Mixed Frobenius Structure and Local A-model

We define the notion of mixed Frobenius structure which is a generalization of the structure of a Frobenius manifold. We construct a mixed Frobenius structure on the cohomology of weak Fano toric surfaces and that of the three dimensional projective space using local Gromov-Witten invariants. This is an analogue of the Frobenius manifold associated to the quantum cohomology in the local Calabi-Yau setting.