nLab Donaldson theory

Contents

Context

Manifolds and cobordisms

Quantum field theory

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.

References

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:

Last revised on June 26, 2024 at 11:28:58. See the history of this page for a list of all contributions to it.