Mixed Frobenius Structure and Local Quantum Cohomology

TL;DR

This paper develops a new mathematical structure called mixed Frobenius structure (MFS) on the cohomology of smooth projective varieties, linking it to local quantum cohomology and equivariant limits, advancing the understanding of quantum cohomology frameworks.

Contribution

It constructs a MFS on cohomology related to non-equivariant limits of twisted quantum products, extending Frobenius manifold theory.

Findings

MFS is obtained as a non-equivariant limit of equivariant Frobenius structures.

The construction generalizes Frobenius manifold structures to a broader setting.

Provides a new perspective on local quantum cohomology and its algebraic structures.

Abstract

This paper is a sequel to arXiv:1209.5550 where the notion of mixed Frobenius structure (MFS) was introduced as a generalization of the structure of a Frobenius manifold. Roughly speaking, the MFS is defined by replacing a metric of the Frobenius manifold with a filtration on the tangent bundle equipped with metrics on its graded quotients. The purpose of the current paper is to construct a MFS on the cohomology of a smooth projective variety whose multiplication is the non-equivariant limit of the quantum product twisted by a concave vector bundle. We show that such a MFS is naturally obtained as the non-equivariant limit of the Frobenius structure in the equivariant setting.