Donaldson theory in nLab

Contents

Contents

Idea

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 GG over the 4-manifold.

This method is named after and was first used by Donaldson 1983 (assuming simply-connected GG) 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.

Properties

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.

• topologically twisted D=4 super Yang-Mills theory

• Seiberg-Witten theory

• Gromov-Witten theory

• Kronheimer-Mrowka basic class

References

• 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

• Edward Witten, Topological quantum field theory, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386 (Euclid)

Review:

• Jan Manschot, Four-Manifold Invariants and Donaldson-Witten Theory, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.14709]