Abstract
We calculate the anomalous dimensions of operators with large global charge J in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a W = Φ3 superpotential. Working in a 1/J expansion, we find that the large-J sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge J is always a scalar operator whose dimension ΔJ satisfies the sum rule
up to corrections that vanish at large J . The spectrum of low-lying excited states is also calculable explcitly: for example, the second-lowest primary operator has spin two and dimension \( {\varDelta}_J+\sqrt{3} \). In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order \( {J}^{+\frac{1}{2}} \). The propagation speeds of the Goldstone waves and heavy fermions are \( \frac{1}{\sqrt{2}} \) and \( \pm \frac{1}{2} \) times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large J.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [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].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
S. Hellerman, S. Maeda, J. Maltz and I. Swanson, Effective String Theory Simplified, JHEP 09 (2014) 183 [arXiv:1405.6197] [INSPIRE].
M. Baker and R. Steinke, Semiclassical quantization of effective string theory and Regge trajectories, Phys. Rev. D 65 (2002) 094042 [hep-th/0201169] [INSPIRE].
L.A. Pando Zayas, J. Sonnenschein and D. Vaman, Regge trajectories revisited in the gauge/string correspondence, Nucl. Phys. B 682 (2004) 3 [hep-th/0311190] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [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].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D Conformal Field Theories with Global Symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [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, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) Archipelago, arXiv:1504.07997 [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
E. Barnes, E. Gorbatov, K.A. Intriligator, M. Sudano and J. Wright, The exact superconformal R-symmetry minimizes tau(RR), Nucl. Phys. B 730 (2005) 210 [hep-th/0507137] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
E. Dyer, M. Mezei, S.S. Pufu and S. Sachdev, Scaling dimensions of monopole operators in the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N_{b-1}} \) theory in 2 + 1 dimensions, JHEP 06 (2015) 037 [arXiv:1504.00368] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, Correlation Functions of Large-N Chern-Simons-Matter Theories and Bosonization in Three Dimensions, JHEP 12 (2012) 028 [arXiv:1207.4593] [INSPIRE].
O. Aharony, P. Narayan and T. Sharma, On monopole operators in supersymmetric Chern-Simons-matter theories, JHEP 05 (2015) 117 [arXiv:1502.00945] [INSPIRE].
E. Dyer, M. Mezei and S.S. Pufu, Monopole Taxonomy in Three-Dimensional Conformal Field Theories, arXiv:1309.1160 [INSPIRE].
D. Dyer and E. Ethan, private communication.
J. Polchinski, Effective field theory and the Fermi surface, hep-th/9210046 [INSPIRE].
P. Kopietz, L. Bartosch and F. Schutz, Introduction to the functional renormalization group, Lect. Notes Phys. 798 (2010) 1 [INSPIRE].
A. Brignole, One loop Kähler potential in non renormalizable theories, Nucl. Phys. B 579 (2000) 101 [hep-th/0001121] [INSPIRE].
M.T. Grisaru, W. Siegel and M. Roček, Improved Methods for Supergraphs, Nucl. Phys. B 159 (1979) 429 [INSPIRE].
L. Fu and C.L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett. 100 (2008) 096407 [INSPIRE].
M.Z. Hasan and C.L. Kane, Topological Insulators, Rev. Mod. Phys. 82 (2010) 3045 [arXiv:1002.3895] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the Asymptotics of 4D Quantum Field Theory, JHEP 01 (2013) 152 [arXiv:1204.5221] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
S. Kovacs, Y. Sato and H. Shimada, Membranes from monopole operators in ABJM theory: Large angular momentum and M-theoretic AdS 4/CFT 3, PTEP 2014 (2014) 093B01 [arXiv:1310.0016] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping the Three-Dimensional Supersymmetric Ising Model, Phys. Rev. Lett. 115 (2015) 051601 [arXiv:1502.04124] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with Four Supercharges, JHEP 08 (2015) 142 [arXiv:1503.02081] [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: 1505.01537
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
Hellerman, S., Orlando, D., Reffert, S. et al. On the CFT operator spectrum at large global charge. J. High Energ. Phys. 2015, 1–34 (2015). https://doi.org/10.1007/JHEP12(2015)071
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/JHEP12(2015)071