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Lecture Note
26/2005
Donaldson and Seiberg-Witten Invariants: The Witten Conjecture
Gregory Naber
Abstract
The Donaldson invariants of a smooth 4-manifold M are subtle probes into the differential topological structure of M. They are defined from the structure of a moduli space of solutions to certain partial differential equations proposed by physicists (Yang-Mills) to model the interactions between elementary particles. Edward Witten found that they can also be regarded as expectation values for certain observables in a topological quantum field theory (TQFT). His insight into the physics of this TQFT (based on joint work with Nathan Seiberg) led him to conjecture that the information contained in the Donaldson invariants can also be retrieved from a much simpler set of equations (Seiberg-Witten). The impact of this conjecture, still not fully proved, but surely "true", has been spectacular. The rather modest objective of these lectures is to provide the background required to understand what the conjecture says.
