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We consider $\mathcal{N}=2$ supersymmetric pure gauge theories on toricK\"ahler manifolds, with particular emphasis on $\mathbb{CP}^2$. By choosing avector generating a $U(1)$ action inside the torus of the manifold, weconstruct equivariantly twisted theories. Then, using localization, we computetheir supersymmetric partition functions. As expected, these receivecontributions from a classical, a one-loop, and an instanton term. It turns outthat the one-loop term is trivial and that the instanton contributions arelocalized at the fixed points of the $U(1)$. In fact the full partitionfunction can be re-written in a factorized form with contributions from each ofthe fixed points. The full significance of this is yet to be understood.

Partition functions for equivariantly twisted $\mathcal{N}=2$ gauge theories on toric K\"ahler manifolds