Elliptic Gromov - Witten invariants and the generalized mirror conjecture

TL;DR

This paper formulates and proves a conjecture relating genus 1 Gromov-Witten invariants to mirror symmetry, extending results to torus-equivariant cases and concave bundles, and connects genus 0 mirror theorems with previous proofs.

Contribution

It introduces a conjecture linking genus 1 Gromov-Witten invariants with mirror theory and proves it for specific classes of manifolds, also relating to existing genus 0 mirror results.

Findings

Proof of the genus 1 mirror conjecture for torus-equivariant invariants

A non-linear Serre duality theorem in genus 0 Gromov-Witten theory

A generalized mirror theorem for concave bundle spaces over toric manifolds

Abstract

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov - Witten invariants of compact K\"ahler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov - Witten theory include: a non-linear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.-T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3-folds and our proof of the same conjecture given two years ago.