Landau-Ginzburg Description of D-branes on ALE Spaces

We study the Landau-Ginzburg (LG) mirror theory of the non-linear sigma model on the ALE space $\mathcal{M}$ obtained by resolving the singularity of the orbifold ${\mathbf{C}}^2\mathrm{/}{\mathbf{Z}}_N$. In the LG description, the data of the BPS spectrum and the lines of marginal stability are encoded in the special Lagrangian submanifolds of the mirror manifold $\widehat{\mathcal{M}}$. Our LG description is quite simple as compared with the quiver gauge theory analysis near the orbifold point. Furthermore our mirror analysis can be applied to any point on the moduli space of $\mathcal{M}$.