Motivic Donaldson-Thomas invariants and McKay correspondence

Abstract

Let $G\subset S{L}_2(C)\subset S{L}_3(C)$ be a finite group. We compute motivic Pandharipande-Thomas and Donaldson-Thomas invariants of the crepant resolution $Hil{b}^G(C^3)$ 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.

23 pages

References in corpus (15)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Calabi-Yau algebras
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• A computational criterion for the Kac conjecture
• Crepant resolutions and brane tilings I: Toric realization
• Wall-crossing of the motivic Donaldson-Thomas invariants
• Counting invariants for the ADE McKay quivers
• Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex
• Curve counting theories via stable objects I. DT/PT correspondence
• Wall-crossing formulas for framed objects
• Motivic invariants of quivers via dimensional reduction
• On the motivic Donaldson-Thomas invariants of quivers with potentials
• Three-dimensional flops and non-commutative rings
• An exponential function on the set of varieties
• On the power structure over the Grothendieck ring of varieties and its applications

Cited by in corpus (12)

• Invariants of moduli spaces of stable sheaves on ruled surfaces
• On cohomological Hall algebras of quivers : Yangians
• Counting Higgs bundles
• BPS Lie algebras and the less perverse filtration on the preprojective CoHA
• Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex
• Crystal Melting, BPS Quivers and Plethystics
• Donaldson-Thomas invariants for 3-Calabi-Yau varieties of dihedral quotient type
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• Attractor invariants, brane tilings and crystals
• Higgs bundles over P^1 and quiver representations
• Commuting matrices and volumes of linear stacks
• On the number of points of the Lusztig nilpotent variety over a finite field