Quasimaps and stable pairs

We prove an equivalence between the Bryan-Steinberg theory of π-stable pairs on Y=Am−1×C and the theory of quasimaps to X=Hilb(Am−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.