.J. Iles and G.M.T. Watts, Characters of the W 3 algebra, JHEP 02 (2014) 009 [arXiv:1307.3771] [INSPIRE].
C.A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [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: 1603.09745
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Benjamin, N., Dyer, E., Fitzpatrick, A.L. et al. Universal bounds on charged states in 2d CFT and 3d gravity. J. High Energ. Phys. 2016, 41 (2016). https://doi.org/10.1007/JHEP08(2016)041
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)041