Extended Operators and Defects
Lines, surfaces, and walls probe information that local insertions can miss. This chapter explains how their support, allowed endpoints, singular boundary conditions, and fusion define extended operators, while distinguishing geometric support from topological invariance.
Enter this chapter
Section titled “Enter this chapter”Begin with Support, Codimension, and Operator Data to specify an insertion completely.
Helpful background. Parallel Transport and Holonomy and Genuine Line Spectra, Discrete Theta Data, and Theory Specification supply parallel transport and the allowed genuine-line spectrum.
Topics
Section titled “Topics”- Support, Codimension, and Operator Data — Separates the geometry of support from the operator’s defining data.
- Wilson Lines and Loops — Constructs parallel-transport insertions and their endpoint transformations.
- Disorder Operators and Singular Boundary Conditions — Defines insertions through prescribed singular configurations.
- Genuine Lines, Screening, and Charge Lattices — Tests surface attachment, screening, and allowed charge lattices.
- Surface Defects and Codimension-Two Monodromy — Specifies monodromy and additional data near codimension-two supports.
- Boundaries, Interfaces, and Domain Walls — Distinguishes one-sided boundaries from interfaces and physical domain walls.
- Fusion, Junctions, and Endpoints — Defines composition through collision limits and junction spaces.
- Linking, Braiding, and Framing — Separates mutual linking, exchange operations, and self-framing information.
How the ideas fit together
Section titled “How the ideas fit together”A closed Wilson trace can be gauge invariant without being topological: its expectation may depend on the curve’s shape. A topological symmetry defect, by contrast, can be deformed through an allowed region without changing the correlator until it crosses charged support. This deformation property underlies generalized symmetry actions; see Bhardwaj et al. 2024, § 2.
The support and codimension diagram shows why dimension alone cannot decide gauge invariance, topologicality, or invertibility. Fusion also needs the appropriate collision and junction data; a formal product of labels does not supply them.
Review the chapter
Section titled “Review the chapter”Why is an open Wilson line usually not a gauge-invariant observable by itself?
Solution
Parallel transport transforms independently at its two endpoints. Taking a trace does not cancel two unrelated endpoint transformations. One must close the curve, attach suitable endpoint fields, or specify boundary data that make the combined insertion invariant.
Where to go next
Section titled “Where to go next”Continue to Higher-Form and Higher-Group Symmetries to interpret topological defects as symmetry generators acting on extended charges.