The flow tree formula for quiver moduli

Let a $K$-node quiver have adjacency matrix $\alpha_{ab}$, dimension vector $\gamma=(N_1,\dots,N_K)$, stability parameters $\zeta=(\zeta_1,\dots,\zeta_K)$, and generic superpotential $\mathcal{W}(\phi_{ab,A,ss'})$. Let $\Omega(\gamma,z,y)$ be the rescaled Poincaré polynomial, let $\overline{\Omega}$ be its rational counterpart, and let $\mu$ be the Möbius function. The quiver flow tree conjecture. The rescaled Poincaré polynomial is conjectured to obey

$$\Omega(\gamma,z,y)=\sum_{m\mid\gamma}\mu(m)\,\frac{y-1/y}{m(y^m-1/y^m)}\,\overline{\Omega}(\gamma/m,z,y^m),$$

with

$$\overline{\Omega}(\gamma,z,y)=\sum_{\gamma=\sum_{i=1}^n\gamma_i}\frac{g_{\rm tr}(\{\gamma_i,c_i\},y)}{|{\rm Aut}\{\gamma_i\}|}\prod_{i=1}^n\overline{\Omega}_*(\gamma_i,y),$$

where the sum is over distinct unordered splittings into vectors with non-negative entries, $g_{\rm tr}$ is the tree index, and $\overline{\Omega}_*(\gamma_i,y)$ is evaluated using $\zeta_{*,a}(\gamma_i)=-\sum_{b=1}^K\alpha_{ab}n_{ib}$. This conjectural formula expresses quiver Poincaré polynomials through attractor indices and flow-tree data; the supplied text does not establish it in full generality.