Curve Counting and Instanton Counting

Intuitively, in four-dimensional gauge theories one counts instantons, and in twodimensional string theory one counts holomorphic curves. These correspond to the instanton counting and curve counting in the title. There is no obvious connection between them, but a remarkable physical idea called geometric engineering [9, 10, 11] dictates that they are indeed related. More precisely, the generating series of suitable integrals on the moduli spaces of instantons and holomorphic curves respectively should be related. By localization calculations of the equivariant Âgenera of the Gieseker compactifications of moduli spaces of SU(N) instantons on C, Nekrasov [16] was led to the following partition function: