Matrix Factorizations And Mirror Symmetry: The Cubic Curve

TL;DR

This paper explores open string mirror symmetry for elliptic curves using matrix factorizations, defining flat coordinates in Landau-Ginzburg models, and computing disk instanton counts to determine Yukawa couplings.

Contribution

It introduces a systematic method to compute non-perturbative Yukawa couplings via matrix factorizations and Fukaya products in the Landau-Ginzburg mirror of elliptic curves.

Findings

Derived flat coordinates intrinsically in LG models.

Computed disk instanton counts for D-brane configurations.

Provided a method to determine Yukawa couplings systematically.

Abstract

We revisit open string mirror symmetry for the elliptic curve, using matrix factorizations for describing D-branes on the B-model side. We show how flat coordinates can be intrinsically defined in the Landau-Ginzburg model, and derive the A-model partition function counting disk instantons that stretch between three D-branes. In mathematical terms, this amounts to computing the simplest Fukaya product m_2 from the LG mirror theory. In physics terms, this gives a systematic method for determining non-perturbative Yukawa couplings for intersecting brane configurations.