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 $¶^1\times S$, 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.

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