Localization of four-dimensional super Yang-Mills theories compactified on Riemann surface

We consider the partition function of super Yang-Mills theories defined on $T^2\times {\mathrm{\Sigma }}_g$. This path integral can be computed by the localization. The one-loop determinant is evaluated by the elliptic genus. This elliptic genus gives trivial result in our calculation. As a result, we obtain a theory defined on the Riemann surface.