Abstract
We study non-invertible duality symmetries by gauging a diagonal subgroup of a non-anomalous U(1)×U(1) global symmetry. In particular, we employ the half-space gauging to c = 2 bosonic torus conformal field theory (CFT) in two dimensions and pure U(1)×U(1) gauge theory in four dimensions. In c = 2 bosonic torus CFT, we show that the non-invertible symmetry obtained from the diagonal gauging becomes emergent on an irrational CFT point. We also calculate the fusion rules concerning the duality defect. We find out that the fusion algebra is non-commutative. We also obtain a similar result in pure U(1)×U(1) gauge theory in four dimensions.
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References
G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, NATO Sci. Ser. B 59 (1980) 135 [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
G.W. Moore and N. Seiberg, Taming the Conformal Zoo, Phys. Lett. B 220 (1989) 422 [INSPIRE].
M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B 495 (1997) 533 [cond-mat/9612187] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. I: Partition functions, Nucl. Phys. B 646 (2002) 353 [hep-th/0204148] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. II: Unoriented world sheets, Nucl. Phys. B 678 (2004) 511 [hep-th/0306164] [INSPIRE].
J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. III: Simple currents, Nucl. Phys. B 694 (2004) 277 [hep-th/0403157] [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. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators IV: Structure constants and correlation functions, Nucl. Phys. B 715 (2005) 539 [hep-th/0412290] [INSPIRE].
J. Fjelstad, J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators. V: Proof of modular invariance and factorisation, Theor. Appl. Categor. 16 (2006) 342 [hep-th/0503194] [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].
A. Feiguin et al., Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98 (2007) 160409 [cond-mat/0612341] [INSPIRE].
J. Fuchs, M.R. Gaberdiel, I. Runkel and C. Schweigert, Topological defects for the free boson CFT, J. Phys. A 40 (2007) 11403 [arXiv:0705.3129] [INSPIRE].
S. Fredenhagen, M.R. Gaberdiel and C. Schmidt-Colinet, Bulk flows in Virasoro minimal models with boundaries, J. Phys. A 42 (2009) 495403 [arXiv:0907.2560] [INSPIRE].
J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Defect lines, dualities, and generalised orbifolds, in the proceedings of the 16th International Congress on Mathematical Physics, Prague, Czechia, August 03–08 (2009) [https://doi.org/10.1142/9789814304634_0056] [arXiv:0909.5013] [INSPIRE].
A. Davydov, L. Kong and I. Runkel, Invertible Defects and Isomorphisms of Rational CFTs, Adv. Theor. Math. Phys. 15 (2011) 43 [arXiv:1004.4725] [INSPIRE].
N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7 (2016) 203 [arXiv:1210.6363] [INSPIRE].
C. Bachas, I. Brunner and D. Roggenkamp, Fusion of Critical Defect Lines in the 2D Ising Model, J. Stat. Mech. 1308 (2013) P08008 [arXiv:1303.3616] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, Orbifolds and topological defects, Commun. Math. Phys. 332 (2014) 669 [arXiv:1307.3141] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, A quick guide to defect orbifolds, Proc. Symp. Pure Math. 88 (2014) 231 [arXiv:1310.0062] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, Commun. Math. Phys. 337 (2015) 429 [arXiv:1404.7497] [INSPIRE].
W.W. Ho et al., Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality, Phys. Rev. B 91 (2015) 125119 [arXiv:1411.6932] [INSPIRE].
M. Hauru et al., Topological conformal defects with tensor networks, Phys. Rev. B 94 (2016) 115125 [arXiv:1512.03846] [INSPIRE].
D. Aasen, R.S.K. Mong and P. Fendley, Topological Defects on the Lattice. I: The Ising model, J. Phys. A 49 (2016) 354001 [arXiv:1601.07185] [INSPIRE].
D. Aasen, P. Fendley and R.S.K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies, arXiv:2008.08598 [INSPIRE].
S.X. Cui, Four dimensional topological quantum field theories from G-crossed braided categories, Quantum Topol. 10 (2019) 593 [arXiv:1610.07628] [INSPIRE].
C.L. Douglas and D.J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds, arXiv:1812.11933 [INSPIRE].
T. Johnson-Freyd, On the Classification of Topological Orders, Commun. Math. Phys. 393 (2022) 989 [arXiv:2003.06663] [INSPIRE].
L. Kong, Y. Tian and Z.-H. Zhang, Defects in the 3-dimensional toric code model form a braided fusion 2-category, JHEP 12 (2020) 078 [arXiv:2009.06564] [INSPIRE].
L. Bhardwaj, L.E. Bottini, S. Schafer-Nameki and A. Tiwari, Non-invertible higher-categorical symmetries, SciPost Phys. 14 (2023) 007 [arXiv:2204.06564] [INSPIRE].
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].
C.-M. Chang et al., Topological Defect Lines and Renormalization Group Flows in Two Dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [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].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
T.-C. Huang et al., Numerical Evidence for a Haagerup Conformal Field Theory, Phys. Rev. Lett. 128 (2022) 231603 [arXiv:2110.03008] [INSPIRE].
R. Vanhove et al., Critical Lattice Model for a Haagerup Conformal Field Theory, Phys. Rev. Lett. 128 (2022) 231602 [arXiv:2110.03532] [INSPIRE].
B. Heidenreich et al., Non-invertible global symmetries and completeness of the spectrum, JHEP 09 (2021) 203 [arXiv:2104.07036] [INSPIRE].
M. Nguyen, Y. Tanizaki and M. Ünsal, Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond N-ality, JHEP 03 (2021) 238 [arXiv:2101.02227] [INSPIRE].
M. Nguyen, Y. Tanizaki and M. Ünsal, Noninvertible 1-form symmetry and Casimir scaling in 2D Yang-Mills theory, Phys. Rev. D 104 (2021) 065003 [arXiv:2104.01824] [INSPIRE].
M. Koide, Y. Nagoya and S. Yamaguchi, Non-invertible topological defects in 4-dimensional ℤ2 pure lattice gauge theory, PTEP 2022 (2022) 013B03 [arXiv:2109.05992] [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].
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].
Y. Hayashi and Y. Tanizaki, Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly, JHEP 08 (2022) 036 [arXiv:2204.07440] [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].
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].
P. Niro, K. Roumpedakis and O. Sela, Exploring non-invertible symmetries in free theories, JHEP 03 (2023) 005 [arXiv:2209.11166] [INSPIRE].
Y. Choi, H.T. Lam and S.-H. Shao, Noninvertible Time-Reversal Symmetry, Phys. Rev. Lett. 130 (2023) 131602 [arXiv:2208.04331] [INSPIRE].
Y. Choi, H.T. Lam and S.-H. Shao, Non-invertible Gauss law and axions, JHEP 09 (2023) 067 [arXiv:2212.04499] [INSPIRE].
Y.-H. Lin, M. Okada, S. Seifnashri and Y. Tachikawa, Asymptotic density of states in 2d CFTs with non-invertible symmetries, JHEP 03 (2023) 094 [arXiv:2208.05495] [INSPIRE].
F. Apruzzi, I. Bah, F. Bonetti and S. Schafer-Nameki, Noninvertible Symmetries from Holography and Branes, Phys. Rev. Lett. 130 (2023) 121601 [arXiv:2208.07373] [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].
J.J. Heckman et al., Top down approach to topological duality defects, Phys. Rev. D 108 (2023) 046015 [arXiv:2212.09743] [INSPIRE].
L. Bhardwaj, S. Schafer-Nameki and J. Wu, Universal Non-Invertible Symmetries, Fortsch. Phys. 70 (2022) 2200143 [arXiv:2208.05973] [INSPIRE].
T. Bartsch, M. Bullimore, A.E.V. Ferrari and J. Pearson, Non-invertible Symmetries and Higher Representation Theory I, arXiv:2208.05993 [INSPIRE].
T. Bartsch, M. Bullimore, A.E.V. Ferrari and J. Pearson, Non-invertible Symmetries and Higher Representation Theory II, arXiv:2212.07393 [INSPIRE].
R. Yokokura, Non-invertible symmetries in axion electrodynamics, arXiv:2212.05001 [INSPIRE].
A. Apte, C. Córdova and H.T. Lam, Obstructions to gapped phases from noninvertible symmetries, Phys. Rev. B 108 (2023) 045134 [arXiv:2212.14605] [INSPIRE].
J. Kaidi, K. Ohmori and Y. Zheng, Symmetry TFTs for Non-invertible Defects, Commun. Math. Phys. 404 (2023) 1021 [arXiv:2209.11062] [INSPIRE].
A. Antinucci et al., The holography of non-invertible self-duality symmetries, arXiv:2210.09146 [INSPIRE].
K. Inamura, Fermionization of fusion category symmetries in 1 + 1 dimensions, JHEP 10 (2023) 101 [arXiv:2206.13159] [INSPIRE].
C. Córdova, S. Hong, S. Koren and K. Ohmori, Neutrino Masses from Generalized Symmetry Breaking, arXiv:2211.07639 [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].
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].
L. Bhardwaj, L.E. Bottini, S. Schafer-Nameki and A. Tiwari, Non-invertible symmetry webs, SciPost Phys. 15 (2023) 160 [arXiv:2212.06842] [INSPIRE].
L. Bhardwaj, S. Schafer-Nameki and A. Tiwari, Unifying constructions of non-invertible symmetries, SciPost Phys. 15 (2023) 122 [arXiv:2212.06159] [INSPIRE].
C. Zhang and C. Córdova, Anomalies of (1 + 1)D categorical symmetries, arXiv:2304.01262 [INSPIRE].
J. Kaidi, E. Nardoni, G. Zafrir and Y. Zheng, Symmetry TFTs and anomalies of non-invertible symmetries, JHEP 10 (2023) 053 [arXiv:2301.07112] [INSPIRE].
Y.-H. Lin and S.-H. Shao, Bootstrapping noninvertible symmetries, Phys. Rev. D 107 (2023) 125025 [arXiv:2302.13900] [INSPIRE].
N. Seiberg and S.-H. Shao, Majorana chain and Ising model — (non-invertible) translations, anomalies, and emanant symmetries, arXiv:2307.02534 [INSPIRE].
W. Cao, L. Li, M. Yamazaki and Y. Zheng, Subsystem non-invertible symmetry operators and defects, SciPost Phys. 15 (2023) 155 [arXiv:2304.09886] [INSPIRE].
L. Bhardwaj and S. Schafer-Nameki, Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT, arXiv:2305.17159 [INSPIRE].
T. Bartsch, M. Bullimore and A. Grigoletto, Representation theory for categorical symmetries, arXiv:2305.17165 [INSPIRE].
Y. Choi, B.C. Rayhaun, Y. Sanghavi and S.-H. Shao, Comments on Boundaries, Anomalies, and Non-Invertible Symmetries, arXiv:2305.09713 [INSPIRE].
M. Koide, Y. Nagoya and S. Yamaguchi, Noninvertible symmetries and boundaries in four dimensions, Phys. Rev. D 108 (2023) 065009 [arXiv:2304.01550] [INSPIRE].
K. Inamura and K. Ohmori, Fusion Surface Models: 2 + 1d Lattice Models from Fusion 2-Categories, arXiv:2305.05774 [INSPIRE].
F. Apruzzi, F. Bonetti, D.S.W. Gould and S. Schafer-Nameki, Aspects of Categorical Symmetries from Branes: SymTFTs and Generalized Charges, arXiv:2306.16405 [INSPIRE].
M. van Beest et al., Monopoles, Scattering, and Generalized Symmetries, arXiv:2306.07318 [INSPIRE].
B. Haghighat and Y. Sun, Topological Defect Lines in bosonized Parafermionic CFTs, arXiv:2306.16555 [INSPIRE].
S. Chen and Y. Tanizaki, Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients, arXiv:2307.00939 [INSPIRE].
C. Córdova and K. Ohmori, Quantum Duality in Electromagnetism and the Fine-Structure Constant, arXiv:2307.12927 [INSPIRE].
J.A. Damia et al., Non-invertible symmetries along 4d RG flows, arXiv:2305.17084 [INSPIRE].
C. Lawrie, X. Yu and H.Y. Zhang, Intermediate Defect Groups, Polarization Pairs, and Non-invertible Duality Defects, arXiv:2306.11783 [INSPIRE].
A. Antinucci et al., Anomalies of non-invertible self-duality symmetries: fractionalization and gauging, arXiv:2308.11707 [INSPIRE].
S.D. Pace, Emergent generalized symmetries in ordered phases, arXiv:2308.05730 [INSPIRE].
C. Córdova, P.-S. Hsin and C. Zhang, Anomalies of Non-Invertible Symmetries in (3 + 1)d, arXiv:2308.11706 [INSPIRE].
V. Benedetti, H. Casini and J.M. Magan, ABJ anomaly as a U(1) symmetry and Noether’s theorem, arXiv:2309.03264 [INSPIRE].
Y. Choi, M. Forslund, H.T. Lam and S.-H. Shao, Quantization of Axion-Gauge Couplings and Non-Invertible Higher Symmetries, arXiv:2309.03937 [INSPIRE].
S. Schafer-Nameki, ICTP Lectures on (Non-)Invertible Generalized Symmetries, arXiv:2305.18296 [INSPIRE].
S.-H. Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry, arXiv:2308.00747 [INSPIRE].
D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209 (1998) 692.
S. Gukov and C. Vafa, Rational conformal field theories and complex multiplication, Commun. Math. Phys. 246 (2004) 181 [hep-th/0203213] [INSPIRE].
A. Kapustin and M. Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 11 (2009) 006 [arXiv:0904.0840] [INSPIRE].
L. Kong and X.-G. Wen, Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions, arXiv:1405.5858 [INSPIRE].
D.V. Else and C. Nayak, Cheshire charge in (3 + 1)-dimensional topological phases, Phys. Rev. B 96 (2017) 045136 [arXiv:1702.02148] [INSPIRE].
D. Gaiotto and T. Johnson-Freyd, Condensations in higher categories, arXiv:1905.09566 [INSPIRE].
L. Kong et al., Algebraic higher symmetry and categorical symmetry — a holographic and entanglement view of symmetry, Phys. Rev. Res. 2 (2020) 043086 [arXiv:2005.14178] [INSPIRE].
T. Johnson-Freyd, (3 + 1)D topological orders with only a ℤ2-charged particle, arXiv:2011.11165 [INSPIRE].
Acknowledgments
We are particularly grateful to Justin Kaidi, Kantaro Ohmori and Satoshi Yamaguchi for many enlightening comments and discussions. We are also grateful to Takamasa Ando, Yuma Furuta, Yui Hayashi, Hiroki Imai, Hayato Kanno, Kohki Kawabata, Ryutaro Matsudo, Tatsuma Nishioka for valuable discussions. Discussions during the YITP workshop on “Strings and Fields 2023” were useful to complete this work. This work of Y. N. was supported by JST SPRING, Grant Number JPMJSP2138. The work of S. S. was supported by JSPS fellowship for young students, Grant Number 23KJ1533.
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Nagoya, Y., Shimamori, S. Non-invertible duality defect and non-commutative fusion algebra. J. High Energ. Phys. 2023, 62 (2023). https://doi.org/10.1007/JHEP12(2023)062
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DOI: https://doi.org/10.1007/JHEP12(2023)062