# The Genus-One Global Mirror Theorem for the Quintic Threefold

## TL;DR

This paper proves a key aspect of the Landau-Ginzburg/Calabi-Yau conjecture at genus one for the quintic threefold, providing evidence for the higher-genus correspondence and illustrating the 'genus zero controls higher genus' principle.

## Contribution

It establishes the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture for the first time for the quintic threefold, demonstrating the principle in non-semisimple theories.

## Key findings

- First proof of genus-one restriction for the conjecture
- Supports higher-genus Landau-Ginzburg/Calabi-Yau correspondence
- Shows genus zero controls higher genus in non-semisimple theories

## Abstract

We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This provides the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the "genus zero controls higher genus" principle, in the sense of Givental's quantization formalism, for non-semisimple cohomological field theories.