Superpotentials for M -theory on a G2 holonomy manifold and Triality symmetry

For M-theory on the G 2 holonomy manifold given by the cone on S 3 × 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.