Instanton counting (survey)

Nekrasov defined the instanton partition function by an equivariant integration of $1$ over moduli spaces of instantons on $\mathbf R^4$. I will survey its relations to various subjects: 1) Its leading part is the Seiberg-Witten prepotential defined via a period integral of hyperelliptic curves. 2) (Geometric engineering) The full partition function, setting one of variables 0, is the generating function of Gromov-Witten invariants of a certain local Calabi-Yau 3-fold so that the leading part corresponds to the genus 0 part. 3) The partition function has a natural deformation integrating Chern classes of natural vector bundles over moduli spaces. They are conjecturally related to Poincare polynomials of link homology groups a la Khovanov.