Abstract
Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function \( Z\left(\tau, \overline{\tau}\right) \) in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of \( Z\left(\tau, \overline{\tau}\right) \), and positivity and integrality of coefficients, which imply positivity and integrality of the 𝔲(1)n × 𝔲(1)n character expansion of \( Z\left(\tau, \overline{\tau}\right) \). Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting \( Z\left(\tau, \overline{\tau}\right) \) is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also “fake” polynomials satisfying all necessary properties, that are not associated with any code. They lead to \( Z\left(\tau, \overline{\tau}\right) \) satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z’s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
S. Hellerman and C. Schmidt-Colinet, Bounds for state degeneracies in 2D conformal field theory, JHEP 2011 (2011) 127 [arXiv:1007.0756].
C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [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 2014 (2014) 91 [arXiv:1312.0038].
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.D. Qualls, Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories, arXiv:1508.00548 [INSPIRE].
H. Kim, P. Kravchuk and H. Ooguri, Reflections on conformal spectra, JHEP 2016 (2016) 184 [arXiv:1510.08772].
Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].
T. Anous, R. Mahajan and E. Shaghoulian, Parity and the modular bootstrap, SciPost Phys. 5 (2018) 022 [arXiv:1803.04938] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast conformal bootstrap and constraints on 3d gravity, JHEP 2019 (2019) 87 [arXiv:1903.06272].
M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere packing and quantum gravity, JHEP 2019 (2019) 48 [arXiv:1905.01319].
N. Afkhami-Jeddi et al., High-dimensional sphere packing and the modular bootstrap, JHEP 12 (2020) 066 [arXiv:2006.02560] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
F. Gliozzi, Modular Bootstrap, Elliptic Points, and Quantum Gravity, Phys. Rev. Res. 2 (2020) 013327 [arXiv:1908.00029] [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 and D. Simmons-Duffin, The conformal bootstrap, Nature Phys. 12 (2016) 535 [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
S. El-Showk et al., Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
A. Dymarsky and A. Shapere, Quantum stabilizer codes, lattices, and CFTs, JHEP 21 (2020) 160 [arXiv:2009.01244] [INSPIRE].
A. Dymarsky and A. Shapere, Solutions of modular bootstrap constraints from quantum codes, Phys. Rev. Lett. 126 (2021) 161602 [arXiv:2009.01236] [INSPIRE].
J. Henriksson, A. Kakkar and B. McPeak, Classical codes and chiral CFTs at higher genus, JHEP 05 (2022) 159 [arXiv:2112.05168] [INSPIRE].
J. Henriksson, A. Kakkar and B. McPeak, Narain CFTs and quantum codes at higher genus, JHEP 04 (2023) 011 [arXiv:2205.00025] [INSPIRE].
M. Buican, A. Dymarsky and R. Radhakrishnan, Quantum codes, CFTs, and defects, JHEP 03 (2023) 017 [arXiv:2112.12162] [INSPIRE].
S. Yahagi, Narain CFTs and error-correcting codes on finite fields, JHEP 08 (2022) 058 [arXiv:2203.10848] [INSPIRE].
N. Angelinos, D. Chakraborty and A. Dymarsky, Optimal Narain CFTs from codes, JHEP 11 (2022) 118 [arXiv:2206.14825] [INSPIRE].
A. Dymarsky and A. Sharon, Non-rational Narain CFTs from codes over F4, JHEP 11 (2021) 016 [arXiv:2107.02816] [INSPIRE].
M.R. Gaberdiel, H.R. Hampapura and S. Mukhi, Cosets of Meromorphic CFTs and Modular Differential Equations, JHEP 04 (2016) 156 [arXiv:1602.01022] [INSPIRE].
A.R. Chandra and S. Mukhi, Towards a Classification of Two-Character Rational Conformal Field Theories, JHEP 04 (2019) 153 [arXiv:1810.09472] [INSPIRE].
A.R. Chandra and S. Mukhi, Curiosities above c = 24, SciPost Phys. 6 (2019) 053 [arXiv:1812.05109] [INSPIRE].
S. Mukhi, Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report, in the proceedings of the Pollica Summer Workshop 2019: Mathematical and Geometric Tools for Conformal Field Theories, (2019) [arXiv:1910.02973] [INSPIRE].
J.H. Conway and N.J.A. Sloane, Low-Dimensional Lattices. I. Quadratic Forms of Small Determinant, Proc. Roy. Soc. Lond. A 418 (1988) 17.
J. Milnor, Curvatures of Left Invariant Metrics on Lie Groups, Adv. Math. 21 (1976) 293 [INSPIRE].
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists, Cambridge University Press (2003) [INSPIRE].
A.N. Schellekens, Meromorphic c = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159 [hep-th/9205072] [INSPIRE].
J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Springer Science & Business Media (1999) [https://doi.org/10.1007/978-1-4757-6568-7].
Acknowledgments
We thank Ofer Aharony for collaboration at the early stages of this project, and for comments on the draft. We also thank Felix Jonas and the Naama Barkai lab for help with computing, and Hiromi Ebisu, Masataka Watanabe, Ohad Mamroud, Adam Schwimmer, Shaul Zemel, Micha Berkooz, Erez Urbach, Adar Sharon, and Shai Chester for useful discussions. AD is grateful to Weizmann Institute of Science for hospitality and acknowledges sabbatical support of the Schwartz/Reisman Institute for Theoretical Physics. AD was supported by the National Science Foundation under Grant No. PHY-2013812. The work of RRK was supported in part by an Israel Science Foundation (ISF) center for excellence grant (grant number 2289/18), by ISF grant no. 2159/22, by Simons Foundation grant 994296 (Simons Collaboration on Confinement and QCD Strings), by grant no. 2018068 from the United States-Israel Binational Science Foundation (BSF), by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by the German Research Foundation through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”, and by a research grant from Martin Eisenstein.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2211.15699
Rights and permissions
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.
About this article
Cite this article
Dymarsky, A., Kalloor, R.R. Fake Z. J. High Energ. Phys. 2023, 43 (2023). https://doi.org/10.1007/JHEP06(2023)043
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2023)043