The 1/2-BPS black-hole index formula for the minimal representation of E8

Let $\varphi_{\min}$ be an automorphic form in the minimal representation $\pi_{\min}$ of $E_8$, and let $(p,q)$ be electric-magnetic charges parametrising the character $\psi_p$ at each finite place. Let $\Omega_{1/2}(p,q)$ denote the index counting charged 1/2-BPS black holes in four-dimensional, $\mathcal{N}=8$ supergravity.

1/2-BPS index conjecture. The index is

$$\Omega_{1/2}(p,q)=\prod_{p<\infty}F_{\psi_p}(\varphi_{\min},1),$$

where $F_{\psi_p}(\varphi_{\min},1)$ is the $p$-adic spherical vector in $\pi_{\min}$.

This identifies the 1/2-BPS index with an Euler product of local Fourier coefficients of the minimal automorphic representation. The source attributes the conjecture to Pioline and collaborators, while its status is not resolved in the supplied text.