Quasimaps and stable pairs

Abstract

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 $X=\mathrm{Hilb}({\mathcal{A}}_{m-1})$, 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.

51 pages, journal version

References in corpus (3)

• Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
• Rationality of capped descendent vertex in $K$-theory
• Characters of tangent spaces at torus fixed points and $3d$-mirror symmetry

Cited by in corpus (2)

• A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface