Anomalies, Inflow, and Matching
An anomaly is a quantum obstruction that survives allowed local repairs. This chapter separates that obstruction from explicit breaking and scheme dependence, then connects local anomaly calculations, global tests, inflow, and constraints on low-energy symmetry realization.
Enter this chapter
Section titled “Enter this chapter”Begin with What Is an Anomaly? before choosing a Jacobian, descent, or global-detection method.
Helpful background. Changes of Variables and Regulated Jacobians and Characteristic Classes and Chern–Weil Theory supply regulated measure changes and curvature-based characteristic classes.
Topics
Section titled “Topics”- What Is an Anomaly? — Tests whether a symmetry failure is genuine or removable.
- Regulated Jacobians and Measure Variation — Computes the measure contribution to a regulated transformation.
- Perturbative Chiral and Gauge Anomalies — Identifies representation-dependent local chiral anomalies.
- Consistent and Covariant Anomalies — Distinguishes currents derived from an action from covariant representatives.
- Wess–Zumino Consistency and Descent — Checks the transformation algebra and constructs local descent.
- Anomaly Polynomials and Inflow — Relates perturbative boundary anomalies to higher-dimensional bulk terms.
- Global and Torsion Anomalies — Detects phases invisible to infinitesimal anomaly polynomials.
- Gravitational, Mixed, Discrete, and Orientation-Reversing Anomalies — Tracks the geometry and background structures required by different symmetries.
- ‘t Hooft Anomaly Matching — Constrains infrared descriptions to reproduce the same global-symmetry obstruction.
How the ideas fit together
Section titled “How the ideas fit together”A local anomaly polynomial captures infinitesimal information, while a global anomaly can survive as a phase around a nontrivial loop of backgrounds. Vanishing polynomial data therefore do not establish complete anomaly freedom. Inflow interprets an anomalous theory relative to a bulk whose variation cancels the boundary failure; see Freed 2014, §§ 1–2.
The obstruction decision diagram separates removability, the role of the symmetry, and how the anomaly is detected. Matching then excludes incompatible infrared proposals but does not select a unique phase among those that match.
Review the chapter
Section titled “Review the chapter”Does an anomaly of a global symmetry always make the quantum theory inconsistent?
Solution
No. It can obstruct gauging that symmetry while the original theory remains consistent. A redundancy treated as a dynamical gauge equivalence cannot retain an uncancelled obstruction of the same kind. The role assigned to the symmetry is part of the diagnosis.
Where to go next
Section titled “Where to go next”Continue to Higher-Form and Higher-Group Symmetries for anomalies involving extended symmetry operators and coupled backgrounds.