Mixed Frobenius Structure and Local A-model

TL;DR

This paper introduces mixed Frobenius structures, generalizing Frobenius manifolds, and constructs such structures on specific geometric spaces using local Gromov-Witten invariants, linking to local Calabi-Yau theories.

Contribution

It defines mixed Frobenius structures and constructs them explicitly on cohomology of certain toric surfaces and projective space using local Gromov-Witten invariants.

Findings

Constructed mixed Frobenius structures on weak Fano toric surfaces.

Constructed mixed Frobenius structures on three-dimensional projective space.

Established an analogue of Frobenius manifolds in the local Calabi-Yau setting.

Abstract

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.