Abstract
AbstractWe 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 onYarising from 3D mirror symmetry for quasimaps toX, including the Donaldson-Thomas crepant resolution conjecture.
Highlights
Tools from DT-like sheaf-counting theories can be applied with great effectiveness to GW theory and vice versa; for example, the proof of the Igusa cusp form conjecture in [35]
We prove it only for the geometry A −1 × P1 for ease of exposition, the correspondence certainly extends beyond the type A case to any ADE bundle over P1
The proof of the BS/quasimaps correspondence involves constructing an isomorphism of torus-fixed loci that respects the tangent-obstruction theories. This is done using the torus-equivariant derived McKay equivalence of Subsection 4.2. That it matches the stability conditions defining -stable pairs and quasimaps is the content of Subsection 4.3, where we show that the isomorphism of fixed
Summary
Given a smooth variety , one can construct many compactifications of the moduli space of smooth curves in. ◦ Viewing a curve as the data of an ideal sheaf I ⊂ O and allowing to degenerate into 1dimensional subschemes in the compactification yields moduli spaces of sheaves; for example, as in Donaldson-Thomas (DT) theory. One expects all enumerative theories of curves in to be equivalent, possibly up to some wallcrossing behaviour: a change of variables, analytic continuation and/or normalisation. One example of such an equivalence is the celebrated GW/DT correspondence [27] [28], proved for all toric 3-folds in [29].
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