Cohomological DT invariants from localization

Given a quiver with potential associated to a toric Calabi-Yau threefold, the numerical Donaldson-Thomas invariants for the moduli space of framed representations can be computed by using toric localization, which reduces the problem to the enumeration of molten crystals. We provide a refinement of this localization procedure, which allows to compute motivic Donaldson-Thomas invariants. Using this approach, we prove a universal formula which gives the BPS invariants of any toric quiver, up to undetermined contributions which are invariant under Poincar\'e duality. When the toric Calabi-Yau threefold has compact divisors, these self-Poincar\'e dual contributions have a complicated dependance on the stability parameters, but explicit computations suggest that they drastically simplify for the self-stability condition (also called attractor chamber). We conjecture a universal formula for the attractor invariants, which applies to any toric Calabi-Yau singularity with compact divisors.