The String Geometry Behind Topological Amplitudes

It is shown that the generating function of $\mathcal{N}=2$ topological strings, in the heterotic weak coupling limit, is identified with the partition function of a six-dimensional Melvin background. This background, which corresponds to an exact CFT, realises in string theory the six-dimensional $\Omega$-background of Nekrasov, in the case of opposite deformation parameters ${ϵ}_1=-{ϵ}_2$, thus providing the known perturbative part of the Nekrasov partition function in the field theory limit. The analysis is performed on both heterotic and type I strings and for the cases of ordinary $\mathcal{N}=2$ and $\mathcal{N}=2^{\ast }$ theories.