Symmetries and Anomalies of (1+1)d Theories: 2-groups and Symmetry Fractionalization

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.

35 pages, 7 figures

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• Symmetries and Strings of Adjoint QCD${}_2$
• Orbifold groupoids
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• Discrete Theta Angles, Symmetries and Anomalies
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• Relative Defects in Relative Theories: Trapped Higher-Form Symmetries and Irregular Punctures in Class S
• The anomaly that was not meant IIB
• Duality Defects in $E_8$
• Anomalies of Generalized Symmetries from Solitonic Defects
• Decomposition, Trivially-Acting Symmetries, and Topological Operators
• Topological Orders in (4+1)-Dimensions
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
• Fusion 3-Categories for Duality Defects
• Decomposition in Chern-Simons theories in three dimensions
• Symmetry fractionalization and duality defects in Maxwell theory
• What bordism-theoretic anomaly cancellation can do for U
• Three-dimensional orbifolds by 2-groups
• Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle
• Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries
• (-1)-Form Symmetries and Anomaly Shifting from SymTFT
• Universal Deformations