Mirror Symmetry for Closed, Open, and Unoriented Gromov-Witten Invariants

TL;DR

This paper develops mirror formulas for genus 0 Gromov-Witten invariants of projective spaces and Calabi-Yau threefolds, confirming Walcher's conjectures for unoriented invariants and describing their structure.

Contribution

It extends mirror symmetry formulas to twisted and unoriented Gromov-Witten invariants, and verifies conjectures for Klein bottle and annulus invariants in Calabi-Yau threefolds.

Findings

Mirror formulas for genus 0 two-point GW invariants derived

Walcher's mirror symmetry conjectures confirmed for Klein bottle and annulus invariants

Structure coefficients and relations between them described

Abstract

In the first part of this paper, we obtain mirror formulas for twisted genus 0 two-point Gromov-Witten (GW) invariants of projective spaces and for the genus 0 two-point GW-invariants of Fano and Calabi-Yau complete intersections. This extends previous results for projective hypersurfaces, following the same approach, but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper. In the second and third parts of this paper, we confirm Walcher's mirror symmetry conjectures for the annulus and Klein bottle GW-invariants of Calabi-Yau complete intersection threefolds; these applications are the main results of this paper. In a separate paper, the genus~0 two-point formulas are used to obtain mirror formulas for the genus~1 GW-invariants of all Calabi-Yau complete intersections.