Extended Operators and Defects

An extended operator is not determined by the name of its support. A curve, surface, wall, or boundary becomes a defined insertion only after its transverse geometry, label and global gauge data, orientation or conditional framing, endpoint and attachment rules, junctions, renormalization, and allowed deformations have been stated. This chapter organizes those data and then asks three different questions: how the insertion is constructed, which charges survive screening and surface attachment, and how protected insertions compose or link.

Choose the definition route for Wilson lines, magnetic disorder data, surface monodromy, boundaries, or interfaces. Choose the charge-lattice route when the global gauge group, dynamical matter, mutual locality, or genuineness is the issue. Choose the composition route when a collision, endpoint, junction, associativity map, braid, or framing dependence is the missing datum. The chapter supplies a common physical grammar; model-specific phase dynamics, conformal or supersymmetric classifications, anyonic platforms, and theorem-first higher-categorical structures belong to the continuations listed below.

Helpful background. Parallel Transport and Holonomy supplies the transport law, path ordering, and endpoint covariance used by Wilson lines. Genuine Line Spectra, Discrete Theta Data, and Theory Specification supplies the global-form and mutually local line-spectrum input needed when a question begins beyond local Lie-algebra data. Neither is required merely to use this overview.

Parent volume. Symmetry and Gauge Structure

Jump to: check preparation · choose a route · read the atlas · open the exact guide · review the chapter

Check the geometry and global gauge data first

For a smooth proper interior support $\Sigma^p\subset M^d$ , codimension $c=d-p\geq 1$ determines the rank of the normal bundle and the local transverse link:

c=d-p\geq 1, \qquad D_x^c\ \text{normal to }\Sigma, \qquad S_x^{c-1}=\partial D_x^c.

This is local information. The normal sphere bundle may twist, an endpoint or junction has a stratified link, and a spacetime boundary has a one-sided collar rather than the two-sided $S^0$ model of an interior interface. These smooth-stratum and collar facts are reviewed in Lee 2013, 2nd ed., Ch. 5, p. 99 and pp. 106–107, Prop. 5.16; Ch. 9, pp. 222–223, Thm. 9.25. The author’s current corrections PDF, updated 3 January 2026 do not change the statements used here.

Geometry also does not choose the operator type. Keep the following distinctions operational:

An extended operator or defect is a positive-dimensional insertion or a locus across which field-theory data change. The terms overlap in common usage.
A topological defect is invariant under a declared class of structure-preserving deformations. Support dimension does not prove that invariance.
A symmetry defect is a topological defect equipped with an action on charged insertions and with fusion and junction data. Invertible group-like symmetry defects are a special class, not the definition of every defect.
A genuine line requires no auxiliary surface in the declared theory. That is different from being unscreened, non-endable, topological, or invertible.
An endpoint or junction is a lower-dimensional operator or theory. A label-conservation equation is necessary but does not construct it.

This vocabulary and the group-like symmetry-defect model are organized in Gaiotto et al. 2015, Introduction and §§ 2–3, arXiv v2, pp. 1–3 and 5–13, especially eqs. (1.1)–(1.2), (2.2)–(2.4), and (3.1)–(3.4), Open PDF. The chapter does not extend those group-like conclusions to every defect.

Choose by the question you need to answer

The routes below are productive reading orders. A page named in the preparation column supplies an actual dependency; the arrows in a route do not assert that every later conclusion follows automatically.

Choose a chapter route from the missing physical datum
Question	Route	Additional preparation	Capability at the end
What data define any extended insertion?	Support and codimension	Hard: local and composite insertions. Helpful: differential forms and Stokes.	Write a complete support and transverse-data sheet
How does holonomy define a line?	Support → Wilson lines	Hard: Support, parallel transport, and gauge redundancy and observables.	Separate closed invariant traces from covariant open transport
How is a singular or monodromy insertion defined?	Support → disorder operators → surface defects	Hard for disorder: Support and local potentials and global bundles. Disorder is helpful, not hard, for the surface page.	State flux, patching, holonomy, localized theory, and counterterms
Which electric, magnetic, or dyonic lines are genuine?	Support → Wilson and disorder lines → genuine lines	Hard: Wilson, disorder, and the complete line spectrum and discrete theta data.	Distinguish labels, screening classes, mutual locality, and attachment
What makes a boundary or interface consistent?	Support → boundaries, interfaces, and walls	Hard: Support and boundary flux and Ward identities. Helpful: edge modes and factorization.	Test geometry, variation, matching, symmetry, anomaly, and transmission
What is needed beyond a fusion or linking rule?	Support → fusion and junctions → linking and braiding	Hard: Support for fusion; fusion plus homotopy and winding for linking. Helpful: genuine-line data.	Type collision limits, junction spaces, associators, braids, and framing

If a proposed line or surface has not yet been defined, begin at the support page even when the eventual goal is a charge lattice or braid. If the geometry and operator definition are already secure, enter at the later page whose missing datum matches the first column.

One data sheet connects the chapter

The atlas below is the chapter’s visual index. Read the upper row as local transverse geometry and the lower row as the additional quantum data. In particular, compare the two-sided interface with the one-sided boundary, then inspect the junction: neither codimension nor label arithmetic supplies a junction operator.

Increasing support dimension lowers the local linking-sphere dimension, but support geometry still leaves labels, orientation, endpoints, junctions, renormalization, and deformation laws unspecified; an interface is two-sided while a boundary is one-sided. — For a smooth interior support $\Sigma^p\subset M^d$ , codimension $c=d-p$ gives the local normal disk $D^c$ and link $S^{c-1}$ . The atlas also records the one-sided boundary exception and a finite-symmetry junction. It is a schematic, not-to-scale classification: the normal sphere bundle can twist, and endpoints, corners, and junctions require stratified data.

Swipe horizontally to inspect the full atlas, or open the vector figure at full size.

The complete nonvisual reading is:

Support geometry and the independent data that complete an insertion
Object	Local geometry	Data still required	What the geometry does not prove
Point	Support dimension zero, codimension d, and a local sphere of dimension d minus 1	Field or composite label, spin, regulator, and normalization	Gauge invariance or a regulator-independent product
Line	Support dimension one, codimension d minus 1, and a local sphere of dimension d minus 2	Orientation, representation or charge, endpoints, and optional framing	Genuineness, topological invariance, or permission to end
Surface	Support dimension two, codimension d minus 2, and a local sphere of dimension d minus 3; in four dimensions this is a normal circle	Embedding, orientation, and localized or transverse data	Invertibility, a symmetry action, or shape independence
Interior interface	Two-sided codimension-one support	Ordered bulk theories, coorientation, matching, and wall degrees	A one-sided boundary condition or transparent transmission
Boundary	One-sided collar at the edge of spacetime	Boundary condition or theory, outward normal, allowed transformations	A second bulk side or the full interface model
Geometric defect	Its embedded support can carry shape, metric, and extrinsic data	Embedding dependence and support-local counterterms	Deformation invariance
Topological defect	Its support may move through an allowed separated deformation domain	Attachments, forbidden crossings, lower strata, and conditional framing	A symmetry action or invertibility
Symmetry defect	Support degree depends on the symmetry degree	Action on charged objects, orientation, fusion, junctions, and invertibility when applicable	That every topological defect is invertible or group-like
Finite-symmetry junction	Sheets labeled a and b meet through J_ab and continue with label a+b modulo N	Signed incidence, the junction operator, normalization, and coherence	Existence from the additive label rule
Endpoint or junction	Lower stratum with incident supports and a decorated link	Signed incidence, allowed operator, multiplicity, normalization, coherence	Existence from label conservation alone
Complete specification	Support and normal data are only its geometric layer	Labels and global form, orientation or conditional framing, attachments, junctions, renormalization, and deformation class	That any omitted consequential datum is trivial

A reusable data sheet therefore records, as applicable:

the ambient theory, spacetime domain, boundary conditions, and global gauge group;
the embedded or stratified support, its dimension, codimension, sides, orientation, and normal data;
the representation, charge, flux, monodromy, or other transverse label;
whether the insertion multiplies the integrand or changes the admitted field domain;
any attached surface, endpoint, boundary termination, or junction operator;
regulator, counterterms, operator mixing, and finite normalization; and
the allowed deformations, crossings, framing transport, and claim of geometric, conformal, or topological behavior.

Omitting an inapplicable entry is legitimate. Omitting a consequential entry and silently assuming it is trivial changes the operator.

A complete definition develops in layers

The chapter’s main reasoning can be read as a sequence of independent tests.

Fix the support and transverse link. State whether the locus is a smooth interior stratum, boundary, or stratified network, and choose the orientation or coorientation used by flux and incidence signs.
Choose order or disorder data. A Wilson line is built from parallel transport. A disorder insertion changes the field domain near its support by a flux, monodromy, or other boundary condition. Dyonic and decorated insertions can carry both kinds of data.
Apply the actual global group. A Wilson representation must descend to the global gauge group, and a magnetic label must define an allowed cocharacter or bundle transition. Local Lie-algebra data are not enough.
Test genuineness and screening separately. Ask whether an auxiliary surface is required, then quotient charge labels by declared dynamical endpoints. Mutual locality and a complete absolute line spectrum are further inputs.
Supply lower strata. Open supports, terminations, fusion vertices, and intersections need endpoint or junction operators. Incidence equations are selection rules, not constructions.
Regulate collision and renormalization. A formal label product becomes fusion only after a controlled collision limit. Junction spaces, multiplicities, basis choices, and associativity maps remain independent data.
State the deformation domain. Linking needs disjoint complementary supports and appropriate homological data; braiding is transport in a configuration space; self-linking needs a push-off or framing. None of these makes an ordinary geometric operator topological by itself.

The order prevents a common circular argument: one cannot use a fusion or linking formula to certify an insertion whose global label, attachment, or renormalized definition has not yet been established.

Two recurring examples keep the distinctions concrete

The leaves reuse two examples as a consistency thread. They are not a new universal classification.

In the chapter’s controlled compact Abelian example, work on a closed oriented spin four-manifold at $\theta=0$ . A faithfully normalized compact connection $\mathfrak a$ , with $f=\mathrm d\mathfrak a$ locally, defines an integer Wilson label

W_n(C)=\exp\!\left(i n\oint_C\mathfrak a\right), \qquad n\in\mathbb Z, \qquad \frac{1}{2\pi}\int_{S^2}f\in\mathbb Z.

Wilson transport, magnetic disorder flux, charged endpoints, and surface attachment can then be examined without changing normalization from page to page. If dynamical electric charges generate $N\mathbb Z$ , $N\geq 2$ , the electric screening classes form $\mathbb Z_N$ . Hold magnetic backgrounds and monopole endpoints outside this electric calculation. When the corresponding electric $\mathbb Z_N$ one-form symmetry is exact in the declared background, the Wilson charge is $r=[n]_N$ . Let $\alpha,\beta,\gamma\in\mathbb Z_N$ , and take $C$ and $\Sigma$ to be oriented, disjoint, closed supports in a region where integer linking is defined. Fix the sign so a positive unit link of $U_1$ with $W_1$ gives $e^{2\pi i/N}$ . The normalized topological surface action is

\begin{aligned} \left\langle U_\alpha(\Sigma)W_n(C)\,\mathcal X\right\rangle &= \exp\!\left[ \frac{2\pi i}{N}\,\alpha r\,\operatorname{Lk}(\Sigma,C) \right] \left\langle W_n(C)\,\mathcal X\right\rangle, \\ U_\alpha\otimes U_\beta &\simeq U_{\alpha+\beta\bmod N}. \end{aligned}

A two-in/one-out surface junction with label $\gamma$ must satisfy $\alpha+\beta-\gamma=0\pmod N$ , but this arithmetic does not supply its operator, normalization, or coherence. Reversing either support negates $\operatorname{Lk}(\Sigma,C)$ and inverts the phase. The screening, character, fusion, and linking ingredients are developed in Bhardwaj et al. 2024, § 2.1, arXiv v2, pp. 6–10, Statement 2.1 and eqs. (2.4)–(2.23); §§ 2.2.1–2.2.2, pp. 12–20, Definition 2.1 and eqs. (2.38)–(2.45) and (2.58)–(2.70); §§ 3.1–3.2.1, pp. 26–31, Definitions 3.1–3.2 and eqs. (3.1)–(3.4) and (3.10)–(3.19), Open PDF. The displayed relation assumes the surface can be normalized and removed after the allowed sweep and that no other insertion or boundary is crossed.

In an independent three-dimensional theory with an exact non-anomalous group-like $\mathbb Z_N$ zero-form symmetry, write $U_a(W)$ for a cooriented symmetry wall. These walls are surfaces, trivalent junctions of wall sheets are lines, and comparisons between resolved junction networks are point operators. A two-dimensional wall in three dimensions has local link $S^0$ ; the two-dimensional one-form symmetry surface $\Sigma$ above has local link $S^1$ in four dimensions. The same signed incidence pattern appears, but the support geometry and physical origin are different. If the non-anomalous hypothesis is dropped, an anomaly can twist or obstruct coherent junction data even when label arithmetic closes Gaiotto et al. 2015, § 2, arXiv v2, pp. 7–8, the paragraph preceding eq. (2.6) and eqs. (2.6)–(2.8), Open PDF. This is why the chapter carries support dimension and symmetry degree alongside every network label.

Diagnose the missing preparation

Use the following checks to enter at the first page that supplies something you do not yet have.

Regulated insertions. You are ready for the support page if you can state the regulated meaning, labels, normalization, and possible mixing of a local or composite insertion. If not, begin with Local and Composite Operator Insertions.

Gauge redundancy and observables. You are ready to call a holonomy an observable only if you can test its finite gauge transformation in the actual field domain and distinguish invariant, covariant, and gauge-dependent data. If not, use Gauge Fields, Redundancy, and Observable Content before the Wilson page.

Transport and gauge covariance. You are ready for the Wilson page if you can derive how parallel transport transforms at both endpoints and why a closed trace is invariant. If not, use Parallel Transport and Holonomy before entering the line route.

Bundles and singular data. You are ready for disorder and monodromy if you can distinguish a patchwise compact connection, its transition function, and its curvature or holonomy. If not, use Local Potentials and Global Gauge Configurations before the disorder page.

Global form and line spectrum. You are ready for the charge-lattice page if the actual global gauge group, honest electric representations, allowed magnetic labels, and chosen genuine spectrum are stated. If only a Lie algebra has been named, use Genuine Line Spectra, Discrete Theta Data, and Theory Specification and then read both the Wilson and disorder pages.

Boundary variation. You are ready for interfaces when the action, admitted variations, outward normals of the cut regions or a declared common coorientation, and boundary flux convention are explicit. If those data are missing, use Boundaries, Flux, and Boundary Ward Identities before the codimension-one page.

Motion and collision. For mutual linking, fix closed disjoint supports, labels, orientations, complementary dimensions, and the homological domain. For braid transport in a resolved sector, also fix the configuration space, fusion channel, junctions or endpoints, and any required framing. The linking page uses the fusion page’s network grammar, but an ordinary linking number does not itself require a fusion channel. If a collision is part of the question and remains undefined, begin with Fusion, Junctions, and Endpoints; if the motion class is unclear, use Homotopy, Degree, Winding, and Covering Spaces.

Exact chapter guide

1. Support, Codimension, and Operator Data

This is the common starting point. It constructs the smooth transverse link, separates a one-sided boundary from a two-sided interface, treats stratified endpoints and junctions, and turns an operator name into a complete data sheet. After it, you can reject claims that infer genuineness, topology, framing, or junction existence from support alone. Continue to the Wilson or disorder page according to how the insertion is defined. It requires Local and Composite Operator Insertions; differential forms and Stokes’ theorem are helpful.

2. Wilson Lines and Loops

This page derives path-ordered transport, its endpoint covariance, the gauge-invariant closed trace, orientation reversal, honest global-group representations, endpoint dressings, line renormalization, and conditional framing. After it, you can distinguish a closed Wilson loop from a bare open holonomy and state which additional data make either a quantum operator. Continue to disorder lines for the complementary magnetic construction or to genuine lines when both electric and magnetic labels are ready. It requires the support page, Parallel Transport and Holonomy, and Gauge Fields, Redundancy, and Observable Content.

3. Disorder Operators and Singular Boundary Conditions

This page defines an insertion by excising a tube and restricting the field domain near its support. Compact Abelian flux, patch transitions, non-Abelian cocharacters, monodromy, counterterms, attachments, and Bianchi constraints are kept explicit. After it, you can tell a physical singular boundary condition from a gauge-presentation string and identify the global data that make a magnetic label admissible. Continue to genuine lines or to the codimension-two surface page. It requires the support page and Local Potentials and Global Gauge Configurations.

4. Genuine Lines, Screening, and Charge Lattices

This page separates the tests for an honest label, a surface-free genuine line, screening by dynamical endpoints, mutual locality, and a complete absolute line spectrum. It derives the finite screening quotient and gives compact Abelian and reduced non-Abelian examples. After it, you can explain why genuine, unscreened, non-endable, and topological are different predicates. Continue to higher-form symmetry for the associated symmetry operators or to fusion when the allowed lines themselves are to be composed. It requires the Wilson and disorder pages plus Genuine Line Spectra, Discrete Theta Data, and Theory Specification.

5. Surface Defects and Codimension-Two Monodromy

This page treats holonomy on the normal circle bundle, affine-Weyl identifications, localized defect theories, topological couplings, counterterms, ordered-meridian junctions, and the distinction between geometric, conformal, and topological surfaces. After it, you can state what a monodromy label fixes and which global, local, and lower-stratum data remain. Continue to fusion for network composition or to a specialized conformal or supersymmetric treatment for additional protected structure. It requires the support page; the disorder page and local/global bundle language are helpful rather than hard prerequisites.

6. Boundaries, Interfaces, and Domain Walls

This page distinguishes a physical boundary, a two-sided interface, a dynamical interpolating wall, and a fiducial cut. It derives the total variational matching condition, folding, current and anomaly compatibility, Maxwell reflection and transmission, finite-symmetry wall junctions, and the requirements for a controlled collision. After it, you can state why flux continuity is not transparency and why orientation reversal is not an inverse. Continue to fusion for wall composition or to the relevant conformal, holographic, or relative-theory specialization. It requires the support page and Boundaries, Flux, and Boundary Ward Identities; edge-mode language is helpful.

7. Fusion, Junctions, and Endpoints

This reference begins with a regulated collision rather than a formal label product. In the protected finite setting it separates channels and multiplicities from junction and endpoint operators, then types the associativity comparison between two resolutions. After it, you can identify the data missing from a fusion coefficient and test whether an endpoint or splitting is actually allowed. Continue to linking and braiding while carrying every orientation, junction, and conditional framing datum. It requires the support page; Chains, Homology, Cohomology, and Exact Sequences is helpful for signed incidence and relative cycles.

8. Linking, Braiding, and Framing

This page defines integer linking in its homological domain, distinguishes a closed link from braid transport, types exchange and full monodromy maps, states the dimensional limits of point-particle braid intuition, and defines self-linking through a framing. After it, you can tell when a result is an integer, a scalar, a matrix, or an ill-posed claim. Continue to higher-form symmetry, topological gauge theory, many-body anyons, or theorem-level braided structures according to the application. It requires the fusion page and Homotopy, Degree, Winding, and Covering Spaces; the genuine-line page is helpful.

Keep the classifications independent

Names that require separate evidence
Claim	Question to test	Does not follow from
Genuine	Can the insertion be defined without an auxiliary surface?	An honest local charge label or a closed support
Unscreened	Can declared dynamical endpoints change its charge class?	Genuineness or the absence of one light field
Topological	Are correlators invariant under the declared deformations?	Codimension, quantized charge, conformality, or symmetry language
Symmetry defect	Are the required topological deformations, action on charged objects, fusion, and junction data defined?	Being topological or carrying a group-like-looking label
Invertible	Does there exist E with both D fused with E and E fused with D equivalent to the identity?	Orientation reversal, an adjoint, or a one-to-one label map
Framed	Does the definition or regulator require transported normal data?	Orientation, spin structure, or line support alone

The same caution applies to fusion and linking. A finite label law is not a junction or associator, and one pairwise linking phase is not a complete link invariant or a proof of topological order.

Boundaries and canonical exits

This chapter stops when the question requires a developed dynamical, application-specific, or theorem-first framework.

Where to continue when the remaining question leaves this chapter
Remaining question	Continue to	New input supplied there
How does a topological operator define a higher-form symmetry?	Higher-Form and Higher-Group Symmetries	Symmetry degree, charged objects, backgrounds, anomalies, and mixing
How can topological fusion define a symmetry without inverses?	Non-Invertible Symmetries	Operational non-invertibility, fusion actions, and construction mechanisms
What are the explicit Chern–Simons, BF, or finite-gauge data?	Topological Gauge Theories and Symmetry TFT	Quantized actions, state spaces, link amplitudes, boundaries, and gluing
How are Wilson lines used in soft or eikonal factorization?	Eikonal Approximation and Wilson Lines	Momentum-space denominators, soft factors, and perturbative kinematics
What does a line diagnose about confinement or screening?	Line Operators, Screening, and Generalized-Symmetry Diagnostics	Phase dynamics, order of limits, matter screening, and diagnostic ceilings
Which defects are conformal and what data classify them?	Conformal Boundaries and Defects	Defect conformal symmetry, displacement data, correlators, and bootstrap input
Which boundaries and surfaces are BPS?	BPS Boundaries and Surface Defects	Supersymmetry conditions, protected couplings, and duality-specific data
How are anyons realized in quantum matter?	Anyons as Quasiparticles in Quantum Matter	Gapped phases, quasiparticle preparation, observables, and platforms
What is the theorem-level composition structure?	Defects on Stratified Spacetimes and Higher-Categorical Composition	Typed higher morphisms, coherence, functoriality, and rigorous comparison

Some continuations are chapter entry points rather than prerequisites. Use them when the question in the first column is actually being asked; their specialized conclusions should not be imported backward into a generic extended insertion.

Review the chapter

Use each outline after choosing your own example.

1. Choose an entry route and repair one missing prerequisite

A request asks whether a spin- $\tfrac12$ Wilson line exists in an $SO(3)$ gauge theory and whether it can be screened. Begin with the Wilson page to test whether the representation descends to the actual global group. Use the global line-spectrum preparation and then the genuine-lines page to separate existence, surface attachment, and screening. The spin- $\tfrac12$ representation does not descend to $SO(3)$ , so no standalone spin- $\tfrac12$ Wilson line exists to be screened. The odd-electric $(1,1)$ dyonic line of $SO(3)_-$ is a different object whose existence is fixed by the complete line spectrum. Starting at fusion would assume the line being composed is already defined.

2. Complete an insertion data sheet

For a proposed codimension-two surface defect, name the ambient theory and global group; the embedded, oriented, or stratified support; its normal circle bundle; the monodromy or singular boundary condition; residual and localized fields; endpoints or junctions; regulator and counterterms; and the allowed deformations. A conjugacy class alone fills only one row of this definition.

3. Separate genuineness from screening

First ask whether the line requires an attached surface. Then identify the charges of declared dynamical endpoint excitations and form the appropriate screening classes. A closed line can be genuine and still screenable; a line can be unscreened but non-genuine because its definition retains an attached surface.

4. Explain what a fusion coefficient omits

A formal channel coefficient does not give the regulated collision limit, a junction operator, its multiplicity basis or normalization, endpoint data, the map comparing two trivalent resolutions, or higher coherence. Outside a protected finite setting, even a discrete channel sum may fail to be the right output description.

5. Diagnose an underspecified linking claim

Demand the ambient and support dimensions, disjointness, orientations, homological or relative data, operator labels, allowed isotopy, and any framing. For braid transport also specify the configuration space and channel space. A closed-link phase need not recover a single exchange map, and self-linking has no naked, choice-free definition.

6. Choose the correct continuation

Use higher-form symmetry for the abstract action of topological surfaces on lines, topological gauge theory for explicit level or link amplitudes, nonperturbative dynamics for phase diagnostics, conformal or supersymmetric treatments for protected specializations, many-body physics for quasiparticle realizations, and Mathematical QFT for theorem-level higher composition.

7. Derive, reverse, and transfer the finite example

Charge- $N$ endpoints identify electric labels $n\sim n+N$ , so the surviving class is $r=[n]_N$ . The character of $U_\alpha$ is $\exp(2\pi i\alpha r/N)$ and repeated oriented linking raises it to the signed linking number. Reversing either support negates that number and inverts the phase. To transfer the reasoning, begin a new codimension-two surface or codimension-one wall example with the data sheet rather than the phase: restate its support, transverse link, label, lower strata, renormalization, and deformation domain. Use the genuine-line page to repair the quotient step and the linking page to repair the orientation or homological domain.

Continue from here

For a complete first pass, read Support, Codimension, and Operator Data followed by Wilson Lines and Loops and Disorder Operators and Singular Boundary Conditions. Then choose the genuine-line, surface, or codimension-one branch before rejoining at Fusion, Junctions, and Endpoints and Linking, Braiding, and Framing. Return to Symmetry and Gauge Structure to choose another chapter.

References

Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
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.
Lee, John M. Introduction to Smooth Manifolds. 2nd ed. Graduate Texts in Mathematics 218. New York: Springer, 2013. DOI. Author corrections PDF, updated 3 January 2026.