Donaldson theory

E860091

gauge theory mathematical theory topological invariant theory

Donaldson theory is a gauge-theoretic framework in four-dimensional differential topology that uses moduli spaces of anti-self-dual connections to define powerful invariants distinguishing smooth structures on 4-manifolds.

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Statements (47)

Predicate	Object
instanceOf	gauge theory ⓘ mathematical theory ⓘ topological invariant theory ⓘ
aimsTo	distinguish smooth structures on 4-manifolds ⓘ
appliesTo	simply connected 4-manifolds ⓘ smooth 4-manifolds ⓘ
basedOn	SO(3) gauge theory ⓘ SU(2) gauge theory ⓘ anti-self-dual Yang–Mills equations ⓘ
contrastsWith	Freedman theory of topological 4-manifolds ⓘ
defines	Donaldson invariants NERFINISHED ⓘ smooth structure invariants of 4-manifolds ⓘ
developedBy	Simon Donaldson NERFINISHED ⓘ
field	four-dimensional differential topology ⓘ gauge theory in mathematics ⓘ low-dimensional topology ⓘ
hasInvariantType	Donaldson polynomial invariants NERFINISHED ⓘ polynomial invariants of 4-manifolds ⓘ
hasKeyEquation	F_A^+ = 0 ⓘ
hasKeyObject	anti-self-dual connection ⓘ instanton moduli space ⓘ
historicalPeriod	1980s ⓘ
implies	diagonalizability of definite unimodular intersection forms over the integers ⓘ restrictions on intersection forms of smooth 4-manifolds ⓘ
influenced	development of geometric analysis ⓘ modern 4-manifold topology ⓘ
inspired	Seiberg–Witten theory NERFINISHED ⓘ
ledTo	classification results for definite 4-manifolds ⓘ
relatedTo	Floer homology NERFINISHED ⓘ Seiberg–Witten invariants NERFINISHED ⓘ gauge-theoretic moduli spaces ⓘ instanton Floer homology NERFINISHED ⓘ
requires	Riemannian metric on a 4-manifold ⓘ principal G-bundle over a 4-manifold ⓘ
shows	existence of exotic smooth structures on 4-manifolds ⓘ
studies	space of connections modulo gauge transformations ⓘ
uses	Fredholm theory NERFINISHED ⓘ Uhlenbeck compactness NERFINISHED ⓘ Yang–Mills gauge theory NERFINISHED ⓘ elliptic partial differential equations ⓘ moduli spaces of anti-self-dual connections ⓘ
usesConcept	moduli space compactification ⓘ orientation of moduli spaces ⓘ transversality in infinite-dimensional manifolds ⓘ
usesTool	Hodge theory on 4-manifolds ⓘ Morse theory on infinite-dimensional spaces ⓘ index theory ⓘ

Referenced by (1)

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Seiberg–Witten theory → relatedTo → Donaldson theory ⓘ