| Résume | In the fall of 1994 Edward Witten announced a "new gauge theory of 4-manifolds", capable of giving results analogous to the earlier theory of Donaldson, but where the computations involved are "at least a thousand times easier" (Taubes). The equations involved in this new gauge theory are well known as the SeibergWitten equations. The equations introduced by Witten led quickly to a revolution in 3- and 4-dimensional differential geometry and they remain at the forefront of research today. Shortly after their appearance, Witten showed how one could count solutions to the equations, defining an invariant of the underlying smooth 4-manifold. Despite a lot of effort, higher-dimensional generalisations of the Seiberg–Witten equations were unknown till now. In this talk we present an elliptic system of equations over a SpinC-manifold of any dimension which generalise the Seiberg–Witten equations in the cases n = 3, 4. If time permits, we describe modified versions of these equations in dimensions 6, 7 and 8, which make sense on manifolds with SU (3), G2 and Spin(7) structures respectively. |
