Short notes on Seiberg Witten and applications Workshop Berlin 2d december Freie Universität

Short notes on Seiberg-Witten and applications Workshop Berlin 2d december, Freie Universität Damien Gayet, ICJ Lyon, Frankreich 17 octobre 2009 Abstract We give here a short introduction to the Seiberg-Witten invariants, and two applications of the theory. Introduction Mathematics and physics have been being historically deeply connected. One of the last exchange was the use in the 80's by Donaldson of the Yang-Mills theory to give some totally new topological invariants. Then Witten reinterpre- ted this theory in 1994 in terms of topological quantum field theory, and give physical arguments to conjecture that it was equivalent to an other sytem of EDPs, studied by him and Seiberg. Immediatly, this equations, from far easier to treat with than Donaldson's, recovered the results of the former theory, but give new one. In this short teaching, we will give an introduction to this equations, the structure of the set of it solutions, and they use to construct topological in- variants, the so-called Seiberg-Witten invariants. We will show that this inva- riants are non trivial on the Kähler manifolds. This will give us examples of homeomorphic but non diffeomorphic manifolds. Moreover, we will establish the Thom's conjecture : in CP 2, the smooth holomorphic curves are genus mi- nimizing amongst real surfaces of the same degree.

• can construct


• spinor bundle


• tx


• seiberg-witten equations


• positive spinor


• bundle


• levi-civita connection


• all positive


• spinc structure