Counting invariants for the ADE McKay quivers

Abstract

We consider the moduli space of the McKay quiver representations associated to the binary polyhedral groups G < SU(2) < SU(3). The derived category of such representations is equivalent to the derived category of coherent sheaves on the corresponding ADE resolution Y=G-Hilb(C^3). Following the ideas of Nagao and Nakajima, by making particular choices of parameters in the space of stability conditions on the equivalent derived categories above, we recover Donaldson-Thomas (DT), Pandharipande-Thomas (PT) and Szendroi (NCDT) moduli spaces. We also compute the Gromov-Witten (GW) partition function of Y directly and express the result in terms of the root system of the associated ADE Dynkin diagram. We then verify the conjectural GW/DT/NCDT-correspondence by assuming the DT/PT-correspondence. The Szendroi invariants are the same as the orbifold Donaldson-Thomas invariants for C^3/G defined by Bryan. This will allow us to verify the Crepant Resolution Conjecture for the orbifold Donaldson-Thomas theory in this case.