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.