Some Applications of Abelianization in Gromov-Witten Theory

TL;DR

This paper explores the use of abelianization techniques in Gromov-Witten theory to compute $I$-functions for certain complete intersections, demonstrating explicit examples including a quantum period related to conjectures.

Contribution

It introduces general methods for applying quasimap formulas to compute $I$-functions in Gromov-Witten theory, with explicit examples and a rigorous derivation of a conjectured quantum period.

Findings

Developed techniques for using quasimap formulas in Gromov-Witten computations

Explicit calculations of $I$-functions for specific complete intersections

Rigorous derivation of a quantum period conjectured by Oneto-Petracci

Abstract

Let $G$ be a complex reductive group and let $X$ and $E$ be two linear representations of $G$. Let $Y$ be a complete intersection in $X$ equal to the zero locus of a $G$-equivariant section of the trivial bundle $E \times X \to X$. We explain some general techniques for using quasimap formulas to compute useful $I$-functions of $Y\mathord{/\mkern-6mu/} G$. We work several explicit examples, including a rigorous derivation of a quantum period computed conjecturally by Oneto-Petracci.