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, doi:10.4310/jdg/1214437665 .
• Donaldson, Simon K.; Kronheimer, Peter B. (1990-09-13) (in en). The Geometry of Four-Manifolds. Oxford Mathematical Monographs. Oxford University Press. doi:10.1093/oso/9780198535539.001.0001. ISBN 978-0-198-53553-9. 
• Uhlenbeck, Karen K.; Freed, Daniel S. (1984) (in en). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. 1. Springer. doi:10.1007/978-1-4613-9703-8. ISBN 978-1-4613-9703-8. .
• Scorpan, A. (2005), The wild world of 4-manifolds, Providence: American Mathematical Society, ISBN 0-8218-3749-4 .
• Naber, Gregory L. (2011) (in en). Topology, Geometry and Gauge Fields. Applied Mathematical Sciences. 141 (Second ed.). Springer. doi:10.1007/978-1-4419-7895-0. ISBN 978-1-4419-7894-3.