Chern–Simons theory

E240804

gauge theory quantum field theory topological quantum field theory

Chern–Simons theory is a topological quantum field theory in three dimensions that plays a central role in modern geometry, topology, and theoretical physics, particularly in the study of knot invariants and gauge fields.

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All labels observed (14)

Label	Occurrences
Chern–Simons theory canonical	4
Chern–Simons forms	2
Chern–Simons–Witten theory	2
Abelian Chern–Simons theory	1
Chern–Simons 3-form	1
Chern–Simons gauge field	1
Chern–Simons topological quantum field theory	1
Chern–Simons–matter theory	1
Mean-field Chern–Simons theory	1
SU(2) Chern–Simons theory	1
SU(N) Chern–Simons theory	1
U(1) Chern–Simons theory	1
non-Abelian Chern–Simons theory	1
supersymmetric Chern–Simons theory	1

Statements (49)

Predicate	Object
instanceOf	gauge theory ⓘ quantum field theory ⓘ topological quantum field theory ⓘ
actionFunctionalDependsOn	Chern–Simons theory self-linksurface differs ⓘ surface form: Chern–Simons 3-form gauge connection ⓘ
appliedIn	3-manifold topology ⓘ M-theory ⓘ quantum Hall effect ⓘ surface form: fractional quantum Hall effect knot theory ⓘ string theory ⓘ
boundaryTheory	Wess–Zumino–Witten model ⓘ surface form: Wess–Zumino–Witten conformal field theory
classicalActionGivenBy	integral of Tr(A∧dA + (2/3)A∧A∧A) ⓘ
definedOn	three-dimensional manifolds ⓘ
describes	anyonic excitations in 2+1 dimensions ⓘ
equationsOfMotionImply	flat gauge connection ⓘ
gaugeInvarianceRequires	level quantization ⓘ
hasMathematicalOriginIn	Chern–Simons forms ⓘ secondary characteristic classes ⓘ
hasParameter	level k ⓘ
hasSpacetimeDimension	3 ⓘ
hasVariant	Chern–Simons theory self-linksurface differs ⓘ surface form: Abelian Chern–Simons theory Chern–Simons theory self-linksurface differs ⓘ surface form: Chern–Simons–matter theory Chern–Simons theory self-linksurface differs ⓘ surface form: non-Abelian Chern–Simons theory Chern–Simons theory self-linksurface differs ⓘ surface form: supersymmetric Chern–Simons theory
influenced	Witten’s work on quantum invariants of 3-manifolds ⓘ
introducedBy	James Harris Simons ⓘ Shiing-Shen Chern ⓘ
isMetricIndependent	true ⓘ
isTopological	true ⓘ
levelQuantizationCondition	k ∈ ℤ for compact simple gauge groups ⓘ
namedAfter	James Harris Simons ⓘ Shiing-Shen Chern ⓘ
pathIntegralLocalizesOn	flat connections ⓘ
produces	knot invariants ⓘ link invariants ⓘ topological invariants of 3-manifolds ⓘ
quantizationLeadsTo	Wess–Zumino–Witten model ⓘ
relatedTo	Atiyah–Segal axioms ⓘ surface form: Atiyah–Segal axioms for TQFT HOMFLY-PT polynomial ⓘ surface form: HOMFLY polynomial Jones polynomial ⓘ Witten–Reshetikhin–Turaev invariant ⓘ surface form: Reshetikhin–Turaev invariants quantum groups at roots of unity ⓘ
specialCase	Chern–Simons theory self-linksurface differs ⓘ surface form: SU(2) Chern–Simons theory Chern–Simons theory self-linksurface differs ⓘ surface form: SU(N) Chern–Simons theory Chern–Simons theory self-linksurface differs ⓘ surface form: U(1) Chern–Simons theory
usedIn	topological quantum computation ⓘ
usedToConstruct	3-dimensional topological invariants via path integrals ⓘ
usesGaugeGroup	compact Lie group ⓘ
yearIntroduced	1974 ⓘ

Referenced by (19)

Full triples — surface form annotated when it differs from this entity's canonical label.

Shiing-Shen Chern → knownFor → Chern–Simons theory ⓘ

Edward Witten → notableWork → Chern–Simons theory ⓘ

this entity surface form: Chern–Simons–Witten theory

topological quantum field theory → example → Chern–Simons theory ⓘ

Green–Schwarz mechanism → relatedTo → Chern–Simons theory ⓘ

this entity surface form: Chern–Simons forms

Chern–Weil theory → relatedTo → Chern–Simons theory ⓘ

Jones polynomial → connectedTo → Chern–Simons theory ⓘ

this entity surface form: Chern–Simons topological quantum field theory

HOMFLY-PT polynomial → relatedTo → Chern–Simons theory ⓘ

Shiing-Shen → knownFor → Chern–Simons theory ⓘ

subject surface form: Shiing-Shen Chern

this entity surface form: Chern–Simons forms

Chern–Simons theory → actionFunctionalDependsOn → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: Chern–Simons 3-form

Chern–Simons theory → specialCase → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: SU(2) Chern–Simons theory

Chern–Simons theory → specialCase → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: SU(N) Chern–Simons theory

Chern–Simons theory → specialCase → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: U(1) Chern–Simons theory

Chern–Simons theory → hasVariant → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: Abelian Chern–Simons theory

Chern–Simons theory → hasVariant → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: non-Abelian Chern–Simons theory

Chern–Simons theory → hasVariant → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: supersymmetric Chern–Simons theory

Chern–Simons theory → hasVariant → Chern–Simons theory self-linksurface differs ⓘ

this entity surface form: Chern–Simons–matter theory

Witten → knownFor → Chern–Simons theory ⓘ

subject surface form: Edward Witten

this entity surface form: Chern–Simons–Witten theory

Composite fermion → associatedWith → Chern–Simons theory ⓘ

this entity surface form: Chern–Simons gauge field

Composite fermion → hasTheoreticalTool → Chern–Simons theory ⓘ

this entity surface form: Mean-field Chern–Simons theory