Abstract: 

We describe higher genus correspondences between open, closed, orbifold, and logarithmic Gromov-Witten invariants that can be associated to a smooth log Calabi Yau pair (X, E) consisting of a toric Fano surface X with a smooth elliptic curve E. We use methods such as the degeneration formula for GW-invariants, Topological Vertex, and constructions from Gross-Siebert mirror symmetry. As an application, we describe an equivalence of quantum periods in the context of mirror symmetry for Fano manifolds and its enumerative significance.