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

TL;DR

This work analyzes (1+1)d quantum field theories with intertwined discrete zero-form and one-form symmetries, clarifying when their interaction forms a 2-group or yields symmetry fractionalization. It develops a formal framework using groupoid-based 2-groups, background gauge fields, and gauging procedures, and employs the Serre spectral sequence and supercohomology to classify anomalies in both bosonic and fermionic cases. The paper shows how decomposition into multiple vacua arises and how gauging the one-form symmetry induces mixed, sector-dependent anomalies controlled by the 2-group data, with the Postnikov invariant encoding the coupling between zero- and one-form sectors. It also explores symmetry fractionalization as a split 2-group characterized by $H^2_ ho(G;A_{[1]})$, analyzes resulting anomalies, and connects discrete torsion to SPT stacking and bulk-boundary perspectives, including explicit computations for $b Z_2$ cases. The results provide a comprehensive, calculable picture of how higher-form symmetries in low dimensions organize subsectors, topological manipulations, and anomaly structures, with implications for coupling to bulk TFTs and for understanding disjoint subtheories within a single $(1+1)d$ framework.

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.

Figures (3)

• Figure 4: The line $a$ is acted on by the zero-form symmetry as it punctures the surfaces, and passing the line $a$ by $\alpha$ results in a braiding.
• Figure 6: The boundary of the bulk theory is denoted $B[\mathcal{T}]$. The resulting theories $\mathcal{T}_{\widehat{i}}$ are separated by walls. When we compose the filter defect in with the boundary, the new boundary becomes that of the bulk theory $\mathcal{T}/\!/\mathcal{A}_{[1]}$.
• Figure 7: A single defect ending on the boundary separates two subsectors by a line.