Gromov-Witten Theory of Non-Convex Complete Intersections

Abstract

This thesis provides a technique to compute the Gromov-Witten invariants of complete intersections in GIT quotient stacks, regardless of any convexity assumptions. In particular, our technique addresses the failure of the Quantum Lefshetz Hyperplane theorem for such targets, and recovers all the invariants one would expect from such a target, if not more.

Our technique revolves around modifying the GIT presentation of the target based on a chosen set of Chen-Ruan cohomology classes. We first do this for toric stacks, where we provide explicit formulas for these modifications through geometric motivations. Then, using the orbifold quasimap theory of Cheong, Ciocan-Fontanine, and Kim, we compute a series associated to this presentation known as an I-function, analogous to the I-functions of Givental. After a mirror transformation, we show that this series lies on Givental's Lagrangian cone, as well as proving that this mirror transformation is invertible. More concretely, we show we are able to obtain explicit values of Gromov-Witten invariants with insertions coming from the classes we extend by, which we illustrate through examples. Notably, these examples recover previously known results, uncover interesting numerical phenomena, and provide cases where invariants with primitive insertions can be computed.

We also extend the above results to non-abelian quotients via Webb's Abelian/Non-abelian Correspondence. We study how the above ideas interact with this correspondence, and prove analogous results for Weyl-invariant Chen-Ruan classes. We then apply these techniques to the example of a stacky del Pezzo to recover its full quantum period, proving a conjecture of Oneto and Petracci.