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

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.

43 pages, 8 figures; section 3.4 revived

References in corpus (17)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Curve counting via stable pairs in the derived category
• Donaldson-Thomas invariants via microlocal geometry
• Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds
• Consistency conditions for brane tilings
• Dimer models and the special McKay correspondence
• Dimer models and Calabi-Yau algebras
• On moduli spaces of quiver representations associated with dimer models
• On the noncommutative Donaldson-Thomas invariants arising from brane tilings
• Counting invariant of perverse coherent sheaves and its wall-crossing
• Computing a pyramid partition generating function with dimer shuffling
• Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions
• Hilbert schemes and stable pairs: GIT and derived category wall crossings
• Perverse coherent sheaves on blow-up. I. a quiver description
• Non-commutative Donaldson-Thomas theory and vertex operators
• Birational Calabi-Yau 3-folds and BPS state counting
• Perverse coherent sheaves on blow-up. II. wall-crossing and Betti numbers formula

Cited by in corpus (33)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Crystal Melting and Toric Calabi-Yau Manifolds
• Wall crossing in local Calabi Yau manifolds
• Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions
• Wall Crossing and M-theory
• Bound state transformation walls
• Open BPS Wall Crossing and M-theory
• Motivic Donaldson-Thomas invariants and McKay correspondence
• Predictions for Gromov-Witten invariants of noncommutative resolutions
• On the noncommutative Donaldson-Thomas invariants arising from brane tilings
• Counting invariant of perverse coherent sheaves and its wall-crossing
• Chamber Structure and Wallcrossing in The ADHM Theory of Curves I
• Wall-crossing of D4-D2-D0 and flop of the conifold
• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Dimer models and exceptional collections
• Geometric engineering of (framed) BPS states
• BPS State Counting in Local Obstructed Curves from Quiver Theory and Seiberg Duality
• Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions
• Hilbert schemes and stable pairs: GIT and derived category wall crossings
• Lectures on Noncommutative Resolutions
• Statistical model and BPS D4-D2-D0 counting
• Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
• Counting invariants for the ADE McKay quivers
• Curve counting theories via stable objects II: DT/ncDT flop formula
• On higher rank Donaldson-Thomas invariants
• Crystal Melting, BPS Quivers and Plethystics
• Non-commutative Donaldson-Thomas theory and vertex operators
• Wall-crossing formulas for framed objects
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Exact Lefschetz fibrations associated with dimer models
• Crepant Resolutions, Quivers and GW/NCDT Duality
• Geometry and Combinatorics of Crystal Melting