Symmetry and Gauge Structure

Symmetry connects conservation laws, selection rules, and the possible phases of quantum field theory. Gauge structure determines which field descriptions represent the same physical situation and which observables distinguish different theories. This volume develops both ideas, from charges and Ward identities to anomalies, extended operators, and topological field theories.

A useful starting example is a complex scalar with a global U(1) symmetry. A constant phase rotation acts on charged operators. Coupling its current to a fixed background gauge field lets us probe that symmetry; integrating over gauge fields, with a specified action, measure, quotient, and allowed sectors, instead constructs a gauged theory. Keeping physical action, background probe, and gauge redundancy distinct is the organizing idea of the volume.

Where to begin

• First study of symmetry: begin with what a QFT symmetry is, then use Chapters 2–3 to connect charges, Ward identities, and background response. Continue to Chapter 5 when the question concerns broken symmetry or Goldstone modes.
• Preparing to calculate in a gauge theory: begin with gauge structure and observables. Follow with Chapter 7 for gauge fixing and Chapter 8 for quantum consistency; choose Chapter 6 when boundaries or asymptotic charges matter.
• Studying lines, defects, or topological response: begin with extended operators after reviewing the global gauge data in Chapter 4. Chapters 10–13 develop the symmetry actions, fusion rules, and topological theories built from these objects.

Helpful background. Group actions, Lie algebras, and representations supply the language of transformations and multiplets. Noether’s construction supplies the starting current calculation; differential forms become useful for fluxes, backgrounds, and topological terms. Each topic states the preparation needed for its calculations.

Chapters

1. Symmetry, Actions, and Redundancy identifies the action on physical states and operators, distinguishing it from gauge redundancy, duality, and a transformation of external parameters.
2. Currents, Charges, and Quantum Ward Identities turns continuous transformations into operator and correlation-function identities, including contact terms and the conditions for conserved charges.
3. Background Fields and Gauging develops symmetry probes and explains what must be added when those probes become dynamical fields.
4. Gauge Structure, Global Form, and Observables connects Gauss constraints and gauge orbits to bundles, the global gauge group, line operators, and physical observables.
5. Symmetry Realization and Breaking distinguishes symmetry of a theory from symmetry of a state, then develops spontaneous breaking, Goldstone results, and gauge-invariant Higgs physics.
6. Boundary Symmetry, Surface Charges, and Edge Modes determines which admissible transformations carry physical boundary charges and how boundary conditions, flux, and subsystem choices enter.
7. Gauge Fixing, BRST, and BV develops local gauge fixing and its cohomological organization, with explicit limits on perturbative methods and global gauge choices.
8. Anomalies, Inflow, and Matching tests quantum symmetry obstructions after allowed counterterms, separating gauge inconsistency from global-symmetry anomalies and their infrared constraints.
9. Extended Operators and Defects defines lines, surfaces, and interfaces through their support, endpoints, screening, fusion, and linking.
10. Higher-Form and Higher-Group Symmetries studies symmetry acting on extended objects and the coupled background transformations that distinguish a higher group.
11. Non-Invertible Symmetries studies topological defects without fusion inverses, their action on physical operators, and the resulting constraints.
12. Topological Terms and Invertible Responses tests global definitions, quantization, and boundary dependence before interpreting an action as a topological response.
13. Topological Gauge Theories and Symmetry TFT computes states and observables in controlled topological models and explains when an auxiliary topological theory can encode symmetry and anomaly data.

Three distinctions to carry through the volume

Gauge redundancy depends on the boundary problem. A gauge transformation that vanishes suitably at a boundary can be redundant, while an admissible transformation with a nontrivial boundary charge can act physically. Maxwell theory makes this distinction concrete through electric flux; the boundary conditions must be specified before deciding which transformations to identify. See Harlow and Wu 2020, § 3.3, pp. 26–27, PDF.

Local equations do not specify all global data. The same gauge algebra can support different global groups and line spectra. For example, four-dimensional SU(2) and SO(3) gauge theories allow different Wilson lines; additional magnetic and dyonic line choices can distinguish theories even after the group is fixed. See Aharony, Seiberg, and Tachikawa 2013, §§ 1.1–1.2, pp. 2–5, PDF.

A global-symmetry anomaly constrains a theory without making it inconsistent. A ’t Hooft anomaly obstructs gauging that symmetry by itself and must be reproduced along symmetry-preserving renormalization-group flows. It does not, by itself, select a unique infrared phase. Higher-form symmetries extend these questions to charged lines and surfaces; they require their own operator and background data. See Gaiotto et al. 2015, § 1, pp. 1–4, PDF.

How the ideas connect

Ordinary symmetry leads to currents and backgrounds, or branches toward symmetry breaking. Gauge structure connects to boundary charges, gauge fixing, extended operators, and topological terms. Together with currents and backgrounds, it prepares the study of anomalies. Extended operators lead to higher symmetries and then non-invertible symmetry. Anomalies and higher symmetries meet in topological responses, which prepare topological gauge theories. These are useful reading paths, not a requirement to study all thirteen chapters in order.

The optional full-size conceptual map shows these connections and selected onward routes. Its solid arrows mark conceptual continuations; dashed arrows lead to applications elsewhere in QFT.org. Foundations and Mathematical Methods supply the starting examples and mathematics. Currents continue into renormalization and EFT; gauge methods into scattering and Standard Model physics; anomalies into curved spacetime; defects into conformal theory and quantum matter; and topological structures into Mathematical QFT. The schematic map is not a complete prerequisite graph.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. DOI. Open PDF, arXiv v5.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI. Open PDF, arXiv v2.
• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, no. 10 (2020): 146. DOI. Open PDF, arXiv v4.