Derived categories of small toric Calabi-Yau 3-folds and counting invariants

Source: arXiv

ABSTRACT We first construct a derived equivalence between a small crepant resolution
of an affine toric Calabi-Yau 3-fold and a certain quiver with a
superpotential. Under this derived equivalence we establish a wall-crossing
formula for the generating function of the counting invariants of perverse
coherent systems. As an application we provide certain equations on
Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we
show that moduli spaces associated with a quiver given by successive mutations
are realized as the moduli spaces associated the original quiver by changing
the stability conditions.

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    ABSTRACT: In this paper, we apply recent methods of localized GLSMs to make predictions for Gromov-Witten invariants of noncommutative resolutions, as defined by e.g. Kontsevich, and use those predictions to examine the connectivity of the SCFT moduli space. Noncommutative spaces, in the present sense, are defined by their sheaves, their B-branes. Examples of abstract CFT's whose B-branes correspond with those defining noncommutative spaces arose in examples of abelian GLSMs describing branched double covers, in which the double cover structure arises nonperturbatively. This note will examine the GLSM for P^7[2,2,2,2], which realizes this phenomenon. Its Landau-Ginzburg point is a noncommutative resolution of a (singular) branched double cover of P^3. Regardless of the complex structure of the large-radius P^7[2,2,2,2], the Landau-Ginzburg point is always a noncommutative resolution of a singular space, which begs the question of whether the noncommutative resolution is connected in SCFT moduli space by a complex structure deformation to a smooth branched double cover. Using recent localization techniques, we make a prediction for the Gromov-Witten invariants of the noncommutative resolution, and find that they do not match those of a smooth branched double cover, telling us that these abstract CFT's are not continuously connected to sigma models on smooth branched double covers.
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