Abstract
We investigate the interactions of discrete zero-form and one-form global symmetries in (1+1)d theories. Focus is put on the interactions that the symmetries can have on each other, which in this low dimension result in 2-group symmetries or symmetry fractionalization. A large part of the discussion will be to understand a major feature in (1+1)d: the multiple sectors into which a theory decomposes. We perform gauging of the one-form symmetry, and remark on the effects this has on our theories, especially in the case when there is a global 2-group symmetry. We also implement the spectral sequence to calculate anomalies for the 2-group theories and symmetry fractionalized theory in the bosonic and fermionic cases. Lastly, we discuss topological manipulations on the operators which implement the symmetries, and draw insights on the (1+1)d effects of such manipulations by coupling to a bulk (2+1)d theory.
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References
D. Gaiotto and T. Johnson-Freyd, Condensations in higher categories, arXiv:1905.09566 [INSPIRE].
D. Gaiotto and J. Kulp, Orbifold groupoids, JHEP 02 (2021) 132 [arXiv:2008.05960] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal, and temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
P.-S. Hsin and H. T. Lam, Discrete theta angles, symmetries and anomalies, SciPost Phys. 10 (2021) 032 [arXiv:2007.05915] [INSPIRE].
E. Sharpe, Undoing decomposition, Int. J. Mod. Phys. A 34 (2020) 1950233 [arXiv:1911.05080] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Topological defect lines and renormalization group flows in two dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].
S. M. Lane, Categories for the working mathematician, Springer, New York, NY, U.S.A. (2013).
A. Debray, Geometry and string theory seminar: spring 2018, https://web.ma.utexas.edu/users/a.debray/lecture_notes/s18_higher_symmetries.pdf, (2018).
Y. Tachikawa, TASI 2019 lectures, https://member.ipmu.jp/yuji.tachikawa/lectures/2019-top-anom.
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].
T. Johnson-Freyd, On the classification of topological orders, arXiv:2003.06663 [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].
L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436 [arXiv:1307.8244] [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].
H. Sati, U. Schreiber and J. Stasheff, L∞ algebra connections and applications to string- and Chern-Simons n-transport, Quantum Field Theory (2009) 303 [arXiv:0801.3480] [INSPIRE].
D. Fiorenza, U. Schreiber and J. Stasheff, Čech cocycles for differential characteristic classes: an ∞-Lie theoretic construction, Adv. Theor. Math. Phys. 16 (2012) 149 [arXiv:1011.4735] [INSPIRE].
D. Fiorenza, H. Sati and U. Schreiber, Extended higher cup-product Chern-Simons theories, J. Geom. Phys. 74 (2013) 130 [arXiv:1207.5449] [INSPIRE].
C. Córdova, T. T. Dumitrescu and K. Intriligator, Exploring 2-group global symmetries, JHEP 02 (2019) 184 [arXiv:1802.04790] [INSPIRE].
C. Córdova, D. S. Freed, H. T. Lam and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications I, SciPost Phys. 8 (2020) 001 [arXiv:1905.09315] [INSPIRE].
Q.-R. Wang and Z.-C. Gu, Towards a complete classification of symmetry-protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory, Phys. Rev. X 8 (2018) 011055 [arXiv:1703.10937] [INSPIRE].
D. Gaiotto and T. Johnson-Freyd, Symmetry protected topological phases and generalized cohomology, JHEP 05 (2019) 007 [arXiv:1712.07950] [INSPIRE].
A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, U.K. (2002).
P.-S. Hsin and S.-H. Shao, Lorentz symmetry fractionalization and dualities in (2 + 1)d, SciPost Phys. 8 (2020) 018 [arXiv:1909.07383] [INSPIRE].
M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100 (2019) 115147 [arXiv:1410.4540] [INSPIRE].
X. Chen, Symmetry fractionalization in two dimensional topological phases, Rev. Phys. 2 (2017) 3 [arXiv:1606.07569] [INSPIRE].
M. Barkeshli and M. Cheng, Relative anomalies in (2 + 1)d symmetry enriched topological states, SciPost Phys. 8 (2020) 028 [arXiv:1906.10691] [INSPIRE].
C. Vafa and E. Witten, On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995) 189 [hep-th/9409188] [INSPIRE].
A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, arXiv:1309.4721 [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
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ArXiv ePrint: 2010.01136v2
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Yu, M. Symmetries and anomalies of (1+1)d theories: 2-groups and symmetry fractionalization. J. High Energ. Phys. 2021, 61 (2021). https://doi.org/10.1007/JHEP08(2021)061
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DOI: https://doi.org/10.1007/JHEP08(2021)061