Tyler Kelly:Open FJRW theory and Mirror Symmetry

Abstract: A Landau-Ginzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function W:X à C from a quasi-projective variety X with a group G acting on X leaving W invariant. An enumerative theory developed by Fan, Jarvis, and Ruan gives FJRW invariants, the analogue of Gromov-Witten invariants for LG models. We define a new open enumerative theory for certain Landau-Ginzburg LG models. Roughly speaking, this involves computing specific integrals on certain moduli of disks with boundary and interior marked points. One can then construct a mirror Landau-Ginzburg model to a Landau-Ginzburg model using these invariants. This allows us to prove a mirror symmetry result analogous to that established by Cho-Oh, Fukaya-Oh-Ohta-Ono, and Gross for mirror symmetry for toric Fano manifolds. If time permits, I will explain some key features that this enumerative geometry enjoys (e.g., open topological recursion relations and wall-crossing phenomena). This is joint work with Mark Gross and Ran Tessler.