Extending the Picard-Fuchs system of local mirror symmetry

TL;DR

This paper introduces an extended set of differential operators for local mirror symmetry, providing a complete description for certain Calabi-Yau manifolds and uncovering new intersection theory conjectures and operators.

Contribution

It proposes a novel extended differential operator framework for local mirror symmetry, applicable to specific Calabi-Yau manifolds, and explores new operators and intersection theory conjectures.

Findings

Operators fully describe mirror symmetry for certain Calabi-Yau manifolds.

New operators identified on examples of type X=K_S.

Open string Picard-Fuchs systems are analyzed.

Abstract

We propose an extended set of differential operators for local mirror symmetry. If $X$ is Calabi-Yau such that $\dim H_4(X,\Z)=0$, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such $X$ is uncovered. We also find new operators on several examples of type $X=K_S$ through similar techniques. In addition, open string PF systems are considered.