Article

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

• ##### Kentaro Nagao
09/2008;
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.

0 Bookmarks
·
89 Views
• Source
##### Article: Wall-crossing, open BPS counting and matrix models
[Hide abstract]
ABSTRACT: We consider wall-crossing phenomena associated to the counting of D2-branes attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both from M-theory and matrix model perspective. Firstly, from M-theory viewpoint, we review that open BPS generating functions in various chambers are given by a restriction of the modulus square of the open topological string partition functions. Secondly, we show that these BPS generating functions can be identified with integrands of matrix models, which naturally arise in the free fermion formulation of corresponding crystal models. A parameter specifying a choice of an open BPS chamber has a natural, geometric interpretation in the crystal model. These results extend previously known relations between open topological string amplitudes and matrix models to include chamber dependence.
Journal of High Energy Physics 11/2010; · 5.62 Impact Factor
• Source
##### Article: Wall-crossing formulas for framed objects
[Hide abstract]
ABSTRACT: We prove wall-crossing formulas for the motivic invariants of the moduli spaces of framed objects in the ind-constructible abelian categories. Developed techniques are applied in the case of the motivic Donaldson-Thomas invariants of quivers with potentials. Another application is a new proof of the formula for the motivic invariants of smooth models of quiver moduli spaces.
The Quarterly Journal of Mathematics 04/2011; · 0.56 Impact Factor
• Source
##### Article: Motivic Donaldson-Thomas invariants and McKay correspondence
[Hide abstract]
ABSTRACT: Let \$G\subset SL_2(C)\subset SL_3(C)\$ be a finite group. We compute motivic Pandharipande-Thomas and Donaldson-Thomas invariants of the crepant resolution \$Hilb^G(C^3)\$ of \$C^3/G\$ generalizing results of Gholampour and Jiang who computed numerical DT/PT invariants using localization techniques. Our formulas rely on the computation of motivic Donaldson-Thomas invariants for a special class of quivers with potentials. We show that these motivic Donaldson-Thomas invariants are closely related to the polynomials counting absolutely indecomposable quiver representations over finite fields introduced by Kac. We formulate a conjecture on the positivity of Donaldson-Thomas invariants for a broad class of quivers with potentials. This conjecture, if true, implies the Kac positivity conjecture for arbitrary quivers.
07/2011;