The Coulomb branch formula for rational BPS indices

Let $\gamma$ be a total charge, let $\gamma_i$ range over decompositions of it, let $g_{\rm C}(\{\gamma_i\},z,y)$ be the Coulomb index, and let $\overline{\Omega}_S(\gamma_i)$ be single-centered rational indices. Define $\Omega_{\rm tot}$ by

$$\Omega_{\rm tot}(\gamma,y)=\Omega_S(\gamma,y)+\sum_{\gamma=\sum_{i=1}^m m_i\beta_i}H_m(\{\beta_i,m_i\},y)\prod_{i=1}^m\Omega_S(\beta_i,y^{m_i}).$$

The Coulomb branch formula. The rational index is conjectured to be

$$\overline{\Omega}(\gamma,z,y)=\sum_{\gamma=\sum_{i=1}^n\gamma_i}\frac{g_{\rm C}(\{\gamma_i\},z,y)}{|{\rm Aut}\{\gamma_i\}|}\prod_{i=1}^n\left\{\sum_{m_i\in\mathbb{Z}\atop m_i\mid\gamma_i}\frac{y-1/y}{m_i(y^{m_i}-y^{-m_i})}\Omega_{\rm tot}(\gamma_i/m_i,y^{m_i})\right\}.$$

Here $\Omega_{\rm tot}$ is determined from the single-centered indices by the preceding formula. The Coulomb branch formula is a conjectural expression for total BPS indices in terms of single-centered data; the source discusses agreement with flow-tree expressions in cases without scaling solutions and differences when oriented loops permit scaling solutions. Its general status remains open in the supplied material.