Instanton Counting Through Non-abelian Localization

In this dissertation we study the problem of calculating equivariant volumes of moduli-spaces of framed instantons. The motivation for this is given by instanton counting, a recent development in theoretical physics that gives a direct approach to the non-perturbative study of certain super-symmetric quantum field theories. We develop a strategy for calculating the integrals using a combination of several techniques in symplectic geometry and equivariant cohomology. Most importantly we use an equivariant version of non-abelian localization, applied to the ADHM-construction of the moduli-spaces. Furthermore, we reduce the problem to a compact setting by means of varying compactifications using symplectic cuts, recovering the original integral over a non-compact space as the limit of integrals over compact spaces. In contrast with previous applications, in our case the contribution at infinity introduced by these compactifications turns out to be of primordial importance. We illustrate this method by explicitly calculating the volumes for moduli-spaces with low instanton number