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, MR 0710056. Donaldson, Simon K.; Kronheimer, Peter B. (1990-09-13). The Geometry of Four-Manifolds. Oxford Mathematical Monographs. Oxford University Press. doi:10.1093/oso/9780198535539.001.0001. ISBN 978-0-198-53553-9. MR 1079726. Uhlenbeck, Karen K.; Freed, Daniel S. (1984). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 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). Topology, Geometry and Gauge Fields. Applied Mathematical Sciences. Vol. 141 (Second ed.). Springer. doi:10.1007/978-1-4419-7895-0. ISBN 978-1-4419-7894-3. ISSN 0066-5452.