On \(Y M_ 2\) measures and area-preserving diffeomorphisms (Q1922661)

It is known that for a given gauge group and compact Riemannian two-manifold, the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to \textit{SDiff}, the group of all area-preserving diffeomorphisms of the surface. The first objective of this work is to study whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. This study needs the construction of an \textit{SDiff}-equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. The coupling of the Yang-Mills measure to determinants of Dirac operators is also considered. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyse the action of \textit{SDiff}.