Anomaly of Continuous Symmetries from Topological Defect Network

We show that the ’t Hooft anomaly of a quantum field theory with continuous flavor symmetry can be detected from rearrangements of the topological defect webs implementing the global symmetry in general spacetime dimension, which is concretized in 2D by the F-moves of the defect lines. Via dualizing the defects to flat background gauge field configurations, we characterize the ’t Hooft anomaly by various cohomological data of the symmetry group, where the cohomology of Lie groups with discrete topology plays the central role. We find that an extra dimension emerges naturally as a consequence of the mathematical description of the ’t Hooft anomaly in the case of flat gauging.

💡 Research Summary

The paper presents a unified framework for detecting ’t Hooft anomalies of quantum field theories (QFTs) that possess continuous global (0‑form) symmetries. The authors show that the anomaly can be read off from the rearrangements of topological defect networks (TDNs) that implement the symmetry, a perspective that has been well‑developed for finite groups but was missing for continuous groups.

The key technical step is the “dualization” of a defect line (or higher‑codimension defect web) into a flat background gauge field. For a defect line labeled by a group element (g_\alpha=e^{i\alpha}) the associated gauge field is (A_L=i\alpha,\delta(x),dx), which satisfies the flatness condition (dA_L+A_L\wedge A_L=0). This establishes an exact equivalence between inserting a topological defect and coupling the conserved current to a flat gauge configuration. The authors point out a subtlety: when (\pi_1(G)\neq0) (e.g. (G=U(1))), flat connections can have non‑zero field strength (F=dA) while still having trivial holonomy around contractible loops; this is crucial for correctly capturing anomalies of abelian groups.

Having identified the gauge fields, the authors compare two defect configurations related by an F‑move. The F‑move corresponds to a gauge transformation (A\to A’ = e^{-i\Lambda} A e^{i\Lambda}+e^{-i\Lambda}de^{i\Lambda}). The difference in the partition functions is encoded in a phase (e^{2\pi i \mathcal A