Symmetries and anomalies of (1+1)d theories: 2-groups and symmetry fractionalization

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.

Automated summary

The study focuses on the interactions of discrete zero-form and one-form global symmetries in (1+1)d theories, resulting in 2-group symmetries or symmetry fractionalization in this low dimension. Emphasis is placed on understanding the decomposition of theories into multiple sectors and gauging of the one-form symmetry, especially in the presence of a global 2-group symmetry. Spectral sequence calculations are used to determine anomalies for the 2-group theories, with discussions on topological manipulations and their effects when coupled to a bulk (2+1)d theory.