manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
Donaldson theory is concerned with 4-manifolds using the moduli space the anti-self-dual Yang-Mills equations (ASDYM equations), which require a principal bundle with a compact gauge group over the 4-manifold.
This method is named after and was first used by Donaldson 1983 (assuming simply-connected ) and Donaldson 1987 (without that restriction) to prove Donaldson's theorem.
Donaldson theory was later surpassed by Seiberg-Witten theory, since Donaldson invariants often give weaker results than Seiberg-Witten invariants and the former often requires an additional compactification of the moduli space. Nonetheless, there are still unsolved problems in Donaldson theory including the Witten conjecture and the Atiyah-Floer conjecture.
A topological FQFT-formulation of Donaldson theory is supposed to be given as a functor from a suitable symplectic category of symplectic manifolds with Lagrangian correspondences between them which sends a symplectic manifold to its Fukaya category. For more on this see at Lagrangian correspondences and category-valued TFT.
Simon Donaldson. An application of gauge theory to four-dimensional topology. In: Journal of Differential Geometry. 18. Jahrgang, Nr. 2, 1. Januar 1983, doi:10.4310/jdg/1214437665
Simon Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. In: Journal of Differential Geometry. 26. Jahrgang, Nr. 3, 1. Januar 1987, doi:10.4310/jdg/1214441485
Wikipedia, Donaldson theory
The relation to the topologically twisted N=2 D=4 super Yang-Mills theory is due to
Review: