Journal of Holography Applications in Physics
Categorical Symmetries and Fiber Functors from Multiple Gaugeable Homomorphisms from 6D N=(2,0) SCFTs
Document Type : Regular article
Author
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai, 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China; Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, UK
Abstract
Exploiting the symmetry topological field theory/topological order correspondence (SymTFT/TO), together with the higher-categorical structure of 6D N=(2,0) SCFTs, we prove that the total quantum dimension of the relative gaugeable algebra leading to intrinsic non-invertible symmetries between class S theories is greater with respect to the non-intrinsic case. From a higher-categorical perspective, this supports the idea that multiplicity is allowed to exceed unity in some superselection sectors.
Keywords
Main Subjects
Article PDF
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[2] E. Witten, “Conformal Field Theory In Four And Six Dimensions,” [arXiv:0712.0157 [math.RT]].
[3] O. Chacaltana, J. Distler and Y. Tachikawa, “Nilpotent orbits and codimensiontwo defects of 6d N=(2,0) theories,” Int. J. Mod. Phys. A 28, 1340006 (2013) DOI:10.1142/S0217751X1340006X [arXiv:1203.2930 [hep-th]].
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[9] G. W. Moore and N. Seiberg, “Classical and Quantum Conformal Field Theory,” Commun. Math. Phys. 123, 177 (1989) DOI:10.1007/BF01238857.
[10] G. W. Moore and N. Seiberg, “Naturality in Conformal Field Theory,” Nucl. Phys. B 313, 16 (1989) DOI:10.1016/0550-3213(89)90511-7.
[11] E. P. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal Field Theory,” Nucl. Phys. B 300, 360 (1988) DOI:10.1016/0550-3213(88)90603-7.
[12] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” JHEP 02, 172 (2015) DOI:10.1007/JHEP02(2015)172 [arXiv:1412.5148 [hep-th]].
[13] D. Gaiotto and J. Kulp, “Orbifold groupoids,” JHEP 02, 132 (2021) DOI:10.1007/JHEP02(2021)132 [arXiv:2008.05960 [hep-th]].
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[15] L. Kong and Z. H. Zhang, “An invitation to topological orders and category theory,” [arXiv:2205.05565 [cond-mat.str-el]].
[16] L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part I,” JHEP 02, 150 (2020) DOI:10.1007/JHEP02(2020)150 [arXiv:1905.04924 [condmat.str-el]].
[17] L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part II,” Nucl. Phys. B 966, 115384 (2021) DOI:10.1016/j.nuclphysb.2021.115384 [arXiv:1912.01760 [cond-mat.str-el]].
[18] L. Kong and H. Zheng, “Categorical computation,” Front. Phys. 18, 21302 (2023) DOI:10.1007/s11467-022-1251-5 [arXiv:2102.04814 [quant-ph]].
[19] D. Gaiotto and T. Johnson-Freyd, “Condensations in higher categories,” [arXiv:1905.09566 [math.CT]].
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[27] V. Bashmakov, M. Del Zotto, A. Hasan and J. Kaidi, “Non-invertible Symmetries of Class S Theories,” [arXiv:2211.05138 [hep-th]].
[28] L. Bhardwaj, S. Schafer-Nameki and A. Tiwari, “Unifying Constructions of NonInvertible Symmetries,” [arXiv:2212.06159 [hep-th]].
[29] L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki and A. Tiwari, “Non-Invertible HigherCategorical Symmetries,” [arXiv:2204.06564 [hep-th]].
[30] J. Kaidi, K. Ohmori and Y. Zheng, “Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,” Phys. Rev. Lett. 128, 111601 (2022) DOI:10.1103/PhysRevLett.128.111601 [arXiv:2111.01141 [hep-th]].
[31] J. Kaidi, G. Zafrir and Y. Zheng, “Non-invertible symmetries of N = 4 SYM and twisted compactification,” JHEP 08, 053 (2022) DOI:10.1007/JHEP08(2022)053 [arXiv:2205.01104 [hep-th]].
[32] J. Kaidi, K. Ohmori and Y. Zheng, “Symmetry TFTs for Non-Invertible Defects,” [arXiv:2209.11062 [hep-th]].
[33] I. M. Burbano, J. Kulp and J. Neuser, “Duality defects in E8,” JHEP 10, 186 (2022) DOI:10.1007/JHEP10(2022)187 [arXiv:2112.14323 [hep-th]].
[34] Y. Choi, C. Cordova, P. S. Hsin, H. T. Lam and S. H. Shao, “Non-invertible gauging, Duality, and Triality Defects in 3+1 Dimensions,” [arXiv:2204.09025 [hep-th]].
[35] Y. Choi, C. Cordova, P. S. Hsin, H. T. Lam and S. H. Shao, “Noninvertible duality defects in 3+1 dimensions,” Phys. Rev. D 105, 125016 (2022) DOI:10.1103/PhysRevD.105.125016 [arXiv:2111.01139 [hep-th]].
[36] A. Antinucci, F. Benini, C. Copetti, G. Galati and G. Rizi, “The holography of noninvertible self-duality symmetries,” [arXiv:2210.09146 [hep-th]].
[37] A. Antinucci, C. Copetti, G. Galati and G. Rizi, “Zoology of non-invertible duality defects: the view from class S,” [arXiv:2212.09549 [hep-th]].
[38] T. Johnson-Freyd, “Operators and higher categories in quantum field theory,” Lecture series.
[39] L. Kong, “Anyon gauging and tensor categories,” Nucl. Phys. B 886, 436 (2014) DOI:10.1016/j.nuclphysb.2014.07.003 [arXiv:1307.8244 [cond-mat.str-el]].
[40] M. Yu, “Gauging Categorical Symmetries in 3d Topological Orders and Bulk Reconstruction,” [arXiv:2111.13697 [hep-th]].
[41] O. Aharony, N. Seiberg and Y. Tachikawa, “Reading between the lines of fourdimensional gauge theories,” JHEP 08, 115 (2013) DOI:10.1007/JHEP08(2013)115 [arXiv:1305.0318 [hep-th]].
[42] W. Ji and X. G. Wen, “Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions,” Phys. Rev. Research. 1, 033054 (2019) DOI:10.1103/PhysRevResearch.1.033054 [arXiv:1905.13279 [cond-mat.str-el]].
[43] J. C. Y. Teo, T. L. Hughes and E. Fradkin, “Theory of Twist Liquids: Gauging an Anyonic Symmetry,” Annals Phys. 360, 3495 (2015) DOI:10.1016/j.aop.2015.05.012 [arXiv:1503.06812 [cond-mat.str-el]].
[44] M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, “Symmetry Fractionalization, Defects, and Gauging of Topological Phases,” Phys. Rev. B 100, 115147 (2019) DOI:10.1103/PhysRevB.100.115147 [arXiv:1410.4540 [cond-mat.str-el]].
[45] V. Pasquarella, “Factorisation Homology for Class S Theories,” [arXiv:2312.06760 [hepth]].
[46] J. J. Heckman, M. Hübner, E. Torres and H. Y. Zhang, “The Branes Behind Generalized Symmetry Operators,” Fortsch. Phys. 71, 2200180 (2023) DOI:10.1002/prop.202200180 [arXiv:2209.03343 [hep-th]].
[47] M. Dierigl, J. J. Heckman, M. Montero and E. Torres, “IIB string theory explored: Reflection 7-branes,” Phys. Rev. D 107, 086015 (2023) DOI:10.1103/PhysRevD.107.086015 [arXiv:2212.05077 [hep-th]].
[2] E. Witten, “Conformal Field Theory In Four And Six Dimensions,” [arXiv:0712.0157 [math.RT]].
[3] O. Chacaltana, J. Distler and Y. Tachikawa, “Nilpotent orbits and codimensiontwo defects of 6d N=(2,0) theories,” Int. J. Mod. Phys. A 28, 1340006 (2013) DOI:10.1142/S0217751X1340006X [arXiv:1203.2930 [hep-th]].
[4] Y. Tachikawa, “On the 6d origin of discrete additional data of 4d gauge theories,” JHEP 05, 020 (2014) DOI:10.1007/JHEP05(2014)020 [arXiv:1309.0697 [hep-th]].
[5] G. Moore, “Applications of the six-dimensional (2,0) theories to Physical Mathematics,” Felix Klein lectures Bonn (2012).
[6] E. Witten, “Geometric Langlands From Six Dimensions,” [arXiv:0905.2720 [hep-th]].
[7] D. Ben-Zvi, “Theory X and Geometric Representation Theory,” talks at Mathematical Aspects of Six-Dimensional Quantum Field Theories IHES 2014, notes by Qiaochu Yuan.
[8] I. Bah, D. S. Freed, G. W. Moore, N. Nekrasov, S. S. Razamat and S. Schafer-Nameki, “A Panorama Of Physical Mathematics c. 2022,” [arXiv:2211.04467 [hep-th]].
[9] G. W. Moore and N. Seiberg, “Classical and Quantum Conformal Field Theory,” Commun. Math. Phys. 123, 177 (1989) DOI:10.1007/BF01238857.
[10] G. W. Moore and N. Seiberg, “Naturality in Conformal Field Theory,” Nucl. Phys. B 313, 16 (1989) DOI:10.1016/0550-3213(89)90511-7.
[11] E. P. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal Field Theory,” Nucl. Phys. B 300, 360 (1988) DOI:10.1016/0550-3213(88)90603-7.
[12] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” JHEP 02, 172 (2015) DOI:10.1007/JHEP02(2015)172 [arXiv:1412.5148 [hep-th]].
[13] D. Gaiotto and J. Kulp, “Orbifold groupoids,” JHEP 02, 132 (2021) DOI:10.1007/JHEP02(2021)132 [arXiv:2008.05960 [hep-th]].
[14] L. Kong, T. Lan, X. G. Wen, Z. H. Zhang and H. Zheng, “Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry,” Phys. Rev. Res. 2, 043086 (2020) DOI:10.1103/PhysRevResearch.2.043086 [arXiv:2005.14178 [cond-mat.str-el]].
[15] L. Kong and Z. H. Zhang, “An invitation to topological orders and category theory,” [arXiv:2205.05565 [cond-mat.str-el]].
[16] L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part I,” JHEP 02, 150 (2020) DOI:10.1007/JHEP02(2020)150 [arXiv:1905.04924 [condmat.str-el]].
[17] L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part II,” Nucl. Phys. B 966, 115384 (2021) DOI:10.1016/j.nuclphysb.2021.115384 [arXiv:1912.01760 [cond-mat.str-el]].
[18] L. Kong and H. Zheng, “Categorical computation,” Front. Phys. 18, 21302 (2023) DOI:10.1007/s11467-022-1251-5 [arXiv:2102.04814 [quant-ph]].
[19] D. Gaiotto and T. Johnson-Freyd, “Condensations in higher categories,” [arXiv:1905.09566 [math.CT]].
[20] T. Johnson-Freyd and M. Yu, “Topological Orders in (4+1)-Dimensions,” SciPost Phys. 13, 068 (2022) DOI:10.21468/SciPostPhys.13.3.068 [arXiv:2104.04534 [hep-th]].
[21] T. Johnson-Freyd, “On the Classification of Topological Orders,” Commun. Math. Phys. 393, 989 (2022) DOI:10.1007/s00220-022-04380-3 [arXiv:2003.06663 [math.CT]].
[22] T. D. Décoppet and M. Yu, “Gauging noninvertible defects: a 2-categorical perspective,” Lett. Math. Phys. 113, 36 (2023) DOI:10.1007/s11005-023-01655-1 [arXiv:2211.08436 [math.CT]].
[23] D. S. Freed and C. Teleman, “Relative quantum field theory,” Commun. Math. Phys. 326, 4596 (2014) DOI:10.1007/s00220-013-1880-1 [arXiv:1212.1692 [hep-th]].
[24] D. S. Freed, G. W. Moore and C. Teleman, “Topological symmetry in quantum field theory,” [arXiv:2209.07471 [hep-th]].
[25] D. S. Freed, “Introduction to topological symmetry in QFT,” [arXiv:2212.00195 [hepth]].
[26] V. Bashmakov, M. Del Zotto and A. Hasan, “On the 6d Origin of Non-invertible Symmetries in 4d,” [arXiv:2206.07073 [hep-th]].
[27] V. Bashmakov, M. Del Zotto, A. Hasan and J. Kaidi, “Non-invertible Symmetries of Class S Theories,” [arXiv:2211.05138 [hep-th]].
[28] L. Bhardwaj, S. Schafer-Nameki and A. Tiwari, “Unifying Constructions of NonInvertible Symmetries,” [arXiv:2212.06159 [hep-th]].
[29] L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki and A. Tiwari, “Non-Invertible HigherCategorical Symmetries,” [arXiv:2204.06564 [hep-th]].
[30] J. Kaidi, K. Ohmori and Y. Zheng, “Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,” Phys. Rev. Lett. 128, 111601 (2022) DOI:10.1103/PhysRevLett.128.111601 [arXiv:2111.01141 [hep-th]].
[31] J. Kaidi, G. Zafrir and Y. Zheng, “Non-invertible symmetries of N = 4 SYM and twisted compactification,” JHEP 08, 053 (2022) DOI:10.1007/JHEP08(2022)053 [arXiv:2205.01104 [hep-th]].
[32] J. Kaidi, K. Ohmori and Y. Zheng, “Symmetry TFTs for Non-Invertible Defects,” [arXiv:2209.11062 [hep-th]].
[33] I. M. Burbano, J. Kulp and J. Neuser, “Duality defects in E8,” JHEP 10, 186 (2022) DOI:10.1007/JHEP10(2022)187 [arXiv:2112.14323 [hep-th]].
[34] Y. Choi, C. Cordova, P. S. Hsin, H. T. Lam and S. H. Shao, “Non-invertible gauging, Duality, and Triality Defects in 3+1 Dimensions,” [arXiv:2204.09025 [hep-th]].
[35] Y. Choi, C. Cordova, P. S. Hsin, H. T. Lam and S. H. Shao, “Noninvertible duality defects in 3+1 dimensions,” Phys. Rev. D 105, 125016 (2022) DOI:10.1103/PhysRevD.105.125016 [arXiv:2111.01139 [hep-th]].
[36] A. Antinucci, F. Benini, C. Copetti, G. Galati and G. Rizi, “The holography of noninvertible self-duality symmetries,” [arXiv:2210.09146 [hep-th]].
[37] A. Antinucci, C. Copetti, G. Galati and G. Rizi, “Zoology of non-invertible duality defects: the view from class S,” [arXiv:2212.09549 [hep-th]].
[38] T. Johnson-Freyd, “Operators and higher categories in quantum field theory,” Lecture series.
[39] L. Kong, “Anyon gauging and tensor categories,” Nucl. Phys. B 886, 436 (2014) DOI:10.1016/j.nuclphysb.2014.07.003 [arXiv:1307.8244 [cond-mat.str-el]].
[40] M. Yu, “Gauging Categorical Symmetries in 3d Topological Orders and Bulk Reconstruction,” [arXiv:2111.13697 [hep-th]].
[41] O. Aharony, N. Seiberg and Y. Tachikawa, “Reading between the lines of fourdimensional gauge theories,” JHEP 08, 115 (2013) DOI:10.1007/JHEP08(2013)115 [arXiv:1305.0318 [hep-th]].
[42] W. Ji and X. G. Wen, “Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions,” Phys. Rev. Research. 1, 033054 (2019) DOI:10.1103/PhysRevResearch.1.033054 [arXiv:1905.13279 [cond-mat.str-el]].
[43] J. C. Y. Teo, T. L. Hughes and E. Fradkin, “Theory of Twist Liquids: Gauging an Anyonic Symmetry,” Annals Phys. 360, 3495 (2015) DOI:10.1016/j.aop.2015.05.012 [arXiv:1503.06812 [cond-mat.str-el]].
[44] M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, “Symmetry Fractionalization, Defects, and Gauging of Topological Phases,” Phys. Rev. B 100, 115147 (2019) DOI:10.1103/PhysRevB.100.115147 [arXiv:1410.4540 [cond-mat.str-el]].
[45] V. Pasquarella, “Factorisation Homology for Class S Theories,” [arXiv:2312.06760 [hepth]].
[46] J. J. Heckman, M. Hübner, E. Torres and H. Y. Zhang, “The Branes Behind Generalized Symmetry Operators,” Fortsch. Phys. 71, 2200180 (2023) DOI:10.1002/prop.202200180 [arXiv:2209.03343 [hep-th]].
[47] M. Dierigl, J. J. Heckman, M. Montero and E. Torres, “IIB string theory explored: Reflection 7-branes,” Phys. Rev. D 107, 086015 (2023) DOI:10.1103/PhysRevD.107.086015 [arXiv:2212.05077 [hep-th]].
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Volume 6, Issue 1
December 2025Pages 10-40
- Receive Date: 17 September 2025
- Revise Date: 08 October 2025
- Accept Date: 23 October 2025