Gauge Structure, Global Form, and Observables
A gauge potential is a local description, not a complete specification of a gauge theory. Physical observables depend on how configurations are identified, how patches are joined, and which matter and line operators are admitted.
Enter this chapter
Section titled “Enter this chapter”Begin with Gauge Fields, Redundancy, and Observable Content before passing to orbits and global configurations.
Helpful background. The Free Maxwell Field and Gauge Redundancy and Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supply the Maxwell example and covariant differentiation.
Topics
Section titled “Topics”- Gauge Fields, Redundancy, and Observable Content — Distinguishes a connection potential from gauge-invariant information.
- Gauge Orbits, Gauss Constraints, and Stabilizers — Relates Gauss constraints to the regular physical quotient and its singular strata.
- Local Potentials and Global Gauge Configurations — Builds global configurations from compatible local potentials.
- Gauge-Invariant and Dressed Observables — Constructs singlet and dressed quantities with physical meaning.
- Global Form, Matter Representations, and the Faithful Gauge Group — Separates the Lie algebra, global group, and permitted representations.
- Genuine Line Spectra, Discrete Theta Data, and Theory Specification — Identifies further line-spectrum and discrete-theta choices.
- Large Gauge Transformations and Topological Sectors — Tests disconnected transformations against topology and boundary conditions.
How the ideas fit together
Section titled “How the ideas fit together”The constraint surface selects allowed initial data; quotienting its proper gauge directions identifies descriptions of the same state. Inspect the two progressions in the figure: imposing a constraint and identifying redundant representatives are distinct operations.
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Classical gauge reduction in one fixed bundle sector, shown schematically and not to scale. Gauss law restricts the canonical phase-space data; quotienting by identifies zero-generator orbit directions; physical observables are constant on those orbits. Boundary-charged or inadmissible transformations are not included merely because they share the same local gauge formula, and enhanced stabilizers can make the quotient stratified.
Global data remain after this local reduction. Gauge groups with the same Lie algebra can permit different bundles and Wilson lines; even fixing the group can leave distinct consistent magnetic and dyonic line spectra. See Aharony, Seiberg, and Tachikawa 2013, § 1.
Review the chapter
Section titled “Review the chapter”Does matter transforming trivially under the center force a quotient gauge group?
Solution
No. It makes those matter representations compatible with that quotient, but the global theory must still be chosen. Bundles and allowed line probes can distinguish the simply connected theory from its quotient even when their local matter transformation laws agree.
Where to go next
Section titled “Where to go next”Continue to Extended Operators and Defects to use line and surface probes, or Boundary Symmetry, Surface Charges, and Edge Modes when the quotient depends on a boundary.