A Non-Perturbative Superpotential With $E_8$ Symmetry
  We compute the non-perturbative superpotential in $F$-theory compactification to four dimensions on a complex three-fold $\P^1\times S$, where $S$ is a rational elliptic surface. In contrast to examples considered previously, the superpotential in this case has interesting modular properties; it is essentially an $E_8$ theta function. 
R. Donagi, A. Grassi, and E. Witten