Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic Landau-Ginzburg model, i.e. a non-compact manifold equipped with the extra data of a ‘stop’ in its boundary at infinity. Most of the talk will focus on a family of Landau-Ginzburg models which provide mirrors to curves in (C*)2 or in toric surfaces (or more generally to hypersurfaces in toric varieties), and their fiberwise wrapped Fukaya categories (joint work with Mohammed Abouzaid). I will then discuss more a speculative way of constructing mirrors of curves without Landau-Ginzburg models, involving a new flavour of Lagrangian Floer theory in trivalent configurations of Riemann surfaces (joint work with Alexander Efimov and Ludmil Katzarkov).
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
