The cyclic open--closed map and variations of Hodge structures

TL;DR

This paper constructs a cyclic open-closed map within the big relative Fukaya category, demonstrating its role as a morphism of polarized variations of semi-infinite Hodge structures and linking Gromov--Witten invariants to homological mirror symmetry in Calabi-Yau cases.

Contribution

It introduces a new cyclic open-closed map for the bulk-deformed relative Fukaya category and provides criteria for its isomorphism, connecting enumerative and homological mirror symmetry.

Findings

The map is a morphism of polarized variations of semi-infinite Hodge structures.

In certain cases, the map is an isomorphism, enabling extraction of Gromov--Witten invariants.

Enumerative mirror symmetry follows from homological mirror symmetry in Calabi-Yau contexts.

Abstract

We construct the cyclic open--closed map for the big (i.e., bulk-deformed) relative Fukaya category, in the semipositive case, and show that it is a morphism of `polarized variations of semi-infinite Hodge structures'. We also give a natural criterion for the map to be an isomorphism, which is verified for example in the context of Batyrev mirror pairs. We conclude in such Calabi-Yau cases that the rational Gromov--Witten invariants can be extracted from the relative Fukaya category, and hence that enumerative mirror symmetry is a consequence of homological mirror symmetry for Calabi--Yau mirror pairs.