Particles and connections on four-manifolds
The author studies the spaces of gauge equivalence classes of based connections on a principal \(G\)-bundle over a compact four-manifold \(X\). The aim is to build a particle model for these spaces (in the spirit of May Milgram models) so that one gets natural operations in homotopy. To perform this, for a simple and simply connected classical gauge group \(G\), the author builds spaces \(P(X,Y)\) homotopy equivalent to spaces \(B(X,Y)\) of gauge equivalence classes of pairs \((G\)-connections on a four manifold \(X\), trivialisations of the \(G\) bundle over \(Y(X)\)). The space consists of configurations of particle like connections, each corresponding to an instanton over the four sphere. It is related to the Atiyah-Jones conjecture (stated originally for the four sphere) that the inclusion of the moduli space of instantons of charge \(k\) to the space of gauge equivalence classes of connections of charge \(k\) induces a homology equivalence through a range that increases with \(k\). Hence, the models in the paper provide a first step in proving the Atiyah-Jones conjecture.
scientific article; zbMATH DE number 1295153