Doran-Harder-Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves

TL;DR

This paper proves the Doran-Harder-Thompson conjecture for elliptic curves using SYZ mirror symmetry, demonstrating how to construct mirror Calabi-Yau manifolds via gluing affine structures and theta functions.

Contribution

It provides the first proof of the conjecture in the elliptic curve case by applying SYZ ideas and explicit constructions.

Findings

Confirmed the conjecture for elliptic curves.

Developed a method to glue affine manifolds for complex structure.

Constructed theta functions from Landau-Ginzburg superpotentials.

Abstract

We prove the Doran-Harder-Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi-Yau manifold $X$ degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi-Yau manifold of $X$ can be constructed by gluing the two mirror Landau-Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau-Ginzburg superpotentials.