Some Applications of Abelianization in Gromov-Witten Theory

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 $YG$. We work several explicit examples, including a rigorous derivation of a quantum period computed conjecturally by Oneto-Petracci.