  Supersymmetric D-brane bound states on a Calabi-Yau threefold $X$ are counted
by generalized Donaldsdon-Thomas invariants $\Omega_Z(\gamma)$, depending on a
Chern character (or electromagnetic charge) $\gamma\in H^*(X)$ and a stability
condition (or central charge) $Z$. Attractor invariants $\Omega_*(\gamma)$ are
special instances of DT invariants, where $Z$ is the attractor stability
condition $Z_\gamma$ (a generic perturbation of self-stability), from which DT
invariants for any other stability condition can be deduced. While difficult to
compute in general, these invariants become tractable when $X$ is a crepant
resolution of a singular toric Calabi-Yau threefold associated to a brane
tiling, and hence to a quiver with potential. We survey some known results and
conjectures about framed and unframed refined DT invariants in this context,
and compute attractor invariants explicitly for a variety of toric Calabi-Yau
threefolds, in particular when $X$ is the total space of the canonical bundle
of a smooth projective surface, or when $X$ is a crepant resolution of $C^3/G$.
We check that in all these cases, $\Omega_*(\gamma)=0$ unless $\gamma$ is the
dimension vector of a simple representation or belongs to the kernel of the
skew-symmetrized Euler form. Based on computations in small dimensions, we
predict the values of all attractor invariants, thus potentially solving the
problem of counting DT invariants of these threefolds in all stability
chambers. We also compute the non-commutative refined DT invariants and verify
that they agree with the counting of molten crystals in the unrefined limit.
