Localization for the Norm-square of the Moment Map and the Two-dimensional Yang-mills Integral

The first seven sections of the paper contain a version of localization for the normsquare of the moment map in equivariant de Rham theory. Similar results are proved by P.-E. Paradan in [30], [31], [32]; our statement and method of proof are somewhat different. The existence of such a formula was first suggested by Witten [39], see also Jeffrey and Kirwan [19]. A K-theory version is given by Vergne [38] and Paradan [32]. A similar result for sheaf cohomology that I learned from C. Teleman is explained in the eighth section. The ninth section contains a definition and computation of the Yang-Mills path integral in two dimensions. The idea is to reverse the logic in Witten’s paper [39], and take the localization formula as the definition of the path integral. In order to compute it we apply a symmetry argument from Teleman-Woodward [37] which reduces the computation to an integration over Jacobians and gives the Migdal formula for the path integral. Taking the large coupling limit gives yet another proof of Witten’s volume formulas for moduli spaces of flat connections, although this proof is not our main goal. Much more general