Motivic Donaldson-Thomas invariants and McKay correspondence

TL;DR

This paper computes motivic Donaldson-Thomas and Pandharipande-Thomas invariants for crepant resolutions of quotient singularities, linking them to quiver representation polynomials and proposing a broad positivity conjecture.

Contribution

It generalizes previous numerical invariants to motivic invariants for specific crepant resolutions and relates these to Kac's polynomials, introducing a new positivity conjecture.

Findings

Formulas for motivic DT/PT invariants of $Hilb^G(C^3)$

Connection between motivic DT invariants and Kac's polynomials

Conjecture on positivity of DT invariants implying Kac's positivity

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.