Abstract
We study generalizations of the non-invertible duality defects present in \( \mathcal{N} \) = 4 SU(N) SYM by studying theories with larger duality groups. We focus on 4d \( \mathcal{N} \) = 2 theories of class \( \mathcal{S} \) obtained by the dimensional reduction of the 6d \( \mathcal{N} \) = (2, 0) theory of AN−1 type on a Riemann surface Σg without punctures. We discuss their non-invertible duality symmetries and provide two ways to compute their fusion algebra: either using discrete topological manipulations or a 5d TQFT description. We also introduce the concept of “rank” of a non-invertible duality symmetry and show how it can be used to (almost) completely fix the fusion algebra with little computational effort.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
K. Roumpedakis, S. Seifnashri and S.-H. Shao, Higher Gauging and Non-invertible Condensation Defects, Commun. Math. Phys. 401 (2023) 3043 [arXiv:2204.02407] [INSPIRE].
L. Bhardwaj, L.E. Bottini, S. Schäfer-Nameki and A. Tiwari, Non-invertible higher-categorical symmetries, SciPost Phys. 14 (2023) 007 [arXiv:2204.06564] [INSPIRE].
G. Arias-Tamargo and D. Rodriguez-Gomez, Non-invertible symmetries from discrete gauging and completeness of the spectrum, JHEP 04 (2023) 093 [arXiv:2204.07523] [INSPIRE].
Y. Choi et al., Non-invertible Condensation, Duality, and Triality Defects in 3 + 1 Dimensions, Commun. Math. Phys. 402 (2023) 489 [arXiv:2204.09025] [INSPIRE].
J. Kaidi, G. Zafrir and Y. Zheng, Non-invertible symmetries of \( \mathcal{N} \) = 4 SYM and twisted compactification, JHEP 08 (2022) 053 [arXiv:2205.01104] [INSPIRE].
Y. Choi, H.T. Lam and S.-H. Shao, Noninvertible Global Symmetries in the Standard Model, Phys. Rev. Lett. 129 (2022) 161601 [arXiv:2205.05086] [INSPIRE].
C. Córdova and K. Ohmori, Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X 13 (2023) 011034 [arXiv:2205.06243] [INSPIRE].
A. Antinucci, G. Galati and G. Rizi, On continuous 2-category symmetries and Yang-Mills theory, JHEP 12 (2022) 061 [arXiv:2206.05646] [INSPIRE].
V. Bashmakov, M. Del Zotto and A. Hasan, On the 6d origin of non-invertible symmetries in 4d, JHEP 09 (2023) 161 [arXiv:2206.07073] [INSPIRE].
J.A. Damia, R. Argurio and L. Tizzano, Continuous Generalized Symmetries in Three Dimensions, JHEP 05 (2023) 164 [arXiv:2206.14093] [INSPIRE].
J.A. Damia, R. Argurio and E. Garcia-Valdecasas, Non-invertible defects in 5d, boundaries and holography, SciPost Phys. 14 (2023) 067 [arXiv:2207.02831] [INSPIRE].
Y. Choi, H.T. Lam and S.-H. Shao, Noninvertible Time-Reversal Symmetry, Phys. Rev. Lett. 130 (2023) 131602 [arXiv:2208.04331] [INSPIRE].
L. Bhardwaj, S. Schäfer-Nameki and J. Wu, Universal Non-Invertible Symmetries, Fortsch. Phys. 70 (2022) 2200143 [arXiv:2208.05973] [INSPIRE].
L. Lin, D.G. Robbins and E. Sharpe, Decomposition, Condensation Defects, and Fusion, Fortsch. Phys. 70 (2022) 2200130 [arXiv:2208.05982] [INSPIRE].
T. Bartsch, M. Bullimore, A.E.V. Ferrari and J. Pearson, Non-invertible Symmetries and Higher Representation Theory I, arXiv:2208.05993 [INSPIRE].
F. Apruzzi, I. Bah, F. Bonetti and S. Schäfer-Nameki, Noninvertible Symmetries from Holography and Branes, Phys. Rev. Lett. 130 (2023) 121601 [arXiv:2208.07373] [INSPIRE].
I. García Etxebarria, Branes and Non-Invertible Symmetries, Fortsch. Phys. 70 (2022) 2200154 [arXiv:2208.07508] [INSPIRE].
J.J. Heckman, M. Hübner, E. Torres and H.Y. Zhang, The Branes Behind Generalized Symmetry Operators, Fortsch. Phys. 71 (2023) 2200180 [arXiv:2209.03343] [INSPIRE].
P. Niro, K. Roumpedakis and O. Sela, Exploring non-invertible symmetries in free theories, JHEP 03 (2023) 005 [arXiv:2209.11166] [INSPIRE].
A. Antinucci et al., The holography of non-invertible self-duality symmetries, arXiv:2210.09146 [INSPIRE].
S. Chen and Y. Tanizaki, Solitonic Symmetry beyond Homotopy: Invertibility from Bordism and Noninvertibility from Topological Quantum Field Theory, Phys. Rev. Lett. 131 (2023) 011602 [arXiv:2210.13780] [INSPIRE].
C. Córdova, S. Hong, S. Koren and K. Ohmori, Neutrino Masses from Generalized Symmetry Breaking, arXiv:2211.07639 [INSPIRE].
I. García Etxebarria and N. Iqbal, A Goldstone theorem for continuous non-invertible symmetries, JHEP 09 (2023) 145 [arXiv:2211.09570] [INSPIRE].
Y. Choi, H.T. Lam and S.-H. Shao, Non-invertible Gauss law and axions, JHEP 09 (2023) 067 [arXiv:2212.04499] [INSPIRE].
R. Yokokura, Non-invertible symmetries in axion electrodynamics, arXiv:2212.05001 [INSPIRE].
L. Bhardwaj, S. Schäfer-Nameki and A. Tiwari, Unifying constructions of non-invertible symmetries, SciPost Phys. 15 (2023) 122 [arXiv:2212.06159] [INSPIRE].
L. Bhardwaj, L.E. Bottini, S. Schäfer-Nameki and A. Tiwari, Non-invertible symmetry webs, SciPost Phys. 15 (2023) 160 [arXiv:2212.06842] [INSPIRE].
T. Bartsch, M. Bullimore, A.E.V. Ferrari and J. Pearson, Non-invertible Symmetries and Higher Representation Theory II, arXiv:2212.07393 [INSPIRE].
N. Mekareeya and M. Sacchi, Mixed anomalies, two-groups, non-invertible symmetries, and 3d superconformal indices, JHEP 01 (2023) 115 [arXiv:2210.02466] [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
C.-M. Chang et al., Topological Defect Lines and Renormalization Group Flows in Two Dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
R. Thorngren and Y. Wang, Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, arXiv:1912.02817 [INSPIRE].
R. Thorngren and Y. Wang, Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond, arXiv:2106.12577 [INSPIRE].
J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [cond-mat/0404051] [INSPIRE].
J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
Y. Choi et al., Noninvertible duality defects in 3 + 1 dimensions, Phys. Rev. D 105 (2022) 125016 [arXiv:2111.01139] [INSPIRE].
J. Kaidi, K. Ohmori and Y. Zheng, Kramers-Wannier-like Duality Defects in (3 + 1)D Gauge Theories, Phys. Rev. Lett. 128 (2022) 111601 [arXiv:2111.01141] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP 05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
L. Bhardwaj, M. Hübner and S. Schäfer-Nameki, 1-form Symmetries of 4d N = 2 Class S Theories, SciPost Phys. 11 (2021) 096 [arXiv:2102.01693] [INSPIRE].
L. Bhardwaj, S. Giacomelli, M. Hübner and S. Schäfer-Nameki, Relative defects in relative theories: Trapped higher-form symmetries and irregular punctures in class S, SciPost Phys. 13 (2022) 101 [arXiv:2201.00018] [INSPIRE].
T. Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press (2000) [ISBN: 9780521798099].
D.S. Freed and C. Teleman, Relative quantum field theory, Commun. Math. Phys. 326 (2014) 459 [arXiv:1212.1692] [INSPIRE].
J. Kaidi, K. Ohmori and Y. Zheng, Symmetry TFTs for Non-invertible Defects, Commun. Math. Phys. 404 (2023) 1021 [arXiv:2209.11062] [INSPIRE].
P.-S. Hsin, H.T. Lam and N. Seiberg, Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d, SciPost Phys. 6 (2019) 039 [arXiv:1812.04716] [INSPIRE].
V. Bashmakov, M. Del Zotto, A. Hasan and J. Kaidi, Non-invertible symmetries of class S theories, JHEP 05 (2023) 225 [arXiv:2211.05138] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
L. Bhardwaj, Y. Lee and Y. Tachikawa, SL(2, ℤ) action on QFTs with ℤ2 symmetry and the Brown-Kervaire invariants, JHEP 11 (2020) 141 [arXiv:2009.10099] [INSPIRE].
F. Benini, C. Córdova and P.-S. Hsin, On 2-Group Global Symmetries and their Anomalies, JHEP 03 (2019) 118 [arXiv:1803.09336] [INSPIRE].
J. Kaidi et al., Higher central charges and topological boundaries in 2 + 1-dimensional TQFTs, SciPost Phys. 13 (2022) 067 [arXiv:2107.13091] [INSPIRE].
F. Benini, C. Copetti and L. Di Pietro, Factorization and global symmetries in holography, SciPost Phys. 14 (2023) 019 [arXiv:2203.09537] [INSPIRE].
F. Apruzzi et al., Symmetry TFTs from String Theory, Commun. Math. Phys. 402 (2023) 895 [arXiv:2112.02092] [INSPIRE].
M. van Beest, D.S.W. Gould, S. Schäfer-Nameki and Y.-N. Wang, Symmetry TFTs for 3d QFTs from M-theory, JHEP 02 (2023) 226 [arXiv:2210.03703] [INSPIRE].
I. Bah, F. Bonetti and R. Minasian, Discrete and higher-form symmetries in SCFTs from wrapped M5-branes, JHEP 03 (2021) 196 [arXiv:2007.15003] [INSPIRE].
S. Monnier, The anomaly field theories of six-dimensional (2, 0) superconformal theories, Adv. Theor. Math. Phys. 22 (2018) 2035 [arXiv:1706.01903] [INSPIRE].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
C.V. Johnson, From M theory to F theory, with branes, Nucl. Phys. B 507 (1997) 227 [hep-th/9706155] [INSPIRE].
T. Wedhorn, Manifolds, Sheaves, and Cohomology, Springer (2016) [https://doi.org/10.1007/978-3-658-10633-1].
Acknowledgments
We thank V. Bashmakov, M. Del Zotto, J. Kaidi, K. Ohmori, S. Schafer-Nameki and Y. Wang for useful discussions. We especially thank F. Benini for discussions and collaboration on a related project. We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University, where some of the research for this paper was performed. The authors are partially supported by the INFN “Iniziativa Specifica ST&FI”. A.A., C.C. and G.R. are supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative dynamics of quantum fields: from new deconfined phases of matter to quantum black holes”, by the MIUR-SIR grant RBSI1471GJ, and by the MIUR-PRIN contract 2015 MP2CX4.
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: 2212.09549
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
Antinucci, A., Copetti, C., Galati, G. et al. “Zoology” of non-invertible duality defects: the view from class \( \mathcal{S} \). J. High Energ. Phys. 2024, 36 (2024). https://doi.org/10.1007/JHEP04(2024)036
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2024)036