In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and especially
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, Donaldson theory is the study of the topology of smooth
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. T ...
s using
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
s of
anti-self-dual instantons. It was started by
Simon Donaldson (1983) who proved
Donaldson's theorem restricting the possible quadratic forms on the second
cohomology group of a compact simply connected 4-manifold. Important consequences of this theorem include the existence of an
Exotic R4 and the failure of the smooth
h-cobordism theorem in 4 dimensions. The results of Donaldson theory depend therefore on the manifold having a differential structure, and are largely false for topological 4-manifolds.
Many of the theorems in Donaldson theory can now be proved more easily using
Seiberg–Witten theory
In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action (for massless degrees of freedom) of a \mathcal = 2 supersymmetric gauge theory—namely the metric of the moduli space of vacua.
S ...
, though there are a number of open problems remaining in Donaldson theory, such as the
Witten conjecture In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper , and generalized in .
Witten's original conjecture was proved by Maxi ...
and the
Atiyah–Floer conjecture.
See also
*
Kronheimer–Mrowka basic class
*
Instanton
*
Floer homology
*
Yang–Mills equations
References
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Geometric topology
4-manifolds
Differential topology
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