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Anomaly of Continuous Symmetries from Topological Defect Network

Qiang Jia, Ran Luo, Jiahua Tian, Yi-Nan Wang, Yi Zhang

Abstract

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.

Anomaly of Continuous Symmetries from Topological Defect Network

Abstract

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.
Paper Structure (8 sections, 71 equations, 5 figures)

This paper contains 8 sections, 71 equations, 5 figures.

Table of Contents

  1. 't Hooft Anomaly of $U(1)$ Global Symmetry and Lie Group Cohomology
  2. Group and Lie algebra cohomology in higher dimensions
  3. Anomalies, cohomology groups and secondary invariants.
  4. F-move and group cohomologies.
  5. Classifying space and secondary invariant.
  6. Transgression and Homotopy fiber.
  7. On $H^3(U(1)^\delta,U(1))$
  8. Anomaly of gauge transformation

Figures (5)

  • Figure 1: The F-move of defect web for $G=\mathbb{Z}_m$ 0-form symmetry group in 2D. $a_i \in \{0,1,\ldots,m-1\}$ and $\overline{m}$ is defined as $m$ mod $\mathbb{Z}$.
  • Figure 2: The deformations of the combined and configuration. The group multiplications in this figure are all taken from the right.
  • Figure 3: The showing of open cover $U_k$$(k=0,1,2)$ of $M_2$ as well as the open cover $\overline{U}_k$$(k=0,1,2)$ of $M_2\times I$.
  • Figure 4: The F-move of defect lines in a theory with $U(1)$ flavor symmetry. A topological line carrying $e^{i\alpha}\in U(1)$ is labeled by $[\alpha] := \alpha \mod \ 2\pi\mathbb{Z}$, with an arrow labeling the direction of gauge transformation in the same sense as in (\ref{['junction']}).
  • Figure 5: The F-move for $U(1)$ global symmetry.