The refined BPS growth conjecture for three-dimensional N=2 theories

Let $a_n$ be the coefficients of a supersymmetric partition function for a three-dimensional $\mathcal{N}=2$ theory. Let $r\in\mathbb{Q}/\mathbb{Z}$ and let $c_{\text{eff}}\in\mathbb{C}$. Refined BPS growth conjecture. The growth of supersymmetric (BPS) states is

$$a_n\sim\operatorname{Re}\left[\exp\left(\sqrt{\frac{2\pi^2}{3}c_{\text{eff}}n}+2\pi i rn\right)\right].$$

This refines the basic BPS Cardy behavior by allowing oscillatory phases and complex effective central charge; its validity beyond the examples motivating the paper remains open.