Abstract

Mirror symmetry relates Fano varieties to Landau-Ginzburg models – non-compact varieties together with a potential function W. Gross-Siebert developed an algorithmic construction of mirrors via algebro-combinatorial objects called scattering diagrams and broken lines. I describe the combinatorial aspects of this construction for the easiest case of a Fano surface – P2 relative to an elliptic curve E. Mirror symmetry predicts a relation between curve counts for (P2,E) and complex structure deformations of its mirror. On the combinatorial level this translates to a correspondence between curve counts for (P2,E) and scattering diagrams resp. broken lines. Such a correspondence can be proved using tropical geometry. If time permits, I will also talk about joint work with Helge Ruddat and Eric Zaslow in which we relate the Landau-Ginzburg potential defined via broken lines to the open mirror map of Aganagic-Vafa branes in framing zero.