Superpotentials for M-theory on a G_2 holonomy manifold and Triality symmetry

For $M$-theory on the $G_2$ holonomy manifold given by the cone on ${\bf S^3}\x {\bf S^3}$ we consider the superpotential generated by membrane instantons and study its transformations properties, especially under monodromy transformations and triality symmetry. We find that the latter symmetry is, essentially, even a symmetry of the superpotential. As in Seiberg/Witten theory, where a flat bundle given by the periods of an universal elliptic curve over the $u$-plane occurs, here a flat bundle related to the Heisenberg group appears and the relevant universal object over the moduli space is related to hyperbolic geometry.

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