Abstract for the talk on 07.12.2022 (15:00 h)
Seminar on Nonlinear AlgebraMaximilian Wiesmann (MPI MiS, Leipzig)
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 \(\mathbb{C}^*\), 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 \(\mathbb{P}_{\mathbb{C}}^1\).