Non-planar topological 3-vertex formalism

Using embeddings of complex curves in the complex projective plane P2, we develop a non-planar topological 3-vertex formalism for topological strings on the family of local Calabi-Yau threefolds X(m,−m,0)=O(m)⊕O(−m)→E(t,∞). The base E(t,∞) stands for the degenerate elliptic curve with Kähler parameter t; but a large complex structure μ; i.e. |μ|→∞. We also give first results regarding A-model topological string amplitudes on X(m,−m,0). The 2D U(1) gauged N=2 supersymmetric sigma models of the degenerate elliptic curve E(t,∞) as well as for the family X(m,−m,0) are studied and the role of D- and F-terms is explicitly exhibited.