Refined Topological Strings on Local ${\mathbb{P}}^2$

We calculate the refined topological string partition function of the Calabi-Yau threefold which is the total space of the canonical bundle on ${\mathbb{P}}^2$ (the local ${\mathbb{P}}^2$). The refined topological vertex formalism can not be directly applied to local ${\mathbb{P}}^2$ therefore we use the properties of the refined Hopf link to define a new two legged vertex which together with the refined vertex gives the partition function of the local ${\mathbb{P}}^2$.