Homological Mirror Symmetry for 1-dimensional Toric Varieties

  • Maximilian Wiesmann (MPI MiS, Leipzig)
E1 05 (Leibniz-Saal)


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 $\mathbb{C}^*$, thereby proving that $\mathbb{C}^*$ 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 $\mathbb{P}_{\mathbb{C}}^1$.

Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail

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