Seiberg–Witten invariants

E861521

gauge-theoretic invariant smooth 4-manifold invariant topological invariant

Seiberg–Witten invariants are powerful topological invariants of smooth four-manifolds derived from solutions to the Seiberg–Witten equations, used to distinguish different smooth structures and study the geometry and topology of 4D spaces.

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Statements (48)

Predicate	Object
instanceOf	gauge-theoretic invariant ⓘ smooth 4-manifold invariant ⓘ topological invariant ⓘ
constructedFrom	count of solutions to Seiberg–Witten equations ⓘ oriented moduli space of solutions ⓘ
definedOn	oriented smooth 4-manifolds ⓘ smooth four-manifolds ⓘ
definedUsing	Seiberg–Witten equations NERFINISHED ⓘ moduli space of monopoles ⓘ solutions of Seiberg–Witten equations ⓘ spin^c structures ⓘ
dependsOn	chamber structure in b2^+ = 1 case ⓘ choice of spin^c structure ⓘ
field	4-manifold topology ⓘ differential topology ⓘ gauge theory ⓘ geometric analysis ⓘ symplectic topology ⓘ
generalizes	earlier gauge-theoretic invariants of 4-manifolds ⓘ
hasVariant	equivariant Seiberg–Witten invariants ⓘ monopole Floer homology ⓘ relative Seiberg–Witten invariants ⓘ
implies	adjunction inequality for embedded surfaces ⓘ
inspiredBy	quantum field theory ⓘ supersymmetric gauge theory ⓘ
introducedBy	Edward Witten NERFINISHED ⓘ Nathan Seiberg NERFINISHED ⓘ
relatedTo	Donaldson invariants ⓘ Floer homology NERFINISHED ⓘ Heegaard Floer homology NERFINISHED ⓘ Yang–Mills gauge theory NERFINISHED ⓘ monopole equations ⓘ
requires	compactness of moduli space ⓘ transversality of moduli space ⓘ
sensitiveTo	smooth structure but not just homeomorphism type ⓘ
simplifiedComputationComparedTo	Donaldson invariants NERFINISHED ⓘ
takesValuesIn	integers ⓘ
usedFor	detecting exotic smooth structures ⓘ distinguishing homeomorphic but non-diffeomorphic 4-manifolds ⓘ distinguishing smooth structures on 4-manifolds ⓘ proving non-existence of metrics of positive scalar curvature ⓘ studying geometry of 4-manifolds ⓘ studying symplectic structures on 4-manifolds ⓘ studying topology of 4-manifolds ⓘ
usedToProve	Thom conjecture for CP^2 NERFINISHED ⓘ constraints on intersection forms of 4-manifolds ⓘ
wellDefinedWhen	b2^+ > 1 ⓘ
yearIntroduced	1994 ⓘ

Referenced by (2)

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

Seiberg–Witten theory → relatedTo → Seiberg–Witten invariants ⓘ

Dirac operator → usedToStudy → Seiberg–Witten invariants ⓘ