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

Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

“…The Hilb/PT correspondence (between (i) and (ii)) was recently established in full generality by Nesterov [67]. For C 2 and resolutions of A n singularities the triangle of correspondences was worked out previously in [84,17,85,58,55,59,50].…”

Exotic Quantum Difference Equations and Integral Solutions

“…The vertex with descendants provides a type of generalized q-hypergeometric function. For explicit computations in this regard, see [9,14,15,16,31,40]. It also plays a key role in the enumerative perspective on 3d mirror symmetry, see [1,4,5,7,8].…”

Euler characteristic of stable envelopes

“…More specifically, we expect that for any two varieties related by 3d mirror symmetry satisfying the conditions listed at the beginning of the introduction, the index vertex and the K-theoretic stable envelopes should be related in the same way as described here. Indeed, the proof of Theorem 2 shows that the relationship between the K-theoretic stable envelopes and the index vertex is really a consequence of a more general conjecture regarding elliptic stable envelopes and vertex functions of 3d mirror dual varieties, see [9], [7], [28], [14] and the introduction of [1]. While a general construction of 3d mirror dual pairs is not presently known, the construction of Coloumb branches in [17] and [2] provides a large class of varieties 3d mirror dual to quiver varieties, and more generally Higgs branches of 3d N = 4 gauge theories.…”