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Operators, defects, and symmetries

Identify local operators, Wilson lines, and ordinary or higher-form symmetry.

Selected local and extended objects. The support, transformation law, and physical role distinguish objects with similar notation.

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  • Higher-form symmetry

    A higher-form symmetry acts on extended charged operators and is characterized only after degree, normalization, topology, and linking data are fixed.

    Applies to: Invertible p-form global symmetries and their charged p-dimensional operators.

    • Spacetime dimension, operator support, current degree, charge normalization, allowed cycles, and global form are explicit.
    Notation caution
    The phrase generalized symmetry also includes non-invertible and higher-group structures that are not equivalent to an ordinary p-form symmetry.
    Defining data
    Form degree, current or topological operator, charged-operator dimension, group, normalization, topology, and linking action.

    Full explanation: Higher-Form Symmetry from Operators and Linking

    Notation and checks
    Also called
    p-form symmetry, generalized global symmetry

  • Local operator

    An insertion localized at a spacetime point in distributional notation, defined through smearing and, for composite products, a specified renormalization prescription.

    Applies to: Continuum QFT with declared operator domains and short-distance prescription.

    Defining data
    Field content, transformation law, engineering or scaling dimension, and composite-operator definition.
    Failure warning
    Writing a product of fields at one point does not by itself define a composite operator.

    Full explanation: Free Wick Products and Point Splitting

  • Wilson line

    A Wilson line is an extended gauge observable or probe whose definition depends on a path, representation, connection, and ordering convention.

    Applies to: Line-supported operators in gauge theory; open and closed lines have distinct endpoint and gauge-invariance requirements.

    • Gauge group, global form, representation, charge normalization, path orientation, and connection convention are specified by the owner.
    Notation caution
    A Wilson loop is a closed Wilson line; lattice Wilson action and Wilsonian renormalization are different uses of the name Wilson.
    Defining data
    Path, orientation, representation, gauge connection, and path ordering.

    Full explanation: Wilson Lines and Loops

    Notation and checks
    Also called
    Wilson operator, holonomy line