Matrix Factorizations And Mirror Symmetry: The Cubic Curve

TL;DR

This work investigates open string mirror symmetry on the cubic elliptic curve using boundary matrix factorizations to describe B-model D-branes. It defines flat coordinates intrinsically in the Landau-Ginzburg model and derives the A-model disk-instanton partition function, i.e., the Fukaya product $m_2$, by computing boundary-changing correlators whose coefficients are encoded by Theta-functions. The authors show that the three-point boundary-changing correlators yield Yukawa couplings counting triangle disk instantons, and these results satisfy a heat equation, matching known A-model data. By connecting the LG boundary data to the Fukaya category, the paper demonstrates how Theta-function identities encode the open-string Fukaya products, offering a concrete realization of open-closed mirror symmetry on an elliptic curve.

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.

Figures (3)

• Figure 1: Shown are the long and short diagonals on the covering space of the torus; note that they correspond to roots and weights of the $SU(3)$ lattice, resp. The long diagonals ${\mathcal{L}}_i$ correspond, via mirror symmetry, to the $3\times3$ matrix factorizarions (\ref{['simpleQQ']}) we discuss in this paper, while the short diagonals ${\mathcal{S}}_i$ correspond to $2\times2$ factorizations.
• Figure 2: Quiver representation of the open string spectrum between the three $D$-branes ${\mathcal{L}}_i$ under consideration. One of our objectives is to find suitable Landau-Ginzburg representatives of all the pictured quantities that continuously depend on the bulk/boundary moduli.
• Figure 3: Shown is the fundamental region of the cubic torus at $\rho=e^{2\pi i/3}$, with the three special lagrangian $D$-branes ${\mathcal{L}}_i$ on top. The triangular world-sheets $\Delta_{abc}$ shown give the leading instanton corrections to the Yukawa couplings $C_{abc}$. Note that we have slightly shifted ${\mathcal{L}}_2$ by setting $u_2\not=0$, so that each of the three triple intersections gets resolved into three pairwise intersections, and the $\Delta_{aaa}$ get a non-vanishing area. The boundary changing open string operators $\Psi_{ij}^{(a)}$ are localized at the corresponding intersection points of the branes ${\mathcal{L}}_j$ and ${\mathcal{L}}_i$ (an example of which is indicated).