Open-String Quantum Lefschetz Formula

Let Y be a symplectic divisor of X, ω. In the Kahler setting, Givental's Quantum Lefschetz formula relates certain Gromov-Witten invariants (encoded by the G function) of X and Y. Given an Lagrangian L in (Y, ω|Y), we can lift it to a Lagrangian L' in neighbourhood NY ⊂X. We will introduce the notion of the potential of a Lagrangian, which encodes information of Maslov index 2 J-holomorphic disks with boundary on it. We will discuss the conditions in which the potential for L' relates with the potential for L according to a lifting formula. In particular, this formula involves counts J-holomorphic spheres with certain tangency on Y (part of relative Gromov-Witten invariants). It generalizes a formula that can be extracted from Biran-Khanevski, under some more restrictive assumptions on Y. As applications, we recover some Lefschetz formulas appearing in the work of Coates-Corti-Galkin-Kasprczyk and show the existence of infinitely many Lagrangian tori in CPn, Quadrics, Cubics, among other symplectic manifolds. This is joint work with Luis Diogo, Dmitry Tonkonog and Weiwei Wu.