Quasimaps and stable pairs

TL;DR

This paper establishes a deep equivalence between two theories related to sheaf counting on specific algebraic varieties, linking combinatorics, mirror symmetry, and conjectures in enumerative geometry.

Contribution

It proves an explicit equivalence between Bryan--Steinberg's $ ext{ extbackslash pi}$-stable pairs theory and quasimaps to the Hilbert scheme, with applications to sheaf-counting and mirror symmetry.

Findings

Equivalence of $ ext{ extbackslash pi}$-stable pairs and quasimaps theories

Explicit combinatorial descriptions via box counting

Implications for sheaf-counting and the DT crepant resolution conjecture

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.