The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations is an important system of equations in the study of genus zero Gromov-Witten invariants. It implies the associativity of the quantum product. The associativity of the quantum product has many important applications including the recursive formula given by Kontsevich and Manin that calculates the Gromov-Witten invariants of the projective plane.  

The system of open WDVV equations plays an important role in the study of open Gromov-Witten invariants. It can be viewed as an extension of the WDVV equation to the open sector. The natural structure that captures the WDVV equation is that of a Frobenius manifold. Similarly, the system of open WDVV equations determines the structure of an F-manifold, a generalization of a Frobenius manifold.  

In this talk, we prove two versions of open WDVV equations for toric Calabi-Yau 3-folds.  The first version leads to the construction of a semi-simple (formal) Frobenius manifold and the second version leads to the construction of a (formal) F-manifold. This is a joint work with Song Yu.