BCOV theory on the elliptic curve and higher genus mirror symmetry

TL;DR

This paper develops a quantum B-model for elliptic curves, showing that its higher genus partition functions are almost holomorphic modular forms and establishing mirror symmetry at all genera for this compact Calabi-Yau example.

Contribution

It introduces the quantum Kodaira-Spencer theory on elliptic curves and proves mirror symmetry at all genera for this case, linking B-model partition functions to Gromov-Witten invariants.

Findings

Partition functions are almost holomorphic modular forms.

Mirror symmetry is established at all genera for elliptic curves.

First compact Calabi-Yau example with full higher genus mirror symmetry.

Abstract

We develop the quantum Kodaira-Spencer theory on the elliptic curve and establish the corresponding higher genus B-model. We show that the partition functions of the higher genus B-model on the elliptic curve are almost holomorphic modular forms, which can be identified with partition functions of descendant Gromov-Witten invariants on the mirror elliptic curve. This gives the first compact Calabi-Yau example where mirror symmetry is established at all genera.