Donaldson theory

Short description: Study in mathematical gauge theory

In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces 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, though there are a number of open problems remaining in Donaldson theory, such as the Witten conjecture and the Atiyah–Floer conjecture.

See also

• Kronheimer–Mrowka basic class
• Instanton
• Floer homology
• Yang–Mills equations

References

• Donaldson, Simon (1983), "An Application of Gauge Theory to Four Dimensional Topology", Journal of Differential Geometry 18 (2): 279–315 .
• Donaldson, S. K.; Kronheimer, P. B. (1997), The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford: Clarendon Press, ISBN 0-19-850269-9 .
• Freed, D. S.; Uhlenbeck, K. K. (1984), Instantons and four-manifolds, New York: Springer, ISBN 0-387-96036-8 .
• Scorpan, A. (2005), The wild world of 4-manifolds, Providence: American Mathematical Society, ISBN 0-8218-3749-4 .