Refined topological strings on local $ {\mathrm{\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 \( {\mathrm{\mathbb{P}}}^2 \) (the local \( {\mathrm{\mathbb{P}}}^2 \)). The refined topological vertex formalism can not be directly applied to local \( {\mathrm{\mathbb{P}}}^2 \) 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 \( {\mathrm{\mathbb{P}}}^2 \).