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

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.