Quasimaps and stable pairs

We prove an equivalence between the Bryan–Steinberg theory of $\pi$-stable pairs on $Y={\mathcal{A}}_{m-1}\times \mathbb{C}$ and the theory of quasimaps to $X=\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}({\mathcal{A}}_{m-1})$, in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on $Y$ arising from 3d mirror symmetry for quasimaps to $X$, including the Donaldson–Thomas crepant resolution conjecture.