SFU Number Theory and Algebraic Geometry Seminar: Tyler Kelly

A Landau-Ginzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function $W:X\to\mathbb{C}$ from a quasi-projective variety $X$ with a group $G$ acting on $X$ leaving $W$ invariant. One can build an analogue of Hodge theory and period integrals associated to an LG model when $G$ is trivial. This involves oscillatory integrals on certain cycles in $X$ (fear not: this is actually cute and will be done in examples!). Mirror symmetry states that period integrals often encode enumerative geometry and this is also the case here. An enumerative theory developed by Fan, Jarvis, and Ruan gives FJRW invariants, the analogue of Gromov-Witten invariants for LG models. These invariants are now called FJRW invariants. A problem is that finding the right deformation period integrals is hard. We define and use a new open enumerative theory for certain Landau-Ginzburg LG models to solve this problem in low dimension. 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 that gives you the right deformation for free. 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. This is joint work with Mark Gross and Ran Tessler.