Homological Mirror Symmetry for 1-dimensional Toric Varieties

Homological Mirror Symmetry (HMS) is a conjecture (proven in some cases) relating the A-model of a manifold with the B-model of its mirror dual manifold; the A-model comprises symplectic geometry whereas the B-model is complex-algebraic. More precisely, the A-model is given by a Fukaya category and the B-model is given by the derived category of coherent sheaves; HMS establishes an equivalence between these two categories.

In this talk we will introduce the concept of Fukaya categories and present a calculation of the wrapped Fukaya category of ${\mathbb{C}}^{\ast }$, thereby proving that ${\mathbb{C}}^{\ast }$ is mirror dual to itself. Moreover, partially wrapped Fukaya categories and their combinatorial descriptions as marked surfaces will be introduced, stating HMS for ${\mathbb{A}}_{\mathbb{C}}^1$ and ${\mathbb{P}}_{\mathbb{C}}^1$. In the end, we will point out connections to matrix factorizations appearing in the B-model of the Landau—Ginzburg model mirror dual to ${\mathbb{P}}_{\mathbb{C}}^1$.