Attractor invariants conjecture for toric Calabi–Yau threefolds

Let $X$ be a toric Calabi–Yau threefold whose toric diagram has $i\geq 1$ internal lattice points. Let $x_i$ denote the variables associated with the quiver vertices, let $\delta$ be the relevant dimension vector, let $b$ be the number of boundary lattice points, and let $\alpha^z_{[kk'[}$ denote the dimension vectors associated with the side $z$ and indices $k\neq k'$. Write $\Omega_\ast(x)$ for the generating function of attractor invariants.

Attractor invariants conjecture. The attractor invariants are given by

$$\Omega_\ast(x)=\sum_i x_i+\left(-\mathbb{L}^{3/2}-(b-3+i)\mathbb{L}^{1/2}-i\mathbb{L}^{-1/2}\right)\sum_{n\geq 1}x^{n\delta}-\mathbb{L}^{1/2}\sum_z\sum_{k\neq k'}\sum_{n\geq 0}x^{n\delta+\alpha^z_{[kk'[}}.$$

This refines conjectures that attractor invariants are supported on simple representations and on dimension vectors in the kernel of the antisymmetrized Euler form. The formula is proposed for toric Calabi–Yau threefolds with compact divisors, where the quiver is asymmetric and the refined invariants are otherwise difficult to compute; its general status is not established.