Abstract
Recently, Harvey and Wu proposed a suitable Hecke operator for vector-valued SL(2, ℤ) modular forms to connect the characters of different 2d rational conformal field theories (RCFTs). We generalize such an operator to the 2d fermionic RCFTs and call it fermionic Hecke operator. The new Hecke operator naturally maps the Neveu-Schwarz (NS) characters of a fermionic theory to the NS characters of another fermionic theory. Mathematically, it is the natural Hecke operator on vector-valued Γθ modular forms of weight zero. We find it can also be extended to \( \overset{\sim }{\textrm{NS}} \) and Ramond (R) sectors by combining the characters of the two sectors together. We systematically study the fermionic Hecke relations among 2d fermionic RCFTs with up to five NS characters and find that almost all known supersymmetric RCFTs can be realized as fermionic Hecke images of some simple theories such as supersymmetric minimal models. We also study the coset relations between fermionic Hecke images with respect to c = 12k holomorphic SCFTs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.-B. Bae et al., Fermionic rational conformal field theories and modular linear differential equations, PTEP 2021 (2021) 08B104 [arXiv:2010.12392] [INSPIRE].
J.-B. Bae et al., Bootstrapping fermionic rational CFTs with three characters, JHEP 01 (2022) 089 [arXiv:2108.01647] [INSPIRE].
Z. Duan, K. Lee, S. Lee and L. Li, On classification of fermionic rational conformal field theories, JHEP 02 (2023) 079 [arXiv:2210.06805] [INSPIRE].
T. Lan, L. Kong and X.-G. Wen, Theory of (2 + 1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries, Phys. Rev. B 94 (2016) 155113 [arXiv:1507.04673] [INSPIRE].
P. Bruillard et al., Fermionic Modular Categories and the 16-fold Way, J. Math. Phys. 58 (2017) 041704 [arXiv:1603.09294] [INSPIRE].
P. Bonderson, E.C. Rowell, Q. Zhang and Z. Wang, Congruence Subgroups and Super-Modular Categories, arXiv:1704.02041 [https://doi.org/10.48550/arXiv.1704.02041].
P. Bruillard, J.Y. Plavnik, E.C. Rowell and Q. Zhang, Classification of super-modular categories, arXiv:1909.09843.
P. Bruillard et al., Classification of super-modular categories by rank, arXiv:1705.05293 [https://doi.org/10.48550/arXiv.1705.05293].
G.Y. Cho, H.-C. Kim, D. Seo and M. You, Classification of fermionic topological orders from congruence representations, Phys. Rev. B 108 (2023) 115103 [arXiv:2210.03681] [INSPIRE].
J.F. Duncan, Super-moonshine for Conway’s largest sporadic group, math/0502267 [https://doi.org/10.48550/arXiv.math/0502267].
T. Creutzig, J.F.R. Duncan and W. Riedler, Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory, J. Phys. A 51 (2018) 034001 [arXiv:1704.03678] [INSPIRE].
T. Johnson-Freyd, Supersymmetry and the Suzuki chain, arXiv:1908.11012 [https://doi.org/10.2140/tunis.2021.3.309] [INSPIRE].
S.M. Harrison, N.M. Paquette, D. Persson and R. Volpato, Fun with F24, JHEP 02 (2021) 039 [arXiv:2009.14710] [INSPIRE].
J. Albert, J. Kaidi and Y.-H. Lin, Topological modularity of supermoonshine, PTEP 2023 (2023) 033B06 [arXiv:2210.14923] [INSPIRE].
Y. Tachikawa, Topological phases and relativistic QFTs, notes of the lectures given in the CERN winter school, February 2018 [https://member.ipmu.jp/yuji.tachikawa/lectures/2018-cern-rikkyo/].
A. Karch, D. Tong and C. Turner, A Web of 2d Dualities: Z2 Gauge Fields and Arf Invariants, SciPost Phys. 7 (2019) 007 [arXiv:1902.05550] [INSPIRE].
I. Runkel and G.M.T. Watts, Fermionic CFTs and classifying algebras, JHEP 06 (2020) 025 [arXiv:2001.05055] [INSPIRE].
C.-T. Hsieh, Y. Nakayama and Y. Tachikawa, Fermionic minimal models, Phys. Rev. Lett. 126 (2021) 195701 [arXiv:2002.12283] [INSPIRE].
J. Kulp, Two More Fermionic Minimal Models, JHEP 03 (2021) 124 [arXiv:2003.04278] [INSPIRE].
J.-B. Bae and S. Lee, Emergent supersymmetry on the edges, SciPost Phys. 11 (2021) 091 [arXiv:2105.02148] [INSPIRE].
K. Kikuchi, Emergent SUSY in two dimensions, arXiv:2204.03247 [INSPIRE].
J.A. Harvey and Y. Wu, Hecke Relations in Rational Conformal Field Theory, JHEP 09 (2018) 032 [arXiv:1804.06860] [INSPIRE].
J.A. Harvey, Y. Hu and Y. Wu, Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories, J. Phys. A 53 (2020) 334003 [arXiv:1912.11955] [INSPIRE].
Y. Wu, Hecke Operators and Galois Symmetry in Rational Conformal Field Theory, Doctoral Dissertation, The University of Chicago (2020).
Z. Duan, K. Lee and K. Sun, Hecke relations, cosets and the classification of 2d RCFTs, JHEP 09 (2022) 202 [arXiv:2206.07478] [INSPIRE].
G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster), arXiv:0706.0236.
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.N. Schellekens, Meromorphic c = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159 [hep-th/9205072] [INSPIRE].
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.8), https://www.sagemath.org (2019).
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
D. Friedan, Z.-A. Qiu and S.H. Shenker, Conformal Invariance, Unitarity and Two-Dimensional Critical Exponents, Phys. Rev. Lett. 52 (1984) 1575 [INSPIRE].
D. Friedan, Z.-A. Qiu and S.H. Shenker, Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model, Phys. Lett. B 151 (1985) 37 [INSPIRE].
M.A. Bershadsky, V.G. Knizhnik and M.G. Teitelman, Superconformal Symmetry in Two-Dimensions, Phys. Lett. B 151 (1985) 31 [INSPIRE].
P. Goddard, A. Kent and D.I. Olive, Unitary Representations of the Virasoro and Supervirasoro Algebras, Commun. Math. Phys. 103 (1986) 105 [INSPIRE].
T. Gannon, Comments on nonunitary conformal field theories, Nucl. Phys. B 670 (2003) 335 [hep-th/0305070] [INSPIRE].
K. Schoutens, Supersymmetry and Factorizable Scattering, Nucl. Phys. B 344 (1990) 665 [INSPIRE].
E. Melzer, Supersymmetric analogs of the Gordon-Andrews identities, and related TBA systems, hep-th/9412154 [INSPIRE].
A. Berkovich, B.M. McCoy and W.P. Orrick, Polynomial identities, indices, and duality for the N = 1 superconformal model SM (2, 4ν), J. Statist. Phys. 83 (1996) 795 [hep-th/9507072] [INSPIRE].
A. Berkovich and B.M. McCoy, Generalizations of the Andrews-Bressoud identities for the N = 1 superconformal model SM (2, 4ν), Math. Comput. Modelling 26 (1997) 37.
D. Kastor, Modular Invariance in Superconformal Models, Nucl. Phys. B 280 (1987) 304 [INSPIRE].
Y. Matsuo and S. Yahikozawa, Superconformal Field Theory With Modular Invariance on a Torus, Phys. Lett. B 178 (1986) 211 [INSPIRE].
A. Cappelli, Modular Invariant Partition Functions of Superconformal Theories, Phys. Lett. B 185 (1987) 82 [INSPIRE].
P. Di Vecchia, J.L. Petersen, M. Yu and H.B. Zheng, Explicit Construction of Unitary Representations of the N = 2 Superconformal Algebra, Phys. Lett. B 174 (1986) 280 [INSPIRE].
Y. Kazama and H. Suzuki, New N = 2 Superconformal Field Theories and Superstring Compactification, Nucl. Phys. B 321 (1989) 232 [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, On the Classification of Rational Conformal Field Theories, Phys. Lett. B 213 (1988) 303 [INSPIRE].
M. Dittmann and H. Wang, Theta blocks related to root systems, arXiv:2006.12967 [https://doi.org/10.48550/arXiv.2006.12967].
N.R. Scheithauer, The Fake monster superalgebra, math/9905113 [INSPIRE].
S.M. Harrison, N.M. Paquette and R. Volpato, A Borcherds-Kac-Moody Superalgebra with Conway Symmetry, Commun. Math. Phys. 370 (2019) 539 [arXiv:1803.10798] [INSPIRE].
K. Sun, H. Wang and B. Williams, On the hyperbolization of affine Kac-Moody algebras, to appear.
P. Bantay, The Kernel of the modular representation and the Galois action in RCFT, Commun. Math. Phys. 233 (2003) 423 [math/0102149] [INSPIRE].
J. Sturm, On the Congruence of Modular Forms, in Number Theory, Lecture Notes in Mathematics, vol. 1240, Springer-Verlag Berlin (1987), p. 275–280 [https://doi.org/10.1007/BFb0072985].
J.-B. Bae, K. Lee and S. Lee, Monster Anatomy, JHEP 07 (2019) 026 [arXiv:1811.12263] [INSPIRE].
J.-B. Bae et al., Conformal Field Theories with Sporadic Group Symmetry, Commun. Math. Phys. 388 (2021) 1 [arXiv:2002.02970] [INSPIRE].
Acknowledgments
We would like to thank Gil Young Cho, Zhihao Duan, Hee-Cheol Kim, Sungjay Lee, Linfeng Li, Minyoung You and Haowu Wang for useful discussions. KL and KS are supported by KIAS Grants PG006904 and QP081001 respectively. KL is also supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. 2017R1D1A1B06034369).
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: 2211.15304
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
Lee, K., Sun, K. Hecke relations among 2d fermionic RCFTs. J. High Energ. Phys. 2023, 44 (2023). https://doi.org/10.1007/JHEP09(2023)044
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)044