Non Planar Topological 3-Vertex Formalism

Using embedding of complex curves in the complex projective plane $\bf{P }^{2}$, we develop a \emph{non planar} topological 3-vertex formalism for topological strings on the family of local Calabi-Yau threefolds $X^{(m,-m,0) }=$\mathcal{O}$(m)\oplus $\mathcal{O}$(-m)\to E^{(t,\infty)}$\mathrm{.}Thebase$E^{(t,\infty)}$standsforthedegenerateellipticcurvewithKahlerparameter$t$;butalargecomplexstructure$\mu $;i\mathrm{.}e$| \mu | \longrightarrow \infty $. We also give first results regarding A-model topological string amplitudes on $X^{(m,-m,0)}$. The 2D $U(1)$ gauged $\mathcal{N}=2$ supersymmetric sigma models of the degenerate elliptic curve $E^{(t,\mathrm{\infty })}$ as well as for the family $X^{(m,-m,0)}$ are studied and the role of D- and F-terms is explicitly exhibited.